66. Optimal Taxation without State-Contingent Debt#
Contents
66.1. Overview#
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 [AMSS02] (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.
66.2. 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 [AMSS02] 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].
66.2.1. 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 \(t\) 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 nonnegative 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 (66.4) yields:
Components of \(z(s^t)\) on the right side depend on \(s^t\), but the left side is required to depend on \(s^{t-1}\) only.
This is what it means for one-period government debt to be risk-free.
Therefore, the sum on the right side of equation (66.5) 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 (66.5) 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:
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 the appropriate versions of right side of (66.7) for \(z(s^{t+j})\) into equation (66.6), we obtain a sequence of implementability constraints on a Ramsey allocation in an AMSS economy.
Expression (66.6) 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 (66.9) must hold for each \(s^t\) for each \(t \geq 1\).
66.2.2. Comparison with Lucas-Stokey Economy#
The expression on the right side of (66.9) in the Lucas-Stokey (1983) economy would equal the present value of a continuation stream of government 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 surpluses in the Lucas-Stokey model must belong to the marketable subspace of the AMSS model.
66.2.3. 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 \(b_0(s^{-1})\).
66.2.3.1. Lagrangian Formulation#
Let \(\gamma_0(s^0)\) be a nonnegative Lagrange multiplier on constraint (66.10).
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 stochastic processes \(\{\gamma_t(s^t)\}_{t=1}^\infty\) of Lagrange multipliers to the implementability constraints (66.11).
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 (66.11), 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.
66.2.4. Some Calculations#
It is helpful to apply two transformations to the Lagrangian.
Multiply constraint (66.10) by \(u_c(s^0)\) and the constraints (66.11) by \(\beta^t u_c(s^{t})\).
Then a Lagrangian for the Ramsey problem can be represented as
where
In (66.12), 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 (66.7) and its derivative \(z_c(s^t)\) into first-order condition (66.14), 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 first-order condition (66.14) 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 first-order condition (66.14) 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 our an earlier lecture on optimal taxation with state-contingent debt sequential allocation implementation:
using LinearAlgebra, Statistics, Random
using QuantEcon, NLsolve, NLopt
import QuantEcon.simulate
mutable struct Model{TF <: AbstractFloat,
TM <: AbstractMatrix{TF},
TV <: AbstractVector{TF}}
beta::TF
Pi::TM
G::TV
Theta::TV
transfers::Bool
U::Function
Uc::Function
Ucc::Function
Un::Function
Unn::Function
n_less_than_one::Bool
end
struct SequentialAllocation{TP <: Model,
TI <: Integer,
TV <: AbstractVector}
model::TP
mc::MarkovChain
S::TI
cFB::TV
nFB::TV
XiFB::TV
zFB::TV
end
function SequentialAllocation(model::Model)
beta, Pi, G, Theta = model.beta, model.Pi, model.G, model.Theta
mc = MarkovChain(Pi)
S = size(Pi, 1) # Number of states
# Now find the first best allocation
cFB, nFB, XiFB, zFB = find_first_best(model, S, 1)
return SequentialAllocation(model, mc, S, cFB, nFB, XiFB, zFB)
end
function find_first_best(model::Model, S::Integer, version::Integer)
if version != 1 && version != 2
throw(ArgumentError("version must be 1 or 2"))
end
beta, Theta, Uc, Un, G, Pi =
model.beta, model.Theta, model.Uc, model.Un, model.G, model.Pi
function res!(out, z)
c = z[1:S]
n = z[S+1:end]
out[1:S] = Theta .* Uc.(c, n) + Un.(c, n)
out[S+1:end] = Theta .* n .- c .- G
end
res = nlsolve(res!, 0.5 * ones(2 * S))
if converged(res) == false
error("Could not find first best")
end
if version == 1
cFB = res.zero[1:S]
nFB = res.zero[S+1:end]
XiFB = Uc(cFB, nFB) # Multiplier on the resource constraint
zFB = vcat(cFB, nFB, XiFB)
return cFB, nFB, XiFB, zFB
elseif version == 2
cFB = res.zero[1:S]
nFB = res.zero[S+1:end]
IFB = Uc(cFB, nFB) .* cFB + Un(cFB, nFB) .* nFB
xFB = \(LinearAlgebra.I - beta * Pi, IFB)
zFB = [vcat(cFB[s], xFB[s], xFB) for s in 1:S]
return cFB, nFB, IFB, xFB, zFB
end
end
function time1_allocation(pas::SequentialAllocation, mu::Real)
model, S = pas.model, pas.S
Theta, beta, Pi, G, Uc, Ucc, Un, Unn =
model.Theta, model.beta, model.Pi, model.G,
model.Uc, model.Ucc, model.Un, model.Unn
function FOC!(out, z::Vector)
c = z[1:S]
n = z[S+1:2S]
Xi = z[2S+1:end]
out[1:S] = Uc.(c, n) - mu * (Ucc.(c, n) .* c + Uc.(c, n)) - Xi # FOC c
out[S+1:2S] = Un.(c, n) - mu * (Unn(c, n) .* n .+ Un.(c, n)) + Theta .* Xi # FOC n
out[2S+1:end] = Theta .* n - c .- G # resource constraint
return out
end
# Find the root of the FOC
res = nlsolve(FOC!, pas.zFB)
if res.f_converged == false
error("Could not find LS allocation.")
