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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

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 $
  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 (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
  4. 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

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

Sequence Implementation

The above steps are implemented in a class called SequentialAllocation

In [1]:
import numpy as np
from scipy.optimize import root
from quantecon import MarkovChain


class SequentialAllocation:

    '''
    Class that takes CESutility or BGPutility object as input returns
    planner's allocation as a function of the multiplier on the
    implementability constraint μ.
    '''

    def __init__(self, model):

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

        # Find the first best allocation
        self.find_first_best()

    def find_first_best(self):
        '''
        Find the first best allocation
        '''
        model = self.model
        S, Θ, G = self.S, self.Θ, self.G
        Uc, Un = model.Uc, model.Un

        def res(z):
            c = z[:S]
            n = z[S:]
            return np.hstack([Θ * Uc(c, n) + Un(c, n), Θ * n - c - G])

        res = root(res, 0.5 * np.ones(2 * S))

        if not res.success:
            raise Exception('Could not find first best')

        self.cFB = res.x[:S]
        self.nFB = res.x[S:]

        # Multiplier on the resource constraint
        self.ΞFB = Uc(self.cFB, self.nFB)
        self.zFB = np.hstack([self.cFB, self.nFB, self.ΞFB])

    def time1_allocation(self, μ):
        '''
        Computes optimal allocation for time t >= 1 for a given μ
        '''
        model = self.model
        S, Θ, G = self.S, self.Θ, self.G
        Uc, Ucc, Un, Unn = model.Uc, model.Ucc, model.Un, model.Unn

        def FOC(z):
            c = z[:S]
            n = z[S:2 * S]
            Ξ = z[2 * S:]
            return np.hstack([Uc(c, n) - μ * (Ucc(c, n) * c + Uc(c, n)) - Ξ,          # FOC of c
                              Un(c, n) - μ * (Unn(c, n) * n + Un(c, n)) + \
                              Θ * Ξ,  # FOC of n
                              Θ * n - c - G])

        # Find the root of the first order condition
        res = root(FOC, self.zFB)
        if not res.success:
            raise Exception('Could not find LS allocation.')
        z = res.x
        c, n, Ξ = z[:S], z[S:2 * S], z[2 * S:]

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

        return c, n, x, Ξ

    def time0_allocation(self, B_, s_0):
        '''
        Finds the optimal allocation given initial government debt B_ and state s_0
        '''
        model, π, Θ, G, β = self.model, self.π, self.Θ, self.G, self.β
        Uc, Ucc, Un, Unn = model.Uc, model.Ucc, model.Un, model.Unn

        # First order conditions of planner's problem
        def FOC(z):
            μ, c, n, Ξ = z
            xprime = self.time1_allocation(μ)[2]
            return np.hstack([Uc(c, n) * (c - B_) + Un(c, n) * n + β * 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]])

        # Find root
        res = root(FOC, np.array(
            [0, self.cFB[s_0], self.nFB[s_0], self.ΞFB[s_0]]))
        if not res.success:
            raise Exception('Could not find time 0 LS allocation.')

        return res.x

    def time1_value(self, μ):
        '''
        Find the value associated with multiplier μ
        '''
        c, n, x, Ξ = self.time1_allocation(μ)
        U = self.model.U(c, n)
        V = np.linalg.solve(np.eye(self.S) - self.β * self.π, U)
        return c, n, x, V

    def Τ(self, c, n):
        '''
        Computes Τ given c, n
        '''
        model = self.model
        Uc, Un = model.Uc(c, n), model.Un(c,  n)

        return 1 + Un / (self.Θ * Uc)

    def simulate(self, B_, s_0, T, sHist=None):
        '''
        Simulates planners policies for T periods
        '''
        model, π, β = self.model, self.π, self.β
        Uc = model.Uc

        if sHist is None:
            sHist = self.mc.simulate(T, s_0)

        cHist, nHist, Bhist, ΤHist, μHist = np.zeros((5, T))
        RHist = np.zeros(T - 1)

        # Time 0
        μ, cHist[0], nHist[0], _ = self.time0_allocation(B_, s_0)
        ΤHist[0] = self.Τ(cHist[0], nHist[0])[s_0]
        Bhist[0] = B_
        μHist[0] = μ

