# 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.

```
using LinearAlgebra, Statistics
```

## 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 QuantEcon, NLsolve, NLopt
import QuantEcon.simulate
mutable struct Model{TF <: AbstractFloat,
TM <: AbstractMatrix{TF},
TV <: AbstractVector{TF}}
β::TF
Π::TM
G::TV
Θ::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
ΞFB::TV
zFB::TV
end
function SequentialAllocation(model::Model)
β, Π, G, Θ = model.β, model.Π, model.G, model.Θ
mc = MarkovChain(Π)
S = size(Π, 1) # Number of states
# Now find the first best allocation
cFB, nFB, ΞFB, zFB = find_first_best(model, S, 1)
return SequentialAllocation(model, mc, S, cFB, nFB, ΞFB, 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
β, Θ, Uc, Un, G, Π =
model.β, model.Θ, model.Uc, model.Un, model.G, model.Π
function res!(out, z)
c = z[1:S]
n = z[S+1:end]
out[1:S] = Θ .* Uc.(c, n) + Un.(c, n)
out[S+1:end] = Θ .* 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]
ΞFB = Uc(cFB, nFB) # Multiplier on the resource constraint
zFB = vcat(cFB, nFB, ΞFB)
return cFB, nFB, ΞFB, 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 - β * Π, 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, μ::Real)
model, S = pas.model, pas.S
Θ, β, Π, G, Uc, Ucc, Un, Unn =
model.Θ, model.β, model.Π, model.G,
model.Uc, model.Ucc, model.Un, model.Unn
function FOC!(out, z::Vector)
c = z[1:S]
n = z[S+1:2S]
Ξ = z[2S+1:end]
out[1:S] = Uc.(c, n) - μ * (Ucc.(c, n) .* c + Uc.(c, n)) - Ξ # FOC c
out[S+1:2S] = Un.(c, n) - μ * (Unn(c, n) .* n .+ Un.(c, n)) + Θ .* Ξ # FOC n
out[2S+1:end] = Θ .* 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, Ξ = 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 - β * model.Π, I)
return c, n, x, Ξ
end
function time0_allocation(pas::SequentialAllocation,
B_::AbstractFloat, s_0::Integer)
model = pas.model
Π, Θ, G, β = model.Π, model.Θ, model.G, model.β
Uc, Ucc, Un, Unn =
model.Uc, model.Ucc, model.Un, model.Unn
# First order conditions of planner's problem
function FOC!(out, z)
μ, c, n, Ξ = z[1], z[2], z[3], z[4]
xprime = time1_allocation(pas, μ)[3]
out .= vcat(
Uc(c, n) .* (c - B_) + Un(c, n) .* n + β * dot(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]
)
end
# Find root
res = nlsolve(FOC!, [0.0, pas.cFB[s_0], pas.nFB[s_0], pas.ΞFB[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, μ::Real)
model = pas.model
c, n, x, Ξ = time1_allocation(pas, μ)
U_val = model.U.(c, n)
V = \(LinearAlgebra.I - model.β*model.Π, U_val)
return c, n, x, V
end
function Τ(model::Model, c::Union{Real,Vector}, n::Union{Real,Vector})
Uc, Un = model.Uc.(c, n), model.Un.(c, n)
return 1. .+ Un./(model.Θ .* Uc)
end
function simulate(pas::SequentialAllocation,
B_::AbstractFloat, s_0::Integer,
T::Integer,
sHist::Union{Vector, Nothing}=nothing)
model = pas.model
Π, β, Uc = model.Π, model.β, model.Uc
if isnothing(sHist)
sHist = QuantEcon.simulate(pas.mc, T, init=s_0)
end
cHist = zeros(T)
nHist = zeros(T)
Bhist = zeros(T)
ΤHist = zeros(T)
μHist = zeros(T)
RHist = zeros(T-1)
# time 0
μ, cHist[1], nHist[1], _ = time0_allocation(pas, B_, s_0)
ΤHist[1] = Τ(pas.model, cHist[1], nHist[1])[s_0]
Bhist[1] = B_
μHist[1] = μ
# time 1 onward
for t in 2:T
c, n, x, Ξ = time1_allocation(pas,μ)
u_c = Uc(c,n)
s = sHist[t]
ΤHist[t] = Τ(pas.model, c, n)[s]
Eu_c = dot(Π[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]) / (β * Eu_c)
μHist[t] = μ
end
return cHist, nHist, Bhist, ΤHist, sHist, μHist, 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.Π, 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
β, Θ, G, Π = model.β, model.Θ, model.G, model.Π
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) + β * dot(Π[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 - β * dot(Π[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
β, Θ, G, Π = model.β, model.Θ, model.G, model.Π
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) + β * dot(Π[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 - β * dot(Π[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

