Advanced Quantitative Economics with Python
Optimal Taxation without State-Contingent Debt
40. Optimal Taxation without State-Contingent Debt¶
Contents
In addition to what’s in Anaconda, this lecture will need the following libraries:
!pip install --upgrade quantecon
40.1. Overview¶
Let’s start with following imports:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.optimize import root, fmin_slsqp
from scipy.interpolate import UnivariateSpline
from quantecon import MarkovChain
In an earlier lecture, we described a model of optimal taxation with state-contingent debt due to Robert E. Lucas, Jr., and Nancy Stokey [LS83].
Aiyagari, Marcet, Sargent, and Seppälä [AMSSeppala02] (hereafter, AMSS) studied optimal taxation in a model without state-contingent debt.
In this lecture, we
describe assumptions and equilibrium concepts
solve the model
implement the model numerically
conduct some policy experiments
compare outcomes with those in a corresponding complete-markets model
We begin with an introduction to the model.
40.2. Competitive Equilibrium with Distorting Taxes¶
Many but not all features of the economy are identical to those of the Lucas-Stokey economy.
Let’s start with things that are identical.
For \(t \geq 0\), a history of the state is represented by \(s^t = [s_t, s_{t-1}, \ldots, s_0]\).
Government purchases \(g(s)\) are an exact time-invariant function of \(s\).
Let \(c_t(s^t)\), \(\ell_t(s^t)\), and \(n_t(s^t)\) denote consumption, leisure, and labor supply, respectively, at history \(s^t\) at time \(t\).
Each period a representative household is endowed with one unit of time that can be divided between leisure \(\ell_t\) and labor \(n_t\):
Output equals \(n_t(s^t)\) and can be divided between consumption \(c_t(s^t)\) and \(g(s_t)\)
Output is not storable.
The technology pins down a pre-tax wage rate to unity for all \(t, s^t\).
A representative household’s preferences over \(\{c_t(s^t), \ell_t(s^t)\}_{t=0}^\infty\) are ordered by
where
\(\pi_t(s^t)\) is a joint probability distribution over the sequence \(s^t\), and
the utility function \(u\) is increasing, strictly concave, and three times continuously differentiable in both arguments.
The government imposes a flat rate tax \(\tau_t(s^t)\) on labor income at time \(t\), history \(s^t\).
Lucas and Stokey assumed that there are complete markets in one-period Arrow securities; also see smoothing models.
It is at this point that AMSS [AMSSeppala02] modify the Lucas and Stokey economy.
AMSS allow the government to issue only one-period risk-free debt each period.
Ruling out complete markets in this way is a step in the direction of making total tax collections behave more like that prescribed in [Bar79] than they do in [LS83].
40.2.1. Risk-free One-Period Debt Only¶
In period \(t\) and history \(s^t\), let
\(b_{t+1}(s^t)\) be the amount of the time \(t+1\) consumption good that at time \(t\), history \(s^t\) the government promised to pay
\(R_t(s^t)\) be the gross interest rate on risk-free one-period debt between periods \(t\) and \(t+1\)
\(T_t(s^t)\) be a non-negative lump-sum transfer to the representative household 1
That \(b_{t+1}(s^t)\) is the same for all realizations of \(s_{t+1}\) captures its risk-free character.
The market value at time \(t\) of government debt maturing at time \(t+1\) equals \(b_{t+1}(s^t)\) divided by \(R_t(s^t)\).
The government’s budget constraint in period \(t\) at history \(s^t\) is
where \(z(s^t)\) is the net-of-interest government surplus.
To rule out Ponzi schemes, we assume that the government is subject to a natural debt limit (to be discussed in a forthcoming lecture).
The consumption Euler equation for a representative household able to trade only one-period risk-free debt with one-period gross interest rate \(R_t(s^t)\) is
Substituting this expression into the government’s budget constraint (40.4) yields:
Components of \(z(s^t)\) on the right side depend on \(s^t\), but the left side is required to depend only on \(s^{t-1}\) .
This is what it means for one-period government debt to be risk-free.
Therefore, the sum on the right side of equation (40.5) also has to depend only on \(s^{t-1}\).
This requirement will give rise to measurability constraints on the Ramsey allocation to be discussed soon.
If we replace \(b_{t+1}(s^t)\) on the right side of equation (40.5) by the right side of next period’s budget constraint (associated with a particular realization \(s_{t}\)) we get
After making similar repeated substitutions for all future occurrences of government indebtedness, and by invoking the natural debt limit, we arrive at:
Notice how the conditioning sets in equation (40.6) differ: they are \(s^{t-1}\) on the left side and \(s^t\) on the right side.
Now let’s
substitute the resource constraint into the net-of-interest government surplus, and
use the household’s first-order condition \(1-\tau^n_t(s^t)= u_{\ell}(s^t) /u_c(s^t)\) to eliminate the labor tax rate
so that we can express the net-of-interest government surplus \(z(s^t)\) as
If we substitute appropriate versions of the right side of (40.7) for \(z(s^{t+j})\) into equation (40.6), we obtain a sequence of implementability constraints on a Ramsey allocation in an AMSS economy.
Expression (40.6) at time \(t=0\) and initial state \(s^0\) was also an implementability constraint on a Ramsey allocation in a Lucas-Stokey economy:
Indeed, it was the only implementability constraint there.
