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TractableBufferStockModel.py
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TractableBufferStockModel.py
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'''
Defines and solves the Tractable Buffer Stock model described in lecture notes
for "A Tractable Model of Buffer Stock Saving" (henceforth, TBS) available at
http://econ.jhu.edu/people/ccarroll/public/lecturenotes/consumption/TractableBufferStock
The model concerns an agent with constant relative risk aversion utility making
decisions over consumption and saving. He is subject to only a very particular
sort of risk: the possibility that he will become permanently unemployed until
the day he dies; barring this, his income is certain and grows at a constant rate.
The model has an infinite horizon, but is not solved by backward iteration in a
traditional sense. Because of the very specific assumptions about risk, it is
possible to find the agent's steady state or target level of market resources
when employed, as well as information about the optimal consumption rule at this
target level. The full consumption function can then be constructed by "back-
shooting", inverting the Euler equation to find what consumption *must have been*
in the previous period. The consumption function is thus constructed by repeat-
edly adding "stable arm" points to either end of a growing list until specified
bounds are exceeded.
Despite the non-standard solution method, the iterative process can be embedded
in the HARK framework, as shown below.
'''
from __future__ import division, print_function
from __future__ import absolute_import
from builtins import str
import numpy as np
# Import the HARK library.
from HARK import AgentType, NullFunc, Solution
from HARK.utilities import warnings # Because of "patch" to warnings modules
from HARK.utilities import CRRAutility, CRRAutilityP, CRRAutilityPP, CRRAutilityPPP, CRRAutilityPPPP, CRRAutilityP_inv, CRRAutility_invP, CRRAutility_inv
from HARK.interpolation import CubicInterp
from HARK.distribution import Lognormal, Bernoulli
from copy import copy
from scipy.optimize import newton, brentq
__all__ = ['TractableConsumerSolution', 'TractableConsumerType']
# If you want to run the "tractable" version of cstwMPC, uncomment the line below
# and have TractableConsumerType inherit from cstwMPCagent rather than AgentType
#from HARK.cstwMPC.cstwMPC import cstwMPCagent
# Define utility function and its derivatives (plus inverses)
utility = CRRAutility
utilityP = CRRAutilityP
utilityPP = CRRAutilityPP
utilityPPP = CRRAutilityPPP
utilityPPPP = CRRAutilityPPPP
utilityP_inv = CRRAutilityP_inv
utility_invP = CRRAutility_invP
utility_inv = CRRAutility_inv
class TractableConsumerSolution(Solution):
'''
A class representing the solution to a tractable buffer saving problem.
Attributes include a list of money points mNrm_list, a list of consumption points
cNrm_list, a list of MPCs MPC_list, a perfect foresight consumption function
while employed, and a perfect foresight consumption function while unemployed.
The solution includes a consumption function constructed from the lists.
'''
def __init__(self, mNrm_list=[], cNrm_list=[], MPC_list=[], cFunc_U=NullFunc, cFunc=NullFunc):
'''
The constructor for a new TractableConsumerSolution object.
Parameters
----------
mNrm_list : [float]
List of normalized market resources points on the stable arm.
cNrm_list : [float]
List of normalized consumption points on the stable arm.
MPC_list : [float]
List of marginal propensities to consume on the stable arm, corres-
ponding to the (mNrm,cNrm) points.
cFunc_U : function
The (linear) consumption function when permanently unemployed.
cFunc : function
The consumption function when employed.
Returns
-------
new instance of TractableConsumerSolution
'''
self.mNrm_list = mNrm_list
self.cNrm_list = cNrm_list
self.MPC_list = MPC_list
self.cFunc_U = cFunc_U
self.cFunc = cFunc
self.distance_criteria = ['PointCount']
# The distance between two solutions is the difference in the number of
# stable arm points in each. This is a very crude measure of distance
# that captures the notion that the process is over when no points are added.
def findNextPoint(DiscFac,Rfree,CRRA,PermGroFacCmp,UnempPrb,Rnrm,Beth,cNext,mNext,MPCnext,PFMPC):
'''
Calculates what consumption, market resources, and the marginal propensity
to consume must have been in the previous period given model parameters and
values of market resources, consumption, and MPC today.
Parameters
----------
DiscFac : float
Intertemporal discount factor on future utility.
Rfree : float
Risk free interest factor on end-of-period assets.
PermGroFacCmp : float
Permanent income growth factor, compensated for the possibility of
permanent unemployment.
