45. Optimal Taxation with State-Contingent Debt#

45.2.1. 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 $\{q_t^0(s^t){\}}_{t=0}^{\mathrm{\infty }}$ to Arrow securities prices $\{p_{t+1}(s_{t+1}|s^t){\}}_{t=0}^{\mathrm{\infty }}$

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.

45.2.2. 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 $\{q_t^0(s^t),{\tau }_t(s^t){\}}_{t=0}^{\mathrm{\infty }}$ as functions of the allocation $\{c_t(s^t),n_t(s^t){\}}_{t=0}^{\mathrm{\infty }}$.

2. 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.

3. Find the allocation that maximizes the utility of the representative household (45.3) subject to the feasibility constraints (45.1) and (45.2) and the implementability condition derived in step 2.

   • This optimal allocation is called the Ramsey allocation.

4. Use the Ramsey allocation together with the formulas from step 1 to find taxes and prices.

45.2.3. The Implementability Constraint#

By sequential substitution of one one-period budget constraint (45.5) into another, we can obtain the household’s present-value budget constraint:

$\{q_t^0(s^t){\}}_{t=1}^{\mathrm{\infty }}$ 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

and

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

Equation (45.9) implies that the Arrow-Debreu price system satisfies

(The stochastic process $\{q_t^0(s^t)\}$ is an instance of what finance economists call a stochastic discount factor process.)

Using the first-order conditions (45.8) and (45.9) to eliminate taxes and prices from (45.7), we derive the implementability condition

The Ramsey problem is to choose a feasible allocation that maximizes

subject to (45.11).

45.2.4. Solution Details#

First, define a “pseudo utility function”

where $\mathrm{\Phi }$ is a Lagrange multiplier on the implementability condition (45.7).

Next form the Lagrangian

where $\{{\theta }_t(s^t);\mathrm{\forall }{s}^t{\}}_{t\ge 0}$ is a sequence of Lagrange multipliers on the feasible conditions (45.2).

Given an initial government debt $b_0$, we want to maximize $J$ with respect to $\{c_t(s^t),n_t(s^t);\mathrm{\forall }{s}^t{\}}_{t\ge 0}$ and to minimize with respect to $\mathrm{\Phi }$ and with respect to $\{\theta (s^t);\mathrm{\forall }{s}^t{\}}_{t\ge 0}$.

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

and

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

It is instructive to use first-order conditions (45.15) for $t\ge 1$ to eliminate the multipliers ${\theta }_t(s^t)$.

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

where we have imposed conditions (45.1) and (45.2).

Equation (45.17) is one equation that can be solved to express the unknown $c$ as a function of the exogenous variable $g$ and the Lagrange multiplier $\mathrm{\Phi }$.

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

Notice that a counterpart to $b_0$ does not appear in (45.17), so $c$ does not directly depend on it for $t\ge 1$.

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

An analogous argument for the $t=0$ equations (45.16) leads to one equation that can be solved for $c_0$ as a function of the pair $(g(s_0),b_0)$ and the Lagrange multiplier $\mathrm{\Phi }$.

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 $t,\tau \ge 0$, i.e., if

then it follows from (45.17) 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 $g$.

45.2.5. The Ramsey Allocation for a Given Multiplier#

Temporarily take $\mathrm{\Phi }$ as given.

We shall compute $c_0(s^0,b_0)$ and $n_0(s^0,b_0)$ from the first-order conditions (45.16).

Evidently, for $t\ge 1$, $c$ 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 $t\ge 1$.

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

Of course, $b_0$ affects the Ramsey allocation for $t\ge 1$ indirectly through its effect on $\mathrm{\Phi }$.

$\mathrm{\Phi }$ 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 $\mathrm{\Phi }$.

45.2.6. 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 $s\in [1,\dots ,S]$ and transition matrix $\mathrm{\Pi }$, where

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.

45.2.7. Determining the Lagrange Multiplier#

We complete the Ramsey plan by computing the Lagrange multiplier $\mathrm{\Phi }$ on the implementability constraint (45.11).