But now we also have a large number of additional implementability constraints
Equation (40.9) must hold for each \(s^t\) for each \(t \geq 1\).
40.2.2. Comparison with Lucas-Stokey Economy¶
The expression on the right side of (40.9) in the Lucas-Stokey (1983) economy would equal the present value of a continuation stream of government net-of-interest surpluses evaluated at what would be competitive equilibrium Arrow-Debreu prices at date \(t\).
In the Lucas-Stokey economy, that present value is measurable with respect to \(s^t\).
In the AMSS economy, the restriction that government debt be risk-free imposes that that same present value must be measurable with respect to \(s^{t-1}\).
In a language used in the literature on incomplete markets models, it can be said that the AMSS model requires that at each \((t, s^t)\) what would be the present value of continuation government net-of-interest surpluses in the Lucas-Stokey model must belong to the marketable subspace of the AMSS model.
40.2.3. Ramsey Problem Without State-contingent Debt¶
After we have substituted the resource constraint into the utility function, we can express the Ramsey problem as being to choose an allocation that solves
where the maximization is subject to
and
given \(b_0(s^{-1})\).
40.2.3.1. Lagrangian Formulation¶
Let \(\gamma_0(s^0)\) be a non-negative Lagrange multiplier on constraint (40.10).
As in the Lucas-Stokey economy, this multiplier is strictly positive when the government must resort to distortionary taxation; otherwise it equals zero.
A consequence of the assumption that there are no markets in state-contingent securities and that a market exists only in a risk-free security is that we have to attach a stochastic process \(\{\gamma_t(s^t)\}_{t=1}^\infty\) of Lagrange multipliers to the implementability constraints (40.11).
Depending on how the constraints bind, these multipliers can be positive or negative:
A negative multiplier \(\gamma_t(s^t)<0\) means that if we could relax constraint (40.11), we would like to increase the beginning-of-period indebtedness for that particular realization of history \(s^t\).
That would let us reduce the beginning-of-period indebtedness for some other history 2.
These features flow from the fact that the government cannot use state-contingent debt and therefore cannot allocate its indebtedness efficiently across future states.
40.2.4. Some Calculations¶
It is helpful to apply two transformations to the Lagrangian.
Multiply constraint (40.10) by \(u_c(s^0)\) and the constraints (40.11) by \(\beta^t u_c(s^{t})\).
Then a Lagrangian for the Ramsey problem can be represented as
where
In (40.12), the second equality uses the law of iterated expectations and Abel’s summation formula (also called summation by parts, see this page).
First-order conditions with respect to \(c_t(s^t)\) can be expressed as
and with respect to \(b_t(s^t)\) as
If we substitute \(z(s^t)\) from (40.7) and its derivative \(z_c(s^t)\) into the first-order condition (40.14), we find two differences from the corresponding condition for the optimal allocation in a Lucas-Stokey economy with state-contingent government debt.
The term involving \(b_t(s^{t-1})\) in the first-order condition (40.14) does not appear in the corresponding expression for the Lucas-Stokey economy.
This term reflects the constraint that beginning-of-period government indebtedness must be the same across all realizations of next period’s state, a constraint that would not be present if government debt could be state contingent.
The Lagrange multiplier \(\Psi_t(s^t)\) in the first-order condition (40.14) may change over time in response to realizations of the state, while the multiplier \(\Phi\) in the Lucas-Stokey economy is time-invariant.
We need some code from an earlier lecture on optimal taxation with state-contingent debt sequential allocation implementation:
import numpy as np
from scipy.optimize import root
from quantecon import MarkovChain
class SequentialAllocation:
'''
Class that takes CESutility or BGPutility object as input returns
planner's allocation as a function of the multiplier on the
implementability constraint μ.
'''
def __init__(self, model):
# Initialize from model object attributes
self.β, self.π, self.G = model.β, model.π, model.G
self.mc, self.Θ = MarkovChain(self.π), model.Θ
self.S = len(model.π) # Number of states
self.model = model
# Find the first best allocation
self.find_first_best()
def find_first_best(self):
'''
Find the first best allocation
'''
model = self.model
S, Θ, G = self.S, self.Θ, self.G
Uc, Un = model.Uc, model.Un
def res(z):
c = z[:S]
n = z[S:]
return np.hstack([Θ * Uc(c, n) + Un(c, n), Θ * n - c - G])
res = root(res, 0.5 * np.ones(2 * S))
if not res.success:
raise Exception('Could not find first best')
self.cFB = res.x[:S]
self.nFB = res.x[S:]
# Multiplier on the resource constraint
self.ΞFB = Uc(self.cFB, self.nFB)
self.zFB = np.hstack([self.cFB, self.nFB, self.ΞFB])
def time1_allocation(self, μ):
'''
Computes optimal allocation for time t >= 1 for a given μ
'''
model = self.model
S, Θ, G = self.S, self.Θ, self.G
Uc, Ucc, Un, Unn = model.Uc, model.Ucc, model.Un, model.Unn
def FOC(z):
c = z[:S]
n = z[S:2 * S]
Ξ = z[2 * S:]
# FOC of c
return np.hstack([Uc(c, n) - μ * (Ucc(c, n) * c + Uc(c, n)) - Ξ,
Un(c, n) - μ * (Unn(c, n) * n + Un(c, n)) \
+ Θ * Ξ, # FOC of n
Θ * n - c - G])
# Find the root of the first-order condition
res = root(FOC, self.zFB)
if not res.success:
raise Exception('Could not find LS allocation.')