UnempPrb : float
Probability of becoming permanently unemployed.
Rnrm : float
Interest factor normalized by compensated permanent income growth factor.
Beth : float
Composite effective discount factor for reverse shooting solution; defined
in appendix "Numerical Solution/The Consumption Function" in TBS
lecture notes
cNext : float
Normalized consumption in the succeeding period.
mNext : float
Normalized market resources in the succeeding period.
MPCnext : float
The marginal propensity to consume in the succeeding period.
PFMPC : float
The perfect foresight MPC; also the MPC when permanently unemployed.
Returns
-------
mNow : float
Normalized market resources this period.
cNow : float
Normalized consumption this period.
MPCnow : float
Marginal propensity to consume this period.
'''
uPP = lambda x : utilityPP(x,gam=CRRA)
cNow = PermGroFacCmp*(DiscFac*Rfree)**(-1.0/CRRA)*cNext*(1 + UnempPrb*((cNext/(PFMPC*(mNext-1.0)))**CRRA-1.0))**(-1.0/CRRA)
mNow = (PermGroFacCmp/Rfree)*(mNext - 1.0) + cNow
cUNext = PFMPC*(mNow-cNow)*Rnrm
# See TBS Appendix "E.1 The Consumption Function"
natural = Beth*Rnrm*(1.0/uPP(cNow))*((1.0-UnempPrb)*uPP(cNext)*MPCnext + UnempPrb*uPP(cUNext)*PFMPC) # Convenience variable
MPCnow = natural / (natural + 1)
return mNow, cNow, MPCnow
def addToStableArmPoints(solution_next,DiscFac,Rfree,CRRA,PermGroFacCmp,UnempPrb,PFMPC,Rnrm,Beth,mLowerBnd,mUpperBnd):
'''
Adds a one point to the bottom and top of the list of stable arm points if
the bounding levels of mLowerBnd (lower) and mUpperBnd (upper) have not yet
been met by a stable arm point in mNrm_list. This acts as the "one period
solver" / solveOnePeriod in the tractable buffer stock model.
Parameters
----------
solution_next : TractableConsumerSolution
The solution object from the previous iteration of the backshooting
procedure. Not the "next period" solution per se.
DiscFac : float
Intertemporal discount factor on future utility.
Rfree : float
Risk free interest factor on end-of-period assets.
CRRA : float
Coefficient of relative risk aversion.
PermGroFacCmp : float
Permanent income growth factor, compensated for the possibility of
permanent unemployment.
UnempPrb : float
Probability of becoming permanently unemployed.
PFMPC : float
The perfect foresight MPC; also the MPC when permanently unemployed.
Rnrm : float
Interest factor normalized by compensated permanent income growth factor.
Beth : float
Damned if I know.
mLowerBnd : float
Lower bound on market resources for the backshooting process. If
min(solution_next.mNrm_list) < mLowerBnd, no new bottom point is found.
mUpperBnd : float
Upper bound on market resources for the backshooting process. If
max(solution_next.mNrm_list) > mUpperBnd, no new top point is found.
Returns:
---------
solution_now : TractableConsumerSolution
A new solution object with new points added to the top and bottom. If
no new points were added, then the backshooting process is about to end.
'''
# Unpack the lists of Euler points
mNrm_list = copy(solution_next.mNrm_list)
cNrm_list = copy(solution_next.cNrm_list)
MPC_list = copy(solution_next.MPC_list)
# Check whether to add a stable arm point to the top
mNext = mNrm_list[-1]
if mNext < mUpperBnd:
# Get the rest of the data for the previous top point
cNext = solution_next.cNrm_list[-1]
MPCNext = solution_next.MPC_list[-1]
# Calculate employed levels of c, m, and MPC from next period's values
mNow, cNow, MPCnow = findNextPoint(DiscFac,Rfree,CRRA,PermGroFacCmp,UnempPrb,Rnrm,Beth,cNext,mNext,MPCNext,PFMPC)
# Add this point to the top of the stable arm list
mNrm_list.append(mNow)
cNrm_list.append(cNow)
MPC_list.append(MPCnow)
# Check whether to add a stable arm point to the bottom
mNext = mNrm_list[0]
if mNext > mLowerBnd:
# Get the rest of the data for the previous bottom point
cNext = solution_next.cNrm_list[0]
MPCNext = solution_next.MPC_list[0]
# Calculate employed levels of c, m, and MPC from next period's values
mNow, cNow, MPCnow = findNextPoint(DiscFac,Rfree,CRRA,PermGroFacCmp,UnempPrb,Rnrm,Beth,cNext,mNext,MPCNext,PFMPC)
# Add this point to the top of the stable arm list
mNrm_list.insert(0,mNow)
cNrm_list.insert(0,cNow)
MPC_list.insert(0,MPCnow)
# Construct and return this period's solution
solution_now = TractableConsumerSolution(mNrm_list=mNrm_list, cNrm_list=cNrm_list, MPC_list=MPC_list)
solution_now.PointCount = len(mNrm_list)
return solution_now
class TractableConsumerType(AgentType):
def __init__(self,cycles=0,**kwds):
'''
Instantiate a new TractableConsumerType with given data.