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

The household’s first-order conditions imply

and the implied one-period Arrow securities prices

Substituting from (45.19), (45.20), and the feasibility condition (45.2) into the recursive version (45.5) of the household budget constraint gives

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 (45.21) 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 $t\ge 1$.

That means that we can express equation (45.21) as

where $s^{\prime }$ denotes a next period value of $s$ and $x^{\prime }(s^{\prime })$ denotes a next period value of $x$.

Given $n(s)$ for $s=1,\dots ,S$, equation (45.22) is easy to solve for $x(s)$ for $s=1,\dots ,S$.

If we let $\overrightarrow{n},\overrightarrow{g},\overrightarrow{x}$ denote $S\times 1$ vectors whose $i$th elements are the respective $n,g$, and $x$ values when $s=i$, and let $\mathrm{\Pi }$ be the transition matrix for the Markov state $s$, then we can express (45.22) as the matrix equation

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

In these equations, by ${\overrightarrow{u}}_c\overrightarrow{n}$, for example, we mean element-by-element multiplication of the two vectors.

After solving for $\overrightarrow{x}$, 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

where division here means an element-by-element division of the respective components of the $S\times 1$ vectors $\overrightarrow{x}$ and ${\overrightarrow{u}}_c$.

Here is a computational algorithm:

1. Start with a guess for the value for $\mathrm{\Phi }$, then use the first-order conditions and the feasibility conditions to compute $c(s_t),n(s_t)$ for $s\in [1,\dots ,S]$ and $c_0(s_0,b_0)$ and $n_0(s_0,b_0)$, given $\mathrm{\Phi }$.

   • these are $2(S+1)$ equations in $2(S+1)$ unknowns.

2. Solve the $S$ equations (45.24) for the $S$ elements of $\overrightarrow{x}$.

   • these depend on $\mathrm{\Phi }$.

3. Find a $\mathrm{\Phi }$ that satisfies

   (45.26)#$$u_{c,0}{b}_0=u_{c,0}(n_0-g_0)-u_{l,0}{n}_0+\beta \sum _{s=1}^S\mathrm{\Pi }(s|s_0)x(s)$$

   by gradually raising $\mathrm{\Phi }$ if the left side of (45.26) exceeds the right side and lowering $\mathrm{\Phi }$ if the left side is less than the right side.

4. After computing a Ramsey allocation, recover the flat tax rate on labor from (45.8) and the implied one-period Arrow securities prices from (45.9).

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 for $S$ components each of $\overrightarrow{c}$, $\overrightarrow{n}$, and $\overrightarrow{x}$ together with $n_0,c_0$, and $\mathrm{\Phi }$.

45.2.8. Time Inconsistency#

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

Then $\{{\tau }_j(s^j){\}}_{j=t}^{\mathrm{\infty }},\{b_{j+1}(s_{j+1}|s^j){\}}_{j=t}^{\mathrm{\infty }}$ 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.

45.2.9. 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) [Aiyagari et al., 2002], 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 Lucas-Stokey paper [Lucas and Stokey, 1983]. We shall soon revert to that specification in a subsequent section.)

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

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

We continue to assume that

We eliminate leisure from the model.

We also eliminate Lucas and Stokey’s restriction that ${\ell }_t+n_t\le 1$.

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

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

With these understandings, equations (45.17) and (45.18) simplify in the case of the CRRA utility function.

They become

and

In equation (45.27), 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 $b_0$:

where ${\tau }_0$ is the time $t=0$ tax rate.

