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
# Model parameters # Parameters that need to be fixed # Relative risk aversion. This is fixed at 2 in order to mantain # the analytical solution that we use, from Carroll (2000) CRRA = 2 # Parameters that can be changed. w = 1 # Deterministic wage per period. willCstFac = 0.35 # Fraction of resources charged by lawyer for writing a will. DiscFac = 0.98 # Time-discount factor. # Define utility (and related) functions u = lambda x: CRRAutility(x,CRRA) uP = lambda x: CRRAutilityP(x, CRRA) uPinv = lambda x: CRRAutilityP_inv(x, CRRA) # Create a grid for market resources mGrid = (aGrid-aGrid[0])*1.5 mGridPlots = np.linspace(w,10*w,100) mGridPlotsC = np.insert(mGridPlots,0,0) # Transformations for value funtion interpolation vTransf = lambda x: np.exp(x) vUntransf = lambda x: np.log(x) # %% [markdown] # # The third (last) period of life # # In the last period of life, the agent's problem is determined by his total amount of resources $m_3$ and a state variable $W$ that indicates whether he wrote a will ($W=1$) or not ($W=0$). #
def update_solution_terminal(self):
"""
Updates the terminal period solution and solves for optimal consumption
and labor when there is no future.
Parameters
----------
None
Returns
-------
None
"""
t = -1
TranShkGrid = self.TranShkGrid[t]
LbrCost = self.LbrCost[t]
WageRte = self.WageRte[t]
bNrmGrid = np.insert(
self.aXtraGrid, 0, 0.0
) # Add a point at b_t = 0 to make sure that bNrmGrid goes down to 0
bNrmCount = bNrmGrid.size # 201
TranShkCount = TranShkGrid.size # = (7,)
bNrmGridTerm = np.tile(
np.reshape(bNrmGrid, (bNrmCount, 1)),
(1, TranShkCount
)) # Replicated bNrmGrid for each transitory shock theta_t
TranShkGridTerm = np.tile(
TranShkGrid, (bNrmCount, 1)
) # Tile the grid of transitory shocks for the terminal solution. (201,7)
# Array of labor (leisure) values for terminal solution
LsrTerm = np.minimum(
(LbrCost / (1.0 + LbrCost)) * (bNrmGridTerm /
(WageRte * TranShkGridTerm) + 1.0),
1.0,
)
LsrTerm[0, 0] = 1.0
LbrTerm = 1.0 - LsrTerm
# Calculate market resources in terminal period, which is consumption
mNrmTerm = bNrmGridTerm + LbrTerm * WageRte * TranShkGridTerm
cNrmTerm = mNrmTerm # Consume everything we have
# Make a bilinear interpolation to represent the labor and consumption functions
LbrFunc_terminal = BilinearInterp(LbrTerm, bNrmGrid, TranShkGrid)
cFunc_terminal = BilinearInterp(cNrmTerm, bNrmGrid, TranShkGrid)
# Compute the effective consumption value using consumption value and labor value at the terminal solution
xEffTerm = LsrTerm**LbrCost * cNrmTerm
vNvrsFunc_terminal = BilinearInterp(xEffTerm, bNrmGrid, TranShkGrid)
vFunc_terminal = ValueFuncCRRA(vNvrsFunc_terminal, self.CRRA)
# Using the envelope condition at the terminal solution to estimate the marginal value function
vPterm = LsrTerm**LbrCost * CRRAutilityP(xEffTerm, gam=self.CRRA)
vPnvrsTerm = CRRAutilityP_inv(
vPterm, gam=self.CRRA
) # Evaluate the inverse of the CRRA marginal utility function at a given marginal value, vP
vPnvrsFunc_terminal = BilinearInterp(vPnvrsTerm, bNrmGrid, TranShkGrid)
vPfunc_terminal = MargValueFuncCRRA(
vPnvrsFunc_terminal, self.CRRA) # Get the Marginal Value function
bNrmMin_terminal = ConstantFunction(
0.0
) # Trivial function that return the same real output for any input
self.solution_terminal = ConsumerLaborSolution(
cFunc=cFunc_terminal,
LbrFunc=LbrFunc_terminal,
vFunc=vFunc_terminal,
vPfunc=vPfunc_terminal,
bNrmMin=bNrmMin_terminal,
)