        # Time 1 onward
        for t in range(1, T):
            c, n, x, Ξ = self.time1_allocation(μ)
            Τ = self.Τ(c, n)
            u_c = Uc(c, 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], nHist[t - 1]) / (β * Eu_c)
            μHist[t] = μ

        return np.array([cHist, nHist, Bhist, ΤHist, sHist, μHist, RHist])

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 class called RecursiveAllocation

In [2]:
from scipy.interpolate import UnivariateSpline
from scipy.optimize import fmin_slsqp


class RecursiveAllocation:

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

    def __init__(self, model, μgrid):

        self.β, self.π, self.G = model.β, model.π, model.G
        self.mc, self.S = MarkovChain(self.π), len(model.π)  # Number of states
        self.Θ, self.model, self.μgrid = model.Θ, model, μgrid

        # Find the first best allocation
        self.solve_time1_bellman()
        self.T.time_0 = True  # Bellman equation now solves time 0 problem


    def solve_time1_bellman(self):
        '''
        Solve the time 1 Bellman equation for calibration model and initial grid μgrid0
        '''
        model, μgrid0 = self.model, self.μgrid
        S = len(model.π)

        # First get initial fit
        PP = SequentialAllocation(model)
        c, n, x, V = map(np.vstack, zip(*map(lambda μ: PP.time1_value(μ), μgrid0)))

        Vf, cf, nf, xprimef = {}, {}, {}, {}
        for s in range(2):
            ind = np.argsort(x[:, s])                    # Sort x
            c, n, x, V = c[ind], n[ind], x[ind], V[ind]  # Sort arrays according to x
            cf[s] = UnivariateSpline(x[:, s], c[:, s])
            nf[s] = UnivariateSpline(x[:, s], n[:, s])
            Vf[s] = UnivariateSpline(x[:, s], V[:, s])
            for sprime in range(S):
                xprimef[s, sprime] = UnivariateSpline(x[:, s], x[:, s])
        policies = [cf, nf, xprimef]

        # Create xgrid
        xbar = [x.min(0).max(), x.max(0).min()]
        xgrid = np.linspace(xbar[0], xbar[1], len(μgrid0))
        self.xgrid = xgrid

        # Now iterate on bellman equation
        T = BellmanEquation(model, xgrid, policies)
        diff = 1
        while diff > 1e-7:
            PF = T(Vf)
            Vfnew, policies = self.fit_policy_function(PF)
            diff = 0
            for s in range(S):
                diff = max(diff, np.abs(
                    (Vf[s](xgrid) - Vfnew[s](xgrid)) / Vf[s](xgrid)).max())
            Vf = Vfnew

        # Store value function policies and Bellman Equations
        self.Vf = Vf
        self.policies = policies
        self.T = T


    def fit_policy_function(self, PF):
        '''
        Fits the policy functions PF using the points xgrid using UnivariateSpline
        '''
        xgrid, S = self.xgrid, self.S

        Vf, cf, nf, xprimef = {}, {}, {}, {}
        for s in range(S):
            PFvec = np.vstack(map(lambda x: PF(x, s), xgrid))
            Vf[s] = UnivariateSpline(xgrid, PFvec[:, 0], s=0)
            cf[s] = UnivariateSpline(xgrid, PFvec[:, 1], s=0, k=1)
            nf[s] = UnivariateSpline(xgrid, PFvec[:, 2], s=0, k=1)
            for sprime in range(S):
                xprimef[s, sprime] = UnivariateSpline(
                    xgrid, PFvec[:, 3 + sprime], s=0, k=1)

        return Vf, [cf, nf, xprimef]


    def Τ(self, c, n):
        '''
        Computes Τ given c, n
        '''
        model = self.model
        Uc, Un = model.Uc(c, n), model.Un(c, n)

        return 1 + Un / (self.Θ * Uc)


    def time0_allocation(self, B_, s0):
        '''
        Finds the optimal allocation given initial government debt B_ and state s_0
        '''
        PF = self.T(self.Vf)
        z0 = PF(B_, s0)
        c0, n0, xprime0 = z0[1], z0[2], z0[3:]
        return c0, n0, xprime0


    def simulate(self, B_, s_0, T, sHist=None):
        '''
        Simulates Ramsey plan for T periods
        '''
        model, π = self.model, self.π
        Uc = model.Uc
        cf, nf, xprimef = self.policies

        if sHist is None:
            sHist = self.mc.simulate(T, s_0)

        cHist, nHist, Bhist, ΤHist, μHist = np.zeros((5, T))
        RHist = np.zeros(T - 1)