end
z = res.zero
c, n, Xi = z[1:S], z[S+1:2S], z[2S+1:end]
# Now compute x
I = Uc(c, n) .* c + Un(c, n) .* n
x = \(LinearAlgebra.I - beta * model.Pi, I)
return c, n, x, Xi
end
function time0_allocation(pas::SequentialAllocation,
B_::AbstractFloat, s_0::Integer)
model = pas.model
Pi, Theta, G, beta = model.Pi, model.Theta, model.G, model.beta
Uc, Ucc, Un, Unn =
model.Uc, model.Ucc, model.Un, model.Unn
# First order conditions of planner's problem
function FOC!(out, z)
mu, c, n, Xi = z[1], z[2], z[3], z[4]
xprime = time1_allocation(pas, mu)[3]
out .= vcat(
Uc(c, n) .* (c - B_) + Un(c, n) .* n + beta * dot(Pi[s_0, :], xprime),
Uc(c, n) .- mu * (Ucc(c, n) .* (c - B_) + Uc(c, n)) .- Xi,
Un(c, n) .- mu * (Unn(c, n) .* n + Un(c, n)) + Theta[s_0] .* Xi,
(Theta .* n .- c .- G)[s_0]
)
end
# Find root
res = nlsolve(FOC!, [0.0, pas.cFB[s_0], pas.nFB[s_0], pas.XiFB[s_0]])
if res.f_converged == false
error("Could not find time 0 LS allocation.")
end
return (res.zero...,)
end
function time1_value(pas::SequentialAllocation, mu::Real)
model = pas.model
c, n, x, Xi = time1_allocation(pas, mu)
U_val = model.U.(c, n)
V = \(LinearAlgebra.I - model.beta*model.Pi, U_val)
return c, n, x, V
end
function Omega(model::Model, c::Union{Real,Vector}, n::Union{Real,Vector})
Uc, Un = model.Uc.(c, n), model.Un.(c, n)
return 1. .+ Un./(model.Theta .* Uc)
end
function simulate(pas::SequentialAllocation,
B_::AbstractFloat, s_0::Integer,
T::Integer,
sHist::Union{Vector, Nothing}=nothing)
model = pas.model
Pi, beta, Uc = model.Pi, model.beta, model.Uc
if isnothing(sHist)
sHist = QuantEcon.simulate(pas.mc, T, init=s_0)
end
cHist = zeros(T)
nHist = zeros(T)
Bhist = zeros(T)
OmegaHist = zeros(T)
muHist = zeros(T)
RHist = zeros(T-1)
# time 0
mu, cHist[1], nHist[1], _ = time0_allocation(pas, B_, s_0)
OmegaHist[1] = Omega(pas.model, cHist[1], nHist[1])[s_0]
Bhist[1] = B_
muHist[1] = mu
# time 1 onward
for t in 2:T
c, n, x, Xi = time1_allocation(pas,mu)
u_c = Uc(c,n)
s = sHist[t]
OmegaHist[t] = Omega(pas.model, c, n)[s]
Eu_c = dot(Pi[sHist[t-1],:], u_c)
cHist[t], nHist[t], Bhist[t] = c[s], n[s], x[s] / u_c[s]
RHist[t-1] = Uc(cHist[t-1], nHist[t-1]) / (beta * Eu_c)
muHist[t] = mu
end
return cHist, nHist, Bhist, OmegaHist, sHist, muHist, RHist
end
mutable struct BellmanEquation{TP <: Model,
TI <: Integer,
TV <: AbstractVector,
TM <: AbstractMatrix{TV},
TVV <: AbstractVector{TV}}
model::TP
S::TI
xbar::TV
time_0::Bool
z0::TM
cFB::TV
nFB::TV
xFB::TV
zFB::TVV
end
function BellmanEquation(model::Model, xgrid::AbstractVector, policies0::Vector)
S = size(model.Pi, 1) # number of states
xbar = [minimum(xgrid), maximum(xgrid)]
time_0 = false
cf, nf, xprimef = policies0
z0 = [vcat(cf[s](x), nf[s](x), [xprimef[s, sprime](x) for sprime in 1:S])
for x in xgrid, s in 1:S]
cFB, nFB, IFB, xFB, zFB = find_first_best(model, S, 2)
return BellmanEquation(model, S, xbar, time_0, z0, cFB, nFB, xFB, zFB)
end
function get_policies_time1(T::BellmanEquation,
i_x::Integer, x::AbstractFloat,
s::Integer, Vf::AbstractArray)
model, S = T.model, T.S
beta, Theta, G, Pi = model.beta, model.Theta, model.G, model.Pi
U, Uc, Un = model.U, model.Uc, model.Un
function objf(z::Vector, grad)
c, xprime = z[1], z[2:end]
n=c+G[s]
Vprime = [Vf[sprime](xprime[sprime]) for sprime in 1:S]
return -(U(c, n) + beta * dot(Pi[s, :], Vprime))
end
function cons(z::Vector, grad)
c, xprime = z[1], z[2:end]
n=c+G[s]
return x - Uc(c, n) * c - Un(c, n) * n - beta * dot(Pi[s, :], xprime)
end
lb = vcat(0, T.xbar[1] * ones(S))
ub = vcat(1 - G[s], T.xbar[2] * ones(S))
