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# Optimal Taxation with State-Contingent Debt¶

## Overview¶

This lecture describes a celebrated model of optimal fiscal policy by Robert E. Lucas, Jr., and Nancy Stokey [LS83].

The model revisits classic issues about how to pay for a war.

Here a war means a more or less temporary surge in an exogenous government expenditure process.

The model features

• a government that must finance an exogenous stream of government expenditures with either

• a flat rate tax on labor, or
• purchases and sales from a full array of Arrow state-contingent securities
• a representative household that values consumption and leisure

• a linear production function mapping labor into a single good
• a Ramsey planner who at time $t=0$ chooses a plan for taxes and trades of Arrow securities for all $t \geq 0$

After first presenting the model in a space of sequences, we shall represent it recursively in terms of two Bellman equations formulated along lines that we encountered in Dynamic Stackelberg models.

As in Dynamic Stackelberg models, to apply dynamic programming we shall define the state vector artfully.

In particular, we shall include forward-looking variables that summarize optimal responses of private agents to a Ramsey plan.

See Optimal taxation for an analysis within a linear-quadratic setting.

### Setup¶

In [1]:
using InstantiateFromURL
github_project("QuantEcon/quantecon-notebooks-julia", version = "0.5.0")
# github_project("QuantEcon/quantecon-notebooks-julia", version = "0.5.0", instantiate = true) # uncomment to force package installation

In [2]:
using LinearAlgebra, Statistics
using QuantEcon, NLsolve, NLopt, Interpolations


## A Competitive Equilibrium with Distorting Taxes¶

For $t \geq 0$, a history $s^t = [s_t, s_{t-1}, \ldots, s_0]$ of an exogenous state $s_t$ has joint probability density $\pi_t(s^t)$.

We begin by assuming that government purchases $g_t(s^t)$ at time $t \geq 0$ depend on $s^t$.

Let $c_t(s^t)$, $\ell_t(s^t)$, and $n_t(s^t)$ denote consumption, leisure, and labor supply, respectively, at history $s^t$ and date $t$.

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

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

Output equals $n_t(s^t)$ and can be divided between $c_t(s^t)$ and $g_t(s^t)$

$$c_t(s^t) + g_t(s^t) = n_t(s^t) \tag{2}$$

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

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

where 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 $s^t$.

There are complete markets in one-period Arrow securities.

One unit of an Arrow security issued at time $t$ at history $s^t$ and promising to pay one unit of time $t+1$ consumption in state $s_{t+1}$ costs $p_{t+1}(s_{t+1}|s^t)$.

The government issues one-period Arrow securities each period.

The government has a sequence of budget constraints whose time $t \geq 0$ component is

$$g_t(s^t) = \tau_t(s^t) n_t(s^t) + \sum_{s_{t+1}} p_{t+1}(s_{t+1} | s^t) b_{t+1}(s_{t+1} | s^t) - b_t(s_t | s^{t-1}) \tag{4}$$

where

• $p_{t+1}(s_{t+1}|s^t)$ is a competitive equilibrium price of one unit of consumption at date $t+1$ in state $s_{t+1}$ at date $t$ and history $s^t$
• $b_t(s_t|s^{t-1})$ is government debt falling due at time $t$, history $s^t$.

Government debt $b_0(s_0)$ is an exogenous initial condition.

The representative household has a sequence of budget constraints whose time $t\geq 0$ component is

$$c_t(s^t) + \sum_{s_{t+1}} p_t(s_{t+1} | s^t) b_{t+1}(s_{t+1} | s^t) = \left[1-\tau_t(s^t)\right] n_t(s^t) + b_t(s_t | s^{t-1}) \quad \forall t \geq 0. \tag{5}$$

A government policy is an exogenous sequence $\{g(s_t)\}_{t=0}^\infty$, a tax rate sequence $\{\tau_t(s^t)\}_{t=0}^\infty$, and a government debt sequence $\{b_{t+1}(s^{t+1})\}_{t=0}^\infty$.

A feasible allocation is a consumption-labor supply plan $\{c_t(s^t), n_t(s^t)\}_{t=0}^\infty$ that satisfies (2) at all $t, s^t$.

A price system is a sequence of Arrow security prices $\{p_{t+1}(s_{t+1} | s^t) \}_{t=0}^\infty$.

The household faces the price system as a price-taker and takes the government policy as given.

The household chooses $\{c_t(s^t), \ell_t(s^t)\}_{t=0}^\infty$ to maximize (3) subject to (5) and (1) for all $t, s^t$.

A competitive equilibrium with distorting taxes is a feasible allocation, a price system, and a government policy such that

• Given the price system and the government policy, the allocation solves the household’s optimization problem.
• Given the allocation, government policy, and price system, the government’s budget constraint is satisfied for all $t, s^t$.

Note: There are many competitive equilibria with distorting taxes.

They are indexed by different government policies.

The Ramsey problem or optimal taxation problem is to choose a competitive equilibrium with distorting taxes that maximizes (3).