In equation (45.29), it is understood that

45.2.10. Sequence Implementation#

The above steps are implemented in a class called SequentialLS

class SequentialLS:

    '''
    Class that takes a preference object, state transition matrix,
    and state contingent government expenditure plan as inputs, and
    solves the sequential allocation problem described above.
    It returns optimal allocations about consumption and labor supply,
    as well as the multiplier on the implementability constraint Φ.
    '''

    def __init__(self,
                 pref,
                 π=np.full((2, 2), 0.5),
                 g=np.array([0.1, 0.2])):

        # Initialize from pref object attributes
        self.β, self.π, self.g = pref.β, π, g
        self.mc = MarkovChain(self.π)
        self.S = len(π)  # Number of states
        self.pref = pref

        # Find the first best allocation
        self.find_first_best()

    def FOC_first_best(self, c, g):
        '''
        First order conditions that characterize
        the first best allocation.
        '''

        pref = self.pref
        Uc, Ul = pref.Uc, pref.Ul

        n = c + g
        l = 1 - n

        return Uc(c, l) - Ul(c, l)

    def find_first_best(self):
        '''
        Find the first best allocation
        '''
        S, g = self.S, self.g

        res = root(self.FOC_first_best, np.full(S, 0.5), args=(g,))

        if (res.fun > 1e-10).any():
            raise Exception('Could not find first best')

        self.cFB = res.x
        self.nFB = self.cFB + g

    def FOC_time1(self, c, Φ, g):
        '''
        First order conditions that characterize
        optimal time 1 allocation problems.
        '''

        pref = self.pref
        Uc, Ucc, Ul, Ull, Ulc = pref.Uc, pref.Ucc, pref.Ul, pref.Ull, pref.Ulc

        n = c + g
        l = 1 - n

        LHS = (1 + Φ) * Uc(c, l) + Φ * (c * Ucc(c, l) - n * Ulc(c, l))
        RHS = (1 + Φ) * Ul(c, l) + Φ * (c * Ulc(c, l) - n * Ull(c, l))

        diff = LHS - RHS

        return diff

    def time1_allocation(self, Φ):
        '''
        Computes optimal allocation for time t >= 1 for a given Φ
        '''
        pref = self.pref
        S, g = self.S, self.g

        # use the first best allocation as intial guess
        res = root(self.FOC_time1, self.cFB, args=(Φ, g))

        if (res.fun > 1e-10).any():
            raise Exception('Could not find LS allocation.')

        c = res.x
        n = c + g
        l = 1 - n

        # Compute x
        I = pref.Uc(c, n) * c - pref.Ul(c, l) * n
        x = np.linalg.solve(np.eye(S) - self.β * self.π, I)

        return c, n, x

    def FOC_time0(self, c0, Φ, g0, b0):
        '''
        First order conditions that characterize
        time 0 allocation problem.
        '''

        pref = self.pref
        Ucc, Ulc = pref.Ucc, pref.Ulc

        n0 = c0 + g0
        l0 = 1 - n0

        diff = self.FOC_time1(c0, Φ, g0)
        diff -= Φ * (Ucc(c0, l0) - Ulc(c0, l0)) * b0

        return diff

    def implementability(self, Φ, b0, s0, cn0_arr):
        '''
        Compute the differences between the RHS and LHS
        of the implementability constraint given Φ,
        initial debt, and initial state.
        '''

        pref, π, g, β = self.pref, self.π, self.g, self.β
        Uc, Ul = pref.Uc, pref.Ul
        g0 = self.g[s0]

        c, n, x = self.time1_allocation(Φ)

        res = root(self.FOC_time0, cn0_arr[0], args=(Φ, g0, b0))
        c0 = res.x
        n0 = c0 + g0
        l0 = 1 - n0

        cn0_arr[:] = c0.item(), n0.item()

        LHS = Uc(c0, l0) * b0
        RHS = Uc(c0, l0) * c0 - Ul(c0, l0) * n0 + β * π[s0] @ x

        return RHS - LHS

    def time0_allocation(self, b0, s0):
        '''
        Finds the optimal time 0 allocation given
        initial government debt b0 and state s0
        '''

        # use the first best allocation as initial guess
        cn0_arr = np.array([self.cFB[s0], self.nFB[s0]])

        res = root(self.implementability, 0., args=(b0, s0, cn0_arr))

        if (res.fun > 1e-10).any():
            raise Exception('Could not find time 0 LS allocation.')