z = res.x
c, n, Ξ = z[:S], z[S:2 * S], z[2 * S:]
# Compute x
I = Uc(c, n) * c + Un(c, n) * n
x = np.linalg.solve(np.eye(S) - self.β * self.π, I)
return c, n, x, Ξ
def time0_allocation(self, B_, s_0):
'''
Finds the optimal allocation given initial government debt B_ and
state s_0
'''
model, π, Θ, G, β = self.model, self.π, self.Θ, self.G, self.β
Uc, Ucc, Un, Unn = model.Uc, model.Ucc, model.Un, model.Unn
# First order conditions of planner's problem
def FOC(z):
μ, c, n, Ξ = z
xprime = self.time1_allocation(μ)[2]
return np.hstack([Uc(c, n) * (c - B_) + Un(c, n) * n + β * π[s_0]
@ xprime,
Uc(c, n) - μ * (Ucc(c, n)
* (c - B_) + Uc(c, n)) - Ξ,
Un(c, n) - μ * (Unn(c, n) * n
+ Un(c, n)) + Θ[s_0] * Ξ,
(Θ * n - c - G)[s_0]])
# Find root
res = root(FOC, np.array(
[0, self.cFB[s_0], self.nFB[s_0], self.ΞFB[s_0]]))
if not res.success:
raise Exception('Could not find time 0 LS allocation.')
return res.x
def time1_value(self, μ):
'''
Find the value associated with multiplier μ
'''
c, n, x, Ξ = self.time1_allocation(μ)
U = self.model.U(c, n)
V = np.linalg.solve(np.eye(self.S) - self.β * self.π, U)
return c, n, x, V
def Τ(self, c, n):
'''
Computes Τ given c, n
'''
model = self.model
Uc, Un = model.Uc(c, n), model.Un(c, n)
return 1 + Un / (self.Θ * Uc)
def simulate(self, B_, s_0, T, sHist=None):
'''
Simulates planners policies for T periods
'''
model, π, β = self.model, self.π, self.β
Uc = model.Uc
if sHist is None:
sHist = self.mc.simulate(T, s_0)
cHist, nHist, Bhist, ΤHist, μHist = np.zeros((5, T))
RHist = np.zeros(T - 1)
# Time 0
μ, cHist[0], nHist[0], _ = self.time0_allocation(B_, s_0)
ΤHist[0] = self.Τ(cHist[0], nHist[0])[s_0]
Bhist[0] = B_
μHist[0] = μ
# Time 1 onward
for t in range(1, T):
c, n, x, Ξ = self.time1_allocation(μ)
Τ = self.Τ(c, n)
u_c = Uc(c, n)
s = sHist[t]
Eu_c = π[sHist[t - 1]] @ u_c
cHist[t], nHist[t], Bhist[t], ΤHist[t] = c[s], n[s], x[s] / u_c[s], \
Τ[s]
RHist[t - 1] = Uc(cHist[t - 1], nHist[t - 1]) / (β * Eu_c)
μHist[t] = μ
return np.array([cHist, nHist, Bhist, ΤHist, sHist, μHist, RHist])
To analyze the AMSS model, we find it useful to adopt a recursive formulation using techniques like those in our lectures on dynamic Stackelberg models and optimal taxation with state-contingent debt.
40.3. Recursive Version of AMSS Model¶
We now describe a recursive formulation of the AMSS economy.
We have noted that from the point of view of the Ramsey planner, the restriction to one-period risk-free securities
leaves intact the single implementability constraint on allocations (40.8) from the Lucas-Stokey economy, but
adds measurability constraints (40.6) on functions of tails of allocations at each time and history
We now explore how these constraints alter Bellman equations for a time \(0\) Ramsey planner and for time \(t \geq 1\), history \(s^t\) continuation Ramsey planners.
40.3.1. Recasting State Variables¶
In the AMSS setting, the government faces a sequence of budget constraints
where \(R_t(s^t)\) is the gross risk-free rate of interest between \(t\) and \(t+1\) at history \(s^t\) and \(T_t(s^t)\) are non-negative transfers.
Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this makes a possibly important difference from AMSS [AMSSeppala02], who restricted transfers to be non-negative).
In this case, the household faces a sequence of budget constraints
The household’s first-order conditions are \(u_{c,t} = \beta R_t \mathbb E_t u_{c,t+1}\) and \((1-\tau_t) u_{c,t} = u_{l,t}\).
Using these to eliminate \(R_t\) and \(\tau_t\) from budget constraint (40.16) gives
or
Now define
and represent the household’s budget constraint at time \(t\), history \(s^t\) as
for \(t \geq 1\).
40.3.2. Measurability Constraints¶
Write equation (40.18) as
The right side of equation (40.21) expresses the time \(t\) value of government debt in terms of a linear combination of terms whose individual components are measurable with respect to \(s^t\).
The sum of terms on the right side of equation (40.21) must equal \(b_t(s^{t-1})\).
That implies that it has to be measurable with respect to \(s^{t-1}\).
Equations (40.21) are the measurability constraints that the AMSS model adds to the single time \(0\) implementation constraint imposed in the Lucas and Stokey model.