        # Time 0
        cHist[0], nHist[0], xprime = self.time0_allocation(B_, s_0)
        ΤHist[0] = self.Τ(cHist[0], nHist[0])[s_0]
        Bhist[0] = B_
        μHist[0] = 0

        # Time 1 onward
        for t in range(1, T):
            s, x = sHist[t], xprime[sHist[t]]
            c, n, xprime = np.empty(self.S), nf[s](x), np.empty(self.S)
            for shat in range(self.S):
                c[shat] = cf[shat](x)
            for sprime in range(self.S):
                xprime[sprime] = xprimef[s, sprime](x)

            Τ = self.Τ(c, n)[s]
            u_c = Uc(c, n)
            Eu_c = π[sHist[t - 1]] @ u_c
            μHist[t] = self.Vf[s](x, 1)

            RHist[t - 1] = Uc(cHist[t - 1], nHist[t - 1]) / (self.β * Eu_c)

            cHist[t], nHist[t], Bhist[t], ΤHist[t] = c[s], n, x / u_c[s], Τ

        return np.array([cHist, nHist, Bhist, ΤHist, sHist, μHist, RHist])


class BellmanEquation:

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

    def __init__(self, model, xgrid, policies0):

        self.β, self.π, self.G = model.β, model.π, model.G
        self.S = len(model.π)  # Number of states
        self.Θ, self.model = model.Θ, model

        self.xbar = [min(xgrid), max(xgrid)]
        self.time_0 = False

        self.z0 = {}
        cf, nf, xprimef = policies0
        for s in range(self.S):
            for x in xgrid:
                xprime0 = np.empty(self.S)
                for sprime in range(self.S):
                    xprime0[sprime] = xprimef[s, sprime](x)
                self.z0[x, s] = np.hstack([cf[s](x), nf[s](x), xprime0])

        self.find_first_best()


    def find_first_best(self):
        '''
        Find the first best allocation
        '''
        model = self.model
        S, Θ, Uc, Un, G = self.S, self.Θ, model.Uc, model.Un, self.G

        def res(z):
            c = z[:S]
            n = z[S:]
            return np.hstack([Θ * Uc(c, n) + Un(c, n), Θ * n - c - G])

        res = root(res, 0.5 * np.ones(2 * S))
        if not res.success:
            raise Exception('Could not find first best')

        self.cFB = res.x[:S]
        self.nFB = res.x[S:]
        IFB = Uc(self.cFB, self.nFB) * self.cFB + Un(self.cFB, self.nFB) * self.nFB
        self.xFB = np.linalg.solve(np.eye(S) - self.β * self.π, IFB)
        self.zFB = {}

        for s in range(S):
            self.zFB[s] = np.hstack([self.cFB[s], self.nFB[s], self.xFB])


    def __call__(self, Vf):
        '''
        Given continuation value function next period return value function this
        period return T(V) and optimal policies
        '''
        if not self.time_0:
            def PF(x, s): return self.get_policies_time1(x, s, Vf)
        else:
            def PF(B_, s0): return self.get_policies_time0(B_, s0, Vf)
        return PF


    def get_policies_time1(self, x, s, Vf):
        '''
        Finds the optimal policies
        '''
        model, β, Θ, = self.model, self.β, self.Θ,
        G, S, π = self.G, self.S, self.π
        U, Uc, Un = model.U, model.Uc, model.Un

        def objf(z):
            c, n, xprime = z[0], z[1], z[2:]
            Vprime = np.empty(S)
            for sprime in range(S):
                Vprime[sprime] = Vf[sprime](xprime[sprime])

            return -(U(c, n) + β * π[s] @ Vprime)

        def cons(z):
            c, n, xprime = z[0], z[1], z[2:]
            return np.hstack([x - Uc(c, n) * c - Un(c, n) * n - β * π[s] @ xprime,
                              (Θ * n - c - G)[s]])

