Code should execute sequentially if run in a Jupyter notebook

# Optimal Taxation without State-Contingent Debt¶

## 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 the assumptions and equilibrium concepts
• solve the model
• implement the model numerically and
• conduct some policy experiments
• compare outcomes with those in a complete-markets model

We begin with an introduction to the model

## A competitive equilibrium with distorting taxes¶

Many but not all features of the economy are identical to those of the Lucas-Stokey economy

For $$t \geq 0$$, the 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 $$t$$

A representative household is endowed with one unit of time each period that can be divided between leisure $$\ell_t$$ and labor $$n_t$$:

(1)$n_t(s^t) + \ell_t(s^t) = 1$

Output equals $$n_t(s^t)$$ and can be divided between consumption $$c_t(s^t)$$ and $$g(s_t)$$

(2)$c_t(s^t) + g(s_t) = n_t(s^t)$

A representative household’s preferences over $$\{c_t(s^t), \ell_t(s^t)\}_{t=0}^\infty$$ are ordered by

(3)$\sum_{t=0}^\infty \sum_{s^t} \beta^t \pi_t(s^t) u[c_t(s^t), \ell_t(s^t)]$

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 technology pins down a pre-tax wage rate to unity for all $$t, s^t$$

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

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

### 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 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

(4)\begin{split}\begin{aligned} b_t(s^{t-1}) & = \tau^n_t(s^t) n_t(s^t) - g_t(s_t) - T_t(s^t) + {b_{t+1}(s^t) \over R_t(s^t )} \\ & \equiv z(s^t) + {b_{t+1}(s^t) \over R_t(s^t )}, \end{aligned}\end{split}

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

${1 \over R_t(s^t)} = \sum_{s^{t+1}\vert s^t} \beta \pi_{t+1}(s^{t+1} | s^t) { u_c(s^{t+1}) \over u_c(s^{t}) }$

Substituting this expression into the government’s budget constraint (4) yields:

(5)$b_t(s^{t-1}) = z(s^t) + \beta \sum_{s^{t+1}\vert s^t} \pi_{t+1}(s^{t+1} | s^t) { u_c(s^{t+1}) \over u_c(s^{t}) } \; b_{t+1}(s^t)$

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 (5) also has to depend only on $$s^{t-1}$$

This feature 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 (5) by the right side of next period’s budget constraint (associated with a particular realization $$s_{t}$$) we get

$b_t(s^{t-1}) = z(s^t) + \sum_{s^{t+1}\vert s^t} \beta \pi_{t+1}(s^{t+1} | s^t) { u_c(s^{t+1}) \over u_c(s^{t}) } \, \left[z(s^{t+1}) + {b_{t+2}(s^{t+1}) \over R_{t+1}(s^{t+1})}\right]$

After similar repeated substitutions for all future occurrences of government indebtedness, and by invoking the natural debt limit, we arrive at:

(6)\begin{aligned} b_t(s^{t-1}) &= \sum_{j=0}^\infty \sum_{s^{t+j} | s^t} \beta^j \pi_{t+j}(s^{t+j} | s^t) { u_c(s^{t+j}) \over u_c(s^{t}) } \;z(s^{t+j}) \end{aligned}

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

(7)$z(s^t) = \left[1 - {u_{\ell}(s^t) \over u_c(s^t)}\right] \left[c_t(s^t)+g_t(s_t)\right] -g_t(s_t) - T_t(s^t)\,.$

If we substitute the appropriate versions of right side of (7) for $$z(s^{t+j})$$ in equation (6), we obtain a sequence of implementability constraints on a Ramsey allocation in an AMSS economy

Expression (6) at time $$t=0$$ and initial state $$s^0$$ was also an implementability constraint on a Ramsey allocation in a Lucas-Stokey economy:

(8)$b_0(s^{-1}) = \EE_0 \sum_{j=0}^\infty \beta^j { u_c(s^{j}) \over u_c(s^{0}) } \;z(s^{j})$

But now we also have a large number of additional implementability constraints

(9)$b_t(s^{t-1}) = \EE_t \sum_{j=0}^\infty \beta^j { u_c(s^{t+j}) \over u_c(s^{t}) } \;z(s^{t+j})$

