def defUtilityFuncs(self):
'''
Defines CRRA utility function for this period (and its derivatives,
and their inverses), saving them as attributes of self for other methods
to use.
Note - the input here is xi*c-v(L), not c itself (GHH preferences with
preference shock xi)
Parameters
----------
none
Returns
-------
none
'''
ConsPerfForesightSolver.defUtilityFuncs(self)
self.uPinv = lambda u: utilityP_inv(u, gam=self.CRRA)
self.uPinvP = lambda u: utilityP_invP(u, gam=self.CRRA)
self.uinvP = lambda u: utility_invP(u, gam=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)