40.3.3. Two Bellman Equations¶
Let \(\Pi(s|s_-)\) be a Markov transition matrix whose entries tell probabilities of moving from state \(s_-\) to state \(s\) in one period.
Let
\(V(x_-, s_-)\) be the continuation value of a continuation Ramsey plan at \(x_{t-1} = x_-, s_{t-1} =s_-\) for \(t \geq 1\)
\(W(b, s)\) be the value of the Ramsey plan at time \(0\) at \(b_0=b\) and \(s_0 = s\)
We distinguish between two types of planners:
For \(t \geq 1\), the value function for a continuation Ramsey planner satisfies the Bellman equation
subject to the following collection of implementability constraints, one for each \(s \in {\cal S}\):
A continuation Ramsey planner at \(t \geq 1\) takes \((x_{t-1}, s_{t-1}) = (x_-, s_-)\) as given and before \(s\) is realized chooses \((n_t(s_t), x_t(s_t)) = (n(s), x(s))\) for \(s \in {\cal S}\).
The Ramsey planner takes \((b_0, s_0)\) as given and chooses \((n_0, x_0)\).
The value function \(W(b_0, s_0)\) for the time \(t=0\) Ramsey planner satisfies the Bellman equation
where maximization is subject to
40.3.4. Martingale Supercedes State-Variable Degeneracy¶
Let \(\mu(s|s_-) \Pi(s|s_-)\) be a Lagrange multiplier on the constraint (40.23) for state \(s\).
After forming an appropriate Lagrangian, we find that the continuation Ramsey planner’s first-order condition with respect to \(x(s)\) is
Applying the envelope theorem to Bellman equation (40.22) gives
Equations (40.26) and (40.27) imply that
Equation (40.28) states that \(V_x(x, s)\) is a risk-adjusted martingale.
Saying that \(V_x(x, s)\) is a risk-adjusted martingale means that \(V_x(x, s)\) is a martingale with respect to the probability distribution over \(s^t\) sequences that are generated by the twisted transition probability matrix:
Exercise: Please verify that \(\check \Pi(s|s_-)\) is a valid Markov transition density, i.e., that its elements are all non-negative and that for each \(s_-\), the sum over \(s\) equals unity.
40.3.5. Absence of State Variable Degeneracy¶
Along a Ramsey plan, the state variable \(x_t = x_t(s^t, b_0)\) becomes a function of the history \(s^t\) and initial government debt \(b_0\).
In Lucas-Stokey model, we found that
a counterpart to \(V_x(x,s)\) is time-invariant and equal to the Lagrange multiplier on the Lucas-Stokey implementability constraint
time invariance of \(V_x(x,s)\) is the source of a key feature of the Lucas-Stokey model, namely, state variable degeneracy in which \(x_t\) is an exact time-invariant function of \(s_t\))
That \(V_x(x,s)\) varies over time according to a twisted martingale means that there is no state-variable degeneracy in the AMSS model.
In the AMSS model, both \(x\) and \(s\) are needed to describe the state.
This property of the AMSS model transmits a twisted martingale component to consumption, employment, and the tax rate.
40.3.6. Digression on Non-negative Transfers¶
Throughout this lecture, we have imposed that transfers \(T_t = 0\).
AMSS [AMSSeppala02] instead imposed a nonnegativity constraint \(T_t\geq 0\) on transfers.
They also considered a special case of quasi-linear preferences, \(u(c,l)= c + H(l)\).
In this case, \(V_x(x,s)\leq 0\) is a non-positive martingale.
By the martingale convergence theorem \(V_x(x,s)\) converges almost surely.
Furthermore, when the Markov chain \(\Pi(s| s_-)\) and the government expenditure function \(g(s)\) are such that \(g_t\) is perpetually random, \(V_x(x, s)\) almost surely converges to zero.
For quasi-linear preferences, the first-order condition for maximizing (40.22) subject to (40.23) with respect to \(n(s)\) becomes
When \(\mu(s|s_-) = \beta V_x(x(s),x)\) converges to zero, in the limit \(u_l(s)= 1 =u_c(s)\), so that \(\tau(x(s),s) =0\).
Thus, in the limit, if \(g_t\) is perpetually random, the government accumulates sufficient assets to finance all expenditures from earnings on those assets, returning any excess revenues to the household as non-negative lump-sum transfers.
40.3.7. Code¶
The recursive formulation is implemented as follows
import numpy as np
from scipy.optimize import fmin_slsqp
from scipy.optimize import root
from quantecon import MarkovChain
class RecursiveAllocationAMSS:
def __init__(self, model, μgrid, tol_diff=1e-4, tol=1e-4):
self.β, self.π, self.G = model.β, model.π, model.G
self.mc, self.S = MarkovChain(self.π), len(model.π) # Number of states
self.Θ, self.model, self.μgrid = model.Θ, model, μgrid
self.tol_diff, self.tol = tol_diff, tol
# Find the first best allocation
self.solve_time1_bellman()
self.T.time_0 = True # Bellman equation now solves time 0 problem
def solve_time1_bellman(self):
'''
Solve the time 1 Bellman equation for calibration model and
initial grid μgrid0
'''
model, μgrid0 = self.model, self.μgrid
π = model.π
S = len(model.π)
