def makeEndOfPrdvFuncCond(self):
'''
Construct the end-of-period value function conditional on next period's
state. NOTE: It might be possible to eliminate this method and replace
it with ConsIndShockSolver.makeEndOfPrdvFunc, but the self.X_cond
variables must be renamed.
Parameters
----------
none
Returns
-------
EndofPrdvFunc_cond : ValueFunc
The end-of-period value function conditional on a particular state
occuring in the next period.
'''
VLvlNext = (self.PermShkVals_temp**(1.0-self.CRRA)*
self.PermGroFac**(1.0-self.CRRA))*self.vFuncNext(self.mNrmNext)
EndOfPrdv_cond = self.DiscFacEff*np.sum(VLvlNext*self.ShkPrbs_temp,axis=0)
EndOfPrdvNvrs_cond = self.uinv(EndOfPrdv_cond)
EndOfPrdvNvrsP_cond = self.EndOfPrdvP_cond*self.uinvP(EndOfPrdv_cond)
EndOfPrdvNvrs_cond = np.insert(EndOfPrdvNvrs_cond,0,0.0)
EndOfPrdvNvrsP_cond = np.insert(EndOfPrdvNvrsP_cond,0,EndOfPrdvNvrsP_cond[0])
aNrm_temp = np.insert(self.aNrm_cond,0,self.BoroCnstNat)
EndOfPrdvNvrsFunc_cond = CubicInterp(aNrm_temp,EndOfPrdvNvrs_cond,EndOfPrdvNvrsP_cond)
EndofPrdvFunc_cond = ValueFunc(EndOfPrdvNvrsFunc_cond,self.CRRA)
return EndofPrdvFunc_cond
def postSolve(self):
self.solution_fast = deepcopy(self.solution)
if self.cycles == 0:
terminal = 1
else:
terminal = self.cycles
self.solution[terminal] = self.solution_terminal_cs
for i in range(terminal):
solution = self.solution[i]
# Construct the consumption function as a linear interpolation.
cFunc = LinearInterp(solution.mNrm, solution.cNrm)
"""
Defines the value and marginal value functions for this period.
Uses the fact that for a perfect foresight CRRA utility problem,
if the MPC in period t is :math:`\kappa_{t}`, and relative risk
aversion :math:`\rho`, then the inverse value vFuncNvrs has a
constant slope of :math:`\kappa_{t}^{-\rho/(1-\rho)}` and
vFuncNvrs has value of zero at the lower bound of market resources
mNrmMin. See PerfForesightConsumerType.ipynb documentation notebook
for a brief explanation and the links below for a fuller treatment.
https://econ.jhu.edu/people/ccarroll/public/lecturenotes/consumption/PerfForesightCRRA/#vFuncAnalytical
https://econ.jhu.edu/people/ccarroll/SolvingMicroDSOPs/#vFuncPF
"""
vFuncNvrs = LinearInterp(
np.array([solution.mNrmMin, solution.mNrmMin + 1.0]),
np.array([0.0, solution.vFuncNvrsSlope]),
)
vFunc = ValueFunc(vFuncNvrs, self.CRRA)
vPfunc = MargValueFunc(cFunc, self.CRRA)
consumer_solution = ConsumerSolution(
cFunc=cFunc,
vFunc=vFunc,
vPfunc=vPfunc,
mNrmMin=solution.mNrmMin,
hNrm=solution.hNrm,
MPCmin=solution.MPCmin,
MPCmax=solution.MPCmax,
)
ExIncNext = 1.0 # Perfect foresight income of 1
# Add mNrmSS to the solution and return it
consumer_solution.mNrmSS = _addSSmNrmNumba(
self.Rfree,
self.PermGroFac[i],
solution.mNrm,
solution.cNrm,
solution.mNrmMin,
ExIncNext,
_searchSSfunc,
)
self.solution[i] = consumer_solution
def makevFunc(self, solution):
"""
Construct the value function for each current state.
Parameters
----------
solution : ConsumerSolution
The solution to the single period consumption-saving problem. Must
have a consumption function cFunc (using cubic or linear splines) as
a list with elements corresponding to the current Markov state. E.g.
solution.cFunc[0] is the consumption function when in the i=0 Markov
state this period.
Returns
-------
vFuncNow : [ValueFunc]
A list of value functions (defined over normalized market resources
m) for each current period Markov state.