opt = Opt(:LN_COBYLA, length(T.z0[i_x, s])-1)
min_objective!(opt, objf)
equality_constraint!(opt, cons)
lower_bounds!(opt, lb)
upper_bounds!(opt, ub)
maxeval!(opt, 300)
maxtime!(opt, 10)
init = vcat(T.z0[i_x, s][1], T.z0[i_x, s][3:end])
for (i, val) in enumerate(init)
if val > ub[i]
init[i] = ub[i]
elseif val < lb[i]
init[i] = lb[i]
end
end
(minf, minx, ret) = NLopt.optimize(opt, init)
T.z0[i_x, s] = vcat(minx[1], minx[1] + G[s], minx[2:end])
return vcat(-minf, T.z0[i_x, s])
end
function get_policies_time0(T::BellmanEquation,
B_::AbstractFloat, s0::Integer, Vf::Array)
model, S = T.model, T.S
beta, Theta, G, Pi = model.beta, model.Theta, model.G, model.Pi
U, Uc, Un = model.U, model.Uc, model.Un
function objf(z, grad)
c, xprime = z[1], z[2:end]
n = c+G[s0]
Vprime = [Vf[sprime](xprime[sprime]) for sprime in 1:S]
return -(U(c, n) + beta * dot(Pi[s0, :], Vprime))
end
function cons(z::Vector, grad)
c, xprime = z[1], z[2:end]
n = c + G[s0]
return -Uc(c, n) * (c - B_) - Un(c, n) * n - beta * dot(Pi[s0, :], xprime)
end
lb = vcat(0, T.xbar[1] * ones(S))
ub = vcat(1-G[s0], T.xbar[2] * ones(S))
opt = Opt(:LN_COBYLA, length(T.zFB[s0])-1)
min_objective!(opt, objf)
equality_constraint!(opt, cons)
lower_bounds!(opt, lb)
upper_bounds!(opt, ub)
maxeval!(opt, 300)
maxtime!(opt, 10)
init = vcat(T.zFB[s0][1], T.zFB[s0][3:end])
for (i, val) in enumerate(init)
if val > ub[i]
init[i] = ub[i]
elseif val < lb[i]
init[i] = lb[i]
end
end
(minf, minx, ret) = NLopt.optimize(opt, init)
return vcat(-minf, vcat(minx[1], minx[1]+G[s0], minx[2:end]))
end
get_policies_time0 (generic function with 1 method)
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.
66.3. 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 (66.8) from the Lucas-Stokey economy, but
adds measurability constraints (66.6) 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.
66.3.1. 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 nonnegative 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 [AMSS02], who restricted transfers to be nonnegative).
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 (66.16) gives
or
Now define
and represent the household’s budget constraint at time \(t\), history \(s^t\) as
for \(t \geq 1\).
66.3.2. Measurability Constraints#
Write equation (66.18) as
The right side of equation (66.21) 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 (66.21) must equal \(b_t(s^{t-1})\).
That implies that it is has to be measurable with respect to \(s^{t-1}\).
Equations (66.21) are the measurablility constraints that the AMSS model adds to the single time \(0\) implementation constraint imposed in the Lucas and Stokey model.
66.3.3. 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
66.3.4. Martingale Supercedes State-Variable Degeneracy#
Let \(\mu(s|s_-) \Pi(s|s_-)\) be a Lagrange multiplier on constraint (66.23) 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 (66.22) gives
Equations (66.26) and (66.27) imply that
Equation (66.28) 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 is 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 nonnegative and that for each \(s_-\), the sum over \(s\) equals unity.
66.3.5. 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 (i.e., \(x_t\) is an exact 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.
66.3.6. Digression on Nonnegative Transfers#
Throughout this lecture we have imposed that transfers \(T_t = 0\).
AMSS [AMSS02] 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 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 nonnegative lump sum transfers.