        out, fx, _, imode, smode = fmin_slsqp(objf, 
                                              self.z0[x, s], 
                                              f_eqcons=cons,
                                              bounds=[(0, 100), (0, 100)] +
                                              [self.xbar] * S,
                                              full_output=True, 
                                              iprint=0, 
                                              acc=1e-10)

        if imode > 0:
            raise Exception(smode)

        self.z0[x, s] = out
        return np.hstack([-fx, out])


    def get_policies_time0(self, B_, s0, Vf):
        '''
        Finds the optimal policies
        '''
        model, β, Θ, = self.model, self.β, self.Θ,
        G, S, π = self.G, self.S, self.π
        U, Uc, Un = model.U, model.Uc, model.Un

        def objf(z):
            c, n, xprime = z[0], z[1], z[2:]
            Vprime = np.empty(S)
            for sprime in range(S):
                Vprime[sprime] = Vf[sprime](xprime[sprime])

            return -(U(c, n) + β * π[s0] @ Vprime)

        def cons(z):
            c, n, xprime = z[0], z[1], z[2:]
            return np.hstack([-Uc(c, n) * (c - B_) - Un(c, n) * n - β * π[s0] @ xprime,
                              (Θ * n - c - G)[s0]])

        out, fx, _, imode, smode = fmin_slsqp(objf, self.zFB[s0], f_eqcons=cons,
                                              bounds=[(0, 100), (0, 100)] +
                                              [self.xbar] * S,
                                              full_output=True, iprint=0, acc=1e-10)

        if imode > 0:
            raise Exception(smode)

        return np.hstack([-fx, out])

Examples

Anticipated One Period War

This example illustrates in a simple setting how a Ramsey planner manages risk

Government expenditures are known for sure in all periods except one

  • For $ t<3 $ and $ t > 3 $ we assume that $ g_t = g_l = 0.1 $
  • At $ t = 3 $ a war occcurs 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 $

We 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

$$ \Pi = \left(\begin{matrix}0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0.5 & 0.5 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 1\end{matrix}\right) $$

Government expenditures at each state are

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

We assume that the representative agent has utility function

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

and set $ \sigma = 2 $, $ \gamma = 2 $, and the discount factor $ \beta = 0.9 $

Note: For convenience in terms of matching our code, we have expressed utility as a function of $ n $ rather than leisure $ l $

This utility function is implemented in the class CRRAutility

In [3]:
import numpy as np


class CRRAutility:

    def __init__(self,
                 β=0.9,
                 σ=2,
                 γ=2,
                 π=0.5*np.ones((2, 2)),
                 G=np.array([0.1, 0.2]),
                 Θ=np.ones(2),
                 transfers=False):

        self.β, self.σ, self.γ = β, σ, γ
        self.π, self.G, self.Θ, self.transfers = π, G, Θ, transfers

    # Utility function
    def U(self, c, n):
        σ = self.σ
        if σ == 1.:
            U = np.log(c)
        else:
            U = (c**(1 - σ) - 1) / (1 - σ)
        return U - n**(1 + self.γ) / (1 + self.γ)

    # Derivatives of utility function
    def Uc(self, c, n):
        return c**(-self.σ)

    def Ucc(self, c, n):
        return -self.σ * c**(-self.σ - 1)

    def Un(self, c, n):
        return -n**self.γ

    def Unn(self, c, n):
        return -self.γ * n**(self.γ - 1)

We set initial government debt $ b_0 = 1 $

We can now plot the Ramsey tax under both realizations of time $ t = 3 $ government expenditures

  • black when $ g_3 = .1 $, and
  • red when $ g_3 = .2 $
In [4]:
import matplotlib.pyplot as plt
%matplotlib inline

time_π = np.array([[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]])

time_G = np.array([0.1, 0.1, 0.1, 0.2, 0.1, 0.1])
time_Θ = np.ones(6)                                      # Θ can in principle be random
time_example = CRRAutility(π=time_π, G=time_G, Θ=time_Θ)

time_allocation = SequentialAllocation(time_example)     # Solve sequential problem
sHist_h = np.array([0, 1, 2, 3, 5, 5, 5])
sHist_l = np.array([0, 1, 2, 4, 5, 5, 5])
sim_seq_h = time_allocation.simulate(1, 0, 7, sHist_h)
sim_seq_l = time_allocation.simulate(1, 0, 7, sHist_l)