"""
vFuncNow = [] # Initialize an empty list of value functions
# Loop over each current period state and construct the value function
for i in range(self.StateCount):
# Make state-conditional grids of market resources and consumption
mNrmMin = self.mNrmMin_list[i]
mGrid = mNrmMin + self.aXtraGrid
cGrid = solution.cFunc[i](mGrid)
aGrid = mGrid - cGrid
# Calculate end-of-period value at each gridpoint
EndOfPrdv_all = np.zeros((self.StateCount, self.aXtraGrid.size))
for j in range(self.StateCount):
if self.possible_transitions[i, j]:
EndOfPrdv_all[j, :] = self.EndOfPrdvFunc_list[j](aGrid)
EndOfPrdv = np.dot(self.MrkvArray[i, :], EndOfPrdv_all)
# Calculate (normalized) value and marginal value at each gridpoint
vNrmNow = self.u(cGrid) + EndOfPrdv
vPnow = self.uP(cGrid)
# Make a "decurved" value function with the inverse utility function
vNvrs = self.uinv(vNrmNow) # value transformed through inverse utility
vNvrsP = vPnow * self.uinvP(vNrmNow)
mNrm_temp = np.insert(mGrid, 0, mNrmMin) # add the lower bound
vNvrs = np.insert(vNvrs, 0, 0.0)
vNvrsP = np.insert(
vNvrsP, 0, self.MPCmaxEff[i] ** (-self.CRRA / (1.0 - self.CRRA))
)
MPCminNvrs = self.MPCminNow[i] ** (-self.CRRA / (1.0 - self.CRRA))
vNvrsFunc_i = CubicInterp(
mNrm_temp, vNvrs, vNvrsP, MPCminNvrs * self.hNrmNow[i], MPCminNvrs
)
# "Recurve" the decurved value function and add it to the list
vFunc_i = ValueFunc(vNvrsFunc_i, self.CRRA)
vFuncNow.append(vFunc_i)
return vFuncNow
def updateSolutionTerminal(self):
"""
Update the terminal period solution. This method should be run when a
new AgentType is created or when CRRA changes.
"""
self.solution_terminal_cs = ConsumerSolution(
cFunc=self.cFunc_terminal_,
vFunc=ValueFunc(self.cFunc_terminal_, self.CRRA),
vPfunc=MargValueFunc(self.cFunc_terminal_, self.CRRA),
vPPfunc=MargMargValueFunc(self.cFunc_terminal_, self.CRRA),
mNrmMin=0.0,
hNrm=0.0,
MPCmin=1.0,
MPCmax=1.0,
)
def updateSolutionTerminal(self):
"""
Solves the terminal period of the portfolio choice problem. The solution is
trivial, as usual: consume all market resources, and put nothing in the risky
asset (because you have nothing anyway).
Parameters
----------
None
Returns
-------
None
"""
# Consume all market resources: c_T = m_T
cFuncAdj_terminal = IdentityFunction()
cFuncFxd_terminal = IdentityFunction(i_dim=0, n_dims=2)
# Risky share is irrelevant-- no end-of-period assets; set to zero
ShareFuncAdj_terminal = ConstantFunction(0.0)
ShareFuncFxd_terminal = IdentityFunction(i_dim=1, n_dims=2)
# Value function is simply utility from consuming market resources
vFuncAdj_terminal = ValueFunc(cFuncAdj_terminal, self.CRRA)
vFuncFxd_terminal = ValueFunc2D(cFuncFxd_terminal, self.CRRA)
# Marginal value of market resources is marg utility at the consumption function
vPfuncAdj_terminal = MargValueFunc(cFuncAdj_terminal, self.CRRA)
dvdmFuncFxd_terminal = MargValueFunc2D(cFuncFxd_terminal, self.CRRA)
dvdsFuncFxd_terminal = ConstantFunction(
0.0
) # No future, no marg value of Share
# Construct the terminal period solution
self.solution_terminal = PortfolioSolution(
cFuncAdj=cFuncAdj_terminal,
ShareFuncAdj=ShareFuncAdj_terminal,
vFuncAdj=vFuncAdj_terminal,
vPfuncAdj=vPfuncAdj_terminal,
cFuncFxd=cFuncFxd_terminal,
ShareFuncFxd=ShareFuncFxd_terminal,
vFuncFxd=vFuncFxd_terminal,
dvdmFuncFxd=dvdmFuncFxd_terminal,
dvdsFuncFxd=dvdsFuncFxd_terminal,
)
def makevFunc(self, solution):
'''
Make the beginning-of-period value function (unconditional on the shock).
Parameters
----------
solution : ConsumerSolution
The solution to this single period problem, which must include the
consumption function.
Returns
-------
vFuncNow : ValueFunc
A representation of the value function for this period, defined over
normalized market resources m: v = vFuncNow(m).