66.3.7. Code#
The recursive formulation is implemented as follows
# Interpolations.jl doesn't support irregular grids for splines
using DataInterpolations
mutable struct BellmanEquation_Recursive{TP <: Model, TI <: Integer, TR <: Real}
model::TP
S::TI
xbar::Array{TR}
time_0::Bool
z0::Array{Array}
cFB::Vector{TR}
nFB::Vector{TR}
xFB::Vector{TR}
zFB::Vector{Vector{TR}}
end
struct RecursiveAllocation{TP <: Model,
TI <: Integer,
TVg <: AbstractVector,
TT <: Tuple}
model::TP
mc::MarkovChain
S::TI
T::BellmanEquation_Recursive
mugrid::TVg
xgrid::TVg
Vf::Array
policies::TT
end
function RecursiveAllocation(model::Model, mugrid::AbstractArray)
G = model.G
S = size(model.Pi, 1) # number of states
mc = MarkovChain(model.Pi)
# now find the first best allocation
Vf, policies, T, xgrid = solve_time1_bellman(model, mugrid)
T.time_0 = true # Bellman equation now solves time 0 problem
return RecursiveAllocation(model, mc, S, T, mugrid, xgrid, Vf, policies)
end
function solve_time1_bellman(model::Model{TR},
mugrid::AbstractArray) where {TR <: Real}
Pi = model.Pi
S = size(model.Pi, 1)
# First get initial fit from lucas stockey solution.
# Need to change things to be ex_ante
PP = SequentialAllocation(model)
function incomplete_allocation(PP::SequentialAllocation,
mu_::AbstractFloat,
s_::Integer)
c, n, x, V = time1_value(PP, mu_)
return c, n, dot(Pi[s_, :], x), dot(Pi[s_, :], V)
end
cf = Array{Function}(undef, S, S)
nf = Array{Function}(undef, S, S)
xprimef = Array{Function}(undef, S, S)
Vf = Vector{Function}(undef, S)
xgrid = Array{TR}(undef, S, length(mugrid))
for s_ in 1:S
c = Array{TR}(undef, length(mugrid), S)
n = Array{TR}(undef, length(mugrid), S)
x = Array{TR}(undef, length(mugrid))
V = Array{TR}(undef, length(mugrid))
for (i_mu, mu) in enumerate(mugrid)
c[i_mu, :], n[i_mu, :], x[i_mu], V[i_mu] = incomplete_allocation(PP,
mu,
s_)
end
xprimes = repeat(x, 1, S)
xgrid[s_, :] = x
for sprime in 1:S
splc = CubicSpline(c[:, sprime][end:-1:1], x[end:-1:1];
extrapolate = true)
spln = CubicSpline(n[:, sprime][end:-1:1], x[end:-1:1];
extrapolate = true)
splx = CubicSpline(xprimes[:, sprime][end:-1:1], x[end:-1:1];
extrapolate = true)
cf[s_, sprime] = y -> splc(y)
nf[s_, sprime] = y -> spln(y)
xprimef[s_, sprime] = y -> splx(y)
end
splV = CubicSpline(V[end:-1:1], x[end:-1:1]; extrapolate = true)
Vf[s_] = y -> splV(y)
end
policies = [cf, nf, xprimef]
# Create xgrid
xbar = [maximum(minimum(xgrid)), minimum(maximum(xgrid))]
xgrid = range(xbar[1], xbar[2], length = length(mugrid))
# Now iterate on Bellman equation
T = BellmanEquation_Recursive(model, xgrid, policies)
diff = 1.0
while diff > 1e-4
PF = (i_x, x, s) -> get_policies_time1(T, i_x, x, s, Vf, xbar)
Vfnew, policies = fit_policy_function(T, PF, xgrid)
diff = 0.0
for s in 1:S
diff = max(diff,
maximum(abs,
(Vf[s].(xgrid) - Vfnew[s].(xgrid)) ./
Vf[s].(xgrid)))
end
println("diff = $diff")
Vf = copy(Vfnew)
end
return Vf, policies, T, xgrid
end
function fit_policy_function(T::BellmanEquation_Recursive,
PF::Function,
xgrid::AbstractVector{TF}) where {
TF <:
AbstractFloat}
S = T.S
# preallocation
PFvec = Array{TF}(undef, 4S + 1, length(xgrid))
cf = Array{Function}(undef, S, S)
nf = Array{Function}(undef, S, S)
xprimef = Array{Function}(undef, S, S)
TTf = Array{Function}(undef, S, S)
Vf = Vector{Function}(undef, S)
# fit policy fuctions
for s_ in 1:S
for (i_x, x) in enumerate(xgrid)
PFvec[:, i_x] = PF(i_x, x, s_)
end
splV = CubicSpline(PFvec[1, :], xgrid)
Vf[s_] = y -> splV(y)
for sprime in 1:S
splc = CubicSpline(PFvec[1 + sprime, :], xgrid)
spln = CubicSpline(PFvec[1 + S + sprime, :], xgrid)
splxprime = CubicSpline(PFvec[1 + 2S + sprime, :], xgrid)
splTT = CubicSpline(PFvec[1 + 3S + sprime, :], xgrid)
cf[s_, sprime] = y -> splc(y)
nf[s_, sprime] = y -> spln(y)