# Government spending paths
sim_seq_l[4] = time_example.G[sHist_l]
sim_seq_h[4] = time_example.G[sHist_h]

# Output paths
sim_seq_l[5] = time_example.Θ[sHist_l] * sim_seq_l[1]
sim_seq_h[5] = time_example.Θ[sHist_h] * sim_seq_h[1]

fig, axes = plt.subplots(3, 2, figsize=(14, 10))
titles = ['Consumption', 'Labor Supply', 'Government Debt',
          'Tax Rate', 'Government Spending', 'Output']

for ax, title, sim_l, sim_h in zip(axes.flatten(), titles, sim_seq_l, sim_seq_h):
    ax.set(title=title)
    ax.plot(sim_l, '-ok', sim_h, '-or', alpha=0.7)
    ax.grid()

plt.tight_layout()
plt.show()

Tax smoothing

  • the tax rate is constant for all $ t\geq 1 $

    • For $ t \geq 1, t \neq 3 $, this is a consequence of $ g_t $ being the same at all those dates
    • For $ t = 3 $, it is a consequence of the special one-period utility function that we have assumed
    • Under other one-period utility functions, the time $ t=3 $ tax rate could be either higher or lower than for dates $ t \geq 1, t \neq 3 $
  • the tax rate is the same at $ t=3 $ for both the high $ g_t $ outcome and the low $ g_t $ outcome

We have assumed that at $ t=0 $, the government owes positive debt $ b_0 $

It sets the time $ t=0 $ tax rate partly with an eye to reducing the value $ u_{c,0} b_0 $ of $ b_0 $

It does this by increasing consumption at time $ t=0 $ relative to consumption in later periods

This has the consequence of raising the time $ t=0 $ value of the gross interest rate for risk-free loans between periods $ t $ and $ t+1 $, which equals

$$ R_t = \frac{u_{c,t}}{\beta\mathbb E_{t}[u_{c,t+1}]} $$

A tax policy that makes time $ t=0 $ consumption be higher than time $ t=1 $ consumption evidently increases the risk-free rate one-period interest rate, $ R_t $, at $ t=0 $

Raising the time $ t=0 $ risk-free interest rate makes time $ t=0 $ consumption goods cheaper relative to consumption goods at later dates, thereby lowering the value $ u_{c,0} b_0 $ of initial government debt $ b_0 $

We see this in a figure below that plots the time path for the risk free interest rate under both realizations of the time $ t=3 $ government expenditure shock

The following plot illustrates how the government lowers the interest rate at time 0 by raising consumption

In [5]:
plt.figure(figsize=(8, 5))
plt.title('Gross Interest Rate')
plt.plot(sim_seq_l[-1], '-ok', sim_seq_h[-1], '-or', alpha=0.7)
plt.grid()
plt.show()

Government Saving

At time $ t=0 $ the government evidently dissaves since $ b_1> b_0 $

  • This is a consequence of it setting a lower tax rate at $ t=0 $, implying more consumption at $ t=0 $

At time $ t=1 $, the government evidently saves since it has set the tax rate sufficiently high to allow it to set $ b_2 < b_1 $

  • Its motive for doing this is that it anticipates a likely war at $ t=3 $

At time $ t=2 $ the government trades state-contingent Arrow securities to hedge against war at $ t=3 $

  • It purchases a security that pays off when $ g_3 = g_h $
  • It sells a security that pays off when $ g_3 = g_l $
  • These purchases are designed in such a way that regardless of whether or not there is a war at $ t=3 $, the government will begin period $ t=4 $ with the same government debt
  • The time $ t=4 $ debt level can be serviced with revenues from the constant tax rate set at times $ t\geq 1 $

At times $ t \geq 4 $ the government rolls over its debt, knowing that the tax rate is set at level required to service the interest payments on the debt and government expenditures

Time 0 Manipulation of Interest Rate

We have seen that when $ b_0>0 $, the Ramsey plan sets the time $ t=0 $ tax rate partly with an eye toward raising a risk-free interest rate for one-period loans between times $ t=0 $ and $ t=1 $

By raising this interest rate, the plan makes time $ t=0 $ goods cheap relative to consumption goods at later times

By doing this, it lowers the value of time $ t=0 $ debt that it has inherited and must finance