'''
# Compute expected value and marginal value on a grid of market resources,
# accounting for all of the discrete preference shocks
PrefShkCount = self.PrefShkVals.size
mNrm_temp = self.mNrmMinNow + self.aXtraGrid
vNrmNow = np.zeros_like(mNrm_temp)
vPnow = np.zeros_like(mNrm_temp)
for j in range(PrefShkCount):
this_shock = self.PrefShkVals[j]
this_prob = self.PrefShkPrbs[j]
cNrmNow = solution.cFunc(mNrm_temp,
this_shock * np.ones_like(mNrm_temp))
aNrmNow = mNrm_temp - cNrmNow
vNrmNow += this_prob * (this_shock * self.u(cNrmNow) +
self.EndOfPrdvFunc(aNrmNow))
vPnow += this_prob * this_shock * self.uP(cNrmNow)
# Construct the beginning-of-period value function
vNvrs = self.uinv(vNrmNow) # value transformed through inverse utility
vNvrsP = vPnow * self.uinvP(vNrmNow)
mNrm_temp = np.insert(mNrm_temp, 0, self.mNrmMinNow)
vNvrs = np.insert(vNvrs, 0, 0.0)
vNvrsP = np.insert(vNvrsP, 0,
self.MPCmaxEff**(-self.CRRA / (1.0 - self.CRRA)))
MPCminNvrs = self.MPCminNow**(-self.CRRA / (1.0 - self.CRRA))
vNvrsFuncNow = CubicInterp(mNrm_temp, vNvrs, vNvrsP,
MPCminNvrs * self.hNrmNow, MPCminNvrs)
vFuncNow = ValueFunc(vNvrsFuncNow, self.CRRA)
return vFuncNow
def solveConsPortfolio(solution_next, ShockDstn, IncomeDstn, RiskyDstn, LivPrb,
DiscFac, CRRA, Rfree, PermGroFac, BoroCnstArt,
aXtraGrid, ShareGrid, vFuncBool, AdjustPrb,
DiscreteShareBool, ShareLimit, IndepDstnBool):
'''
Solve the one period problem for a portfolio-choice consumer.
Parameters
----------
solution_next : PortfolioSolution
Solution to next period's problem.
ShockDstn : [np.array]
List with four arrays: discrete probabilities, permanent income shocks,
transitory income shocks, and risky returns. This is only used if the
input IndepDstnBool is False, indicating that income and return distributions
can't be assumed to be independent.
IncomeDstn : [np.array]
List with three arrays: discrete probabilities, permanent income shocks,
and transitory income shocks. This is only used if the input IndepDsntBool
is True, indicating that income and return distributions are independent.
RiskyDstn : [np.array]
List with two arrays: discrete probabilities and risky asset returns. This
is only used if the input IndepDstnBool is True, indicating that income
and return distributions are independent.
LivPrb : float
Survival probability; likelihood of being alive at the beginning of
the succeeding period.
DiscFac : float
Intertemporal discount factor for future utility.
CRRA : float
Coefficient of relative risk aversion.
Rfree : float
Risk free interest factor on end-of-period assets.
PermGroFac : float
Expected permanent income growth factor at the end of this period.
BoroCnstArt: float or None
Borrowing constraint for the minimum allowable assets to end the
period with. In this model, it is *required* to be zero.
aXtraGrid: np.array
Array of "extra" end-of-period asset values-- assets above the
absolute minimum acceptable level.
ShareGrid : np.array
Array of risky portfolio shares on which to define the interpolation
of the consumption function when Share is fixed.
vFuncBool: boolean
An indicator for whether the value function should be computed and
included in the reported solution.
AdjustPrb : float
Probability that the agent will be able to update his portfolio share.
DiscreteShareBool : bool
Indicator for whether risky portfolio share should be optimized on the
continuous [0,1] interval using the FOC (False), or instead only selected
from the discrete set of values in ShareGrid (True). If True, then
vFuncBool must also be True.
ShareLimit : float
Limiting lower bound of risky portfolio share as mNrm approaches infinity.
IndepDstnBool : bool
Indicator for whether the income and risky return distributions are in-
dependent of each other, which can speed up the expectations step.
Returns
-------
solution_now : PortfolioSolution
The solution to the single period consumption-saving with portfolio choice
problem. Includes two consumption and risky share functions: one for when
the agent can adjust his portfolio share (Adj) and when he can't (Fxd).
'''
# Make sure the individual is liquidity constrained. Allowing a consumer to
# borrow *and* invest in an asset with unbounded (negative) returns is a bad mix.
if BoroCnstArt != 0.0:
raise ValueError('PortfolioConsumerType must have BoroCnstArt=0.0!')
# Make sure that if risky portfolio share is optimized only discretely, then
# the value function is also constructed (else this task would be impossible).
if (DiscreteShareBool and (not vFuncBool)):
raise ValueError(
'PortfolioConsumerType requires vFuncBool to be True when DiscreteShareBool is True!'