xprimef[s_, sprime] = y -> splxprime(y)
TTf[s_, sprime] = y -> splTT(y)
end
end
policies = (cf, nf, xprimef, TTf)
return Vf, policies
end
function Tau(pab::RecursiveAllocation,
c::AbstractArray,
n::AbstractArray)
model = pab.model
Uc, Un = model.Uc(c, n), model.Un(c, n)
return 1.0 .+ Un ./ (model.Theta .* Uc)
end
Tau(pab::RecursiveAllocation, c::Real, n::Real) = Tau(pab, [c], [n])
function time0_allocation(pab::RecursiveAllocation, B_::Real, s0::Integer)
T, Vf = pab.T, pab.Vf
xbar = T.xbar
z0 = get_policies_time0(T, B_, s0, Vf, xbar)
c0, n0, xprime0, T0 = z0[2], z0[3], z0[4], z0[5]
return c0, n0, xprime0, T0
end
function simulate(pab::RecursiveAllocation,
B_::TF, s_0::Integer, T::Integer,
sHist::Vector = simulate(pab.mc, T, init = s_0)) where {
TF <:
AbstractFloat
}
model, mc, Vf, S = pab.model, pab.mc, pab.Vf, pab.S
Pi, Uc = model.Pi, model.Uc
cf, nf, xprimef, TTf = pab.policies
cHist = Array{TF}(undef, T)
nHist = Array{TF}(undef, T)
Bhist = Array{TF}(undef, T)
xHist = Array{TF}(undef, T)
TauHist = Array{TF}(undef, T)
THist = Array{TF}(undef, T)
muHist = Array{TF}(undef, T)
#time0
cHist[1], nHist[1], xHist[1], THist[1] = time0_allocation(pab, B_, s_0)
TauHist[1] = Tau(pab, cHist[1], nHist[1])[s_0]
Bhist[1] = B_
muHist[1] = Vf[s_0](xHist[1])
#time 1 onward
for t in 2:T
s_, x, s = sHist[t - 1], xHist[t - 1], sHist[t]
c = Array{TF}(undef, S)
n = Array{TF}(undef, S)
xprime = Array{TF}(undef, S)
TT = Array{TF}(undef, S)
for sprime in 1:S
c[sprime], n[sprime], xprime[sprime], TT[sprime] = cf[s_, sprime](x),
nf[s_, sprime](x),
xprimef[s_,
sprime](x),
TTf[s_, sprime](x)
end
Tau_val = Tau(pab, c, n)[s]
u_c = Uc(c, n)
Eu_c = dot(Pi[s_, :], u_c)
muHist[t] = Vf[s](xprime[s])
cHist[t], nHist[t], Bhist[t], TauHist[t] = c[s], n[s], x / Eu_c, Tau_val
xHist[t], THist[t] = xprime[s], TT[s]
end
return cHist, nHist, Bhist, xHist, TauHist, THist, muHist, sHist
end
function BellmanEquation_Recursive(model::Model{TF},
xgrid::AbstractVector{TF},
policies0::Array) where {TF <: AbstractFloat}
S = size(model.Pi, 1) # number of states
xbar = [minimum(xgrid), maximum(xgrid)]
time_0 = false
z0 = Array{Array}(undef, length(xgrid), S)
cf, nf, xprimef = policies0[1], policies0[2], policies0[3]
for s in 1:S
for (i_x, x) in enumerate(xgrid)
cs = Array{TF}(undef, S)
ns = Array{TF}(undef, S)
xprimes = Array{TF}(undef, S)
for j in 1:S
cs[j], ns[j], xprimes[j] = cf[s, j](x), nf[s, j](x),
xprimef[s, j](x)
end
z0[i_x, s] = vcat(cs, ns, xprimes, zeros(S))
end
end
cFB, nFB, IFB, xFB, zFB = find_first_best(model, S, 2)
return BellmanEquation_Recursive(model, S, xbar, time_0, z0, cFB, nFB, xFB,
zFB)
end
function get_policies_time1(T::BellmanEquation_Recursive,
i_x::Integer,
x::Real,
s_::Integer,
Vf::AbstractArray{Function},
xbar::AbstractVector)
model, S = T.model, T.S
beta, Theta, G, Pi = model.beta, model.Theta, model.G, model.Pi
U, Uc, Un = model.U, model.Uc, model.Un
S_possible = sum(Pi[s_, :] .> 0)
sprimei_possible = findall(Pi[s_, :] .> 0)
function objf(z, grad)
c, xprime = z[1:S_possible], z[(S_possible + 1):(2S_possible)]
n = (c .+ G[sprimei_possible]) ./ Theta[sprimei_possible]
Vprime = [Vf[sprimei_possible[si]](xprime[si]) for si in 1:S_possible]
return -dot(Pi[s_, sprimei_possible], U.(c, n) + beta * Vprime)
end
function cons(out, z, grad)
c, xprime, TT = z[1:S_possible], z[(S_possible + 1):(2S_possible)],
z[(2S_possible + 1):(3S_possible)]
n = (c .+ G[sprimei_possible]) ./ Theta[sprimei_possible]
u_c = Uc.(c, n)
Eu_c = dot(Pi[s_, sprimei_possible], u_c)
out .= x * u_c / Eu_c - u_c .* (c - TT) - Un(c, n) .* n - beta * xprime
end
function cons_no_trans(out, z, grad)
c, xprime = z[1:S_possible], z[(S_possible + 1):(2S_possible)]
n = (c .+ G[sprimei_possible]) ./ Theta[sprimei_possible]