### Arrow-Debreu Version of Price System¶

We find it convenient sometimes to work with the Arrow-Debreu price system that is implied by a sequence of Arrow securities prices.

Let $q_t^0(s^t)$ be the price at time $0$, measured in time $0$ consumption goods, of one unit of consumption at time $t$, history $s^t$.

The following recursion relates Arrow-Debreu prices $\{q_t^0(s^t)\}_{t=0}^\infty$ to Arrow securities prices $\{p_{t+1}(s_{t+1}|s^t)\}_{t=0}^\infty$

$$q^0_{t+1}(s^{t+1}) = p_{t+1}(s_{t+1}|s^t) q^0_t(s^t) \quad s.t. \quad q_0^0(s^0) = 1 \tag{6}$$

Arrow-Debreu prices are useful when we want to compress a sequence of budget constraints into a single intertemporal budget constraint, as we shall find it convenient to do below.

### Primal Approach¶

We apply a popular approach to solving a Ramsey problem, called the primal approach.

The idea is to use first-order conditions for household optimization to eliminate taxes and prices in favor of quantities, then pose an optimization problem cast entirely in terms of quantities.

After Ramsey quantities have been found, taxes and prices can then be unwound from the allocation.

The primal approach uses four steps:

1. Obtain first-order conditions of the household’s problem and solve them for $\{q^0_t(s^t), \tau_t(s^t)\}_{t=0}^\infty$ as functions of the allocation $\{c_t(s^t), n_t(s^t)\}_{t=0}^\infty$.
1. Substitute these expressions for taxes and prices in terms of the allocation into the household’s present-value budget constraint.
• This intertemporal constraint involves only the allocation and is regarded as an implementability constraint.
1. Find the allocation that maximizes the utility of the representative household (3) subject to the feasibility constraints (1) and (2) and the implementability condition derived in step 2.
• This optimal allocation is called the Ramsey allocation.
1. Use the Ramsey allocation together with the formulas from step 1 to find taxes and prices.

### The Implementability Constraint¶

By sequential substitution of one one-period budget constraint (5) into another, we can obtain the household’s present-value budget constraint:

$$\sum_{t=0}^\infty \sum_{s^t} q^0_t(s^t) c_t(s^t) = \sum_{t=0}^\infty \sum_{s^t} q^0_t(s^t) [1-\tau_t(s^t)] n_t(s^t) + b_0 \tag{7}$$

$\{q^0_t(s^t)\}_{t=1}^\infty$ can be interpreted as a time $0$ Arrow-Debreu price system.

To approach the Ramsey problem, we study the household’s optimization problem.

First-order conditions for the household’s problem for $\ell_t(s^t)$ and $b_t(s_{t+1}| s^t)$, respectively, imply

$$(1 - \tau_t(s^t)) = {\frac{u_l(s^t)}{u_c(s^t)}} \tag{8}$$

and

$$p_{t+1}(s_{t+1}| s^t) = \beta \pi(s_{t+1} | s^t) \left({\frac{u_c(s^{t+1})}{u_c({s^t})}} \right) \tag{9}$$

where $\pi(s_{t+1} | s^t)$ is the probability distribution of $s_{t+1}$ conditional on history $s^t$.

Equation (9) implies that the Arrow-Debreu price system satisfies

$$q_t^0(s^t) = \beta^{t} \pi_{t}(s^{t}) {u_c(s^{t}) \over u_c(s^0)} \tag{10}$$

Using the first-order conditions (8) and (9) to eliminate taxes and prices from (7), we derive the implementability condition

$$\sum_{t=0}^\infty \sum_{s^t} \beta^t \pi_t(s^t) [u_c(s^t) c_t(s^t) - u_\ell(s^t) n_t(s^t)] - u_c(s^0) b_0 = 0. \tag{11}$$

The Ramsey problem is to choose a feasible allocation that maximizes

$$\sum_{t=0}^\infty \sum_{s^t} \beta^t \pi_t(s^t) u[c_t(s^t), 1 - n_t(s^t)] \tag{12}$$

subject to (11).

### Solution Details¶

First define a “pseudo utility function”

$$V\left[c_t(s^t), n_t(s^t), \Phi\right] = u[c_t(s^t),1-n_t(s^t)] + \Phi \left[ u_c(s^t) c_t(s^t) - u_\ell(s^t) n_t(s^t) \right] \tag{13}$$

where $\Phi$ is a Lagrange multiplier on the implementability condition (7).

Next form the Lagrangian

$$J = \sum_{t=0}^\infty \sum_{s^t} \beta^t \pi_t(s^t) \Bigl\{ V[c_t(s^t), n_t(s^t), \Phi] + \theta_t(s^t) \Bigl[ n_t(s^t) - c_t(s^t) - g_t(s_t) \Bigr] \Bigr\} - \Phi u_c(0) b_0 \tag{14}$$

where $\{\theta_t(s^t); \forall s^t\}_{t\geq0}$ is a sequence of Lagrange multipliers on the feasible conditions (2).