)
# Define temporary functions for utility and its derivative and inverse
u = lambda x: utility(x, CRRA)
uP = lambda x: utilityP(x, CRRA)
uPinv = lambda x: utilityP_inv(x, CRRA)
n = lambda x: utility_inv(x, CRRA)
nP = lambda x: utility_invP(x, CRRA)
# Unpack next period's solution
vPfuncAdj_next = solution_next.vPfuncAdj
dvdmFuncFxd_next = solution_next.dvdmFuncFxd
dvdsFuncFxd_next = solution_next.dvdsFuncFxd
vFuncAdj_next = solution_next.vFuncAdj
vFuncFxd_next = solution_next.vFuncFxd
# Major method fork: (in)dependent risky asset return and income distributions
if IndepDstnBool: # If the distributions ARE independent...
# Unpack the shock distribution
IncPrbs_next = IncomeDstn.pmf
PermShks_next = IncomeDstn.X[0]
TranShks_next = IncomeDstn.X[1]
Rprbs_next = RiskyDstn.pmf
Risky_next = RiskyDstn.X
zero_bound = (
np.min(TranShks_next) == 0.
) # Flag for whether the natural borrowing constraint is zero
RiskyMax = np.max(Risky_next)
# bNrm represents R*a, balances after asset return shocks but before income.
# This just uses the highest risky return as a rough shifter for the aXtraGrid.
if zero_bound:
aNrmGrid = aXtraGrid
bNrmGrid = np.insert(RiskyMax * aXtraGrid, 0,
np.min(Risky_next) * aXtraGrid[0])
else:
aNrmGrid = np.insert(aXtraGrid, 0,
0.0) # Add an asset point at exactly zero
bNrmGrid = RiskyMax * np.insert(aXtraGrid, 0, 0.0)
# Get grid and shock sizes, for easier indexing
aNrm_N = aNrmGrid.size
bNrm_N = bNrmGrid.size
Share_N = ShareGrid.size
Income_N = IncPrbs_next.size
Risky_N = Rprbs_next.size
# Make tiled arrays to calculate future realizations of mNrm and Share when integrating over IncomeDstn
bNrm_tiled = np.tile(np.reshape(bNrmGrid, (bNrm_N, 1, 1)),
(1, Share_N, Income_N))
Share_tiled = np.tile(np.reshape(ShareGrid, (1, Share_N, 1)),
(bNrm_N, 1, Income_N))
IncPrbs_tiled = np.tile(np.reshape(IncPrbs_next, (1, 1, Income_N)),
(bNrm_N, Share_N, 1))
PermShks_tiled = np.tile(np.reshape(PermShks_next, (1, 1, Income_N)),
(bNrm_N, Share_N, 1))
TranShks_tiled = np.tile(np.reshape(TranShks_next, (1, 1, Income_N)),
(bNrm_N, Share_N, 1))
# Calculate future realizations of market resources
mNrm_next = bNrm_tiled / (PermShks_tiled * PermGroFac) + TranShks_tiled
Share_next = Share_tiled
# Evaluate realizations of marginal value of market resources next period
dvdmAdj_next = vPfuncAdj_next(mNrm_next)
if AdjustPrb < 1.:
dvdmFxd_next = dvdmFuncFxd_next(mNrm_next, Share_next)
dvdm_next = AdjustPrb * dvdmAdj_next + (
1. -
AdjustPrb) * dvdmFxd_next # Combine by adjustment probability
else: # Don't bother evaluating if there's no chance that portfolio share is fixed
dvdm_next = dvdmAdj_next
# Evaluate realizations of marginal value of risky share next period
dvdsAdj_next = np.zeros_like(
mNrm_next) # No marginal value of Share if it's a free choice!