u_c = Uc.(c, n)
Eu_c = dot(Pi[s_, sprimei_possible], u_c)
out .= x * u_c / Eu_c - u_c .* c - Un(c, n) .* n - beta * xprime
end
if model.transfers == true
lb = vcat(zeros(S_possible), ones(S_possible) * xbar[1],
zeros(S_possible))
if model.n_less_than_one == true
ub = vcat(ones(S_possible) - G[sprimei_possible],
ones(S_possible) * xbar[2], ones(S_possible))
else
ub = vcat(100 * ones(S_possible),
ones(S_possible) * xbar[2],
100 * ones(S_possible))
end
init = vcat(T.z0[i_x, s_][sprimei_possible],
T.z0[i_x, s_][2S .+ sprimei_possible],
T.z0[i_x, s_][3S .+ sprimei_possible])
opt = Opt(:LN_COBYLA, 3S_possible)
equality_constraint!(opt, cons, zeros(S_possible))
else
lb = vcat(zeros(S_possible), ones(S_possible) * xbar[1])
if model.n_less_than_one == true
ub = vcat(ones(S_possible) - G[sprimei_possible],
ones(S_possible) * xbar[2])
else
ub = vcat(ones(S_possible), ones(S_possible) * xbar[2])
end
init = vcat(T.z0[i_x, s_][sprimei_possible],
T.z0[i_x, s_][2S .+ sprimei_possible])
opt = Opt(:LN_COBYLA, 2S_possible)
equality_constraint!(opt, cons_no_trans, zeros(S_possible))
end
init[init .> ub] = ub[init .> ub]
init[init .< lb] = lb[init .< lb]
min_objective!(opt, objf)
lower_bounds!(opt, lb)
upper_bounds!(opt, ub)
maxeval!(opt, 10000000)
maxtime!(opt, 10)
ftol_rel!(opt, 1e-8)
ftol_abs!(opt, 1e-8)
(minf, minx, ret) = NLopt.optimize(opt, init)
if ret != :SUCCESS && ret != :ROUNDOFF_LIMITED && ret != :MAXEVAL_REACHED &&
ret != :FTOL_REACHED && ret != :MAXTIME_REACHED
error("optimization failed: ret = $ret")
end
T.z0[i_x, s_][sprimei_possible] = minx[1:S_possible]
T.z0[i_x, s_][S .+ sprimei_possible] = minx[1:S_possible] .+
G[sprimei_possible]
T.z0[i_x, s_][2S .+ sprimei_possible] = minx[(S_possible .+ 1):(2S_possible)]
if model.transfers == true
T.z0[i_x, s_][3S .+ sprimei_possible] = minx[(2S_possible + 1):(3S_possible)]
else
T.z0[i_x, s_][3S .+ sprimei_possible] = zeros(S)
end
return vcat(-minf, T.z0[i_x, s_])
end
function get_policies_time0(T::BellmanEquation_Recursive,
B_::Real,
s0::Integer,
Vf::AbstractArray{Function},
xbar::AbstractVector)
model = T.model
beta, Theta, G = model.beta, model.Theta, model.G
U, Uc, Un = model.U, model.Uc, model.Un
function objf(z, grad)
if any(isnan, z)
return -Inf
end
c, xprime = z[1], z[2]
n = (c + G[s0]) / Theta[s0]
return -(U(c, n) + beta * Vf[s0](xprime))
end
function cons(z, grad)
if any(isnan, z)
return -Inf
end
c, xprime, TT = z[1], z[2], z[3]
n = (c + G[s0]) / Theta[s0]
return -Uc(c, n) * (c - B_ - TT) - Un(c, n) * n - beta * xprime
end
cons_no_trans(z, grad) = cons(vcat(z, 0), grad)
if model.transfers == true
lb = [0.0, xbar[1], 0.0]
if model.n_less_than_one == true
ub = [1 - G[s0], xbar[2], 100]
else
ub = [100.0, xbar[2], 100.0]
end
init = vcat(T.zFB[s0][1], T.zFB[s0][3], T.zFB[s0][4])
init = [0.95124922, -1.15926816, 0.0]
opt = Opt(:LN_COBYLA, 3)
equality_constraint!(opt, cons)
else
lb = [0.0, xbar[1]]
if model.n_less_than_one == true
ub = [1 - G[s0], xbar[2]]
else
ub = [100, xbar[2]]
end
init = vcat(T.zFB[s0][1], T.zFB[s0][3])
init = [0.95124922, -1.15926816]
opt = Opt(:LN_COBYLA, 2)
equality_constraint!(opt, cons_no_trans)
end
init[init .> ub] = ub[init .> ub]
init[init .< lb] = lb[init .< lb]
min_objective!(opt, objf)
lower_bounds!(opt, lb)
upper_bounds!(opt, ub)
maxeval!(opt, 100000000)
maxtime!(opt, 30)
(minf, minx, ret) = NLopt.optimize(opt, init)
if ret != :SUCCESS && ret != :ROUNDOFF_LIMITED && ret != :MAXEVAL_REACHED &&
ret != :FTOL_REACHED
error("optimization failed: ret = $ret")
end
if model.transfers == true
return -minf, minx[1], minx[1] + G[s0], minx[2], minx[3]
else
return -minf, minx[1], minx[1] + G[s0], minx[2], 0
end
end
get_policies_time0 (generic function with 2 methods)
66.4. Examples#
We now turn to some examples.
