def use_points_for_interpolation(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 = MargValueFuncCRRA(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 make_linear_cFunc(self, mLvl, pLvl, cLvl):
"""
Makes a quasi-bilinear interpolation to represent the (unconstrained)
consumption function.
Parameters
----------
mLvl : np.array
Market resource points for interpolation.
pLvl : np.array
Persistent income level points for interpolation.
cLvl : np.array
Consumption points for interpolation.
Returns
-------
cFuncUnc : LinearInterp
The unconstrained consumption function for this period.
"""
cFunc_by_pLvl_list = [] # list of consumption functions for each pLvl
for j in range(pLvl.shape[0]):
pLvl_j = pLvl[j, 0]
m_temp = mLvl[j, :] - self.BoroCnstNat(pLvl_j)
c_temp = cLvl[
j, :] # Make a linear consumption function for this pLvl
if pLvl_j > 0:
cFunc_by_pLvl_list.append(
LinearInterp(
m_temp,
c_temp,
lower_extrap=True,
slope_limit=self.MPCminNow,
intercept_limit=self.MPCminNow * self.hLvlNow(pLvl_j),
))
else:
cFunc_by_pLvl_list.append(
LinearInterp(m_temp, c_temp, lower_extrap=True))
pLvl_list = pLvl[:, 0]
cFuncUncBase = LinearInterpOnInterp1D(
cFunc_by_pLvl_list, pLvl_list) # Combine all linear cFuncs
cFuncUnc = VariableLowerBoundFunc2D(
cFuncUncBase, self.BoroCnstNat
) # Re-adjust for natural borrowing constraint (as lower bound)
return cFuncUnc
def make_cubic_cFunc(self, mLvl, pLvl, cLvl):
"""
Makes a quasi-cubic spline interpolation of the unconstrained consumption
function for this period. Function is cubic splines with respect to mLvl,
but linear in pLvl.
Parameters
----------
mLvl : np.array
Market resource points for interpolation.
pLvl : np.array
Persistent income level points for interpolation.
cLvl : np.array
Consumption points for interpolation.
Returns
-------
cFuncUnc : CubicInterp
The unconstrained consumption function for this period.
"""
# Calculate the MPC at each gridpoint
EndOfPrdvPP = (self.DiscFacEff * self.Rfree * self.Rfree * np.sum(
self.vPPfuncNext(self.mLvlNext, self.pLvlNext) * self.ShkPrbs_temp,
axis=0,
))
dcda = EndOfPrdvPP / self.uPP(np.array(cLvl[1:, 1:]))
MPC = dcda / (dcda + 1.0)
MPC = np.concatenate((np.reshape(MPC[:, 0], (MPC.shape[0], 1)), MPC),
axis=1)
# Stick an extra MPC value at bottom; MPCmax doesn't work
MPC = np.concatenate((self.MPCminNow * np.ones(
(1, self.aXtraGrid.size + 1)), MPC),
axis=0)
# Make cubic consumption function with respect to mLvl for each persistent income level
cFunc_by_pLvl_list = [] # list of consumption functions for each pLvl
for j in range(pLvl.shape[0]):
pLvl_j = pLvl[j, 0]
m_temp = mLvl[j, :] - self.BoroCnstNat(pLvl_j)
c_temp = cLvl[
j, :] # Make a cubic consumption function for this pLvl
MPC_temp = MPC[j, :]
if pLvl_j > 0:
cFunc_by_pLvl_list.append(
CubicInterp(
m_temp,
c_temp,
MPC_temp,
lower_extrap=True,
slope_limit=self.MPCminNow,
intercept_limit=self.MPCminNow * self.hLvlNow(pLvl_j),
))
else: # When pLvl=0, cFunc is linear
cFunc_by_pLvl_list.append(
LinearInterp(m_temp, c_temp, lower_extrap=True))
pLvl_list = pLvl[:, 0]
cFuncUncBase = LinearInterpOnInterp1D(
cFunc_by_pLvl_list, pLvl_list) # Combine all linear cFuncs
cFuncUnc = VariableLowerBoundFunc2D(cFuncUncBase, self.BoroCnstNat)
# Re-adjust for lower bound of natural borrowing constraint
return cFuncUnc
def make_vFunc(self, solution):
"""
Creates the value function for this period, defined over market resources
m and persistent income p. self must have the attribute EndOfPrdvFunc in
order to execute.
Parameters
----------
solution : ConsumerSolution
The solution to this single period problem, which must include the
consumption function.
Returns
-------
vFuncNow : ValueFuncCRRA
A representation of the value function for this period, defined over
market resources m and persistent income p: v = vFuncNow(m,p).
