def makeEndOfPrdvPfuncCond(self):
'''
Construct the end-of-period marginal value function conditional on next
period's state.
Parameters
----------
None
Returns
-------
EndofPrdvPfunc_cond : MargValueFunc
The end-of-period marginal value function conditional on a particular
state occuring in the succeeding period.
'''
# Get data to construct the end-of-period marginal value function (conditional on next state)
self.aNrm_cond = self.prepareToCalcEndOfPrdvP()
self.EndOfPrdvP_cond= self.calcEndOfPrdvPcond()
EndOfPrdvPnvrs_cond = self.uPinv(self.EndOfPrdvP_cond) # "decurved" marginal value
if self.CubicBool:
EndOfPrdvPP_cond = self.calcEndOfPrdvPP()
EndOfPrdvPnvrsP_cond = EndOfPrdvPP_cond*self.uPinvP(self.EndOfPrdvP_cond) # "decurved" marginal marginal value
# Construct the end-of-period marginal value function conditional on the next state.
if self.CubicBool:
EndOfPrdvPnvrsFunc_cond = CubicInterp(self.aNrm_cond,EndOfPrdvPnvrs_cond,
EndOfPrdvPnvrsP_cond,lower_extrap=True)
else:
EndOfPrdvPnvrsFunc_cond = LinearInterp(self.aNrm_cond,EndOfPrdvPnvrs_cond,
lower_extrap=True)
EndofPrdvPfunc_cond = MargValueFunc(EndOfPrdvPnvrsFunc_cond,self.CRRA) # "recurve" the interpolated marginal value function
return EndofPrdvPfunc_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 usePointsForInterpolation(self, cNrm, mNrm, interpolator):
"""
Make a basic solution object with a consumption function and marginal
value function (unconditional on the preference shock).
Parameters
----------
cNrm : np.array
Consumption points for interpolation.
mNrm : np.array
Corresponding market resource points for interpolation.
interpolator : function
A function that constructs and returns a consumption function.
Returns
-------
solution_now : ConsumerSolution
The solution to this period's consumption-saving problem, with a
consumption function, marginal value function, and minimum m.
"""
# Make the preference-shock specific consumption functions
PrefShkCount = self.PrefShkVals.size
cFunc_list = []
for j in range(PrefShkCount):
MPCmin_j = self.MPCminNow * self.PrefShkVals[j]**(1.0 / self.CRRA)
cFunc_this_shock = LowerEnvelope(
LinearInterp(
mNrm[j, :],
cNrm[j, :],
intercept_limit=self.hNrmNow * MPCmin_j,
slope_limit=MPCmin_j,
),
self.cFuncNowCnst,
)
cFunc_list.append(cFunc_this_shock)
# Combine the list of consumption functions into a single interpolation
cFuncNow = LinearInterpOnInterp1D(cFunc_list, self.PrefShkVals)
# Make the ex ante marginal value function (before the preference shock)
m_grid = self.aXtraGrid + self.mNrmMinNow
vP_vec = np.zeros_like(m_grid)
for j in range(
PrefShkCount): # numeric integration over the preference shock
vP_vec += (self.uP(cFunc_list[j](m_grid)) * self.PrefShkPrbs[j] *
self.PrefShkVals[j])
vPnvrs_vec = self.uPinv(vP_vec)
vPfuncNow = MargValueFunc(LinearInterp(m_grid, vPnvrs_vec), self.CRRA)
# Store the results in a solution object and return it
solution_now = ConsumerSolution(cFunc=cFuncNow,
vPfunc=vPfuncNow,
mNrmMin=self.mNrmMinNow)
return solution_now
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 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 solveConsMarkovALT(solution_next,IncomeDstn,LivPrb,DiscFac,CRRA,Rfree,PermGroFac,uPfac,
MrkvArray,BoroCnstArt,aXtraGrid,vFuncBool,CubicBool):
'''
Solves a single period consumption-saving problem with risky income and
stochastic transitions between discrete states, in a Markov fashion. Has
identical inputs as solveConsIndShock, except for a discrete
Markov transitionrule MrkvArray. Markov states can differ in their interest
factor, permanent growth factor, and income distribution, so the inputs Rfree,
PermGroFac, and IncomeDstn are arrays or lists specifying those values in each
(succeeding) Markov state.
Parameters
----------
solution_next : ConsumerSolution
The solution to next period's one period problem.
IncomeDstn : DiscreteDistribution
A representation of permanent and transitory income shocks that might
arrive at the beginning of next period.
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 : np.array
Risk free interest factor on end-of-period assets for each Markov
state in the succeeding period.
PermGroFac : np.array
Expected permanent income growth factor at the end of this period
for each Markov state in the succeeding period.
uPfac : np.array
Scaling factor for (marginal) utility in each current Markov state.