Equation (9) must hold for each $$s^t$$ for each $$t \geq 1$$

### Comparison with Lucas-Stokey economy¶

The expression on the right side of (9) would in the Lucas-Stokey (1983) economy equal the present value of a continuation stream of government surpluses evaluated at what would be competitive equilbrium 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 same present value 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 what would be the present value of continuation government surpluses in the Lucas-Stokey model has to be the marketable subspace of the AMSS model

### Ramsey problem without state-contingent debt¶

After we have substituted the resource constraint into the utility function, we can express the Ramsey problem as choosing an allocation that solves

$\max_{\{c_t(s^t),b_{t+1}(s^t)\}} \EE_0 \sum_{t=0}^\infty \beta^t u\left(c_t(s^t),1-c_t(s^t)-g_t(s_t)\right)$

where the maximization is subject to

(10)$\EE_{0} \sum_{j=0}^\infty \beta^j { u_c(s^{j}) \over u_c(s^{0}) } \;z(s^{j}) \geq b_0(s^{-1})$

and

(11)$\EE_{t} \sum_{j=0}^\infty \beta^j { u_c(s^{t+j}) \over u_c(s^{t}) } \; z(s^{t+j}) = b_t(s^{t-1}) \quad \forall \, s^t$

given $$b_0(s^{-1})$$

#### Lagrangian formulation¶

Let $$\gamma_0(s^0)$$ be a nonnegative Lagrange multiplier on constraint (10)

As in the Lucas-Stokey economy, this multiplier is strictly positive if the government must resort to distortionary taxation; otherwise it equals zero

A consequence of the assumption that markets in state-contingent securities have been shut down but that a market in a risk-free security remains is that we have to attach stochastic processes $$\{\gamma_t(s^t)\}_{t=1}^\infty$$ of Lagrange multipliers to the implementability constraints (11)

Depending on how the constraints bind, these multipliers might be positive or negative,

\begin{split}\begin{aligned} \gamma_t(s^t) &\;\geq\; (\leq)\;\, 0 \quad \text{if the constraint binds in this direction } \\ & \EE_{t} \sum_{j=0}^\infty \beta^j { u_c(s^{t+j}) \over u_c(s^{t}) } \;z(s^{t+j}) \;\geq \;(\leq)\;\, b_t(s^{t-1}). \end{aligned}\end{split}

A negative multiplier $$\gamma_t(s^t)<0$$ means that if we could relax constraint , 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

### Some calculations¶

Apply two transformations to the Lagrangian

Multiply constraint by $$u_c(s^0)$$ and the constraints by $$\beta^t u_c(s^{t})$$

Then a Lagrangian for the Ramsey problem can be represented as

(12)\begin{split}\begin{aligned} J &= \EE_{0} \sum_{t=0}^\infty \beta^t \biggl\{ u\left(c_t(s^t), 1-c_t(s^t)-g_t(s_t)\right)\\ & \qquad + \gamma_t(s^t) \Bigl[ \EE_{t} \sum_{j=0}^\infty \beta^j u_c(s^{t+j}) \,z(s^{t+j}) - u_c(s^{t}) \,b_t(s^{t-1}) \biggr\} \\ &= \EE_{0} \sum_{t=0}^\infty \beta^t \biggl\{ u\left(c_t(s^t), 1-c_t(s^t)-g_t(s_t)\right) \\ & \qquad + \Psi_t(s^t)\, u_c(s^{t}) \,z(s^{t}) - \gamma_t(s^t)\, u_c(s^{t}) \, b_t(s^{t-1}) \biggr\} \end{aligned}\end{split}

where

(13)$\Psi_t(s^t)=\Psi_{t-1}(s^{t-1})+\gamma_t(s^t) \quad \text{and} \quad \Psi_{-1}(s^{-1})=0$

In (12), the second equality uses the law of iterated expectations and Abel’s summation formula

First-order conditions with respect to $$c_t(s^t)$$ can be expressed as

(14)\begin{split}\begin{aligned} u_c(s^t)-u_{\ell}(s^t) &+ \Psi_t(s^t)\left\{ \left[ u_{cc}(s^t) - u_{c\ell}(s^{t})\right]z(s^{t}) + u_{c}(s^{t})\,z_c(s^{t}) \right\} \\ & \hspace{35mm} - \gamma_t(s^t)\left[ u_{cc}(s^{t}) - u_{c\ell}(s^{t})\right]b_t(s^{t-1}) =0 \end{aligned}\end{split}

and with respect to $$b_t(s^t)$$ as

(15)$\EE_{t} \left[\gamma_{t+1}(s^{t+1})\,u_c(s^{t+1})\right] = 0$

If we substitute $$z(s^t)$$ from (7) and its derivative $$z_c(s^t)$$ into first-order condition (14), we find two differences from the corresponding condition for the optimal allocation in a Lucas-Stokey economy with state-contingent government debt