Given an initial government debt $b_0$, we want to maximize $J$ with respect to $\{c_t(s^t), n_t(s^t); \forall s^t \}_{t\geq0}$ and to minimize with respect to $\{\theta(s^t); \forall s^t \}_{t\geq0}$.

The first-order conditions for the Ramsey problem for periods $t \geq 1$ and $t=0$, respectively, are

\begin{aligned} c_t(s^t)\rm{:} & \; (1+\Phi) u_c(s^t) + \Phi \left[u_{cc}(s^t) c_t(s^t) - u_{\ell c}(s^t) n_t(s^t) \right] - \theta_t(s^t) = 0, \quad t \geq 1 \\ n_t(s^t)\rm{:} & \; -(1+\Phi) u_{\ell}(s^t) - \Phi \left[u_{c\ell}(s^t) c_t(s^t) - u_{\ell \ell}(s^t) n_t(s^t) \right] + \theta_t(s^t) = 0, \quad t \geq 1 \end{aligned} \tag{15}

and

\begin{aligned} c_0(s^0, b_0)\rm{:} & \; (1+\Phi) u_c(s^0, b_0) + \Phi \left[u_{cc}(s^0, b_0) c_0(s^0, b_0) - u_{\ell c}(s^0, b_0) n_0(s^0, b_0) \right] - \theta_0(s^0, b_0) \\ & \quad \quad \quad \quad \quad \quad - \Phi u_{cc}(s^0, b_0) b_0 = 0 \\ n_0(s^0, b_0)\rm{:} & \; -(1+\Phi) u_{\ell}(s^0, b_0) - \Phi \left[u_{c\ell}(s^0, b_0) c_0(s^0, b_0) - u_{\ell \ell}(s^0, b_0) n_0(s^0, b_0) \right] + \theta_0(s^0, b_0) \\ & \quad \quad \quad \quad \quad \quad + \Phi u_{c \ell}(s^0, b_0) b_0 = 0 \end{aligned} \tag{16}

Please note how these first-order conditions differ between $t=0$ and $t \geq 1$.

It is instructive to use first-order conditions (15) for $t \geq 1$ to eliminate the multipliers $\theta_t(s^t)$.

For convenience, we suppress the time subscript and the index $s^t$ and obtain

\begin{aligned} (1+\Phi) &u_c(c,1-c-g) + \Phi \bigl[c u_{cc}(c,1-c-g) - (c+g) u_{\ell c}(c,1-c-g) \bigr] \\ &= (1+\Phi) u_{\ell}(c,1-c-g) + \Phi \bigl[c u_{c\ell}(c,1-c-g) - (c+g) u_{\ell \ell}(c,1-c-g) \bigr] \end{aligned} \tag{17}

where we have imposed conditions (1) and (2).

Equation (17) is one equation that can be solved to express the unknown $c$ as a function of the exogenous variable $g$.

We also know that time $t=0$ quantities $c_0$ and $n_0$ satisfy

\begin{aligned} (1+\Phi) &u_c(c,1-c-g) + \Phi \bigl[c u_{cc}(c,1-c-g) - (c+g) u_{\ell c}(c,1-c-g) \bigr] \\ &= (1+\Phi) u_{\ell}(c,1-c-g) + \Phi \bigl[c u_{c\ell}(c,1-c-g) - (c+g) u_{\ell \ell}(c,1-c-g) \bigr] + \Phi (u_{cc} - u_{c,\ell}) b_0 \end{aligned} \tag{18}

Notice that a counterpart to $b_0$ does not appear in (17), so $c$ does not depend on it for $t \geq 1$.

But things are different for time $t=0$.

An analogous argument for the $t=0$ equations (16) leads to one equation that can be solved for $c_0$ as a function of the pair $(g(s_0), b_0)$.

These outcomes mean that the following statement would be true even when government purchases are history-dependent functions $g_t(s^t)$ of the history of $s^t$.

Proposition: If government purchases are equal after two histories $s^t$ and $\tilde s^\tau$ for $t,\tau\geq0$, i.e., if

$$g_t(s^t) = g^\tau(\tilde s^\tau) = g$$

then it follows from (17) that the Ramsey choices of consumption and leisure, $(c_t(s^t),\ell_t(s^t))$ and $(c_j(\tilde s^\tau),\ell_j(\tilde s^\tau))$, are identical.

The proposition asserts that the optimal allocation is a function of the currently realized quantity of government purchases $g$ only and does not depend on the specific history that preceded that realization of $g$.

### The Ramsey Allocation for a Given $\Phi$¶

Temporarily take $\Phi$ as given.

We shall compute $c_0(s^0, b_0)$ and $n_0(s^0, b_0)$ from the first-order conditions (16).

Evidently, for $t \geq 1$, $c$ and $n$ depend on the time $t$ realization of $g$ only.

But for $t=0$, $c$ and $n$ depend on both $g_0$ and the government’s initial debt $b_0$.

Thus, while $b_0$ influences $c_0$ and $n_0$, there appears no analogous variable $b_t$ that influences $c_t$ and $n_t$ for $t \geq 1$.