66.4.1. 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 constructor
function CRRAModel(;
beta = 0.9,
sigma = 2.0,
gamma = 2.0,
Pi = 0.5 * ones(2, 2),
G = [0.1, 0.2],
Theta = ones(Float64, 2),
transfers = false)
function U(c, n)
if sigma == 1.0
U = log(c)
else
U = (c .^ (1.0 - sigma) - 1.0) / (1.0 - sigma)
end
return U - n .^ (1 + gamma) / (1 + gamma)
end
# Derivatives of utility function
Uc(c, n) = c .^ (-sigma)
Ucc(c, n) = -sigma * c .^ (-sigma - 1.0)
Un(c, n) = -n .^ gamma
Unn(c, n) = -gamma * n .^ (gamma - 1.0)
n_less_than_one = false
return Model(beta, Pi, G, Theta, transfers,
U, Uc, Ucc, Un, Unn, n_less_than_one)
end
CRRAModel (generic function with 1 method)
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.
time_example = CRRAModel(;G = [0.1, 0.1, 0.1, 0.2, 0.1, 0.1],
Theta = ones(6)) # Theta can in principle be random
time_example.Pi = [0.0 1.0 0.0 0.0 0.0 0.0;
0.0 0.0 1.0 0.0 0.0 0.0;
0.0 0.0 0.0 0.5 0.5 0.0;
0.0 0.0 0.0 0.0 0.0 1.0;
0.0 0.0 0.0 0.0 0.0 1.0;
0.0 0.0 0.0 0.0 0.0 1.0]
# Initialize mugrid for value function iteration
mugrid = range(-0.7, 0.01, length = 200)
time_example.transfers = true # Government can use transfers
time_sequential = SequentialAllocation(time_example) # Solve sequential problem
time_bellman = RecursiveAllocation(time_example, mugrid)
sHist_h = [1, 2, 3, 4, 6, 6, 6]
sHist_l = [1, 2, 3, 5, 6, 6, 6]
sim_seq_h = simulate(time_sequential, 1.0, 1, 7, sHist_h)
sim_bel_h = simulate(time_bellman, 1.0, 1, 7, sHist_h)
sim_seq_l = simulate(time_sequential, 1.0, 1, 7, sHist_l)
sim_bel_l = simulate(time_bellman, 1.0, 1, 7, sHist_l)
using Plots
titles = hcat("Consumption", "Labor Supply", "Government Debt",
"Tax Rate", "Government Spending", "Output")
sim_seq_l_plot = hcat(sim_seq_l[1:3]..., sim_seq_l[4],
time_example.G[sHist_l],
time_example.Theta[sHist_l] .* sim_seq_l[2])
sim_bel_l_plot = hcat(sim_bel_l[1:3]..., sim_bel_l[5],
time_example.G[sHist_l],
time_example.Theta[sHist_l] .* sim_bel_l[2])
sim_seq_h_plot = hcat(sim_seq_h[1:3]..., sim_seq_h[4],
time_example.G[sHist_h],
time_example.Theta[sHist_h] .* sim_seq_h[2])
sim_bel_h_plot = hcat(sim_bel_h[1:3]..., sim_bel_h[5],
time_example.G[sHist_h],
time_example.Theta[sHist_h] .* sim_bel_h[2])
p = plot(size = (700, 700), layout = (3, 2),
xaxis = (0:6), grid = false, titlefont = Plots.font("sans-serif", 10))
plot!(p, title = titles)
for i in 1:6
plot!(p[i], 0:6, sim_seq_l_plot[:, i], marker = :circle, color = :black,
lab = "")
plot!(p[i], 0:6, sim_bel_l_plot[:, i], marker = :circle, color = :red,
lab = "")
plot!(p[i], 0:6, sim_seq_h_plot[:, i], marker = :utriangle, color = :black,
lab = "")
plot!(p[i], 0:6, sim_bel_h_plot[:, i], marker = :utriangle, color = :red,
lab = "")
end
p
diff = 0.019541811403379114
diff = 0.02029660452240845
diff = 0.018252901676510072
diff = 0.01732330163767122
diff = 0.0031325418440225337
diff = 0.0009645419766248225
diff = 0.00042579228255786673
diff = 0.00036342521821958824
diff = 0.00032679255217972085
diff = 0.00029387229120929646
diff = 0.00026447739141347324
diff = 0.00023803650678640254
diff = 0.0002142287734397143
diff = 0.00019279780504228292
diff = 0.0001735091765435803
diff = 0.00015615908664984014
diff = 0.00014054590902726154
diff = 0.00012649574833402226
diff = 0.00011397774745936076
diff = 0.0001025633783083926
diff = 9.25715117615316e-5
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.