if AdjustPrb < 1.:
dvdsFxd_next = dvdsFuncFxd_next(mNrm_next, Share_next)
dvds_next = AdjustPrb * dvdsAdj_next + (
1. -
AdjustPrb) * dvdsFxd_next # Combine by adjustment probability
else: # Don't bother evaluating if there's no chance that portfolio share is fixed
dvds_next = dvdsAdj_next
# If the value function has been requested, evaluate realizations of value
if vFuncBool:
vAdj_next = vFuncAdj_next(mNrm_next)
if AdjustPrb < 1.:
vFxd_next = vFuncFxd_next(mNrm_next, Share_next)
v_next = AdjustPrb * vAdj_next + (1. - AdjustPrb) * vFxd_next
else: # Don't bother evaluating if there's no chance that portfolio share is fixed
v_next = vAdj_next
else:
v_next = np.zeros_like(dvdm_next) # Trivial array
# Calculate intermediate marginal value of bank balances by taking expectations over income shocks
temp_fac_A = uP(PermShks_tiled *
PermGroFac) # Will use this in a couple places
dvdb_intermed = np.sum(IncPrbs_tiled * temp_fac_A * dvdm_next, axis=2)
dvdbNvrs_intermed = uPinv(dvdb_intermed)
dvdbNvrsFunc_intermed = BilinearInterp(dvdbNvrs_intermed, bNrmGrid,
ShareGrid)
dvdbFunc_intermed = MargValueFunc2D(dvdbNvrsFunc_intermed, CRRA)
# Calculate intermediate value by taking expectations over income shocks
temp_fac_B = (PermShks_tiled * PermGroFac)**(1. - CRRA
) # Will use this below
if vFuncBool:
v_intermed = np.sum(IncPrbs_tiled * temp_fac_B * v_next, axis=2)
vNvrs_intermed = n(v_intermed)
vNvrsFunc_intermed = BilinearInterp(vNvrs_intermed, bNrmGrid,
ShareGrid)
vFunc_intermed = ValueFunc2D(vNvrsFunc_intermed, CRRA)
# Calculate intermediate marginal value of risky portfolio share by taking expectations
dvds_intermed = np.sum(IncPrbs_tiled * temp_fac_B * dvds_next, axis=2)
dvdsFunc_intermed = BilinearInterp(dvds_intermed, bNrmGrid, ShareGrid)
# Make tiled arrays to calculate future realizations of bNrm and Share when integrating over RiskyDstn
aNrm_tiled = np.tile(np.reshape(aNrmGrid, (aNrm_N, 1, 1)),
(1, Share_N, Risky_N))
Share_tiled = np.tile(np.reshape(ShareGrid, (1, Share_N, 1)),
(aNrm_N, 1, Risky_N))
Rprbs_tiled = np.tile(np.reshape(Rprbs_next, (1, 1, Risky_N)),
(aNrm_N, Share_N, 1))
Risky_tiled = np.tile(np.reshape(Risky_next, (1, 1, Risky_N)),
(aNrm_N, Share_N, 1))
# Calculate future realizations of bank balances bNrm
Share_next = Share_tiled
Rxs = Risky_tiled - Rfree
Rport = Rfree + Share_next * Rxs
bNrm_next = Rport * aNrm_tiled
# Evaluate realizations of value and marginal value after asset returns are realized
dvdb_next = dvdbFunc_intermed(bNrm_next, Share_next)
dvds_next = dvdsFunc_intermed(bNrm_next, Share_next)
if vFuncBool:
v_next = vFunc_intermed(bNrm_next, Share_next)
else:
v_next = np.zeros_like(dvdb_next)
# Calculate end-of-period marginal value of assets by taking expectations
EndOfPrddvda = DiscFac * LivPrb * np.sum(
Rprbs_tiled * Rport * dvdb_next, axis=2)
EndOfPrddvdaNvrs = uPinv(EndOfPrddvda)
# Calculate end-of-period value by taking expectations
if vFuncBool:
EndOfPrdv = DiscFac * LivPrb * np.sum(Rprbs_tiled * v_next, axis=2)
EndOfPrdvNvrs = n(EndOfPrdv)
# Calculate end-of-period marginal value of risky portfolio share by taking expectations
EndOfPrddvds = DiscFac * LivPrb * np.sum(
Rprbs_tiled * (Rxs * aNrm_tiled * dvdb_next + dvds_next), axis=2)
else: # If the distributions are NOT independent...
# Unpack the shock distribution
ShockPrbs_next = ShockDstn[0]
PermShks_next = ShockDstn[1]
TranShks_next = ShockDstn[2]
Risky_next = ShockDstn[3]
zero_bound = (
np.min(TranShks_next) == 0.
) # Flag for whether the natural borrowing constraint is zero
# Make tiled arrays to calculate future realizations of mNrm and Share; dimension order: mNrm, Share, shock
if zero_bound:
aNrmGrid = aXtraGrid
else:
aNrmGrid = np.insert(aXtraGrid, 0,
0.0) # Add an asset point at exactly zero
aNrm_N = aNrmGrid.size
Share_N = ShareGrid.size
Shock_N = ShockPrbs_next.size
aNrm_tiled = np.tile(np.reshape(aNrmGrid, (aNrm_N, 1, 1)),
(1, Share_N, Shock_N))
Share_tiled = np.tile(np.reshape(ShareGrid, (1, Share_N, 1)),
(aNrm_N, 1, Shock_N))
ShockPrbs_tiled = np.tile(np.reshape(ShockPrbs_next, (1, 1, Shock_N)),
(aNrm_N, Share_N, 1))
PermShks_tiled = np.tile(np.reshape(PermShks_next, (1, 1, Shock_N)),
(aNrm_N, Share_N, 1))
TranShks_tiled = np.tile(np.reshape(TranShks_next, (1, 1, Shock_N)),
(aNrm_N, Share_N, 1))
Risky_tiled = np.tile(np.reshape(Risky_next, (1, 1, Shock_N)),
(aNrm_N, Share_N, 1))
# Calculate future realizations of market resources
Rport = (1. - Share_tiled) * Rfree + Share_tiled * Risky_tiled
mNrm_next = Rport * aNrm_tiled / (PermShks_tiled *
PermGroFac) + TranShks_tiled
Share_next = Share_tiled
# Evaluate realizations of marginal value of market resources next period
dvdmAdj_next = vPfuncAdj_next(mNrm_next)
if AdjustPrb < 1.:
dvdmFxd_next = dvdmFuncFxd_next(mNrm_next, Share_next)
dvdm_next = AdjustPrb * dvdmAdj_next + (
1. -
AdjustPrb) * dvdmFxd_next # Combine by adjustment probability
else: # Don't bother evaluating if there's no chance that portfolio share is fixed
dvdm_next = dvdmAdj_next
# Evaluate realizations of marginal value of risky share next period
dvdsAdj_next = np.zeros_like(
mNrm_next) # No marginal value of Share if it's a free choice!