The absence of $b_t$ as a determinant of the Ramsey allocation for $t \geq 1$ and its presence for $t=0$ is a symptom of the time-inconsistency of a Ramsey plan.

$\Phi$ has to take a value that assures that the household and the government’s budget constraints are both satisfied at a candidate Ramsey allocation and price system associated with that $\Phi$.

### Further Specialization¶

At this point, it is useful to specialize the model in the following ways.

We assume that $s$ is governed by a finite state Markov chain with states $s\in [1, \ldots, S]$ and transition matrix $\Pi$, where

$$\Pi(s'|s) = {\rm Prob}(s_{t+1} = s'| s_t =s)$$

Also, assume that government purchases $g$ are an exact time-invariant function $g(s)$ of $s$.

We maintain these assumptions throughout the remainder of this lecture.

### Determining $\Phi$¶

We complete the Ramsey plan by computing the Lagrange multiplier $\Phi$ on the implementability constraint (11).

Government budget balance restricts $\Phi$ via the following line of reasoning.

The household’s first-order conditions imply

$$(1 - \tau_t(s^t)) = {\frac{u_l(s^t)}{u_c(s^t)} } \tag{19}$$

and the implied one-period Arrow securities prices

$$p_{t+1}(s_{t+1}| s^t) = \beta \Pi(s_{t+1} | s_t) {\frac{u_c(s^{t+1})}{u_c({s^t})}} \tag{20}$$

Substituting from (19), (20), and the feasibility condition (2) into the recursive version (5) of the household budget constraint gives

\begin{aligned} u_c(s^t) [ n_t(s^t) - g_t(s^t)] + \beta \sum_{s_{t+1}} \Pi (s_{t+1}| s_t) u_c(s^{t+1}) b_{t+1}(s_{t+1} | s^t) = \\ u_l (s^t) n_t(s^t) + u_c(s^t) b_t(s_t | s^{t-1}) \end{aligned} \tag{21}

Define $x_t(s^t) = u_c(s^t) b_t(s_t | s^{t-1})$.

Notice that $x_t(s^t)$ appears on the right side of (21) while $\beta$ times the conditional expectation of $x_{t+1}(s^{t+1})$ appears on the left side.

Hence the equation shares much of the structure of a simple asset pricing equation with $x_t$ being analogous to the price of the asset at time $t$.

We learned earlier that for a Ramsey allocation $c_t(s^t), n_t(s^t)$ and $b_t(s_t|s^{t-1})$, and therefore also $x_t(s^t)$, are each functions of $s_t$ only, being independent of the history $s^{t-1}$ for $t \geq 1$.

That means that we can express equation (21) as

$$u_c(s) [ n(s) - g(s)] + \beta \sum_{s'} \Pi(s' | s) x'(s') = u_l(s) n(s) + x(s) \tag{22}$$

where $s'$ denotes a next period value of $s$ and $x'(s')$ denotes a next period value of $x$.

Equation (22) is easy to solve for $x(s)$ for $s = 1, \ldots , S$.

If we let $\vec n, \vec g, \vec x$ denote $S \times 1$ vectors whose $i$th elements are the respective $n, g$, and $x$ values when $s=i$, and let $\Pi$ be the transition matrix for the Markov state $s$, then we can express (22) as the matrix equation

$$\vec u_c(\vec n - \vec g) + \beta \Pi \vec x = \vec u_l \vec n + \vec x \tag{23}$$

This is a system of $S$ linear equations in the $S \times 1$ vector $x$, whose solution is

$$\vec x= (I - \beta \Pi )^{-1} [ \vec u_c (\vec n-\vec g) - \vec u_l \vec n] \tag{24}$$

In these equations, by $\vec u_c \vec n$, for example, we mean element-by-element multiplication of the two vectors.

After solving for $\vec x$, we can find $b(s_t|s^{t-1})$ in Markov state $s_t=s$ from $b(s) = {\frac{x(s)}{u_c(s)}}$ or the matrix equation

$$\vec b = {\frac{ \vec x }{\vec u_c}} \tag{25}$$

where division here means element-by-element division of the respective components of the $S \times 1$ vectors $\vec x$ and $\vec u_c$.

Here is a computational algorithm:

1. Start with a guess for the value for $\Phi$, then use the first-order conditions and the feasibility conditions to compute $c(s_t), n(s_t)$ for $s \in [1,\ldots, S]$ and $c_0(s_0,b_0)$ and $n_0(s_0, b_0)$, given $\Phi$
• these are $2 (S+1)$ equations in $2 (S+1)$ unknowns
2. Solve the $S$ equations (24) for the $S$ elements of $\vec x$
• these depend on $\Phi$
3. Find a $\Phi$ that satisfies

$$u_{c,0} b_0 = u_{c,0} (n_0 - g_0) - u_{l,0} n_0 + \beta \sum_{s=1}^S \Pi(s | s_0) x(s) \tag{26}$$

by gradually raising $\Phi$ if the left side of (26) exceeds the right side and lowering $\Phi$ if the left side is less than the right side.

