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Ramsey plans, Time Inconsistency, Sustainable Plans

Co-author: Sebastian Graves

Overview

This lecture describes a linear-quadratic version of a model that Guillermo Calvo [Cal78] used to illustrate the time inconsistency of optimal government plans

Like Chang [Cha98], we use the model as a laboratory in which to explore consequences of different timing protocols for government decision making

The model focuses attention on intertemporal tradeoffs between

  • welfare benefits that anticipated deflation generates by increasing a representative agent’s liquidity as measured by his or her real money balances, and
  • costs associated with distorting taxes that must be used to withdraw money from the economy in order to generate anticipated deflation

The model features

  • rational expectations
  • costly government actions at all dates $ t \geq 1 $ that increase household utilities at dates before $ t $
  • two Bellman equations, one that expresses the private sector’s expectation of future inflation as a function of current and future government actions, another that describes the value function of a planner

A theme of this lecture is that timing protocols affect outcomes

We’ll use ideas from papers by Cagan [Cag56], Calvo [Cal78], Stokey [Sto89], [Sto91], Chari and Kehoe [CK90], Chang [Cha98], and Abreu [Abr88] as well as from chapter 19 of [LS18]

In addition, we’ll use ideas from linear-quadratic dynamic programming described in Linear Quadratic Control as applied to Ramsey problems in Stackelberg problems

In particular, we have specified the model in a way that allows us to use linear-quadratic dynamic programming to compute an optimal government plan under a timing protocol in which a government chooses an infinite sequence of money supply growth rates once and for all at time $ 0 $

The Model

There is no uncertainty

Let:

  • $ p_t $ be the log of the price level
  • $ m_t $ be the log of nominal money balances
  • $ \theta_t = p_{t+1} - p_t $ be the rate of inflation between $ t $ and $ t+1 $
  • $ \mu_t = m_{t+1} - m_t $ be the rate of growth of nominal balances

The demand for real balances is governed by a perfect foresight version of the Cagan [Cag56] demand function:

$$ m_t - p_t = -\alpha(p_{t+1} - p_t) \: , \: \alpha > 0 \tag{1} $$

for $ t \geq 0 $

Equation (1) asserts that the demand for real balances is inversely related to the public’s expected rate of inflation, which here equals the actual rate of inflation

(When there is no uncertainty, an assumption of rational expectations simplifies to perfect foresight)

(See [Sar77] for a rational expectations version of the model when there is uncertainty)

Subtracting the demand function at time $ t $ from the demand function at $ t+1 $ gives:

$$ \mu_t - \theta_t = -\alpha \theta_{t+1} + \alpha \theta_t $$

or

$$ \theta_t = \frac{\alpha}{1+\alpha} \theta_{t+1} + \frac{1}{1+\alpha} \mu_t \tag{2} $$

Because $ \alpha > 0 $, $ 0 < \frac{\alpha}{1+\alpha} < 1 $

Definition: For a scalar $ x_t $, let $ L^2 $ be the space of sequences $ \{x_t\}_{t=0}^\infty $ satisfying

$$ \sum_{t=0}^\infty x_t^2 < +\infty $$

We say that a sequence that belongs to $ L^2 $ is square summable

When we assume that the sequence $ \vec \mu = \{\mu_t\}_{t=0}^\infty $ is square summable and we require that the sequence $ \vec \theta = \{\theta_t\}_{t=0}^\infty $ is square summable, the linear difference equation (2) can be solved forward to get:

$$ \theta_t = \frac{1}{1+\alpha} \sum_{j=0}^\infty \left(\frac{\alpha}{1+\alpha}\right)^j \mu_{t+j} \tag{3} $$

Insight: In the spirit of Chang [Cha98], note that equations (1) and (3) show that $ \theta_t $ intermediates how choices of $ \mu_{t+j}, \ j=0, 1, \ldots $ impinge on time $ t $ real balances $ m_t - p_t = -\alpha \theta_t $

We shall use this insight to help us simplify and analyze government policy problems

That future rates of money creation influence earlier rates of inflation creates optimal government policy problems in which timing protocols matter

We can rewrite the model as:

$$ \begin{bmatrix} 1 \\ \theta_{t+1} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & \frac{1+\alpha}{\alpha} \end{bmatrix} \begin{bmatrix} 1 \\ \theta_{t} \end{bmatrix} + \begin{bmatrix} 0 \\ -\frac{1}{\alpha} \end{bmatrix} \mu_t $$

or

$$ x_{t+1} = A x_t + B \mu_t \tag{4} $$

We write the model in the state-space form (4) even though $ \theta_0 $ is to be determined and so is not an initial condition as it ordinarily would be in the state-space model described in Linear Quadratic Control