# First get initial fit from Lucas Stokey solution.
# Need to change things to be ex ante
pp = SequentialAllocation(model)
interp = interpolator_factory(2, None)
def incomplete_allocation(μ_, s_):
c, n, x, V = pp.time1_value(μ_)
return c, n, π[s_] @ x, π[s_] @ V
cf, nf, xgrid, Vf, xprimef = [], [], [], [], []
for s_ in range(S):
c, n, x, V = zip(*map(lambda μ: incomplete_allocation(μ, s_), μgrid0))
c, n = np.vstack(c).T, np.vstack(n).T
x, V = np.hstack(x), np.hstack(V)
xprimes = np.vstack([x] * S)
cf.append(interp(x, c))
nf.append(interp(x, n))
Vf.append(interp(x, V))
xgrid.append(x)
xprimef.append(interp(x, xprimes))
cf, nf, xprimef = fun_vstack(cf), fun_vstack(nf), fun_vstack(xprimef)
Vf = fun_hstack(Vf)
policies = [cf, nf, xprimef]
# Create xgrid
x = np.vstack(xgrid).T
xbar = [x.min(0).max(), x.max(0).min()]
xgrid = np.linspace(xbar[0], xbar[1], len(μgrid0))
self.xgrid = xgrid
# Now iterate on Bellman equation
T = BellmanEquation(model, xgrid, policies, tol=self.tol)
diff = 1
while diff > self.tol_diff:
PF = T(Vf)
Vfnew, policies = self.fit_policy_function(PF)
diff = np.abs((Vf(xgrid) - Vfnew(xgrid)) / Vf(xgrid)).max()
print(diff)
Vf = Vfnew
# Store value function policies and Bellman Equations
self.Vf = Vf
self.policies = policies
self.T = T
def fit_policy_function(self, PF):
'''
Fits the policy functions
'''
S, xgrid = len(self.π), self.xgrid
interp = interpolator_factory(3, 0)
cf, nf, xprimef, Tf, Vf = [], [], [], [], []
for s_ in range(S):
PFvec = np.vstack([PF(x, s_) for x in self.xgrid]).T
Vf.append(interp(xgrid, PFvec[0, :]))
cf.append(interp(xgrid, PFvec[1:1 + S]))
nf.append(interp(xgrid, PFvec[1 + S:1 + 2 * S]))
xprimef.append(interp(xgrid, PFvec[1 + 2 * S:1 + 3 * S]))
Tf.append(interp(xgrid, PFvec[1 + 3 * S:]))
policies = fun_vstack(cf), fun_vstack(
nf), fun_vstack(xprimef), fun_vstack(Tf)
Vf = fun_hstack(Vf)
return Vf, policies
def Τ(self, c, n):
'''
Computes Τ given c and n
'''
model = self.model
Uc, Un = model.Uc(c, n), model.Un(c, n)
return 1 + Un / (self.Θ * Uc)
def time0_allocation(self, B_, s0):
'''
Finds the optimal allocation given initial government debt B_ and
state s_0
'''
PF = self.T(self.Vf)
z0 = PF(B_, s0)
c0, n0, xprime0, T0 = z0[1:]
return c0, n0, xprime0, T0
def simulate(self, B_, s_0, T, sHist=None):
'''
Simulates planners policies for T periods
'''
model, π = self.model, self.π
Uc = model.Uc
cf, nf, xprimef, Tf = self.policies
if sHist is None:
sHist = simulate_markov(π, s_0, T)
cHist, nHist, Bhist, xHist, ΤHist, THist, μHist = np.zeros((7, T))
# Time 0
cHist[0], nHist[0], xHist[0], THist[0] = self.time0_allocation(B_, s_0)
ΤHist[0] = self.Τ(cHist[0], nHist[0])[s_0]
Bhist[0] = B_
μHist[0] = self.Vf[s_0](xHist[0])
# Time 1 onward
for t in range(1, T):
s_, x, s = sHist[t - 1], xHist[t - 1], sHist[t]
c, n, xprime, T = cf[s_, :](x), nf[s_, :](
x), xprimef[s_, :](x), Tf[s_, :](x)
Τ = self.Τ(c, n)[s]
u_c = Uc(c, n)
Eu_c = π[s_, :] @ u_c
μHist[t] = self.Vf[s](xprime[s])
cHist[t], nHist[t], Bhist[t], ΤHist[t] = c[s], n[s], x / Eu_c, Τ
xHist[t], THist[t] = xprime[s], T[s]
return np.array([cHist, nHist, Bhist, ΤHist, THist, μHist, sHist, xHist])
class BellmanEquation:
'''
Bellman equation for the continuation of the Lucas-Stokey Problem
'''
def __init__(self, model, xgrid, policies0, tol, maxiter=1000):
self.β, self.π, self.G = model.β, model.π, model.G
self.S = len(model.π) # Number of states
self.Θ, self.model, self.tol = model.Θ, model, tol
self.maxiter = maxiter
self.xbar = [min(xgrid), max(xgrid)]
self.time_0 = False
self.z0 = {}
cf, nf, xprimef = policies0
for s_ in range(self.S):
for x in xgrid:
self.z0[x, s_] = np.hstack([cf[s_, :](x),
nf[s_, :](x),
xprimef[s_, :](x),
np.zeros(self.S)])
self.find_first_best()
def find_first_best(self):
'''
Find the first best allocation
'''
model = self.model
S, Θ, Uc, Un, G = self.S, self.Θ, model.Uc, model.Un, self.G
def res(z):
c = z[:S]
n = z[S:]
return np.hstack([Θ * Uc(c, n) + Un(c, n), Θ * n - c - G])
res = root(res, 0.5 * np.ones(2 * S))
if not res.success:
raise Exception('Could not find first best')
self.cFB = res.x[:S]
self.nFB = res.x[S:]
IFB = Uc(self.cFB, self.nFB) * self.cFB + \
Un(self.cFB, self.nFB) * self.nFB
self.xFB = np.linalg.solve(np.eye(S) - self.β * self.π, IFB)
self.zFB = {}
for s in range(S):
self.zFB[s] = np.hstack(
[self.cFB[s], self.nFB[s], self.π[s] @ self.xFB, 0.])