"""
mSize = self.aXtraGrid.size
pSize = self.pLvlGrid.size
# Compute expected value and marginal value on a grid of market resources
pLvl_temp = np.tile(self.pLvlGrid,
(mSize, 1)) # Tile pLvl across m values
mLvl_temp = (np.tile(self.mLvlMinNow(self.pLvlGrid), (mSize, 1)) +
np.tile(np.reshape(self.aXtraGrid, (mSize, 1)),
(1, pSize)) * pLvl_temp)
cLvlNow = solution.cFunc(mLvl_temp, pLvl_temp)
aLvlNow = mLvl_temp - cLvlNow
vNow = self.u(cLvlNow) + self.EndOfPrdvFunc(aLvlNow, pLvl_temp)
vPnow = self.uP(cLvlNow)
# Calculate pseudo-inverse value and its first derivative (wrt mLvl)
vNvrs = self.uinv(vNow) # value transformed through inverse utility
vNvrsP = vPnow * self.uinvP(vNow)
# Add data at the lower bound of m
mLvl_temp = np.concatenate((np.reshape(self.mLvlMinNow(self.pLvlGrid),
(1, pSize)), mLvl_temp),
axis=0)
vNvrs = np.concatenate((np.zeros((1, pSize)), vNvrs), axis=0)
vNvrsP = np.concatenate((np.reshape(vNvrsP[0, :],
(1, vNvrsP.shape[1])), vNvrsP),
axis=0)
# Add data at the lower bound of p
MPCminNvrs = self.MPCminNow**(-self.CRRA / (1.0 - self.CRRA))
m_temp = np.reshape(mLvl_temp[:, 0], (mSize + 1, 1))
mLvl_temp = np.concatenate((m_temp, mLvl_temp), axis=1)
vNvrs = np.concatenate((MPCminNvrs * m_temp, vNvrs), axis=1)
vNvrsP = np.concatenate((MPCminNvrs * np.ones((mSize + 1, 1)), vNvrsP),
axis=1)
# Construct the pseudo-inverse value function
vNvrsFunc_list = []
for j in range(pSize + 1):
pLvl = np.insert(self.pLvlGrid, 0, 0.0)[j]
vNvrsFunc_list.append(
CubicInterp(
mLvl_temp[:, j] - self.mLvlMinNow(pLvl),
vNvrs[:, j],
vNvrsP[:, j],
MPCminNvrs * self.hLvlNow(pLvl),
MPCminNvrs,
))
vNvrsFuncBase = LinearInterpOnInterp1D(
vNvrsFunc_list, np.insert(self.pLvlGrid, 0,
0.0)) # Value function "shifted"
vNvrsFuncNow = VariableLowerBoundFunc2D(vNvrsFuncBase, self.mLvlMinNow)
# "Re-curve" the pseudo-inverse value function into the value function
vFuncNow = ValueFuncCRRA(vNvrsFuncNow, self.CRRA)
return vFuncNow
def make_EndOfPrdvFunc(self, EndOfPrdvP):
"""
Construct the end-of-period value function for this period, storing it
as an attribute of self for use by other methods.
Parameters
----------
EndOfPrdvP : np.array
Array of end-of-period marginal value of assets corresponding to the
asset values in self.aLvlNow x self.pLvlGrid.
Returns
-------
none
"""
vLvlNext = self.vFuncNext(
self.mLvlNext,
self.pLvlNext) # value in many possible future states
EndOfPrdv = self.DiscFacEff * np.sum(
vLvlNext * self.ShkPrbs_temp,
axis=0) # expected value, averaging across states
EndOfPrdvNvrs = self.uinv(
EndOfPrdv) # value transformed through inverse utility
EndOfPrdvNvrsP = EndOfPrdvP * self.uinvP(EndOfPrdv)
# Add points at mLvl=zero
EndOfPrdvNvrs = np.concatenate((np.zeros(
(self.pLvlGrid.size, 1)), EndOfPrdvNvrs),
axis=1)
if hasattr(self, "MedShkDstn"):
EndOfPrdvNvrsP = np.concatenate((np.zeros(
(self.pLvlGrid.size, 1)), EndOfPrdvNvrsP),
axis=1)
else:
EndOfPrdvNvrsP = np.concatenate(
(
np.reshape(EndOfPrdvNvrsP[:, 0], (self.pLvlGrid.size, 1)),
EndOfPrdvNvrsP,
),
axis=1,
)
# This is a very good approximation, vNvrsPP = 0 at the asset minimum
aLvl_temp = np.concatenate(
(
np.reshape(self.BoroCnstNat(self.pLvlGrid),
(self.pLvlGrid.size, 1)),
self.aLvlNow,
),
axis=1,
)
# Make an end-of-period value function for each persistent income level in the grid
EndOfPrdvNvrsFunc_list = []
for p in range(self.pLvlGrid.size):
EndOfPrdvNvrsFunc_list.append(
CubicInterp(
aLvl_temp[p, :] - self.BoroCnstNat(self.pLvlGrid[p]),
EndOfPrdvNvrs[p, :],
EndOfPrdvNvrsP[p, :],
))
EndOfPrdvNvrsFuncBase = LinearInterpOnInterp1D(EndOfPrdvNvrsFunc_list,
self.pLvlGrid)
# Re-adjust the combined end-of-period value function to account for the natural borrowing constraint shifter
EndOfPrdvNvrsFunc = VariableLowerBoundFunc2D(EndOfPrdvNvrsFuncBase,
self.BoroCnstNat)
self.EndOfPrdvFunc = ValueFuncCRRA(EndOfPrdvNvrsFunc, self.CRRA)
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 solve_ConsLaborIntMarg(
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.flatten()
PermShkPrbs = PermShkDstn.pmf
PermShkVals = PermShkDstn.X.flatten()
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 = MargValueFuncCRRA(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 = MargValueFuncCRRA(vPnvrsFuncNow, CRRA)
# Make a solution object for this period and return it
solution = ConsumerLaborSolution(cFunc=cFuncNow,
LbrFunc=LbrFuncNow,
vPfunc=vPfuncNow,
bNrmMin=bNrmMinNow)
return solution