We write the model in the form (4) because we want to apply an approach described in Stackelberg problems

Assume that a representative household’s utility of real balances at time $ t $ is:

$$ U(m_t - p_t) = a_0 + a_1 (m_t - p_t) - \frac{a_2}{2} (m_t - p_t)^2, \quad a_0 > 0, a_1 > 0, a_2 > 0 \tag{5} $$

The “bliss level” of real balances is then $ \frac{a_1}{a_2} $

The money demand function (1) and the utility function (5) imply that the bliss level of real balances is attained when:

$$ \theta_t = \theta^* = -\frac{a_1}{a_2 \alpha} $$

Below, we introduce the discount factor $ \beta \in (0,1) $ that a representative household and a benevolent government both use to discount future utilities

(If we set parameters so that $ \theta^* = \log(\beta) $, then we can regard a recommendation to set $ \theta_t = \theta^* $ as a “poor man’s Friedman rule” that attains Milton Friedman’s optimal quantity of money)

Via equation (3), a government plan $ \vec \mu = \{\mu_t \}_{t=0}^\infty $ leads to an equilibrium sequence of inflation outcomes $ \vec \theta = \{ \theta_t \}_{t=0}^\infty $

We assume that social costs $ \frac{c}{2} \mu_t^2 $ are incurred at $ t $ when the government changes the stock of nominal money balances at rate $ \mu_t $

Therefore, the one-period welfare function of a benevolent government is:

$$ -s(\theta_t, \mu_t) \equiv - r(x_t,\mu_t) = \begin{bmatrix} 1 \\ \theta_t \end{bmatrix}' \begin{bmatrix} a_0 & -\frac{a_1 \alpha}{2} \\ -\frac{a_1 \alpha}{2} & -\frac{a_2 \alpha^2}{2} \end{bmatrix} \begin{bmatrix} 1 \\ \theta_t \end{bmatrix} - \frac{c}{2} \mu_t^2 = - x_t'Rx_t - Q \mu_t^2 \tag{6} $$

Household welfare is summarized by:

$$ v_0 = - \sum_{t=0}^\infty \beta^t r(x_t,\mu_t) = - \sum_{t=0}^\infty \beta^t s(\theta_t,\mu_t) \tag{7} $$

We can represent $ v_0 $ recursively via

$$ v_t = s(\theta_t, \mu_t) + \beta v_{t+1} \tag{8} $$

Structure

The following structure is induced by private agents’ behavior as summarized by the demand function for money (1) that leads to equation (3) that tells how future settings of $ \mu $ affect the current value of $ \theta $

Equation (3) maps a policy sequence of money growth rates $ \vec \mu =\{\mu_t\}_{t=0}^\infty \in L^2 $ into an inflation sequence $ \vec \theta = \{\theta_t\}_{t=0}^\infty \in L^2 $

These in turn induce a discounted value to a government sequence $ \vec v = \{v_t\}_{t=0}^\infty \in L^2 $ that satisfies the recursion

$$ v_t = s(\theta_t,\mu_t) + \beta v_{t+1} $$

where we have called $ s(\theta_t, \mu_t) = r(x_t, \mu_t) $ as above

Thus, we have a triple of sequences $ \vec \mu, \vec \theta, \vec v $ associated with a $ \vec \mu \in L^2 $

At this point $ \vec \mu $ is an arbitrary exogenous policy

To make $ \vec \mu $ endogenous, we require a theory of government decisions

Intertemporal Influences

Criterion function (7) and constraint system (4) exhibit the following structure:

  • Setting $ \mu_t \neq 0 $ imposes costs $ \frac{c}{2} \mu_t^2 $ at time $ t $ and at no other times; but
  • The money growth rate $ \mu_t $ affects the representative household’s one-period utilities at all dates $ s = 0, 1, \ldots, t $

That settings of $ \mu $ at one date affect household utilities at earlier dates sets the stage for the emergence of a time-inconsistent optimal government plan under a Ramsey (also called a Stackelberg) timing protocol

Four Models of Government Policy

We consider four models of policy makers that differ in

  • what a policy maker is allowed to choose, either a sequence $ \vec \mu $ or just a single period $ \mu_t $
  • when a policy maker chooses, either at time $ 0 $ or at times $ t \geq 0 $
  • what a policy maker assumes about how its choice of $ \mu_t $ affects private agents’ expectations about earlier and later inflation rates

In two of our models, there is a single policy maker that chooses a sequence $ \{\mu_t\}_{t=0}^\infty $ once and for all, taking into account how $ \mu_t $‘s at each date affect earlier components of household utilities