def __call__(self, Vf):
'''
Given continuation value function next period return value function this
period return T(V) and optimal policies
'''
if not self.time_0:
def PF(x, s): return self.get_policies_time1(x, s, Vf)
else:
def PF(B_, s0): return self.get_policies_time0(B_, s0, Vf)
return PF
def get_policies_time1(self, x, s_, Vf):
'''
Finds the optimal policies
'''
model, β, Θ, G, S, π = self.model, self.β, self.Θ, self.G, self.S, self.π
U, Uc, Un = model.U, model.Uc, model.Un
def objf(z):
c, n, xprime = z[:S], z[S:2 * S], z[2 * S:3 * S]
Vprime = np.empty(S)
for s in range(S):
Vprime[s] = Vf[s](xprime[s])
return -π[s_] @ (U(c, n) + β * Vprime)
def cons(z):
c, n, xprime, T = z[:S], z[S:2 * S], z[2 * S:3 * S], z[3 * S:]
u_c = Uc(c, n)
Eu_c = π[s_] @ u_c
return np.hstack([
x * u_c / Eu_c - u_c * (c - T) - Un(c, n) * n - β * xprime,
Θ * n - c - G])
if model.transfers:
bounds = [(0., 100)] * S + [(0., 100)] * S + \
[self.xbar] * S + [(0., 100.)] * S
else:
bounds = [(0., 100)] * S + [(0., 100)] * S + \
[self.xbar] * S + [(0., 0.)] * S
out, fx, _, imode, smode = fmin_slsqp(objf, self.z0[x, s_],
f_eqcons=cons, bounds=bounds,
full_output=True, iprint=0,
acc=self.tol, iter=self.maxiter)
if imode > 0:
raise Exception(smode)
self.z0[x, s_] = out
return np.hstack([-fx, out])
def get_policies_time0(self, B_, s0, Vf):
'''
Finds the optimal policies
'''
model, β, Θ, G = self.model, self.β, self.Θ, self.G
U, Uc, Un = model.U, model.Uc, model.Un
def objf(z):
c, n, xprime = z[:-1]
return -(U(c, n) + β * Vf[s0](xprime))
def cons(z):
c, n, xprime, T = z
return np.hstack([
-Uc(c, n) * (c - B_ - T) - Un(c, n) * n - β * xprime,
(Θ * n - c - G)[s0]])
if model.transfers:
bounds = [(0., 100), (0., 100), self.xbar, (0., 100.)]
else:
bounds = [(0., 100), (0., 100), self.xbar, (0., 0.)]
out, fx, _, imode, smode = fmin_slsqp(objf, self.zFB[s0], f_eqcons=cons,
bounds=bounds, full_output=True,
iprint=0)
if imode > 0:
raise Exception(smode)
return np.hstack([-fx, out])
40.4. Examples¶
We now turn to some examples.
We will first build some useful functions for solving the model
import numpy as np
from scipy.interpolate import UnivariateSpline
class interpolate_wrapper:
def __init__(self, F):
self.F = F
def __getitem__(self, index):
return interpolate_wrapper(np.asarray(self.F[index]))
def reshape(self, *args):
self.F = self.F.reshape(*args)
return self
def transpose(self):
self.F = self.F.transpose()
def __len__(self):
return len(self.F)
def __call__(self, xvec):
x = np.atleast_1d(xvec)
shape = self.F.shape
if len(x) == 1:
fhat = np.hstack([f(x) for f in self.F.flatten()])
return fhat.reshape(shape)
else:
fhat = np.vstack([f(x) for f in self.F.flatten()])
return fhat.reshape(np.hstack((shape, len(x))))
class interpolator_factory:
def __init__(self, k, s):
self.k, self.s = k, s
def __call__(self, xgrid, Fs):
shape, m = Fs.shape[:-1], Fs.shape[-1]
Fs = Fs.reshape((-1, m))
F = []
xgrid = np.sort(xgrid) # Sort xgrid
for Fhat in Fs:
F.append(UnivariateSpline(xgrid, Fhat, k=self.k, s=self.s))
return interpolate_wrapper(np.array(F).reshape(shape))
def fun_vstack(fun_list):
Fs = [IW.F for IW in fun_list]
return interpolate_wrapper(np.vstack(Fs))
def fun_hstack(fun_list):
Fs = [IW.F for IW in fun_list]
return interpolate_wrapper(np.hstack(Fs))
def simulate_markov(π, s_0, T):
sHist = np.empty(T, dtype=int)
sHist[0] = s_0
S = len(π)
for t in range(1, T):
sHist[t] = np.random.choice(np.arange(S), p=π[sHist[t - 1]])
return sHist
40.4.1. Anticipated One-Period War¶
In our lecture on optimal taxation with state contingent debt we studied how the government manages uncertainty in a simple setting.