1. The term involving $$b_t(s^{t-1})$$ in first-order condition (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

2. The Lagrange multiplier $$\Psi_t(s^t)$$ in first-order condition (14) may change over time in response to realizations of the state, while the multiplier $$\Phi$$ in the Lucas-Stokey economy is time invariant

Using some code from our an earlier lecture on optimal taxation with state-contingent debt sequential allocation implementation is as follows

using QuantEcon
using NLsolve
using NLopt

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

"""
Class returns planner's allocation as a function of the multiplier
on the implementability constraint mu
"""
struct SequentialAllocation{TP <: Model,
TI <: Integer,
TV <: AbstractVector}
model::TP
mc::MarkovChain
S::TI
cFB::TV
nFB::TV
XiFB::TV
zFB::TV
end

"""
Initializes the class from the calibration model
"""
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

"""
Find the first best allocation
"""
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!(z, out)
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 = $$eye(S) - beta*Pi, IFB) zFB = [vcat(cFB[s], xFB[s], xFB) for s in 1:S] return cFB, nFB, IFB, xFB, zFB end end """ Computes optimal allocation for time t\geq 1 for a given \mu """ 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!(z::Vector, out) 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 = \(eye(S) - beta*model.Pi, I) return c, n, x, Xi end """ Finds the optimal allocation given initial government debt B_ and state s_0 """ 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!(z, out) 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 """ Find the value associated with multiplier mu """ 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 = \(eye(pas.S) - model.beta*model.Pi, U_val) return c, n, x, V end """ Computes Tau given c, n """ function Tau(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 """ Simulates planners policies for T periods """ function simulate(pas::SequentialAllocation, B_::AbstractFloat, s_0::Integer, T::Integer, sHist::Union{Vector, Void}=nothing) model = pas.model Pi, beta, Uc = model.Pi, model.beta, model.Uc if sHist == nothing sHist = QuantEcon.simulate(pas.mc, T, init=s_0) end cHist = zeros(T) nHist = zeros(T) Bhist = zeros(T) TauHist = zeros(T) muHist = zeros(T) RHist = zeros(T-1) #time0 mu, cHist[1], nHist[1], _ = time0_allocation(pas, B_, s_0) TauHist[1] = Tau(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] TauHist[t] = Tau(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, TauHist, sHist, muHist, RHist end """ Bellman equation for the continuation of the Lucas-Stokey Problem """ 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 """ Initializes the class from the calibration model """ 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 """ Finds the optimal policies """ 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) = 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 """ Finds the optimal policies """ 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) = optimize(opt, init) return vcat(-minf, vcat(minx[1], minx[1]+G[s0], minx[2:end])) end  To analyze the AMSS model, we find it useful to adopt a recursive formulation using techniques like those in the lectures on dynamic stackelberg models and optimal taxation with state-contingent debt ## A recursive version of AMSS model¶ We now describe a recursive version 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 (8) from the Lucas-Stokey economy, but • adds measurability constraints (6) on functions of tails of the 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

### Recasting state variables¶

In the AMSS setting, the government faces a sequence of budget constraints

$\tau_t(s^t) n_t(s^t) + T_t(s^t) + b_{t+1}(s^t)/ R_t (s^t) = g_t + b_t(s^{t-1})$

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

In most of the remainder of this lecture, we shall set transfers to zero

In this case, the household faces a sequence of budget constraints

(16)$b_t(s^{t-1}) + (1-\tau_t(s^t)) n_t(s^t) = c_t(s^t) + b_{t+1}(s^t)/R_t(s^t)$

The household’s first-order conditions are $$u_{c,t} = \beta R_t \EE_t u_{c,t+1}$$ and $$(1-\tau_t) u_{c,t} = u_{l,t}$$

Using these to eliminate $$R_t$$ and $$\tau_t$$ from budget constraint (16) gives