if AdjustPrb < 1.:
dvdsFxd_next = dvdsFuncFxd_next(mNrm_next, Share_next)
dvds_next = AdjustPrb * dvdsAdj_next + (
1. -
AdjustPrb) * dvdsFxd_next # Combine by adjustment probability
else: # Don't bother evaluating if there's no chance that portfolio share is fixed
dvds_next = dvdsAdj_next
# If the value function has been requested, evaluate realizations of value
if vFuncBool:
vAdj_next = vFuncAdj_next(mNrm_next)
if AdjustPrb < 1.:
vFxd_next = vFuncFxd_next(mNrm_next, Share_next)
v_next = AdjustPrb * vAdj_next + (1. - AdjustPrb) * vFxd_next
else: # Don't bother evaluating if there's no chance that portfolio share is fixed
v_next = vAdj_next
else:
v_next = np.zeros_like(dvdm_next) # Trivial array
# Calculate end-of-period marginal value of assets by taking expectations
temp_fac_A = uP(PermShks_tiled *
PermGroFac) # Will use this in a couple places
EndOfPrddvda = DiscFac * LivPrb * np.sum(
ShockPrbs_tiled * Rport * temp_fac_A * dvdm_next, axis=2)
EndOfPrddvdaNvrs = uPinv(EndOfPrddvda)
# Calculate end-of-period value by taking expectations
temp_fac_B = (PermShks_tiled * PermGroFac)**(1. - CRRA
) # Will use this below
if vFuncBool:
EndOfPrdv = DiscFac * LivPrb * np.sum(
ShockPrbs_tiled * temp_fac_B * v_next, axis=2)
EndOfPrdvNvrs = n(EndOfPrdv)
# Calculate end-of-period marginal value of risky portfolio share by taking expectations
Rxs = Risky_tiled - Rfree
EndOfPrddvds = DiscFac * LivPrb * np.sum(
ShockPrbs_tiled * (Rxs * aNrm_tiled * temp_fac_A * dvdm_next +
temp_fac_B * dvds_next),
axis=2)
# Major method fork: discrete vs continuous choice of risky portfolio share
if DiscreteShareBool: # Optimization of Share on the discrete set ShareGrid
opt_idx = np.argmax(EndOfPrdv, axis=1)
Share_now = ShareGrid[
opt_idx] # Best portfolio share is one with highest value
cNrmAdj_now = EndOfPrddvdaNvrs[np.arange(
aNrm_N), opt_idx] # Take cNrm at that index as well
if not zero_bound:
Share_now[
0] = 1. # aNrm=0, so there's no way to "optimize" the portfolio
cNrmAdj_now[0] = EndOfPrddvdaNvrs[
0, -1] # Consumption when aNrm=0 does not depend on Share
else: # Optimization of Share on continuous interval [0,1]
# For values of aNrm at which the agent wants to put more than 100% into risky asset, constrain them
FOC_s = EndOfPrddvds
Share_now = np.zeros_like(
aNrmGrid) # Initialize to putting everything in safe asset
cNrmAdj_now = np.zeros_like(aNrmGrid)
constrained = FOC_s[:,
-1] > 0. # If agent wants to put more than 100% into risky asset, he is constrained
Share_now[constrained] = 1.0
if not zero_bound:
Share_now[
0] = 1. # aNrm=0, so there's no way to "optimize" the portfolio
cNrmAdj_now[0] = EndOfPrddvdaNvrs[
0, -1] # Consumption when aNrm=0 does not depend on Share
cNrmAdj_now[constrained] = EndOfPrddvdaNvrs[
constrained, -1] # Get consumption when share-constrained
# For each value of aNrm, find the value of Share such that FOC-Share == 0.
# This loop can probably be eliminated, but it's such a small step that it won't speed things up much.
crossing = np.logical_and(FOC_s[:, 1:] <= 0., FOC_s[:, :-1] >= 0.)