MrkvArray : np.array
An NxN array representing a Markov transition matrix between discrete
states. The i,j-th element of MrkvArray is the probability of
moving from state i in period t to state j in period t+1.
BoroCnstArt: float or None
Borrowing constraint for the minimum allowable assets to end the
period with. If it is less than the natural borrowing constraint,
then it is irrelevant; BoroCnstArt=None indicates no artificial bor-
rowing constraint.
aXtraGrid: np.array
Array of "extra" end-of-period asset values-- assets above the
absolute minimum acceptable level.
vFuncBool: boolean
An indicator for whether the value function should be computed and
included in the reported solution. Not used.
CubicBool: boolean
An indicator for whether the solver should use cubic or linear inter-
polation. Not used.
Returns
-------
solution : ConsumerSolution
The solution to the single period consumption-saving problem. Includes
a consumption function cFunc (using cubic or linear splines), a marg-
inal value function vPfunc, a minimum acceptable level of normalized
market resources mNrmMin. All of these attributes are lists or arrays,
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.
'''
# Get sizes of grids
aCount = aXtraGrid.size
StateCount = MrkvArray.shape[0]
# Loop through next period's states, assuming we reach each one at a time.
# Construct EndOfPrdvP_cond functions for each state.
BoroCnstNat_cond = []
EndOfPrdvPfunc_cond = []
for j in range(StateCount):
# Unpack next period's solution
vPfuncNext = solution_next.vPfunc[j]
mNrmMinNext = solution_next.mNrmMin[j]
# Unpack the income shocks
ShkPrbsNext = IncomeDstn[j].pmf
PermShkValsNext = IncomeDstn[j].X[0]
TranShkValsNext = IncomeDstn[j].X[1]
ShkCount = ShkPrbsNext.size
aXtra_tiled = np.tile(np.reshape(aXtraGrid, (aCount, 1)), (1, ShkCount))
# Make tiled versions of the income shocks
# Dimension order: aNow, Shk
ShkPrbsNext_tiled = np.tile(np.reshape(ShkPrbsNext, (1, ShkCount)), (aCount, 1))
PermShkValsNext_tiled = np.tile(np.reshape(PermShkValsNext, (1, ShkCount)), (aCount, 1))
TranShkValsNext_tiled = np.tile(np.reshape(TranShkValsNext, (1, ShkCount)), (aCount, 1))
# Find the natural borrowing constraint
aNrmMin_candidates = PermGroFac[j]*PermShkValsNext_tiled/Rfree[j]*(mNrmMinNext - TranShkValsNext_tiled[0, :])
aNrmMin = np.max(aNrmMin_candidates)
BoroCnstNat_cond.append(aNrmMin)
# Calculate market resources next period (and a constant array of capital-to-labor ratio)
aNrmNow_tiled = aNrmMin + aXtra_tiled
mNrmNext_array = Rfree[j]*aNrmNow_tiled/PermShkValsNext_tiled + TranShkValsNext_tiled
# Find marginal value next period at every income shock realization and every aggregate market resource gridpoint
vPnext_array = Rfree[j]*PermShkValsNext_tiled**(-CRRA)*vPfuncNext(mNrmNext_array)
# Calculate expectated marginal value at the end of the period at every asset gridpoint
EndOfPrdvP = DiscFac*np.sum(vPnext_array*ShkPrbsNext_tiled, axis=1)
# Make the conditional end-of-period marginal value function
EndOfPrdvPnvrs = EndOfPrdvP**(-1./CRRA)
EndOfPrdvPnvrsFunc = LinearInterp(np.insert(aNrmMin + aXtraGrid, 0, aNrmMin), np.insert(EndOfPrdvPnvrs, 0, 0.0))
EndOfPrdvPfunc_cond.append(MargValueFunc(EndOfPrdvPnvrsFunc, CRRA))