(17)$b_t(s^{t-1}) + \frac{u_{l,t}(s^t)}{u_{c,t}(s^t)} n_t(s^t) = c_t(s^t) + {\frac{\beta (\EE_t u_{c,t+1}) b_{t+1}(s^t)}{u_{c,t}(s^t)}}$

or

(18)$u_{c,t}(s^t) b_t(s^{t-1}) + u_{l,t}(s^t) n_t(s^t) = u_{c,t}(s^t) c_t(s^t) + \beta (\EE_t u_{c,t+1}) b_{t+1}(s^t)$

Now define

(19)$x_t \equiv \beta b_{t+1}(s^t) \EE_t u_{c,t+1} = u_{c,t} (s^t) {\frac{b_{t+1}(s^t)}{R_t(s^t)}}$

and represent the household’s budget constraint at time $$t$$, history $$s^t$$ as

(20)${\frac{u_{c,t} x_{t-1}}{\beta \EE_{t-1} u_{c,t}}} = u_{c,t} c_t - u_{l,t} n_t + x_t$

for $$t \geq 1$$

### Measurability constraints¶

Write equation (18) as

(21)$b_t(s^{t-1}) = c_t(s^t) - { \frac{u_{l,t}(s^t)}{u_{c,t}(s^t)}} n_t(s^t) + {\frac{\beta (\EE_t u_{c,t+1}) b_{t+1}(s^t)}{u_{c,t}}}$

The right side of equation (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 (21) must equal $$b_t(s^{t-1})$$

That means that is has to be measurable with respect to $$s^{t-1}$$

Equations (21) are the measurablility constraints that the AMSS model adds to the single time $$0$$ implementation constraint imposed in the Lucas and Stokey model

### 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

(22)$V(x_-,s_-) = \max_{\{n(s), x(s)\}} \sum_s \Pi(s|s_-) \left[ u(n(s) - g(s), 1-n(s)) + \beta V(x(s),s) \right]$

subject to the following collection of implementability constraints, one for each $$s \in {\cal S}$$:

(23)${\frac{u_c(s) x_- }{\beta \sum_{\tilde s} \Pi(\tilde s|s_-) u_c(\tilde s) }} = u_c(s) (n(s) - g(s)) - u_l(s) n(s) + x(s)$

A continuation Ramsey planner at $$t \geq 1$$ takes $$(x_{t-1}, s_{t-1}) = (x_-, s_-)$$ as given and chooses $$(n_t(s_t), x_t(s_t)) = (n(s), x(s))$$ for $$s \in {\cal S}$$ before $$s$$ is realized

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

(24)$W(b_0, s_0) = \max_{n_0, x_0} u(n_0 - g_0, 1-n_0) + \beta V(x_0,s_0)$

where maximization is subject to

(25)$u_{c,0} b_0 = u_{c,0} (n_0-g_0) - u_{l,0} n_0 + x_0$

### Martingale supercedes state-variable degeneracy¶

Let $$\mu(s|s_-) \Pi(s|s_-)$$ be a Lagrange multiplier on constraint (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

(26)$\beta V_x(x(s),s) = \mu(s|s_-)$

Applying the envelope theorem to Bellman equation (22) gives

(27)$V_x(x_-,s_-) = \sum_s \Pi(s|s_-) \mu(s|s_-) {\frac{u_c(s)}{\beta \sum_{\tilde s} \Pi(\tilde s|s_-) u_c(\tilde s) }}$

Equations (26) and (27) imply that

(28)$V_x(x_-, s_-) = \sum_{s} \left( \Pi(s|s_-) {\frac{u_c(s)}{\sum_{\tilde s} \Pi(\tilde s| s_-) u_c(\tilde s)}} \right) V_x(x(s), s)$

Equation (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 generated by the twisted transition probability matrix:

$\check \Pi(s|s_-) \equiv \Pi(s|s_-) {\frac{u_c(s)}{\sum_{\tilde s} \Pi(\tilde s| s_-) u_c(\tilde s)}}$

Exercise: Please verify that $$\check \Pi(s|s_-)$$ is a valid Markov transition density, in particular, that its elements are all nonnegative and that for each $$s_-$$, the sum over $$s$$ equals unity

### Nonnegative transfers¶

Suppose that instead of imposing $$T_t = 0$$, we impose a nonnegativity constraint $$T_t\geq 0$$ on transfers

We also consider the special case of quasi-linear preferences studied by AMSS, $$u(c,l)= c + H(l)$$