for j in range(aNrm_N):
if Share_now[j] == 0.:
try:
idx = np.argwhere(crossing[j, :])[0][0]
bot_s = ShareGrid[idx]
top_s = ShareGrid[idx + 1]
bot_f = FOC_s[j, idx]
top_f = FOC_s[j, idx + 1]
bot_c = EndOfPrddvdaNvrs[j, idx]
top_c = EndOfPrddvdaNvrs[j, idx + 1]
alpha = 1. - top_f / (top_f - bot_f)
Share_now[j] = (1. - alpha) * bot_s + alpha * top_s
cNrmAdj_now[j] = (1. - alpha) * bot_c + alpha * top_c
except:
print('No optimal controls found for a=' +
str(aNrmGrid[j]))
# Calculate the endogenous mNrm gridpoints when the agent adjusts his portfolio
mNrmAdj_now = aNrmGrid + cNrmAdj_now
# Construct the risky share function when the agent can adjust
if DiscreteShareBool:
mNrmAdj_mid = (mNrmAdj_now[1:] + mNrmAdj_now[:-1]) / 2
mNrmAdj_plus = mNrmAdj_mid * (1. + 1e-12)
mNrmAdj_comb = (np.transpose(np.vstack(
(mNrmAdj_mid, mNrmAdj_plus)))).flatten()
mNrmAdj_comb = np.append(np.insert(mNrmAdj_comb, 0, 0.0),
mNrmAdj_now[-1])
Share_comb = (np.transpose(np.vstack(
(Share_now, Share_now)))).flatten()
ShareFuncAdj_now = LinearInterp(mNrmAdj_comb, Share_comb)
else:
if zero_bound:
Share_lower_bound = ShareLimit
else:
Share_lower_bound = 1.0
Share_now = np.insert(Share_now, 0, Share_lower_bound)
ShareFuncAdj_now = LinearInterp(np.insert(mNrmAdj_now, 0, 0.0),
Share_now,
intercept_limit=ShareLimit,
slope_limit=0.0)
# Construct the consumption function when the agent can adjust
cNrmAdj_now = np.insert(cNrmAdj_now, 0, 0.0)
cFuncAdj_now = LinearInterp(np.insert(mNrmAdj_now, 0, 0.0), cNrmAdj_now)
# Construct the marginal value (of mNrm) function when the agent can adjust
vPfuncAdj_now = MargValueFunc(cFuncAdj_now, CRRA)
# Construct the consumption function when the agent *can't* adjust the risky share, as well
# as the marginal value of Share function
cFuncFxd_by_Share = []
dvdsFuncFxd_by_Share = []
for j in range(Share_N):
cNrmFxd_temp = EndOfPrddvdaNvrs[:, j]
mNrmFxd_temp = aNrmGrid + cNrmFxd_temp
cFuncFxd_by_Share.append(
LinearInterp(np.insert(mNrmFxd_temp, 0, 0.0),
np.insert(cNrmFxd_temp, 0, 0.0)))
dvdsFuncFxd_by_Share.append(
LinearInterp(np.insert(mNrmFxd_temp, 0, 0.0),
np.insert(EndOfPrddvds[:, j], 0, EndOfPrddvds[0, j])))
cFuncFxd_now = LinearInterpOnInterp1D(cFuncFxd_by_Share, ShareGrid)
dvdsFuncFxd_now = LinearInterpOnInterp1D(dvdsFuncFxd_by_Share, ShareGrid)
# The share function when the agent can't adjust his portfolio is trivial
ShareFuncFxd_now = IdentityFunction(i_dim=1, n_dims=2)
# Construct the marginal value of mNrm function when the agent can't adjust his share
dvdmFuncFxd_now = MargValueFunc2D(cFuncFxd_now, CRRA)
# If the value function has been requested, construct it now
if vFuncBool:
# First, make an end-of-period value function over aNrm and Share
EndOfPrdvNvrsFunc = BilinearInterp(EndOfPrdvNvrs, aNrmGrid, ShareGrid)
EndOfPrdvFunc = ValueFunc2D(EndOfPrdvNvrsFunc, CRRA)
# Construct the value function when the agent can adjust his portfolio
mNrm_temp = aXtraGrid # Just use aXtraGrid as our grid of mNrm values
cNrm_temp = cFuncAdj_now(mNrm_temp)
aNrm_temp = mNrm_temp - cNrm_temp
Share_temp = ShareFuncAdj_now(mNrm_temp)
v_temp = u(cNrm_temp) + EndOfPrdvFunc(aNrm_temp, Share_temp)
vNvrs_temp = n(v_temp)
vNvrsP_temp = uP(cNrm_temp) * nP(v_temp)
vNvrsFuncAdj = CubicInterp(
np.insert(mNrm_temp, 0, 0.0), # x_list
np.insert(vNvrs_temp, 0, 0.0), # f_list
np.insert(vNvrsP_temp, 0, vNvrsP_temp[0])) # dfdx_list
vFuncAdj_now = ValueFunc(
vNvrsFuncAdj, CRRA) # Re-curve the pseudo-inverse value function