# Now loop through *this* period's discrete states, calculating end-of-period
# marginal value (weighting across state transitions), then construct consumption
# and marginal value function for each state.
cFuncNow = []
vPfuncNow = []
mNrmMinNow = []
for i in range(StateCount):
# Find natural borrowing constraint for this state
aNrmMin_candidates = np.zeros(StateCount) + np.nan
for j in range(StateCount):
if MrkvArray[i, j] > 0.: # Irrelevant if transition is impossible
aNrmMin_candidates[j] = BoroCnstNat_cond[j]
aNrmMin = np.nanmax(aNrmMin_candidates)
# Find the minimum allowable market resources
if BoroCnstArt is not None:
mNrmMin = np.maximum(BoroCnstArt, aNrmMin)
else:
mNrmMin = aNrmMin
mNrmMinNow.append(mNrmMin)
# Make tiled grid of aNrm
aNrmNow = aNrmMin + aXtraGrid
# Loop through feasible transitions and calculate end-of-period marginal value
EndOfPrdvP = np.zeros(aCount)
for j in range(StateCount):
if MrkvArray[i, j] > 0.:
temp = MrkvArray[i, j]*EndOfPrdvPfunc_cond[j](aNrmNow)
EndOfPrdvP += temp
EndOfPrdvP *= LivPrb[i] # Account for survival out of the current state
# Calculate consumption and the endogenous mNrm gridpoints for this state
cNrmNow = (EndOfPrdvP/uPfac[i])**(-1./CRRA)
mNrmNow = aNrmNow + cNrmNow
# Make a piecewise linear consumption function
c_temp = np.insert(cNrmNow, 0, 0.0) # Add point at bottom
m_temp = np.insert(mNrmNow, 0, aNrmMin)
cFuncUnc = LinearInterp(m_temp, c_temp)
cFuncCnst = LinearInterp(np.array([mNrmMin, mNrmMin+1.0]), np.array([0.0, 1.0]))
cFuncNow.append(LowerEnvelope(cFuncUnc,cFuncCnst))
# Construct the marginal value function using the envelope condition
m_temp = aXtraGrid + mNrmMin
c_temp = cFuncNow[i](m_temp)
uP = uPfac[i]*c_temp**(-CRRA)
vPnvrs = uP**(-1./CRRA)
vPnvrsFunc = LinearInterp(np.insert(m_temp, 0, mNrmMin), np.insert(vPnvrs, 0, 0.0))
vPfuncNow.append(MargValueFunc(vPnvrsFunc, CRRA))
# Pack up and return the solution
solution_now = ConsumerSolution(cFunc=cFuncNow, vPfunc=vPfuncNow, mNrmMin=mNrmMinNow)
return solution_now
def makeSolution(self,cNrm,mNrm):
'''
Construct an object representing the solution to this period's problem.
Parameters
----------
cNrm : np.array
Array of normalized consumption values for interpolation. Each row
corresponds to a Markov state for this period.
mNrm : np.array
Array of normalized market resource values for interpolation. Each
row corresponds to a Markov state for this period.
Returns
-------
solution : ConsumerSolution
The solution to the single period consumption-saving problem. Includes
a consumption function cFunc (using cubic or linear splines), a marg-
inal value function vPfunc, a minimum acceptable level of normalized
market resources mNrmMin, normalized human wealth hNrm, and bounding
MPCs MPCmin and MPCmax. It might also have a value function vFunc
and marginal marginal value function vPPfunc. All of these attributes
are lists or arrays, 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.
'''
solution = ConsumerSolution() # An empty solution to which we'll add state-conditional solutions
# Calculate the MPC at each market resource gridpoint in each state (if desired)
if self.CubicBool:
dcda = self.EndOfPrdvPP/self.uPP(np.array(self.cNrmNow))
MPC = dcda/(dcda+1.0)
self.MPC_temp = np.hstack((np.reshape(self.MPCmaxNow,(self.StateCount,1)),MPC))
interpfunc = self.makeCubiccFunc
else:
interpfunc = self.makeLinearcFunc
# Loop through each current period state and add its solution to the overall solution
for i in range(self.StateCount):
# Set current-period-conditional human wealth and MPC bounds
self.hNrmNow_j = self.hNrmNow[i]
self.MPCminNow_j = self.MPCminNow[i]
if self.CubicBool:
self.MPC_temp_j = self.MPC_temp[i,:]
# Construct the consumption function by combining the constrained and unconstrained portions
self.cFuncNowCnst = LinearInterp([self.mNrmMin_list[i], self.mNrmMin_list[i]+1.0],
[0.0,1.0])
cFuncNowUnc = interpfunc(mNrm[i,:],cNrm[i,:])
cFuncNow = LowerEnvelope(cFuncNowUnc,self.cFuncNowCnst)
# Make the marginal value function and pack up the current-state-conditional solution
vPfuncNow = MargValueFunc(cFuncNow,self.CRRA)
solution_cond = ConsumerSolution(cFunc=cFuncNow, vPfunc=vPfuncNow,
mNrmMin=self.mNrmMinNow)
if self.CubicBool: # Add the state-conditional marginal marginal value function (if desired)
solution_cond = self.addvPPfunc(solution_cond)
# Add the current-state-conditional solution to the overall period solution
solution.appendSolution(solution_cond)
# Add the lower bounds of market resources, MPC limits, human resources,
# and the value functions to the overall solution
solution.mNrmMin = self.mNrmMin_list
solution = self.addMPCandHumanWealth(solution)
if self.vFuncBool:
vFuncNow = self.makevFunc(solution)
solution.vFunc = vFuncNow
# Return the overall solution to this period
return solution
def solveConsRepAgentMarkov(solution_next,MrkvArray,DiscFac,CRRA,IncomeDstn,CapShare,DeprFac,PermGroFac,aXtraGrid):
'''
Solve one period of the simple representative agent consumption-saving model.