Time 0 and Time-Inconsistency

In the preceding example, the Ramsey tax rate at time 0 differs from its value at time 1

To explore what is going on here, let’s simplify things by removing the possibility of war at time $ t=3 $

The Ramsey problem then includes no randomness because $ g_t = g_l $ for all $ t $

The figure below plots the Ramsey tax rates and gross interest rates at time $ t=0 $ and time $ t\geq1 $ as functions of the initial government debt (using the sequential allocation solution and a CRRA utility function defined above)

In [6]:
tax_sequence = SequentialAllocation(CRRAutility(G=0.15,
                                                π=np.ones((1, 1)),
                                                Θ=np.ones(1)))

n = 100
tax_policy = np.empty((n, 2))
interest_rate = np.empty((n, 2))
gov_debt = np.linspace(-1.5, 1, n)

for i in range(n):
    tax_policy[i] = tax_sequence.simulate(gov_debt[i], 0, 2)[3]
    interest_rate[i] = tax_sequence.simulate(gov_debt[i], 0, 3)[-1]

fig, axes = plt.subplots(2, 1, figsize=(10,8), sharex=True)
titles = ['Tax Rate', 'Gross Interest Rate']

for ax, title, plot in zip(axes, titles, [tax_policy, interest_rate]):
    ax.plot(gov_debt, plot[:, 0], gov_debt, plot[:, 1], lw=2)
    ax.set(title=title, xlim=(min(gov_debt), max(gov_debt)))
    ax.grid()

axes[0].legend(('Time $t=0$', 'Time $t \geq 1$'))
axes[1].set_xlabel('Initial Government Debt')

fig.tight_layout()
plt.show()

The figure indicates that if the government enters with positive debt, it sets a tax rate at $ t=0 $ that is less than all later tax rates

By setting a lower tax rate at $ t = 0 $, the government raises consumption, which reduces the value $ u_{c,0} b_0 $ of its initial debt

It does this by increasing $ c_0 $ and thereby lowering $ u_{c,0} $

Conversely, if $ b_{0} < 0 $, the Ramsey planner sets the tax rate at $ t=0 $ higher than in subsequent periods

A side effect of lowering time $ t=0 $ consumption is that it raises the one-period interest rate at time 0 above that of subsequent periods

There are only two values of initial government debt at which the tax rate is constant for all $ t \geq 0 $

The first is $ b_{0} = 0 $

  • Here the government can’t use the $ t=0 $ tax rate to alter the value of the initial debt

The second occurs when the government enters with sufficiently large assets that the Ramsey planner can achieve first best and sets $ \tau_t = 0 $ for all $ t $

It is only for these two values of initial government debt that the Ramsey plan is time-consistent

Another way of saying this is that, except for these two values of initial government debt, a continuation of a Ramsey plan is not a Ramsey plan

To illustrate this, consider a Ramsey planner who starts with an initial government debt $ b_1 $ associated with one of the Ramsey plans computed above

Call $ \tau_1^R $ the time $ t=0 $ tax rate chosen by the Ramsey planner confronting this value for initial government debt government

The figure below shows both the tax rate at time 1 chosen by our original Ramsey planner and what a new Ramsey planner would choose for its time $ t=0 $ tax rate

In [7]:
tax_sequence = SequentialAllocation(CRRAutility(G=0.15,
                                                π=np.ones((1, 1)),
                                                Θ=np.ones(1)))

n = 100
tax_policy = np.empty((n, 2))
τ_reset = np.empty((n, 2))
gov_debt = np.linspace(-1.5, 1, n)

for i in range(n):
    tax_policy[i] = tax_sequence.simulate(gov_debt[i], 0, 2)[3]
    τ_reset[i] = tax_sequence.simulate(gov_debt[i], 0, 1)[3]

fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(gov_debt, tax_policy[:, 1], gov_debt, τ_reset, lw=2)
ax.set(xlabel='Initial Government Debt', title='Tax Rate',
       xlim=(min(gov_debt), max(gov_debt)))
ax.legend((r'$\tau_1$', r'$\tau_1^R$'))
ax.grid()

fig.tight_layout()
plt.show()

The tax rates in the figure are equal for only two values of initial government debt