```
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
μgrid::TVg
xgrid::TVg
Vf::Array
policies::TT
end
function RecursiveAllocation(model::Model, μgrid::AbstractArray)
G = model.G
S = size(model.Π, 1) # number of states
mc = MarkovChain(model.Π)
# now find the first best allocation
Vf, policies, T, xgrid = solve_time1_bellman(model, μgrid)
T.time_0 = true # Bellman equation now solves time 0 problem
return RecursiveAllocation(model, mc, S, T, μgrid, xgrid, Vf, policies)
end
function solve_time1_bellman(model::Model{TR}, μgrid::AbstractArray) where {TR <: Real}
Π = model.Π
S = size(model.Π, 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,
μ_::AbstractFloat,
s_::Integer)
c, n, x, V = time1_value(PP, μ_)
return c, n, dot(Π[s_, :], x), dot(Π[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(μgrid))
for s_ in 1:S
c = Array{TR}(undef, length(μgrid), S)
n = Array{TR}(undef, length(μgrid), S)
x = Array{TR}(undef, length(μgrid))
V = Array{TR}(undef, length(μgrid))
for (i_μ, μ) in enumerate(μgrid)
c[i_μ, :], n[i_μ, :], x[i_μ], V[i_μ] =
incomplete_allocation(PP, μ, s_)
end
xprimes = repeat(x, 1, S)
xgrid[s_, :] = x
for sprime = 1:S
splc = CubicSpline(c[:, sprime][end:-1:1], x[end:-1:1])
spln = CubicSpline(n[:, sprime][end:-1:1], x[end:-1:1])
splx = CubicSpline(xprimes[:, sprime][end:-1:1], x[end:-1:1])
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])
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(μgrid))
# 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=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=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. .+ Un ./ (model.Θ .* 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
Π, Uc = model.Π, 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)
μHist = 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_
μHist[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=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(Π[s_, :], u_c)
μHist[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, μHist, sHist
end
function BellmanEquation_Recursive(model::Model{TF},
xgrid::AbstractVector{TF},
policies0::Array) where {TF <: AbstractFloat}
S = size(model.Π, 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 = 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
β, Θ, G, Π = model.β, model.Θ, model.G, model.Π
U,Uc,Un = model.U, model.Uc, model.Un
S_possible = sum(Π[s_, :].>0)
sprimei_possible = findall(Π[s_, :].>0)
function objf(z, grad)
c, xprime = z[1:S_possible], z[S_possible+1:2S_possible]
n = (c .+ G[sprimei_possible]) ./ Θ[sprimei_possible]
Vprime = [Vf[sprimei_possible[si]](xprime[si]) for si in 1:S_possible]
return -dot(Π[s_, sprimei_possible], U.(c, n) + β * 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]) ./ Θ[sprimei_possible]
u_c = Uc.(c, n)
Eu_c = dot(Π[s_, sprimei_possible], u_c)
out .= x * u_c/Eu_c - u_c .* (c - TT) - Un(c, n) .* n - β * 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]) ./ Θ[sprimei_possible]
u_c = Uc.(c, n)
Eu_c = dot(Π[s_, sprimei_possible], u_c)
out .= x * u_c / Eu_c - u_c .* c - Un(c, n) .* n - β * 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
β, Θ, G = model.β, model.Θ, model.G
U, Uc, Un = model.U, model.Uc, model.Un
function objf(z, grad)
c, xprime = z[1], z[2]
n = (c + G[s0]) / Θ[s0]
return -(U(c, n) + β * Vf[s0](xprime))
end
function cons(z,grad)
c, xprime, TT = z[1], z[2], z[3]
n = (c + G[s0]) / Θ[s0]
return -Uc(c, n) * (c - B_ - TT) - Un(c, n) * n - β * 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 crra_utility(;
β = 0.9,
σ = 2.0,
γ = 2.0,
Π = 0.5 * ones(2, 2),
G = [0.1, 0.2],
Θ = ones(Float64, 2),
transfers = false
)
function U(c, n)
if σ == 1.0
U = log(c)
else
U = (c.^(1.0 - σ) - 1.0) / (1.0 - σ)
end
return U - n.^(1 + γ) / (1 + γ)
end
# Derivatives of utility function
Uc(c,n) = c.^(-σ)
Ucc(c,n) = -σ * c.^(-σ - 1.0)
Un(c,n) = -n.^γ
Unn(c,n) = -γ * n.^(γ - 1.0)