4. After computing a Ramsey allocation, recover the flat tax rate on labor from (8) and the implied one-period Arrow securities prices from (9).

In summary, when $g_t$ is a time invariant function of a Markov state $s_t$, a Ramsey plan can be constructed by solving $3S +3$ equations in $S$ components each of $\vec c$, $\vec n$, and $\vec x$ together with $n_0, c_0$, and $\Phi$.

### Time Inconsistency¶

Let $\{\tau_t(s^t)\}_{t=0}^\infty, \{b_{t+1}(s_{t+1}| s^t)\}_{t=0}^\infty$ be a time $0$, state $s_0$ Ramsey plan.

Then $\{\tau_j(s^j)\}_{j=t}^\infty, \{b_{j+1}(s_{j+1}| s^j)\}_{j=t}^\infty$ is a time $t$, history $s^t$ continuation of a time $0$, state $s_0$ Ramsey plan.

A time $t$, history $s^t$ Ramsey plan is a Ramsey plan that starts from initial conditions $s^t, b_t(s_t|s^{t-1})$.

A time $t$, history $s^t$ continuation of a time $0$, state $0$ Ramsey plan is not a time $t$, history $s^t$ Ramsey plan.

The means that a Ramsey plan is not time consistent.

Another way to say the same thing is that a Ramsey plan is time inconsistent.

The reason is that a continuation Ramsey plan takes $u_{ct} b_t(s_t|s^{t-1})$ as given, not $b_t(s_t|s^{t-1})$.

We shall discuss this more below.

### Specification with CRRA Utility¶

In our calculations below and in a subsequent lecture based on an extension of the Lucas-Stokey model by Aiyagari, Marcet, Sargent, and Seppälä (2002) [AMSS02], we shall modify the one-period utility function assumed above.

(We adopted the preceding utility specification because it was the one used in the original [LS83] paper)

We will modify their specification by instead assuming that the representative agent has utility function

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

where $\sigma > 0$, $\gamma >0$.

We continue to assume that

$$c_t + g_t = n_t$$

We eliminate leisure from the model.

We also eliminate Lucas and Stokey’s restriction that $\ell_t + n_t \leq 1$.

We replace these two things with the assumption that labor $n_t \in [0, +\infty]$.

With these adjustments, the analysis of Lucas and Stokey prevails once we make the following replacements

\begin{aligned} u_\ell(c, \ell) &\sim - u_n(c, n) \\ u_c(c,\ell) &\sim u_c(c,n) \\ u_{\ell,\ell}(c,\ell) &\sim u_{nn}(c,n) \\ u_{c,c}(c,\ell)& \sim u_{c,c}(c,n) \\ u_{c,\ell} (c,\ell) &\sim 0 \\ \end{aligned}

With these understandings, equations (17) and (18) simplify in the case of the CRRA utility function.

They become

$$(1+\Phi) [u_c(c) + u_n(c+g)] + \Phi[c u_{cc}(c) + (c+g) u_{nn}(c+g)] = 0 \tag{27}$$

and

$$(1+\Phi) [u_c(c_0) + u_n(c_0+g_0)] + \Phi[c_0 u_{cc}(c_0) + (c_0+g_0) u_{nn}(c_0+g_0)] - \Phi u_{cc}(c_0) b_0 = 0 \tag{28}$$

In equation (27), it is understood that $c$ and $g$ are each functions of the Markov state $s$.

In addition, the time $t=0$ budget constraint is satisfied at $c_0$ and initial government debt $b_0$:

$$b_0 + g_0 = \tau_0 (c_0 + g_0) + \frac{\bar b}{R_0} \tag{29}$$

where $R_0$ is the gross interest rate for the Markov state $s_0$ that is assumed to prevail at time $t =0$ and $\tau_0$ is the time $t=0$ tax rate.

In equation (29), it is understood that

\begin{aligned} \tau_0 = 1 - \frac{u_{l,0}}{u_{c,0}} \\ R_0 = \beta \sum_{s=1}^S \Pi(s | s_0) \frac{u_c(s)}{u_{c,0}} \end{aligned}

### Sequence Implementation¶

The above steps are implemented in a type called SequentialAllocation

In [3]:
using QuantEcon, NLsolve, NLopt, LinearAlgebra, Interpolations

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)
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, S, version)
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 = \(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)
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)
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
Inv  = Uc(c, n) .* c +  Un(c, n) .* n
x = \(I - β * model.Π, Inv)
return c, n, x, Ξ
end

function time0_allocation(pas::SequentialAllocation, B_, s_0)
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)
model = pas.model
c, n, x, Ξ = time1_allocation(pas, μ)
U_val = model.U.(c, n)
V = \(I - model.β*model.Π, U_val)
return c, n, x, V
end

function Τ(model, c, n)
Uc, Un = model.Uc.(c, n), model.Un.(c, n)
return 1 .+ Un ./ (model.Θ .* Uc)
end

function simulate(pas::SequentialAllocation, B_, s_0, T, sHist = 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 = similar(cHist)
Bhist = similar(cHist)
ΤHist = similar(cHist)
μHist = similar(cHist)
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, xgrid, policies0)
S = size(model.Π, 1) # Number of states
xbar = collect(extrema(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, i_x, x, s, Vf)
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