Tax Smoothing and non-CRRA Preferences

The complete tax smoothing for $ t \geq 1 $ in the preceding example is a consequence of our having assumed CRRA preferences

To see what is driving this outcome, we begin by noting that the Ramsey tax rate for $ t\geq 1 $ is a time invariant function $ \tau(\Phi,g) $ of the Lagrange multiplier on the implementability constraint and government expenditures

For CRRA preferences, we can exploit the relations $ U_{cc}c = -\sigma U_c $ and $ U_{nn} n = \gamma U_n $ to derive

$$ \frac{(1+(1-\sigma)\Phi)U_c}{(1+(1-\gamma)\Phi)U_n} = 1 $$

from the first-order conditions

This equation immediately implies that the tax rate is constant

For other preferences, the tax rate may not be constant

For example, let the period utility function be

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

We will create a new class LogUtility to represent this utility function

In [8]:
class LogUtility:

    def __init__(self,
                 β=0.9,
                 ψ=0.69,
                 π=0.5*np.ones((2, 2)),
                 G=np.array([0.1, 0.2]),
                 Θ=np.ones(2),
                 transfers=False):

        self.β, self.ψ, self.π = β, ψ, π
        self.G, self.Θ, self.transfers = G, Θ, transfers

    # Utility function
    def U(self, c, n):
        return np.log(c) + self.ψ * np.log(1 - n)

    # Derivatives of utility function
    def Uc(self, c, n):
        return 1 / c

    def Ucc(self, c, n):
        return -c**(-2)

    def Un(self, c, n):
        return -self.ψ / (1 - n)

    def Unn(self, c, n):
        return -self.ψ / (1 - n)**2

Also suppose that $ g_t $ follows a two state i.i.d. process with equal probabilities attached to $ g_l $ and $ g_h $

To compute the tax rate, we will use both the sequential and recursive approaches described above

The figure below plots a sample path of the Ramsey tax rate

In [9]:
log_example = LogUtility()
seq_log = SequentialAllocation(log_example)         # Solve sequential problem

# Initialize grid for value function iteration and solve
μ_grid = np.linspace(-0.6, 0.0, 200)
bel_log = RecursiveAllocation(log_example, μ_grid)  # Solve recursive problem

T = 20
sHist = np.array([0, 0, 0, 0, 0, 0, 0,
                  0, 1, 1, 0, 0, 0, 1,
                  1, 1, 1, 1, 1, 0])

# Simulate
sim_seq = seq_log.simulate(0.5, 0, T, sHist)
sim_bel = bel_log.simulate(0.5, 0, T, sHist)

# Government spending paths
sim_seq[4] = log_example.G[sHist]
sim_bel[4] = log_example.G[sHist]

# Output paths
sim_seq[5] = log_example.Θ[sHist] * sim_seq[1]
sim_bel[5] = log_example.Θ[sHist] * sim_bel[1]

fig, axes = plt.subplots(3, 2, figsize=(14, 10))
titles = ['Consumption', 'Labor Supply', 'Government Debt',
          'Tax Rate', 'Government Spending', 'Output']

for ax, title, sim_s, sim_b in zip(axes.flatten(), titles, sim_seq, sim_bel):
    ax.plot(sim_s, '-ob', sim_b, '-xk', alpha=0.7)
    ax.set(title=title)
    ax.grid()

axes.flatten()[0].legend(('Sequential', 'Recursive'))
fig.tight_layout()
plt.show()
/home/quantecon/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:16: RuntimeWarning: divide by zero encountered in log
  app.launch_new_instance()
/home/quantecon/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:20: RuntimeWarning: divide by zero encountered in double_scalars

As should be expected, the recursive and sequential solutions produce almost identical allocations

Unlike outcomes with CRRA preferences, the tax rate is not perfectly smoothed

Instead the government raises the tax rate when $ g_t $ is high

Further Comments

A related lecture describes an extension of the Lucas-Stokey model by Aiyagari, Marcet, Sargent, and Seppälä (2002) [AMSS02]

In th AMSS economy, only a risk-free bond is traded

That lecture compares the recursive representation of the Lucas-Stokey model presented in this lecture with one for an AMSS economy

By comparing these recursive formulations, we shall glean a sense in which the dimension of the state is lower in the Lucas Stokey model

Accompanying that difference in dimension will be different dynamics of government debt