66.4.1.1. 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
function log_utility(; beta = 0.9,
psi = 0.69,
Pi = 0.5 * ones(2, 2),
G = [0.1, 0.2],
Theta = ones(2),
transfers = false)
# Derivatives of utility function
U(c, n) = log(c) + psi * log(1 - n)
Uc(c, n) = 1 ./ c
Ucc(c, n) = -c .^ (-2.0)
Un(c, n) = -psi ./ (1.0 .- n)
Unn(c, n) = -psi ./ (1.0 .- n) .^ 2.0
n_less_than_one = true
return Model(beta, Pi, G, Theta, transfers,
U, Uc, Ucc, Un, Unn, n_less_than_one)
end
log_utility (generic function with 1 method)
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 = log_utility()
log_example.transfers = true # Government can use transfers
log_sequential = SequentialAllocation(log_example) # Solve sequential problem
log_bellman = RecursiveAllocation(log_example, mugrid) # Solve recursive problem
T = 20
sHist = [1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1]
#simulate
sim_seq = simulate(log_sequential, 0.5, 1, T, sHist)
sim_bel = simulate(log_bellman, 0.5, 1, T, sHist)
sim_seq_plot = hcat(sim_seq[1:3]...,
sim_seq[4], log_example.G[sHist],
log_example.Theta[sHist] .* sim_seq[2])
sim_bel_plot = hcat(sim_bel[1:3]...,
sim_bel[5], log_example.G[sHist],
log_example.Theta[sHist] .* sim_bel[2])
#plot policies
p = plot(size = (700, 700), layout = grid(3, 2),
xaxis = (0:T), grid = false, titlefont = Plots.font("sans-serif", 10))
labels = fill(("", ""), 6)
labels[3] = ("Complete Market", "Incomplete Market")
plot!(p, title = titles)
for i in vcat(collect(1:4), 6)
plot!(p[i], sim_seq_plot[:, i], marker = :circle, color = :black,
lab = labels[i][1])
plot!(p[i], sim_bel_plot[:, i], marker = :utriangle, color = :blue,
lab = labels[i][2],
legend = :bottomright)
end
plot!(p[5], sim_seq_plot[:, 5], marker = :circle, color = :blue, lab = "")
diff = 0.0007972381260195522
diff = 0.0006422953131037616
diff = 0.0005516976988532969
diff = 0.00048551723998322815
diff = 0.0004226764995022081
diff = 0.0003755566349948464
diff = 0.00032946155458158543
diff = 0.00029344519610484003
diff = 0.00025864074006475533
diff = 0.00023053953302378002
diff = 0.0002037962884294243
diff = 0.000181692546858797
diff = 0.00016095099490449685
diff = 0.00014347930226203394
diff = 0.00012729620276108014
diff = 0.0001134603872880407
diff = 0.00010076436382712763
diff = 8.97855842666275e-5
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_long = 200
sim_seq_long = simulate(log_sequential, 0.5, 1, T_long)
sHist_long = sim_seq_long[end - 2]
sim_bel_long = simulate(log_bellman, 0.5, 1, T_long, sHist_long)
sim_seq_long_plot = hcat(sim_seq_long[1:4]...,
log_example.G[sHist_long],
log_example.Theta[sHist_long] .* sim_seq_long[2])
sim_bel_long_plot = hcat(sim_bel_long[1:3]..., sim_bel_long[5],
log_example.G[sHist_long],
log_example.Theta[sHist_long] .* sim_bel_long[2])
p = plot(size = (700, 700), layout = (3, 2), xaxis = (0:50:T_long),
grid = false,
titlefont = Plots.font("sans-serif", 10))
plot!(p, title = titles)
for i in 1:6
plot!(p[i], sim_seq_long_plot[:, i], color = :black, linestyle = :solid,
lab = labels[i][1])
plot!(p[i], sim_bel_long_plot[:, i], color = :blue, linestyle = :dot,
lab = labels[i][2],
legend = :bottomright)
end
p