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
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) = 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, B_, s0, Vf)
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
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
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) = optimize(opt, init)
return vcat(-minf, vcat(minx[1], minx[1]+G[s0], minx[2:end]))
end

Out[3]:
get_policies_time0 (generic function with 1 method)

## Recursive Formulation of the Ramsey problem¶

$x_t(s^t) = u_c(s^t) b_t(s_t | s^{t-1})$ in equation (21) appears to be a purely “forward-looking” variable.

But $x_t(s^t)$ is a also a natural candidate for a state variable in a recursive formulation of the Ramsey problem.

### Intertemporal Delegation¶

To express a Ramsey plan recursively, we imagine that a time $0$ Ramsey planner is followed by a sequence of continuation Ramsey planners at times $t = 1, 2, \ldots$.

A “continuation Ramsey planner” has a different objective function and faces different constraints than a Ramsey planner.

A key step in representing a Ramsey plan recursively is to regard the marginal utility scaled government debts $x_t(s^t) = u_c(s^t) b_t(s_t|s^{t-1})$ as predetermined quantities that continuation Ramsey planners at times $t \geq 1$ are obligated to attain.

Continuation Ramsey planners do this by choosing continuation policies that induce the representative household to make choices that imply that $u_c(s^t) b_t(s_t|s^{t-1})= x_t(s^t)$.

A time $t\geq 1$ continuation Ramsey planner delivers $x_t$ by choosing a suitable $n_t, c_t$ pair and a list of $s_{t+1}$-contingent continuation quantities $x_{t+1}$ to bequeath to a time $t+1$ continuation Ramsey planner.

A time $t \geq 1$ continuation Ramsey planner faces $x_t, s_t$ as state variables.

But the time $0$ Ramsey planner faces $b_0$, not $x_0$, as a state variable.

Furthermore, the Ramsey planner cares about $(c_0(s_0), \ell_0(s_0))$, while continuation Ramsey planners do not.

The time $0$ Ramsey planner hands $x_1$ as a function of $s_1$ to a time $1$ continuation Ramsey planner.

These lines of delegated authorities and responsibilities across time express the continuation Ramsey planners’ obligations to implement their parts of the original Ramsey plan, designed once-and-for-all at time $0$.

### Two Bellman Equations¶

After $s_t$ has been realized at time $t \geq 1$, the state variables confronting the time $t$ continuation Ramsey planner are $(x_t, s_t)$.

• Let $V(x, s)$ be the value of a continuation Ramsey plan at $x_t = x, s_t =s$ for $t \geq 1$.
• Let $W(b, s)$ be the value of a Ramsey plan at time $0$ at $b_0=b$ and $s_0 = s$.

We work backwards by presenting a Bellman equation for $V(x,s)$ first, then a Bellman equation for $W(b,s)$.

### The Continuation Ramsey Problem¶

The Bellman equation for a time $t \geq 1$ continuation Ramsey planner is

$$V(x, s) = \max_{n, \{x'(s')\}} u(n-g(s), 1-n) + \beta \sum_{s'\in S} \Pi(s'| s) V(x', s') \tag{30}$$

where maximization over $n$ and the $S$ elements of $x'(s')$ is subject to the single implementability constraint for $t \geq 1$

$$x = u_c(n-g(s)) - u_l n + \beta \sum_{s' \in {\cal S}} \Pi(s' | s) x'(s') \tag{31}$$

Here $u_c$ and $u_l$ are today’s values of the marginal utilities.

For each given value of $x, s$, the continuation Ramsey planner chooses $n$ and an $x'(s')$ for each $s' \in {\cal S}$.

Associated with a value function $V(x,s)$ that solves Bellman equation (30) are $S+1$ time-invariant policy functions

\begin{aligned} n_t & = f(x_t, s_t), \quad t \geq 1 \\ x_{t+1}(s_{t+1}) & = h(s_{t+1}; x_t, s_t), \, s_{t+1} \in {\cal S}, \, t \geq 1 \end{aligned} \tag{32}

### The Ramsey Problem¶

The Bellman equation for the time $0$ Ramsey planner is

$$W(b_0, s_0) = \max_{n_0, \{x'(s_1)\}} u(n_0 - g_0, 1 - n_0) + \beta \sum_{s_1 \in {\cal S}} \Pi(s_1| s_0) V( x'(s_1), s_1) \tag{33}$$

where maximization over $n_0$ and the $S$ elements of $x'(s_1)$ is subject to the time $0$ implementability constraint

$$u_{c,0} b_0 = u_{c,0} (n_0 - g_0) - u_{l,0} n_0 + \beta \sum_{s_1\in {\cal S}} \Pi(s_1 | s_0) x'(s_1) \tag{34}$$

coming from restriction (26).