n_less_than_one = false
return Model(β, Π, G, Θ, transfers,
U, Uc, Ucc, Un, Unn, n_less_than_one)
end
```

```
crra_utility (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 = crra_utility(G=[0.1, 0.1, 0.1, 0.2, 0.1, 0.1],
Θ = ones(6)) # Θ can in principle be random
time_example.Π = [ 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 μgrid for value function iteration
μgrid = 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, μgrid)
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., 1, 7, sHist_h)
sim_bel_h = simulate(time_bellman, 1., 1, 7, sHist_h)
sim_seq_l = simulate(time_sequential, 1., 1, 7, sHist_l)
sim_bel_l = simulate(time_bellman, 1., 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.Θ[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.Θ[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.Θ[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.Θ[sHist_h] .* sim_bel_h[2])
p = plot(size = (920, 750), layout =(3, 2),
xaxis=(0:6), grid=false, titlefont=Plots.font("sans-serif", 10))
plot!(p, title = titles)
for i=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 = 0.05545573932892626
```

```
diff = 0.05894820861417962
```

```
diff = 0.05298804217786332
```

```
diff = 0.05437935197266899
```

```
diff = 0.0018462201896328808
```

```
diff = 0.0017084870909002004
```

```
diff = 0.0006059670356647582
```

```
diff = 0.0005455174588738608
```

```
diff = 0.0004911912886524149
```

```
diff = 0.0004420464883467868
```

```
diff = 0.0003977969812743899
```

```
diff = 0.00035796887638022047
```

```
diff = 0.00032212344256885754
```

```
diff = 0.00028986712036026377
```

```
diff = 0.0002608134900674362
```

```
diff = 0.00023466965302054163
```

```
diff = 0.00021113598212874472
```

```
diff = 0.00018998371865285555
```

```
diff = 0.00017094147083400183
```

```
diff = 0.00015380899495526682
```

```
diff = 0.0001384070099291766
```

```
diff = 0.00012454848077247024
```

```
diff = 0.00011207896725530902
```

```
diff = 0.00010084259614588416
```

```
diff = 9.076645849624518e-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(;β = 0.9,
ψ = 0.69,
Π = 0.5 * ones(2, 2),
G = [0.1, 0.2],
Θ = ones(2),
transfers = false)
# Derivatives of utility function
U(c,n) = log(c) + ψ * log(1 - n)
Uc(c,n) = 1 ./ c
Ucc(c,n) = -c.^(-2.0)
Un(c,n) = -ψ ./ (1.0 .- n)
Unn(c,n) = -ψ ./ (1.0 .- n).^2.0
n_less_than_one = true
return Model(β, Π, G, Θ, 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, μgrid) # 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.Θ[sHist] .* sim_seq[2])
sim_bel_plot = hcat(sim_bel[1:3]...,
sim_bel[5], log_example.G[sHist], log_example.Θ[sHist] .* sim_bel[2])
#plot policies
p = plot(size = (920, 750), 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 = 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 = 0.0007972383286751266
```

```
diff = 0.0006422479721426341
```

```
diff = 0.0005516714642152489
```

```
diff = 0.000485538815688431
```

```
diff = 0.0004226122736966621
```

```
diff = 0.0003754551259341305
```

```
diff = 0.00032934357539644437
```

```
diff = 0.0002932414314594749
```

```
diff = 0.00025845347938131005
```

```
diff = 0.0002302657709037761
```

```
diff = 0.0002035780400847809
```

```
diff = 0.00018139957013859378
```

```
diff = 0.0001607151961719876
```

```
diff = 0.00014318438885313562
```

```
diff = 0.0001270615310969653
```

```
diff = 0.00011255971952117789
```

```
diff = 0.00010088438997328722
```

```
diff = 8.957635941783829e-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.Θ[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.Θ[sHist_long] .* sim_bel_long[2])
p = plot(size = (920, 750), layout = (3, 2), xaxis=(0:50:T_long), grid=false,
titlefont=Plots.font("sans-serif", 10))
plot!(p, title = titles)
for i = 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
```

- 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 (66.11) takes a positive value, so \(\gamma_t(\tilde s^t)>0\).