Parameters
----------
cycles : int
Number of times the sequence of periods should be solved.
Returns:
-----------
New instance of TractableConsumerType.
'''
# Initialize a basic AgentType
AgentType.__init__(self,
cycles=cycles,
pseudo_terminal=True,**kwds)
# Add consumer-type specific objects, copying to create independent versions
self.time_vary = []
self.time_inv = ['DiscFac','Rfree','CRRA','PermGroFacCmp','UnempPrb','PFMPC','Rnrm','Beth','mLowerBnd','mUpperBnd']
self.shock_vars = ['eStateNow']
self.poststate_vars = ['aLvlNow','eStateNow'] # For simulation
self.solveOnePeriod = addToStableArmPoints # set correct solver
def preSolve(self):
'''
Calculates all of the solution objects that can be obtained before con-
ducting the backshooting routine, including the target levels, the per-
fect foresight solution, (marginal) consumption at m=0, and the small
perturbations around the steady state.
Parameters
----------
none
Returns
-------
none
'''
# Define utility functions
uPP = lambda x : utilityPP(x,gam=self.CRRA)
uPPP = lambda x : utilityPPP(x,gam=self.CRRA)
uPPPP = lambda x : utilityPPPP(x,gam=self.CRRA)
# Define some useful constants from model primitives
self.PermGroFacCmp = self.PermGroFac/(1.0-self.UnempPrb) #"uncertainty compensated" wage growth factor
self.Rnrm = self.Rfree/self.PermGroFacCmp # net interest factor (Rfree normalized by wage growth)
self.PFMPC= 1.0-(self.Rfree**(-1.0))*(self.Rfree*self.DiscFac)**(1.0/self.CRRA) # MPC for a perfect forsight consumer
self.Beth = self.Rnrm*self.DiscFac*self.PermGroFacCmp**(1.0-self.CRRA)
# Verify that this consumer is impatient
PatFacGrowth = (self.Rfree*self.DiscFac)**(1.0/self.CRRA)/self.PermGroFacCmp
PatFacReturn = (self.Rfree*self.DiscFac)**(1.0/self.CRRA)/self.Rfree
if PatFacReturn >= 1.0:
raise Exception("Employed consumer not return impatient, cannot solve!")
if PatFacGrowth >= 1.0:
raise Exception("Employed consumer not growth impatient, cannot solve!")
# Find target money and consumption
# See TBS Appendix "B.2 A Target Always Exists When Human Wealth Is Infinite"
Pi = (1+(PatFacGrowth**(-self.CRRA)-1.0)/self.UnempPrb)**(1/self.CRRA)
self.h = (1.0/(1.0-self.PermGroFac/self.Rfree))
zeta = self.Rnrm*self.PFMPC*Pi # See TBS Appendix "C The Exact Formula for target m"
self.mTarg = 1.0+(self.Rfree/(self.PermGroFacCmp+zeta*self.PermGroFacCmp-self.Rfree))
self.cTarg = (1.0-self.Rnrm**(-1.0))*self.mTarg+self.Rnrm**(-1.0)
mTargU = (self.mTarg - self.cTarg)*self.Rnrm
cTargU = mTargU*self.PFMPC
self.SSperturbance = self.mTarg*0.1
# Find the MPC, MMPC, and MMMPC at the target
mpcTargFixedPointFunc = lambda k : k*uPP(self.cTarg) - self.Beth*((1.0-self.UnempPrb)*(1.0-k)*k*self.Rnrm*uPP(self.cTarg)+self.PFMPC*self.UnempPrb*(1.0-k)*self.Rnrm*uPP(cTargU))
self.MPCtarg = newton(mpcTargFixedPointFunc,0)
mmpcTargFixedPointFunc = lambda kk : kk*uPP(self.cTarg) + self.MPCtarg**2.0*uPPP(self.cTarg) - self.Beth*(-(1.0 - self.UnempPrb)*self.MPCtarg*kk*self.Rnrm*uPP(self.cTarg)+(1.0-self.UnempPrb)*(1.0 - self.MPCtarg)**2.0*kk*self.Rnrm**2.0*uPP(self.cTarg)-self.PFMPC*self.UnempPrb*kk*self.Rnrm*uPP(cTargU)+(1.0-self.UnempPrb)*(1.0-self.MPCtarg)**2.0*self.MPCtarg**2.0*self.Rnrm**2.0*uPPP(self.cTarg)+self.PFMPC**2.0*self.UnempPrb*(1.0-self.MPCtarg)**2.0*self.Rnrm**2.0*uPPP(cTargU))