This version supports a discrete Markov process.
Parameters
----------
solution_next : ConsumerSolution
Solution to the next period's problem (i.e. previous iteration).
MrkvArray : np.array
Markov transition array between this period and next period.
DiscFac : float
Intertemporal discount factor for future utility.
CRRA : float
Coefficient of relative risk aversion.
IncomeDstn : [[np.array]]
A list of lists containing three arrays of floats, representing a discrete
approximation to the income process between the period being solved
and the one immediately following (in solution_next). Order: event
probabilities, permanent shocks, transitory shocks.
CapShare : float
Capital's share of income in Cobb-Douglas production function.
DeprFac : float
Depreciation rate of capital.
PermGroFac : [float]
Expected permanent income growth factor for each state we could be in
next period.
aXtraGrid : np.array
Array of "extra" end-of-period asset values-- assets above the
absolute minimum acceptable level. In this model, the minimum acceptable
level is always zero.
Returns
-------
solution_now : ConsumerSolution
Solution to this period's problem (new iteration).
'''
# Define basic objects
StateCount = MrkvArray.shape[0]
aNrmNow = aXtraGrid
aNrmCount = aNrmNow.size
EndOfPrdvP_cond = np.zeros((StateCount,aNrmCount)) + np.nan
# Loop over *next period* states, calculating conditional EndOfPrdvP
for j in range(StateCount):
# Define next-period-state conditional objects
vPfuncNext = solution_next.vPfunc[j]
ShkPrbsNext = IncomeDstn[j].pmf
PermShkValsNext = IncomeDstn[j].X[0]
TranShkValsNext = IncomeDstn[j].X[1]
# Make tiled versions of end-of-period assets, shocks, and probabilities
ShkCount = ShkPrbsNext.size
aNrm_tiled = np.tile(np.reshape(aNrmNow,(aNrmCount,1)),(1,ShkCount))
# Tile arrays of the income shocks and put them into useful shapes
PermShkVals_tiled = np.tile(np.reshape(PermShkValsNext,(1,ShkCount)),(aNrmCount,1))
TranShkVals_tiled = np.tile(np.reshape(TranShkValsNext,(1,ShkCount)),(aNrmCount,1))
ShkPrbs_tiled = np.tile(np.reshape(ShkPrbsNext,(1,ShkCount)),(aNrmCount,1))
# Calculate next period's capital-to-permanent-labor ratio under each combination
# of end-of-period assets and shock realization
kNrmNext = aNrm_tiled/(PermGroFac[j]*PermShkVals_tiled)
# Calculate next period's market resources
KtoLnext = kNrmNext/TranShkVals_tiled
RfreeNext = 1. - DeprFac + CapShare*KtoLnext**(CapShare-1.)
wRteNext = (1.-CapShare)*KtoLnext**CapShare
mNrmNext = RfreeNext*kNrmNext + wRteNext*TranShkVals_tiled
# Calculate end-of-period marginal value of assets for the RA
vPnext = vPfuncNext(mNrmNext)
EndOfPrdvP_cond[j,:] = DiscFac*np.sum(RfreeNext*(PermGroFac[j]*PermShkVals_tiled)**(-CRRA)*vPnext*ShkPrbs_tiled,axis=1)
# Apply the Markov transition matrix to get unconditional end-of-period marginal value
EndOfPrdvP = np.dot(MrkvArray,EndOfPrdvP_cond)
# Construct the consumption function and marginal value function for each discrete state
cFuncNow_list = []
vPfuncNow_list = []
for i in range(StateCount):
# Invert the first order condition to get consumption, then find endogenous gridpoints
cNrmNow = EndOfPrdvP[i,:]**(-1./CRRA)
mNrmNow = aNrmNow + cNrmNow
# Construct the consumption function and the marginal value function
cFuncNow_list.append(LinearInterp(np.insert(mNrmNow,0,0.0),np.insert(cNrmNow,0,0.0)))
vPfuncNow_list.append(MargValueFunc(cFuncNow_list[-1],CRRA))
# Construct and return the solution for this period
solution_now = ConsumerSolution(cFunc=cFuncNow_list,vPfunc=vPfuncNow_list)
return solution_now
def solveConsRepAgent(solution_next,DiscFac,CRRA,IncomeDstn,CapShare,DeprFac,PermGroFac,aXtraGrid):
'''
Solve one period of the simple representative agent consumption-saving model.
Parameters
----------
solution_next : ConsumerSolution
Solution to the next period's problem (i.e. previous iteration).