Associated with a value function $W(b_0, n_0)$ that solves Bellman equation (33) are $S +1$ time $0$ policy functions

\begin{aligned} n_0 & = f_0(b_0, s_0) \cr x_1(s_1) & = h_0(s_1; b_0, s_0) \end{aligned} \tag{35}

Notice the appearance of state variables $(b_0, s_0)$ in the time $0$ policy functions for the Ramsey planner as compared to $(x_t, s_t)$ in the policy functions (32) for the time $t \geq 1$ continuation Ramsey planners.

The value function $V(x_t, s_t)$ of the time $t$ continuation Ramsey planner equals $E_t \sum_{\tau = t}^\infty \beta^{\tau - t} u(c_t, l_t)$, where the consumption and leisure processes are evaluated along the original time $0$ Ramsey plan.

### First-Order Conditions¶

Attach a Lagrange multiplier $\Phi_1(x,s)$ to constraint (31) and a Lagrange multiplier $\Phi_0$ to constraint (26).

Time $t \geq 1$: the first-order conditions for the time $t \geq 1$ constrained maximization problem on the right side of the continuation Ramsey planner’s Bellman equation (30) are

$$\beta \Pi(s' | s) V_x (x', s') - \beta \Pi(s' | s) \Phi_1 = 0 \tag{36}$$

for $x'(s')$ and

$$(1 + \Phi_1) (u_c - u_l ) + \Phi_1 \left[ n (u_{ll} - u_{lc}) + (n-g(s)) (u_{cc} - u_{lc}) \right] = 0 \tag{37}$$

for $n$.

Given $\Phi_1$, equation (37) is one equation to be solved for $n$ as a function of $s$ (or of $g(s)$).

Equation (36) implies $V_x(x', s')= \Phi_1$, while an envelope condition is $V_x(x,s) = \Phi_1$, so it follows that

$$V_x(x', s') = V_x(x,s) = \Phi_1(x,s) \tag{38}$$

Time $t=0$: For the time $0$ problem on the right side of the Ramsey planner’s Bellman equation (33), first-order conditions are

$$V_x(x(s_1), s_1) = \Phi_0 \tag{39}$$

for $x(s_1), s_1 \in {\cal S}$, and

\begin{aligned} (1 + \Phi_0) (u_{c,0} - u_{n,0}) & + \Phi_0 \bigl[ n_0 (u_{ll,0} - u_{lc,0} ) + (n_0 - g(s_0)) (u_{cc,0} - u_{cl,0}) \Bigr] \\ & \quad \quad \quad - \Phi_0 (u_{cc,0} - u_{cl,0}) b_0 = 0 \end{aligned} \tag{40}

Notice similarities and differences between the first-order conditions for $t \geq 1$ and for $t=0$.

An additional term is present in (40) except in three special cases

• $b_0 = 0$, or
• $u_c$ is constant (i.e., preferences are quasi-linear in consumption), or
• initial government assets are sufficiently large to finance all government purchases with interest earnings from those assets, so that $\Phi_0= 0$

Except in these special cases, the allocation and the labor tax rate as functions of $s_t$ differ between dates $t=0$ and subsequent dates $t \geq 1$.

Naturally, the first-order conditions in this recursive formulation of the Ramsey problem agree with the first-order conditions derived when we first formulated the Ramsey plan in the space of sequences.

### State Variable Degeneracy¶

Equations (39) and (40) imply that $\Phi_0 = \Phi_1$ and that

$$V_x(x_t, s_t) = \Phi_0 \tag{41}$$

for all $t \geq 1$.

When $V$ is concave in $x$, this implies state-variable degeneracy along a Ramsey plan in the sense that for $t \geq 1$, $x_t$ will be a time-invariant function of $s_t$.

Given $\Phi_0$, this function mapping $s_t$ into $x_t$ can be expressed as a vector $\vec x$ that solves equation (34) for $n$ and $c$ as functions of $g$ that are associated with $\Phi = \Phi_0$.

### Manifestations of Time Inconsistency¶

While the marginal utility adjusted level of government debt $x_t$ is a key state variable for the continuation Ramsey planners at $t \geq 1$, it is not a state variable at time $0$.

The time $0$ Ramsey planner faces $b_0$, not $x_0 = u_{c,0} b_0$, as a state variable.

The discrepancy in state variables faced by the time $0$ Ramsey planner and the time $t \geq 1$ continuation Ramsey planners captures the differing obligations and incentives faced by the time $0$ Ramsey planner and the time $t \geq 1$ continuation Ramsey planners.

• The time $0$ Ramsey planner is obligated to honor government debt $b_0$ measured in time $0$ consumption goods.
• The time $0$ Ramsey planner can manipulate the value of government debt as measured by $u_{c,0} b_0$.
• In contrast, time $t \geq 1$ continuation Ramsey planners are obligated not to alter values of debt, as measured by $u_{c,t} b_t$, that they inherit from a preceding Ramsey planner or continuation Ramsey planner.