self.MMPCtarg = newton(mmpcTargFixedPointFunc,0)
mmmpcTargFixedPointFunc = lambda kkk : kkk * uPP(self.cTarg) + 3 * self.MPCtarg * self.MMPCtarg * uPPP(self.cTarg) + self.MPCtarg**3 * uPPPP(self.cTarg) - self.Beth * (-(1 - self.UnempPrb) * self.MPCtarg * kkk * self.Rnrm * uPP(self.cTarg) - 3 * (1 - self.UnempPrb) * (1 - self.MPCtarg) * self.MMPCtarg**2 * self.Rnrm**2 * uPP(self.cTarg) + (1 - self.UnempPrb) * (1 - self.MPCtarg)**3 * kkk * self.Rnrm**3 * uPP(self.cTarg) - self.PFMPC * self.UnempPrb * kkk * self.Rnrm * uPP(cTargU) - 3 * (1 - self.UnempPrb) * (1 - self.MPCtarg) * self.MPCtarg**2 * self.MMPCtarg * self.Rnrm**2 * uPPP(self.cTarg) + 3 * (1 - self.UnempPrb) * (1 - self.MPCtarg)**3 * self.MPCtarg * self.MMPCtarg * self.Rnrm**3 * uPPP(self.cTarg) - 3 * self.PFMPC**2 * self.UnempPrb * (1 - self.MPCtarg) * self.MMPCtarg * self.Rnrm**2 * uPPP(cTargU) + (1 - self.UnempPrb) * (1 - self.MPCtarg)**3 * self.MPCtarg**3 * self.Rnrm**3 * uPPPP(self.cTarg) + self.PFMPC**3 * self.UnempPrb * (1 - self.MPCtarg)**3 * self.Rnrm**3 * uPPPP(cTargU))
self.MMMPCtarg = newton(mmmpcTargFixedPointFunc,0)
# Find the MPC at m=0
f_temp = lambda k : self.Beth*self.Rnrm*self.UnempPrb*(self.PFMPC*self.Rnrm*((1.0-k)/k))**(-self.CRRA-1.0)*self.PFMPC
mpcAtZeroFixedPointFunc = lambda k : k - f_temp(k)/(1 + f_temp(k))
#self.MPCmax = newton(mpcAtZeroFixedPointFunc,0.5)
self.MPCmax = brentq(mpcAtZeroFixedPointFunc,self.PFMPC,0.99,xtol=0.00000001,rtol=0.00000001)
# Make the initial list of Euler points: target and perturbation to either side
mNrm_list = [self.mTarg-self.SSperturbance, self.mTarg, self.mTarg+self.SSperturbance]
c_perturb_lo = self.cTarg - self.SSperturbance*self.MPCtarg + 0.5*self.SSperturbance**2.0*self.MMPCtarg - (1.0/6.0)*self.SSperturbance**3.0*self.MMMPCtarg
c_perturb_hi = self.cTarg + self.SSperturbance*self.MPCtarg + 0.5*self.SSperturbance**2.0*self.MMPCtarg + (1.0/6.0)*self.SSperturbance**3.0*self.MMMPCtarg
cNrm_list = [c_perturb_lo, self.cTarg, c_perturb_hi]
MPC_perturb_lo = self.MPCtarg - self.SSperturbance*self.MMPCtarg + 0.5*self.SSperturbance**2.0*self.MMMPCtarg
MPC_perturb_hi = self.MPCtarg + self.SSperturbance*self.MMPCtarg + 0.5*self.SSperturbance**2.0*self.MMMPCtarg
MPC_list = [MPC_perturb_lo, self.MPCtarg, MPC_perturb_hi]
# Set bounds for money (stable arm construction stops when these are exceeded)
self.mLowerBnd = 1.0
self.mUpperBnd = 2.0*self.mTarg
# Make the terminal period solution
solution_terminal = TractableConsumerSolution(mNrm_list=mNrm_list,cNrm_list=cNrm_list,MPC_list=MPC_list)
self.solution_terminal = solution_terminal
# Make two linear steady state functions
self.cSSfunc = lambda m : m*((self.Rnrm*self.PFMPC*Pi)/(1.0+self.Rnrm*self.PFMPC*Pi))
self.mSSfunc = lambda m : (self.PermGroFacCmp/self.Rfree)+(1.0-self.PermGroFacCmp/self.Rfree)*m
def postSolve(self):
'''
This method adds consumption at m=0 to the list of stable arm points,
then constructs the consumption function as a cubic interpolation over
those points. Should be run after the backshooting routine is complete.