# Construct the value function when the agent *can't* adjust his portfolio
mNrm_temp = np.tile(np.reshape(aXtraGrid, (aXtraGrid.size, 1)),
(1, Share_N))
Share_temp = np.tile(np.reshape(ShareGrid, (1, Share_N)),
(aXtraGrid.size, 1))
cNrm_temp = cFuncFxd_now(mNrm_temp, Share_temp)
aNrm_temp = mNrm_temp - cNrm_temp
v_temp = u(cNrm_temp) + EndOfPrdvFunc(aNrm_temp, Share_temp)
vNvrs_temp = n(v_temp)
vNvrsP_temp = uP(cNrm_temp) * nP(v_temp)
vNvrsFuncFxd_by_Share = []
for j in range(Share_N):
vNvrsFuncFxd_by_Share.append(
CubicInterp(
np.insert(mNrm_temp[:, 0], 0, 0.0), # x_list
np.insert(vNvrs_temp[:, j], 0, 0.0), # f_list
np.insert(vNvrsP_temp[:, j], 0,
vNvrsP_temp[j, 0]))) #dfdx_list
vNvrsFuncFxd = LinearInterpOnInterp1D(vNvrsFuncFxd_by_Share, ShareGrid)
vFuncFxd_now = ValueFunc2D(vNvrsFuncFxd, CRRA)
else: # If vFuncBool is False, fill in dummy values
vFuncAdj_now = None
vFuncFxd_now = None
# Create and return this period's solution
return PortfolioSolution(cFuncAdj=cFuncAdj_now,
ShareFuncAdj=ShareFuncAdj_now,
vPfuncAdj=vPfuncAdj_now,
vFuncAdj=vFuncAdj_now,
cFuncFxd=cFuncFxd_now,
ShareFuncFxd=ShareFuncFxd_now,
dvdmFuncFxd=dvdmFuncFxd_now,
dvdsFuncFxd=dvdsFuncFxd_now,
vFuncFxd=vFuncFxd_now)
def postSolve(self):
self.solution_fast = deepcopy(self.solution)
if self.cycles == 0:
cycles = 1
else:
cycles = self.cycles
self.solution[-1] = self.solution_terminal_cs
for i in range(cycles):
for j in range(self.T_cycle):
solution = self.solution[i * self.T_cycle + j]
# Define the borrowing constraint (limiting consumption function)
cFuncNowCnst = LinearInterp(
np.array([solution.mNrmMin, solution.mNrmMin + 1]),
np.array([0.0, 1.0]),
)
"""
Constructs a basic solution for this period, including the consumption
function and marginal value function.
"""
if self.CubicBool:
# Makes a cubic spline interpolation of the unconstrained consumption
# function for this period.
cFuncNowUnc = CubicInterp(
solution.mNrm,
solution.cNrm,
solution.MPC,
solution.cFuncLimitIntercept,
solution.cFuncLimitSlope,
)
else:
# Makes a linear interpolation to represent the (unconstrained) consumption function.
# Construct the unconstrained consumption function
cFuncNowUnc = LinearInterp(
solution.mNrm,
solution.cNrm,
solution.cFuncLimitIntercept,
solution.cFuncLimitSlope,
)
# Combine the constrained and unconstrained functions into the true consumption function
cFuncNow = LowerEnvelope(cFuncNowUnc, cFuncNowCnst)
# Make the marginal value function and the marginal marginal value function
vPfuncNow = MargValueFunc(cFuncNow, self.CRRA)
# Pack up the solution and return it
consumer_solution = ConsumerSolution(
cFunc=cFuncNow,
vPfunc=vPfuncNow,
mNrmMin=solution.mNrmMin,
hNrm=solution.hNrm,
MPCmin=solution.MPCmin,
MPCmax=solution.MPCmax,
)
if self.vFuncBool:
vNvrsFuncNow = CubicInterp(
solution.mNrmGrid,
solution.vNvrs,
solution.vNvrsP,
solution.MPCminNvrs * solution.hNrm,
solution.MPCminNvrs,
)
vFuncNow = ValueFunc(vNvrsFuncNow, self.CRRA)
consumer_solution.vFunc = vFuncNow
if self.CubicBool or self.vFuncBool:
_searchFunc = (_searchSSfuncCubic
if self.CubicBool else _searchSSfuncLinear)
# Add mNrmSS to the solution and return it
consumer_solution.mNrmSS = _addSSmNrmIndNumba(
self.PermGroFac[j],
self.Rfree,
solution.ExIncNext,
solution.mNrmMin,
solution.mNrm,
solution.cNrm,
solution.MPC,
solution.MPCmin,
solution.hNrm,
_searchFunc,
)
self.solution[i * self.T_cycle + j] = consumer_solution