As in that lecture, we assume the one-period utility function
Note
For convenience in matching our computer code, we have expressed utility as a function of \(n\) rather than leisure \(l\).
We consider the same government expenditure process studied in the lecture on optimal taxation with state contingent debt.
Government expenditures are known for sure in all periods except one.
For \(t<3\) or \(t > 3\) we assume that \(g_t = g_l = 0.1\).
At \(t = 3\) a war occurs with probability 0.5.
If there is war, \(g_3 = g_h = 0.2\).
If there is no war \(g_3 = g_l = 0.1\).
A useful trick is to define components of the state vector as the following six \((t,g)\) pairs:
We think of these 6 states as corresponding to \(s=1,2,3,4,5,6\).
The transition matrix is
The government expenditure at each state is
We assume the same utility parameters as in the Lucas-Stokey economy.
This utility function is implemented in the following class.
import numpy as np
class CRRAutility:
def __init__(self,
β=0.9,
σ=2,
γ=2,
π=0.5*np.ones((2, 2)),
G=np.array([0.1, 0.2]),
Θ=np.ones(2),
transfers=False):
self.β, self.σ, self.γ = β, σ, γ
self.π, self.G, self.Θ, self.transfers = π, G, Θ, transfers
# Utility function
def U(self, c, n):
σ = self.σ
if σ == 1.:
U = np.log(c)
else:
U = (c**(1 - σ) - 1) / (1 - σ)
return U - n**(1 + self.γ) / (1 + self.γ)
# Derivatives of utility function
def Uc(self, c, n):
return c**(-self.σ)
def Ucc(self, c, n):
return -self.σ * c**(-self.σ - 1)
def Un(self, c, n):
return -n**self.γ
def Unn(self, c, n):
return -self.γ * n**(self.γ - 1)
The following figure plots the Ramsey plan under both complete and incomplete markets for both possible realizations of the state at time \(t=3\).
Optimal policies when the government has access to state contingent debt are represented by black lines, while the optimal policies when there is only a risk-free bond are in red.
Paths with circles are histories in which there is peace, while those with triangle denote war.
# Initialize μgrid for value function iteration
μ_grid = np.linspace(-0.7, 0.01, 200)
time_example = CRRAutility()
time_example.π = np.array([[0, 1, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0],
[0, 0, 0, 0.5, 0.5, 0],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1]])
time_example.G = np.array([0.1, 0.1, 0.1, 0.2, 0.1, 0.1])
time_example.Θ = np.ones(6) # Θ can in principle be random
time_example.transfers = True # Government can use transfers
# Solve sequential problem
time_sequential = SequentialAllocation(time_example)
# Solve recursive problem
time_bellman = RecursiveAllocationAMSS(time_example, μ_grid)
sHist_h = np.array([0, 1, 2, 3, 5, 5, 5])
sHist_l = np.array([0, 1, 2, 4, 5, 5, 5])
sim_seq_h = time_sequential.simulate(1, 0, 7, sHist_h)
sim_bel_h = time_bellman.simulate(1, 0, 7, sHist_h)
sim_seq_l = time_sequential.simulate(1, 0, 7, sHist_l)
sim_bel_l = time_bellman.simulate(1, 0, 7, sHist_l)
# Government spending paths
sim_seq_l[4] = time_example.G[sHist_l]
sim_seq_h[4] = time_example.G[sHist_h]
sim_bel_l[4] = time_example.G[sHist_l]
sim_bel_h[4] = time_example.G[sHist_h]
# Output paths
sim_seq_l[5] = time_example.Θ[sHist_l] * sim_seq_l[1]
sim_seq_h[5] = time_example.Θ[sHist_h] * sim_seq_h[1]
sim_bel_l[5] = time_example.Θ[sHist_l] * sim_bel_l[1]
sim_bel_h[5] = time_example.Θ[sHist_h] * sim_bel_h[1]
fig, axes = plt.subplots(3, 2, figsize=(14, 10))
titles = ['Consumption', 'Labor Supply', 'Government Debt',
'Tax Rate', 'Government Spending', 'Output']
for ax, title, sim_l, sim_h, bel_l, bel_h in zip(axes.flatten(), titles,
sim_seq_l, sim_seq_h,
sim_bel_l, sim_bel_h):
ax.plot(sim_l, '-ok', sim_h, '-^k', bel_l, '-or', bel_h, '-^r', alpha=0.7)
ax.set(title=title)
ax.grid()
plt.tight_layout()
plt.show()
How a Ramsey planner responds to war depends on the structure of the asset market.
If it is able to trade state-contingent debt, then at time \(t=2\)
the government purchases an Arrow security that pays off when \(g_3 = g_h\)
the government sells an Arrow security that pays off when \(g_3 = g_l\)
these purchases are designed in such a way that regardless of whether or not there is a war at \(t=3\), the government will begin period \(t=4\) with the same government debt
This pattern facilities smoothing tax rates across states.
The government without state contingent debt cannot do this.
Instead, it must enter time \(t=3\) with the same level of debt falling due whether there is peace or war at \(t=3\).