DiscFac : float
Intertemporal discount factor for future utility.
CRRA : float
Coefficient of relative risk aversion.
IncomeDstn : [np.array]
A list containing three arrays of floats, representing a discrete
approximation to the income process between the period being solved
and the one immediately following (in solution_next). Order: event
probabilities, permanent shocks, transitory shocks.
CapShare : float
Capital's share of income in Cobb-Douglas production function.
DeprFac : float
Depreciation rate of capital.
PermGroFac : float
Expected permanent income growth factor at the end of this period.
aXtraGrid : np.array
Array of "extra" end-of-period asset values-- assets above the
absolute minimum acceptable level. In this model, the minimum acceptable
level is always zero.
Returns
-------
solution_now : ConsumerSolution
Solution to this period's problem (new iteration).
'''
# Unpack next period's solution and the income distribution
vPfuncNext = solution_next.vPfunc
ShkPrbsNext = IncomeDstn.pmf
PermShkValsNext = IncomeDstn.X[0]
TranShkValsNext = IncomeDstn.X[1]
# Make tiled versions of end-of-period assets, shocks, and probabilities
aNrmNow = aXtraGrid
aNrmCount = aNrmNow.size
ShkCount = ShkPrbsNext.size
aNrm_tiled = np.tile(np.reshape(aNrmNow,(aNrmCount,1)),(1,ShkCount))
# Tile arrays of the income shocks and put them into useful shapes
PermShkVals_tiled = np.tile(np.reshape(PermShkValsNext,(1,ShkCount)),(aNrmCount,1))
TranShkVals_tiled = np.tile(np.reshape(TranShkValsNext,(1,ShkCount)),(aNrmCount,1))
ShkPrbs_tiled = np.tile(np.reshape(ShkPrbsNext,(1,ShkCount)),(aNrmCount,1))
# Calculate next period's capital-to-permanent-labor ratio under each combination
# of end-of-period assets and shock realization
kNrmNext = aNrm_tiled/(PermGroFac*PermShkVals_tiled)
# Calculate next period's market resources
KtoLnext = kNrmNext/TranShkVals_tiled
RfreeNext = 1. - DeprFac + CapShare*KtoLnext**(CapShare-1.)
wRteNext = (1.-CapShare)*KtoLnext**CapShare
mNrmNext = RfreeNext*kNrmNext + wRteNext*TranShkVals_tiled
# Calculate end-of-period marginal value of assets for the RA
vPnext = vPfuncNext(mNrmNext)
EndOfPrdvP = DiscFac*np.sum(RfreeNext*(PermGroFac*PermShkVals_tiled)**(-CRRA)*vPnext*ShkPrbs_tiled,axis=1)
# Invert the first order condition to get consumption, then find endogenous gridpoints
cNrmNow = EndOfPrdvP**(-1./CRRA)
mNrmNow = aNrmNow + cNrmNow
# Construct the consumption function and the marginal value function
cFuncNow = LinearInterp(np.insert(mNrmNow,0,0.0),np.insert(cNrmNow,0,0.0))
vPfuncNow = MargValueFunc(cFuncNow,CRRA)
# Construct and return the solution for this period
solution_now = ConsumerSolution(cFunc=cFuncNow,vPfunc=vPfuncNow)
return solution_now
def makeBasicSolution(self, EndOfPrdvP, aNrm, wageShkVals, prefShkVals):
'''
Given end of period assets and end of period marginal value, construct
the basic solution for this period.
Parameters
----------
EndOfPrdvP : np.array
Array of end-of-period marginal values.
aNrm : np.array
Array of end-of-period asset values that yield the marginal values
in EndOfPrdvP.
wageShkVals : np.array
Array of this period transitory wage shock values.
prefShkVals : np.array
Array of this period preference shock values.
Returns
-------
solution_now : ConsumerSolution
The solution to this period's consumption-saving problem, with a
consumption function, marginal value function.