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

$(1-\mu(s|s_-) ) (1 - u_l(s)) + \mu(s|s_-) n(s) u_{ll}(s) =0$

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

### 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 also the initial government debt $$b_0$$

In our recursive formulation of the Lucas-Stokey, we found that

• the counterpart to $$V_x(x,s)$$ is time invariant and equal to the Lagrange multiplier on the Lucas-Stokey implementability constraint
• the 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 is what transmits a twisted martingale-like component to consumption, employment, and the tax rate

## Special case of AMSS model¶

That the Ramsey allocation for the AMSS model differs from the Ramsey allocation of the Lucas-Stokey model is a symptom that the measurability constraints (6) bind

Following Bhandari, Evans, Golosov, and Sargent [BEGS13] (henceforth BEGS), we now consider a special case of the AMSS model in which these constraints don’t bind

Here a Ramsey planner would choose not to issue state-contingent debt even if he were free to do so

The environment is one in which fluctuations in the risk-free interest rate provide all of the insurance that the Ramsey planner wants

Following BEGS, we set $$S=2$$ and assume that the state $$s_t$$ is i.i.d., so that the transition matrix $$\Pi(s'|s) = \Pi(s')$$ for $$s=1,2$$

Following BEGS, it is useful to consider the following special case of the implementability constraints evaluated at the constant value of the state variable $$x_-= x(s) = \check x$$:

(29)${\frac{u_c(s) \check x}{\beta \sum_{\tilde s} \Pi(\tilde s) u_c(\tilde s) }} = u_c(s) (n(s) - g(s) ) - u_l(s) n(s) + \check x, \quad s=1,2$

We guess and verify that the scaled Lagrange multiplier $$\mu(s)=\mu$$ is a constant that is independent of $$s$$

At a fixed $$x$$, because $$V_x(x, s)$$ must be independent of $$s_-$$, the risk-adjusted martingale equation (28) becomes

$V_x(\check x) = \sum_{s} \left( \Pi(s) {\frac{u_c(s)}{\sum_{\tilde s} \Pi(\tilde s) u_c(\tilde s)}} \right) V_x(\check x) = V_x(\check x)$

This confirms that $$\mu(s) = \beta V_x$$ is constant at some value $$\mu$$

For the continuation Ramsey planner facing implementability constraints (29), the first-order conditions with respect to $$n(s)$$ become

(30)\begin{aligned} ( u_c(s) - u_l(s) ) \mu \left\{ \frac{\check x}{\beta \sum_{\tilde s} \Pi(\tilde s) u_c(\tilde s)} (u_{cc}(s) - u_{cl}(s) ) - u_c(s) - n(s) (u_{cc}(s) - u_{cl}(s)) - u_{lc}(s) n(s) + u_l(s) \right\} = 0 \end{aligned}

Equations (30) and are four equations in the four unknowns $$\check x$$, and $$\mu$$ and $$n(s), s=1,2$$

Under some regularity conditions on the period utility function $$u(c,l)$$, BEGS show that these equations have a unique solution that features a negative value of $$\check x$$

Consumption $$c(s)$$ and the flat-rate tax on labor $$\tau(s)$$ can then be constructed as history-independent functions of $$s$$

In this AMSS economy, $$\check x = x(s) = u_c(s) {\frac{b_{t+1}(s)}{R_t(s)}}$$

The risk-free interest rate, the tax rate, and the marginal utility of consumption fluctuate with $$s$$, but $$x$$ does not and neither does $$\mu(s)$$

The labor tax rate and the allocation depend only on the current value of $$s$$

For this special AMSS economy to be in a steady state from time $$0$$ onward, it is necessary that initial debt $$b_0$$ satisfy the time $$0$$ implementability constraint at the value $$\check x$$ and the realized value of $$s_0$$

We can solve for this level of $$b_0$$ by plugging the $$n(s_0)$$ and $$\check x$$ that solve our four equation system into

$u_{c,0} b_0 = u_{c,0} (n_0-g_0) - u_{l,0} n_0 + \check x$

and solving for $$b_0$$

This $$b_0$$ assures that a steady state $$\check x$$ prevails from time $$0$$ on

Here is a modified version of the RecursiveAllocation class from an earlier lecture

using QuantEcon
using NLopt
using NLsolve
using Dierckx

"""
Bellman equation for the continuation of the Lucas-Stokey Problem
"""
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

"""
Compute the planner's allocation by solving Bellman
equation.
"""
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

"""
Initializes the type from the calibration Model
"""
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