When government expenditures $g_t$ are a time invariant function of a Markov state $s_t$, a Ramsey plan and associated Ramsey allocation feature marginal utilities of consumption $u_c(s_t)$ that, given $\Phi$, for $t \geq 1$ depend only on $s_t$, but that for $t=0$ depend on $b_0$ as well.

This means that $u_c(s_t)$ will be a time invariant function of $s_t$ for $t \geq 1$, but except when $b_0 = 0$, a different function for $t=0$.

This in turn means that prices of one period Arrow securities $p_{t+1}(s_{t+1} | s_t) = p(s_{t+1}|s_t)$ will be the same time invariant functions of $(s_{t+1}, s_t)$ for $t \geq 1$, but a different function $p_0(s_1|s_0)$ for $t=0$, except when $b_0=0$.

The differences between these time $0$ and time $t \geq 1$ objects reflect the Ramsey planner’s incentive to manipulate Arrow security prices and, through them, the value of initial government debt $b_0$.

### Recursive Implementation¶

The above steps are implemented in a type called RecursiveAllocation

In [4]:
struct RecursiveAllocation{TP <: Model, TI <: Integer,
TVg <: AbstractVector, TVv <: AbstractVector,
TVp <: AbstractArray}
model::TP
mc::MarkovChain
S::TI
T::BellmanEquation
μgrid::TVg
xgrid::TVg
Vf::TVv
policies::TVp
end

function RecursiveAllocation(model, μgrid)
mc = MarkovChain(model.Π)
G = model.G
S = size(model.Π, 1) # Number of states
# 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, μgrid)
μgrid0 = μgrid
S = size(model.Π, 1)
# First get initial fit
PP = SequentialAllocation(model)
c = zeros(length(μgrid), 2)
n = similar(c)
x = similar(c)
V = similar(c)
for (i, μ) in enumerate(μgrid0)
c[i, :], n[i, :], x[i, :], V[i, :] = time1_value(PP, μ)
end
Vf = Vector{AbstractInterpolation}(undef, 2)
cf = similar(Vf)
nf = similar(Vf)
xprimef = similar(Vf, 2, S)
for s in 1:2
cf[s] = LinearInterpolation(x[:, s][end:-1:1], c[:, s][end:-1:1])
nf[s] = LinearInterpolation(x[:, s][end:-1:1], n[:, s][end:-1:1])
Vf[s] = LinearInterpolation(x[:, s][end:-1:1], V[:, s][end:-1:1])
for sprime in 1:S
xprimef[s, sprime] = LinearInterpolation(x[:, s][end:-1:1], x[:, s][end:-1:1])
end
end
policies = [cf, nf, xprimef]
# Create xgrid
xbar = [maximum(minimum(x, dims = 1)), minimum(maximum(x, dims = 1))]
xgrid = range(xbar[1],  xbar[2], length = length(μgrid0))
# Now iterate on bellman equation
T = BellmanEquation(model, xgrid, policies)
diff = 1.0
while diff > 1e-6
if T.time_0 == false
Vfnew, policies =
fit_policy_function(PP,
(i_x, x, s) -> get_policies_time1(T, i_x, x, s, Vf), xgrid)
elseif T.time_0 == true
Vfnew, policies =
fit_policy_function(PP,
(i_x, B_, s0) -> get_policies_time0(T, i_x, B_, s0, Vf), xgrid)
else
error("T.time_0 is $(T.time_0), which is invalid") end diff = 0.0 for s in 1:S diff = max(diff, maximum(abs, (Vf[s].(xgrid)-Vfnew[s].(xgrid))./Vf[s].(xgrid))) end print("diff =$diff \n")
Vf = Vfnew
end
# Store value function policies and Bellman Equations
return Vf, policies, T, xgrid
end

function fit_policy_function(PP, PF, xgrid)
S = PP.S
Vf = Vector{AbstractInterpolation}(undef, S)
cf = similar(Vf)
nf = similar(Vf)
xprimef = similar(Vf, S, S)
for s in 1:S
PFvec = zeros(length(xgrid), 3+S)
for (i_x, x) in enumerate(xgrid)
PFvec[i_x, :] = PF(i_x, x, s)
end
Vf[s] = LinearInterpolation(xgrid, PFvec[:, 1])
cf[s] = LinearInterpolation(xgrid, PFvec[:, 2])
nf[s] = LinearInterpolation(xgrid, PFvec[:, 3])
for sprime in 1:S
xprimef[s, sprime] = LinearInterpolation(xgrid, PFvec[:, 3+sprime])
end
end
return Vf, [cf, nf, xprimef]
end

function time0_allocation(pab::RecursiveAllocation, B_, s0)
xgrid = pab.xgrid
if pab.T.time_0 == false
z0 = get_policies_time1(pab.T, i_x, x, s, pab.Vf)
elseif pab.T.time_0 == true
z0 = get_policies_time0(pab.T, B_, s0, pab.Vf)
else