'''
num_pref_shocks = len(prefShkVals)
num_wage_shocks = len(wageShkVals)
cFuncBaseByPref_list = []
vPFuncBaseByPref_list = []
lFuncBaseByPref_list = []
for i in range(num_wage_shocks):
cFuncBaseByPref_list.append([])
vPFuncBaseByPref_list.append([])
lFuncBaseByPref_list.append([])
for j in range(num_pref_shocks):
c_temp = self.uPinv(EndOfPrdvP / prefShkVals[j])
l_temp = self.LabSupply(wageShkVals[i] * EndOfPrdvP)
b_temp = c_temp + aNrm - l_temp * wageShkVals[i]
if wageShkVals[i] == 0.0:
c_temp = np.insert(c_temp, 0, 0., axis=-1)
l_temp = np.insert(l_temp, 0, 0.0, axis=-1)
b_temp = np.insert(b_temp, 0, 0.0, axis=-1)
lFuncBaseByPref_list[i].append(
LinearInterp(b_temp, l_temp, lower_extrap=True))
cFunc1 = LinearInterp(b_temp, c_temp, lower_extrap=True)
cFunc2 = LinearInterp(b_temp,
l_temp * wageShkVals[i] + b_temp,
lower_extrap=True)
cFuncBaseByPref_list[i].append(LowerEnvelope(cFunc1, cFunc2))
pseudo_inverse_vPfunc1 = LinearInterp(
b_temp,
prefShkVals[j]**(-1.0 / self.CRRA) * c_temp,
lower_extrap=True)
pseudo_inverse_vPfunc2 = LinearInterp(
b_temp,
prefShkVals[j]**(-1.0 / self.CRRA) *
(l_temp * wageShkVals[i] + b_temp),
lower_extrap=True)
pseudo_inverse_vPfunc = LowerEnvelope(pseudo_inverse_vPfunc1,
pseudo_inverse_vPfunc2)
vPFuncBaseByPref_list[i].append(
MargValueFunc(pseudo_inverse_vPfunc, self.CRRA))
cFuncNow = BilinearInterpOnInterp1D(cFuncBaseByPref_list, wageShkVals,
prefShkVals)
vPfuncNow = BilinearInterpOnInterp1D(vPFuncBaseByPref_list,
wageShkVals, prefShkVals)
lFuncNow = BilinearInterpOnInterp1D(lFuncBaseByPref_list, wageShkVals,
prefShkVals)
# Pack up and return the solution
solution_now = ConsumerSolution(cFunc=cFuncNow, vPfunc=vPfuncNow)
solution_now.lFunc = lFuncNow
return solution_now
def solveConsLaborIntMarg(
solution_next,
PermShkDstn,
TranShkDstn,
LivPrb,
DiscFac,
CRRA,
Rfree,
PermGroFac,
BoroCnstArt,
aXtraGrid,
TranShkGrid,
vFuncBool,
CubicBool,
WageRte,
LbrCost,
):
"""
Solves one period of the consumption-saving model with endogenous labor supply
on the intensive margin by using the endogenous grid method to invert the first
order conditions for optimal composite consumption and between consumption and
leisure, obviating any search for optimal controls.
Parameters
----------
solution_next : ConsumerLaborSolution
The solution to the next period's problem; must have the attributes
vPfunc and bNrmMinFunc representing marginal value of bank balances and
minimum (normalized) bank balances as a function of the transitory shock.
PermShkDstn: [np.array]
Discrete distribution of permanent productivity shocks.
TranShkDstn: [np.array]
Discrete distribution of transitory productivity shocks.
LivPrb : float
Survival probability; likelihood of being alive at the beginning of
the succeeding period.
DiscFac : float
Intertemporal discount factor.
CRRA : float
Coefficient of relative risk aversion over the composite good.
Rfree : float
Risk free interest rate on assets retained at the end of the period.
PermGroFac : float
Expected permanent income growth factor for next period.
BoroCnstArt: float or None
Borrowing constraint for the minimum allowable assets to end the
period with. Currently not handled, must be None.
aXtraGrid: np.array
Array of "extra" end-of-period asset values-- assets above the
absolute minimum acceptable level.
TranShkGrid: np.array
Grid of transitory shock values to use as a state grid for interpolation.
vFuncBool: boolean
An indicator for whether the value function should be computed and
included in the reported solution. Not yet handled, must be False.
CubicBool: boolean
An indicator for whether the solver should use cubic or linear interpolation.
Cubic interpolation is not yet handled, must be False.
WageRte: float
Wage rate per unit of labor supplied.
LbrCost: float
Cost parameter for supplying labor: u_t = U(x_t), x_t = c_t*z_t^LbrCost,
where z_t is leisure = 1 - Lbr_t.
Returns
-------
solution_now : ConsumerLaborSolution
The solution to this period's problem, including a consumption function
cFunc, a labor supply function LbrFunc, and a marginal value function vPfunc;
each are defined over normalized bank balances and transitory prod shock.
Also includes bNrmMinNow, the minimum permissible bank balances as a function
of the transitory productivity shock.
"""
# Make sure the inputs for this period are valid: CRRA > LbrCost/(1+LbrCost)
# and CubicBool = False. CRRA condition is met automatically when CRRA >= 1.
frac = 1.0 / (1.0 + LbrCost)
if CRRA <= frac * LbrCost:
print(
"Error: make sure CRRA coefficient is strictly greater than alpha/(1+alpha)."