"""
Solve the time  1 Bellman equation for calibration Model and initial grid mugrid
"""
function solve_time1_bellman{TR <: Real}(model::Model{TR}, mugrid::AbstractArray)
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}(S, S)
nf = Array{Function}(S, S)
xprimef = Array{Function}(S, S)
Vf = Vector{Function}(S)
xgrid = Array{TR}(S, length(mugrid))
for s_ in 1:S
c = Array{TR}(length(mugrid), S)
n = Array{TR}(length(mugrid), S)
x = Array{TR}(length(mugrid))
V = Array{TR}(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 = repmat(x, 1, S)
xgrid[s_, :] = x
for sprime = 1:S
splc = Spline1D(x[end:-1:1], c[:, sprime][end:-1:1], k=3)
spln = Spline1D(x[end:-1:1], n[:, sprime][end:-1:1], k=3)
splx = Spline1D(x[end:-1:1], xprimes[:, sprime][end:-1:1], k=3)
cf[s_, sprime] = y -> splc(y)
nf[s_, sprime] = y -> spln(y)
xprimef[s_, sprime] = y -> splx(y)
# cf[s_, sprime] = LinInterp(x[end:-1:1], c[:, sprime][end:-1:1])
# nf[s_, sprime] = LinInterp(x[end:-1:1], n[:, sprime][end:-1:1])
# xprimef[s_, sprime] = LinInterp(x[end:-1:1], xprimes[:, sprime][end:-1:1])
end
splV = Spline1D(x[end:-1:1], V[end:-1:1], k=3)
Vf[s_] = y -> splV(y)
# Vf[s_] = LinInterp(x[end:-1:1], V[end:-1:1])
end

policies = [cf, nf, xprimef]

#create xgrid
xbar = [maximum(minimum(xgrid)), minimum(maximum(xgrid))]
xgrid = linspace(xbar[1], xbar[2], length(mugrid))

#Now iterate on Bellman equation
T = BellmanEquation_Recursive(model, xgrid, policies)
diff = 1.0
while diff > 1e-6
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 """ Fits the policy functions """ function fit_policy_function{TF<:AbstractFloat}( T::BellmanEquation_Recursive, PF::Function, xgrid::AbstractVector{TF}) S = T.S # preallocation PFvec = Array{TF}(4S+1, length(xgrid)) cf = Array{Function}(S, S) nf = Array{Function}(S, S) xprimef = Array{Function}(S, S) TTf = Array{Function}(S, S) Vf = Vector{Function}(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 = Spline1D(xgrid, PFvec[1,:], k=3) Vf[s_] = y -> splV(y) # Vf[s_] = LinInterp(xgrid, PFvec[1, :]) for sprime=1:S splc = Spline1D(xgrid, PFvec[1+sprime,:], k=3) spln = Spline1D(xgrid, PFvec[1+S+sprime,:], k=3) splxprime = Spline1D(xgrid, PFvec[1+2S+sprime,:], k=3) splTT = Spline1D(xgrid, PFvec[1+3S+sprime,:], k=3) 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 """ Computes Tau given c,n """ 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.Theta .* Uc) end Tau(pab::RecursiveAllocation, c::Real, n::Real) = Tau(pab, [c], [n]) """ Finds the optimal allocation given initial government debt B_ and state s_0 """ 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 """ Simulates planners policies for T periods """ function simulate{TF <: AbstractFloat}( pab::RecursiveAllocation, B_::TF, s_0::Integer, T::Integer, sHist::Vector=QuantEcon.simulate(pab.mc, T, init=s_0)) 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}(T) nHist=Array{TF}(T) Bhist=Array{TF}(T) xHist=Array{TF}(T) TauHist=Array{TF}(T) THist=Array{TF}(T) muHist=Array{TF}(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}(S) n = Array{TF}(S) xprime = Array{TF}(S) TT = Array{TF}(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(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 """ Initializes the class from the calibration Model """ function BellmanEquation_Recursive{TF <: AbstractFloat}(model::Model{TF}, xgrid::AbstractVector{TF}, policies0::Array) S = size(model.Pi, 1) # number of states xbar = [minimum(xgrid), maximum(xgrid)] time_0 = false z0 = Array{Array}(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}(S) ns=Array{TF}(S) xprimes=Array{TF}(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 """ Finds the optimal policies """ 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 = find(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) = 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

"""
Finds the optimal policies
"""
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

c, xprime = z[1], z[2]
n=(c+G[s0])/Theta[s0]
return -(U(c, n)+beta*Vf[s0](xprime))
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