        Φ = res.x[0]
        c0, n0 = cn0_arr

        return Φ, c0, n0

    def τ(self, c, n):
        '''
        Computes τ given c, n
        '''
        pref = self.pref
        Uc, Ul = pref.Uc, pref.Ul

        return 1 - Ul(c, 1-n) / Uc(c, 1-n)

    def simulate(self, b0, s0, T, sHist=None):
        '''
        Simulates planners policies for T periods
        '''
        pref, π, β = self.pref, self.π, self.β
        Uc = pref.Uc

        if sHist is None:
            sHist = self.mc.simulate(T, s0)

        cHist, nHist, Bhist, τHist, ΦHist = np.empty((5, T))
        RHist = np.empty(T-1)

        # Time 0
        Φ, cHist[0], nHist[0] = self.time0_allocation(b0, s0)
        τHist[0] = self.τ(cHist[0], nHist[0])
        Bhist[0] = b0
        ΦHist[0] = Φ

        # Time 1 onward
        for t in range(1, T):
            c, n, x = self.time1_allocation(Φ)
            τ = self.τ(c, n)
            u_c = Uc(c, 1-n)
            s = sHist[t]
            Eu_c = π[sHist[t-1]] @ u_c
            cHist[t], nHist[t], Bhist[t], τHist[t] = c[s], n[s], x[s] / u_c[s], τ[s]
            RHist[t-1] = Uc(cHist[t-1], 1-nHist[t-1]) / (β * Eu_c)
            ΦHist[t] = Φ

        gHist = self.g[sHist]
        yHist = nHist

        return [cHist, nHist, Bhist, τHist, gHist, yHist, sHist, ΦHist, RHist]

45.3.1. 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 $t=1,2,\dots$.

A “continuation Ramsey planner” at time $t\ge 1$ has a different objective function and faces different constraints and state variables than does the Ramsey planner at time $t=0$.

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 $t\ge 1$ 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 $t\ge 1$ continuation Ramsey planner faces $x_t,s_t$ as state variables.

A time $t\ge 1$ 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.

While a time $t\ge 1$ continuation Ramsey planner faces $x_t,s_t$ as state variables, 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 a state-contingent function that make $x_1$ a function of $s_1$ to a time $1$, state $s_1$ continuation Ramsey planner.

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

45.3.2. Two Bellman Equations#

After $s_t$ has been realized at time $t\ge 1$, 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 $t\ge 1$.

• Let $W(b,s)$ be the value of a Ramsey plan at time $0$ at $b_0=b$ and $s_0=s$.

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

45.3.3. The Continuation Ramsey Problem#

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

where maximization over $n$ and the $S$ elements of $x^{\prime }(s^{\prime })$ is subject to the single implementability constraint for $t\ge 1$:

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 $x^{\prime }(s^{\prime })$ for each $s^{\prime }\in \mathcal{S}$.

Associated with a value function $V(x,s)$ that solves Bellman equation (45.30) are $S+1$ time-invariant policy functions

45.3.4. The Ramsey Problem#

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

where maximization over $n_0$ and the $S$ elements of $x^{\prime }(s_1)$ is subject to the time $0$ implementability constraint

coming from restriction (45.26).

Associated with a value function $W(b_0,n_0)$ that solves Bellman equation (45.33) are $S+1$ time $0$ policy functions

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 (45.32) for the time $t\ge 1$ continuation Ramsey planners.

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

45.3.5. First-Order Conditions#

Attach a Lagrange multiplier ${\mathrm{\Phi }}_1(x,s)$ to constraint (45.31) and a Lagrange multiplier ${\mathrm{\Phi }}_0$ to constraint (45.26).

Time $t\ge 1$: First-order conditions for the time $t\ge 1$ constrained maximization problem on the right side of the continuation Ramsey planner’s Bellman equation (45.30) are

for $x^{\prime }(s^{\prime })$ and