Parameters
----------
none
Returns
-------
none
'''
# Add bottom point to the stable arm points
self.solution[0].mNrm_list.insert(0,0.0)
self.solution[0].cNrm_list.insert(0,0.0)
self.solution[0].MPC_list.insert(0,self.MPCmax)
# Construct an interpolation of the consumption function from the stable arm points
self.solution[0].cFunc = CubicInterp(self.solution[0].mNrm_list,self.solution[0].cNrm_list,self.solution[0].MPC_list,self.PFMPC*(self.h-1.0),self.PFMPC)
self.solution[0].cFunc_U = lambda m : self.PFMPC*m
def update():
'''
This method does absolutely nothing, but should remain here for compati-
bility with cstwMPC when doing the "tractable" version.
'''
return None
def simBirth(self,which_agents):
'''
Makes new consumers for the given indices. Initialized variables include aNrm, as
well as time variables t_age and t_cycle. Normalized assets are drawn from a lognormal
distributions given by aLvlInitMean and aLvlInitStd.
Parameters
----------
which_agents : np.array(Bool)
Boolean array of size self.AgentCount indicating which agents should be "born".
Returns
-------
None
'''
# Get and store states for newly born agents
N = np.sum(which_agents) # Number of new consumers to make
self.aLvlNow[which_agents] = Lognormal(self.aLvlInitMean,
sigma=self.aLvlInitStd).draw(N,seed=self.RNG.randint(0,2**31-1))
self.eStateNow[which_agents] = 1.0 # Agents are born employed
self.t_age[which_agents] = 0 # How many periods since each agent was born
self.t_cycle[which_agents] = 0 # Which period of the cycle each agent is currently in
return None
def simDeath(self):
'''
Trivial function that returns boolean array of all False, as there is no death.
Parameters
----------
None
Returns
-------
which_agents : np.array(bool)
Boolean array of size AgentCount indicating which agents die.
'''
# Nobody dies in this model
which_agents = np.zeros(self.AgentCount,dtype=bool)
return which_agents
def getShocks(self):
'''
Determine which agents switch from employment to unemployment. All unemployed agents remain
unemployed until death.
Parameters
----------
None
Returns
-------
None
'''
employed = self.eStateNow == 1.0
N = int(np.sum(employed))
newly_unemployed = Bernoulli(self.UnempPrb).draw(N,
seed=self.RNG.randint(0,2**31-1))
self.eStateNow[employed] = 1.0 - newly_unemployed
def getStates(self):
'''
Calculate market resources for all agents this period.
Parameters
----------
None
Returns
-------
None
'''
self.bLvlNow = self.Rfree*self.aLvlNow
self.mLvlNow = self.bLvlNow + self.eStateNow
def getControls(self):
'''
Calculate consumption for each agent this period.
Parameters
----------
None
Returns
-------
None
'''
employed = self.eStateNow == 1.0
unemployed = np.logical_not(employed)
cLvlNow = np.zeros(self.AgentCount)
cLvlNow[employed] = self.solution[0].cFunc(self.mLvlNow[employed])
cLvlNow[unemployed] = self.solution[0].cFunc_U(self.mLvlNow[unemployed])
self.cLvlNow = cLvlNow
def getPostStates(self):
'''
Calculates end-of-period assets for each consumer of this type.
Parameters
----------
None
Returns
-------
None
'''
self.aLvlNow = self.mLvlNow - self.cLvlNow
return None