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) = 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


### Relationship to a Lucas-Stokey economy¶

The constant value of the Lagrange multiplier $$\mu(s)$$ in the Ramsey plan for our special AMSS economy is a tell tale sign that the measurability restrictions imposed on the Ramsey allocation by the requirement that government debt must be risk free are slack

When they bind, those measurability restrictions cause the AMSS tax policy and allocation to be history dependent — that’s what activates flucations in the risk-adjusted martingale

Because those measurability conditions are slack in this special AMSS economy, we can also view this as a Lucas-Stokey economy that starts from a particular initial government debt

The setting of $$b_0$$ for the corresponding Lucas-Stokey implementability constraint solves

$u_{c,0} b_0 = u_{c,0}(n_0 - g_0) - u_{l,0} + \beta \check x$

In this Lucas-Stokey economy, although the Ramsey planner is free to issue state-contingent debt, it chooses not to and instead issues only risk-free debt

It achieves the risk-sharing it wants with the private sector by altering the amounts of one-period risk-free debt it issues at each current state, while understanding the equilibrium interest rate fluctuations that its tax policy induces

### Convergence to the special case¶

In an i.i.d., $$S=2$$ AMSS economy in which the initial $$b_0$$ does not equal the special value described in the previous subsection, $$x$$ fluctuates and is history-dependent

The Lagrange multiplier $$\mu_s(s^t)$$ is a non trivial risk-adjusted martingale and the allocation and distorting tax rate are both history dependent, as is true generally in an AMSS economy

However, BEGS describe conditions under which such an i.i.d., $$S=2$$ AMSS economy in which the representative agent dislikes consumption risk converges to a Lucas-Stokey economy in the sense that $$x_t \rightarrow \check x$$ as $$t \rightarrow \infty$$

The following subsection displays a numerical example that exhibits convergence

## Examples¶

We now turn to some examples

### 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

$u(c,n) = {\frac{c^{1-\sigma}}{1-\sigma}} - {\frac{n^{1+\gamma}}{1+\gamma}}$

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 the components of the state vector as the following six $$(t,g)$$ pairs:

$(0,g_l), (1,g_l), (2,g_l), (3,g_l), (3,g_h), (t\geq 4,g_l)$

We think of these 6 states as corresponding to $$s=1,2,3,4,5,6$$

The transition matrix is

$\begin{split}P = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0.5 & 0.5 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}\end{split}$

The government expenditure at each state is

$\begin{split}g = \left(\begin{matrix} 0.1\\0.1\\0.1\\0.1\\0.2\\0.1 \end{matrix}\right)\end{split}$

We assume the same utility parameters as the Lucas-Stokey economy

This utility function is implemented in the following class

function crra_utility(;
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 utiltiy 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


The following figure plots the Ramsey tax policy under both complete and incomplete markets for both possible realizations of the state at time $$t=3$$

Optimal policies when the goverhment 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],
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
muvec = linspace(-0.7, 0.01, 200)

time_example.transfers = true #Government can use transfers
time_sequential = SequentialAllocation(time_example) #solve sequential problem

time_bellman = RecursiveAllocation(time_example, muvec)

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
pyplot()
titles = hcat("Consumption", "Labor", "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, 500), 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


How the 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 $$t=2$$, it increases its debt burden in the states when there is peace

That helps it to reduce the debt burden when there is war

This pattern facilities smoothing tax rates across states

The government without state contingent debt cannot do this

Instead, it must enter with the same level of debt at all possible states of the world at time $$t=3$$

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

History dependence occurs more dramatically in a case where 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

$u(c,n) = \log(c) + 0.69 \log(1-n)$

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 utiltiy 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


With these preferences, Ramsey tax rates will vary even in the Lucas-Stokey model with state-contingent debt

The figure below plots the 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, muvec) #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, 500), layout = grid(3, 2),
xaxis=(0:T), grid=false, titlefont=Plots.font("sans-serif", 10))
labels = fill(("", ""), 6)
labels[1] = ("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][1])
end
plot!(p[5], sim_seq_plot[:, 5], marker=:circle, color=:black, lab="")


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 must finance 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 received

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, 500), 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])
end
p


Footnotes

 [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? 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] We will soon see from the first-order conditions of the Ramsey problem, there would then exist 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 takes on a positive value $$\gamma_t(\tilde s^t)>0$$.
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