It responds to this constraint by smoothing tax rates across time.
To finance a war it raises taxes and issues more debt.
To service the additional debt burden, it raises taxes in all future periods.
The absence of state contingent debt leads to an important difference in the optimal tax policy.
When the Ramsey planner has access to state contingent debt, the optimal tax policy is history independent
the tax rate is a function of the current level of government spending only, given the Lagrange multiplier on the implementability constraint
Without state contingent debt, the optimal tax rate is history dependent.
A war at time \(t=3\) causes a permanent increase in the tax rate.
40.4.1.1. Perpetual War Alert¶
History dependence occurs more dramatically in a case in which the government perpetually faces the prospect of war.
This case was studied in the final example of the lecture on optimal taxation with state-contingent debt.
There, each period the government faces a constant probability, \(0.5\), of war.
In addition, this example features the following preferences
In accordance, we will re-define our utility function.
import numpy as np
class LogUtility:
def __init__(self,
β=0.9,
ψ=0.69,
π=0.5*np.ones((2, 2)),
G=np.array([0.1, 0.2]),
Θ=np.ones(2),
transfers=False):
self.β, self.ψ, self.π = β, ψ, π
self.G, self.Θ, self.transfers = G, Θ, transfers
# Utility function
def U(self, c, n):
return np.log(c) + self.ψ * np.log(1 - n)
# Derivatives of utility function
def Uc(self, c, n):
return 1 / c
def Ucc(self, c, n):
return -c**(-2)
def Un(self, c, n):
return -self.ψ / (1 - n)
def Unn(self, c, n):
return -self.ψ / (1 - n)**2
With these preferences, Ramsey tax rates will vary even in the Lucas-Stokey model with state-contingent debt.
The figure below plots optimal tax policies for both the economy with state contingent debt (circles) and the economy with only a risk-free bond (triangles).
log_example = LogUtility()
log_example.transfers = True # Government can use transfers
log_sequential = SequentialAllocation(log_example) # Solve sequential problem
log_bellman = RecursiveAllocationAMSS(log_example, μ_grid)
T = 20
sHist = np.array([0, 0, 0, 0, 0, 0, 0, 0, 1, 1,
0, 0, 0, 1, 1, 1, 1, 1, 1, 0])
# Simulate
sim_seq = log_sequential.simulate(0.5, 0, T, sHist)
sim_bel = log_bellman.simulate(0.5, 0, T, sHist)
titles = ['Consumption', 'Labor Supply', 'Government Debt',
'Tax Rate', 'Government Spending', 'Output']
# Government spending paths
sim_seq[4] = log_example.G[sHist]
sim_bel[4] = log_example.G[sHist]
# Output paths
sim_seq[5] = log_example.Θ[sHist] * sim_seq[1]
sim_bel[5] = log_example.Θ[sHist] * sim_bel[1]
fig, axes = plt.subplots(3, 2, figsize=(14, 10))
for ax, title, seq, bel in zip(axes.flatten(), titles, sim_seq, sim_bel):
ax.plot(seq, '-ok', bel, '-^b')
ax.set(title=title)
ax.grid()
axes[0, 0].legend(('Complete Markets', 'Incomplete Markets'))
plt.tight_layout()
plt.show()
When the government experiences a prolonged period of peace, it is able to reduce government debt and set permanently lower tax rates.
However, the government finances a long war by borrowing and raising taxes.
This results in a drift away from policies with state contingent debt that depends on the history of shocks.
This is even more evident in the following figure that plots the evolution of the two policies over 200 periods.
T = 200 # Set T to 200 periods
sim_seq_long = log_sequential.simulate(0.5, 0, T)
sHist_long = sim_seq_long[-3]
sim_bel_long = log_bellman.simulate(0.5, 0, T, sHist_long)
titles = ['Consumption', 'Labor Supply', 'Government Debt',
'Tax Rate', 'Government Spending', 'Output']
# Government spending paths
sim_seq_long[4] = log_example.G[sHist_long]
sim_bel_long[4] = log_example.G[sHist_long]
# Output paths
sim_seq_long[5] = log_example.Θ[sHist_long] * sim_seq_long[1]
sim_bel_long[5] = log_example.Θ[sHist_long] * sim_bel_long[1]
fig, axes = plt.subplots(3, 2, figsize=(14, 10))
for ax, title, seq, bel in zip(axes.flatten(), titles, sim_seq_long, \
sim_bel_long):
ax.plot(seq, '-k', bel, '-.b', alpha=0.5)
ax.set(title=title)
ax.grid()
axes[0, 0].legend(('Complete Markets','Incomplete Markets'))
plt.tight_layout()
plt.show()
- 1
In an allocation that solves the Ramsey problem and that levies distorting taxes on labor, why would the government ever want to hand revenues back to the private sector? It would not in an economy with state-contingent debt, since any such allocation could be improved by lowering distortionary taxes rather than handing out lump-sum transfers. But, without state-contingent debt there can be circumstances when a government would like to make lump-sum transfers to the private sector.
- 2
From the first-order conditions for the Ramsey problem, there exists another realization \(\tilde s^t\) with the same history up until the previous period, i.e., \(\tilde s^{t-1}= s^{t-1}\), but where the multiplier on constraint (40.11) takes a positive value, so \(\gamma_t(\tilde s^t)>0\).