)
sys.exit()
if BoroCnstArt is not None:
print(
"Error: Model cannot handle artificial borrowing constraint yet. ")
sys.exit()
if vFuncBool or CubicBool is True:
print("Error: Model cannot handle cubic interpolation yet.")
sys.exit()
# Unpack next period's solution and the productivity shock distribution, and define the inverse (marginal) utilty function
vPfunc_next = solution_next.vPfunc
TranShkPrbs = TranShkDstn.pmf
TranShkVals = TranShkDstn.X
PermShkPrbs = PermShkDstn.pmf
PermShkVals = PermShkDstn.X
TranShkCount = TranShkPrbs.size
PermShkCount = PermShkPrbs.size
uPinv = lambda X: CRRAutilityP_inv(X, gam=CRRA)
# Make tiled versions of the grid of a_t values and the components of the shock distribution
aXtraCount = aXtraGrid.size
bNrmGrid = aXtraGrid # Next period's bank balances before labor income
# Replicated axtraGrid of b_t values (bNowGrid) for each transitory (productivity) shock
bNrmGrid_rep = np.tile(np.reshape(bNrmGrid, (aXtraCount, 1)),
(1, TranShkCount))
# Replicated transitory shock values for each a_t state
TranShkVals_rep = np.tile(np.reshape(TranShkVals, (1, TranShkCount)),
(aXtraCount, 1))
# Replicated transitory shock probabilities for each a_t state
TranShkPrbs_rep = np.tile(np.reshape(TranShkPrbs, (1, TranShkCount)),
(aXtraCount, 1))
# Construct a function that gives marginal value of next period's bank balances *just before* the transitory shock arrives
# Next period's marginal value at every transitory shock and every bank balances gridpoint
vPNext = vPfunc_next(bNrmGrid_rep, TranShkVals_rep)
# Integrate out the transitory shocks (in TranShkVals direction) to get expected vP just before the transitory shock
vPbarNext = np.sum(vPNext * TranShkPrbs_rep, axis=1)
# Transformed marginal value through the inverse marginal utility function to "decurve" it
vPbarNvrsNext = uPinv(vPbarNext)
# Linear interpolation over b_{t+1}, adding a point at minimal value of b = 0.
vPbarNvrsFuncNext = LinearInterp(np.insert(bNrmGrid, 0, 0.0),
np.insert(vPbarNvrsNext, 0, 0.0))
# "Recurve" the intermediate marginal value function through the marginal utility function
vPbarFuncNext = MargValueFunc(vPbarNvrsFuncNext, CRRA)
# Get next period's bank balances at each permanent shock from each end-of-period asset values
# Replicated grid of a_t values for each permanent (productivity) shock
aNrmGrid_rep = np.tile(np.reshape(aXtraGrid, (aXtraCount, 1)),
(1, PermShkCount))
# Replicated permanent shock values for each a_t value
PermShkVals_rep = np.tile(np.reshape(PermShkVals, (1, PermShkCount)),
(aXtraCount, 1))
# Replicated permanent shock probabilities for each a_t value
PermShkPrbs_rep = np.tile(np.reshape(PermShkPrbs, (1, PermShkCount)),
(aXtraCount, 1))
bNrmNext = (Rfree / (PermGroFac * PermShkVals_rep)) * aNrmGrid_rep
# Calculate marginal value of end-of-period assets at each a_t gridpoint
# Get marginal value of bank balances next period at each shock
vPbarNext = (PermGroFac *
PermShkVals_rep)**(-CRRA) * vPbarFuncNext(bNrmNext)
# Take expectation across permanent income shocks
EndOfPrdvP = (DiscFac * Rfree * LivPrb *
np.sum(vPbarNext * PermShkPrbs_rep, axis=1, keepdims=True))
# Compute scaling factor for each transitory shock
TranShkScaleFac_temp = (frac * (WageRte * TranShkGrid)**(LbrCost * frac) *
(LbrCost**(-LbrCost * frac) + LbrCost**frac))
# Flip it to be a row vector
TranShkScaleFac = np.reshape(TranShkScaleFac_temp, (1, TranShkGrid.size))
# Use the first order condition to compute an array of "composite good" x_t values corresponding to (a_t,theta_t) values
xNow = (np.dot(EndOfPrdvP,
TranShkScaleFac))**(-1.0 / (CRRA - LbrCost * frac))
# Transform the composite good x_t values into consumption c_t and leisure z_t values
TranShkGrid_rep = np.tile(np.reshape(TranShkGrid, (1, TranShkGrid.size)),
(aXtraCount, 1))
xNowPow = xNow**frac # Will use this object multiple times in math below
# Find optimal consumption from optimal composite good
cNrmNow = ((
(WageRte * TranShkGrid_rep) / LbrCost)**(LbrCost * frac)) * xNowPow
# Find optimal leisure from optimal composite good
LsrNow = (LbrCost / (WageRte * TranShkGrid_rep))**frac * xNowPow
# The zero-th transitory shock is TranShk=0, and the solution is to not work: Lsr = 1, Lbr = 0.
cNrmNow[:, 0] = uPinv(EndOfPrdvP.flatten())
LsrNow[:, 0] = 1.0
# Agent cannot choose to work a negative amount of time. When this occurs, set
# leisure to one and recompute consumption using simplified first order condition.
# Find where labor would be negative if unconstrained
violates_labor_constraint = (LsrNow > 1.0)
EndOfPrdvP_temp = np.tile(np.reshape(EndOfPrdvP, (aXtraCount, 1)),
(1, TranShkCount))
cNrmNow[violates_labor_constraint] = uPinv(
EndOfPrdvP_temp[violates_labor_constraint])
LsrNow[violates_labor_constraint] = 1.0 # Set up z=1, upper limit
# Calculate the endogenous bNrm states by inverting the within-period transition
aNrmNow_rep = np.tile(np.reshape(aXtraGrid, (aXtraCount, 1)),
(1, TranShkGrid.size))
bNrmNow = aNrmNow_rep - WageRte * TranShkGrid_rep + cNrmNow + WageRte * TranShkGrid_rep * LsrNow
# Add an extra gridpoint at the absolute minimal valid value for b_t for each TranShk;
# this corresponds to working 100% of the time and consuming nothing.
bNowArray = np.concatenate((np.reshape(-WageRte * TranShkGrid,
(1, TranShkGrid.size)), bNrmNow),
axis=0)
# Consume nothing
cNowArray = np.concatenate((np.zeros((1, TranShkGrid.size)), cNrmNow),
axis=0)
# And no leisure!
LsrNowArray = np.concatenate((np.zeros((1, TranShkGrid.size)), LsrNow),
axis=0)
LsrNowArray[0, 0] = 1.0 # Don't work at all if TranShk=0, even if bNrm=0
LbrNowArray = 1.0 - LsrNowArray # Labor is the complement of leisure
# Get (pseudo-inverse) marginal value of bank balances using end of period
# marginal value of assets (envelope condition), adding a column of zeros
# zeros on the left edge, representing the limit at the minimum value of b_t.
vPnvrsNowArray = np.concatenate((np.zeros(
(1, TranShkGrid.size)), uPinv(EndOfPrdvP_temp)))
# Construct consumption and marginal value functions for this period
bNrmMinNow = LinearInterp(TranShkGrid, bNowArray[0, :])
# Loop over each transitory shock and make a linear interpolation to get lists
# of optimal consumption, labor and (pseudo-inverse) marginal value by TranShk
cFuncNow_list = []
LbrFuncNow_list = []
vPnvrsFuncNow_list = []
for j in range(TranShkGrid.size):
# Adjust bNrmNow for this transitory shock, so bNrmNow_temp[0] = 0
bNrmNow_temp = (bNowArray[:, j] - bNowArray[0, j])
# Make consumption function for this transitory shock
cFuncNow_list.append(LinearInterp(bNrmNow_temp, cNowArray[:, j]))
# Make labor function for this transitory shock
LbrFuncNow_list.append(LinearInterp(bNrmNow_temp, LbrNowArray[:, j]))
# Make pseudo-inverse marginal value function for this transitory shock
vPnvrsFuncNow_list.append(
LinearInterp(bNrmNow_temp, vPnvrsNowArray[:, j]))
# Make linear interpolation by combining the lists of consumption, labor and marginal value functions
cFuncNowBase = LinearInterpOnInterp1D(cFuncNow_list, TranShkGrid)
LbrFuncNowBase = LinearInterpOnInterp1D(LbrFuncNow_list, TranShkGrid)
vPnvrsFuncNowBase = LinearInterpOnInterp1D(vPnvrsFuncNow_list, TranShkGrid)
# Construct consumption, labor, pseudo-inverse marginal value functions with
# bNrmMinNow as the lower bound. This removes the adjustment in the loop above.
cFuncNow = VariableLowerBoundFunc2D(cFuncNowBase, bNrmMinNow)
LbrFuncNow = VariableLowerBoundFunc2D(LbrFuncNowBase, bNrmMinNow)
vPnvrsFuncNow = VariableLowerBoundFunc2D(vPnvrsFuncNowBase, bNrmMinNow)
# Construct the marginal value function by "recurving" its pseudo-inverse
vPfuncNow = MargValueFunc2D(vPnvrsFuncNow, CRRA)
# Make a solution object for this period and return it
solution = ConsumerLaborSolution(cFunc=cFuncNow,
LbrFunc=LbrFuncNow,
vPfunc=vPfuncNow,
bNrmMin=bNrmMinNow)
return solution
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