  • these two models thus employ a Ramsey or Stackelberg timing protocol

In two other models, there is a sequence of policy makers, each of whom sets $ \mu_t $ at one $ t $ only

  • Each such policy maker ignores effects that its choice of $ \mu_t $ has on household one-period utilities at dates $ s = 0, 1, \ldots, t-1 $

The four models differ with respect to timing protocols, constraints on government policy, and government policy makers’ beliefs about how their decisions affect private agents’ beliefs about future government decisions

The models are

  • A single Ramsey planner chooses a sequence $ \{\mu_t\}_{t=0}^\infty $ once and for all at time $ 0 $
  • A single Ramsey planner chooses a sequence $ \{\mu_t\}_{t=0}^\infty $ once and for all at time $ 0 $ subject to the constraint that $ \mu_t = \mu $ for all $ t \geq 0 $
  • A sequence of separate policy makers chooses $ \mu_t $ for $ t =0, 1, 2, \ldots $

    • a time $ t $ policy maker chooses $ \mu_t $ only and forecasts that future government decisions are unaffected by its choice
  • A sequence of separate policy makers chooses $ \mu_t $ for $ t =0, 1, 2, \ldots $

    • a time $ t $ policy maker chooses only $ \mu_t $ but believes that its choice of $ \mu_t $ shapes private agents’ beliefs about future rates of money creation and inflation, and through them, future government actions

A Ramsey Planner

First we consider a Ramsey planner that chooses $ \{\mu_t, \theta_t\}_{t=0}^\infty $ to maximize (7) subject to the law of motion (4)

We can split this problem into two stages, as in Stackelberg problems and [LS18] Chapter 19

In the first stage, we take the initial inflation rate $ \theta_0 $ as given, and then solve the resulting LQ dynamic programming problem

In the second stage, we maximize over the initial inflation rate $ \theta_0 $

Define a feasible set of $ (\overrightarrow x_1, \overrightarrow \mu_0) $ sequences:

$$ \Omega(x_0) = \left \lbrace ( \overrightarrow x_1, \overrightarrow \mu_0) : x_{t+1} = A x_t + B \mu_t \: , \: \forall t \geq 0 \right \rbrace $$

Subproblem 1

The value function

$$ J(x_0) = \max_{(\overrightarrow x_1, \overrightarrow \mu_0) \in \Omega(x_0)} \sum_{t=0}^\infty \beta^t r(x_t,\mu_t) $$

satisfies the Bellman equation

$$ J(x) = \max_{\mu,x'}\{-r(x,\mu) + \beta J(x')\} $$

subject to:

$$ x' = Ax + B\mu $$

As in Stackelberg problems, we map this problem into a linear-quadratic control problem and then carefully use the optimal value function associated with it

Guessing that $ J(x) = - x'Px $ and substituting into the Bellman equation gives rise to the algebraic matrix Riccatti equation:

$$ P = R + \beta A'PA - \beta^2 A'PB(Q + \beta B'PB)^{-1} B'PA $$

and the optimal decision rule

$$ \mu_t = - F x_t $$

where

$$ F = \beta (Q + \beta B'PB)^{-1} B'PA $$

The QuantEcon LQ class solves for $ F $ and $ P $ given inputs $ Q, R, A, B $, and $ \beta $

Subproblem 2

The value of the Ramsey problem is

$$ V = \max_{x_0} J(x_0) $$

The value function

$$ J(x_0) = -\begin{bmatrix} 1 & \theta_0 \end{bmatrix} \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} 1 \\ \theta_0 \end{bmatrix} = -P_{11} - 2 P_{21} \theta_0 - P_{22} \theta_0^2 $$

Maximizing this with respect to $ \theta_0 $ yields the FOC:

$$ - 2 P_{21} - 2 P_{22} \theta_0 =0 $$

which implies

$$ \theta_0^* = - \frac{P_{21}}{P_{22}} $$

Representation of Ramsey Plan

The preceding calculations indicate that we can represent a Ramsey plan $ \vec \mu $ recursively with the following system created in the spirit of Chang [Cha98]:

$$ \begin{aligned} \theta_0 & = \theta_0^* \\ \mu_t & = b_0 + b_1 \theta_t \\ \theta_{t+1} & = d_0 + d_1 \theta_t \end{aligned} \tag{9} $$

To interpret this system, think of the sequence $ \{\theta_t\}_{t=0}^\infty $ as a sequence of synthetic promised inflation rates that are just tools for generating a sequence $ \vec\mu $ of money growth rates that are to be substituted into equation (3) to form actual rates of inflation

It can verified that if we substitute a plan $ \vec \mu = \{\mu_t\}_{t=0}^\infty $ that satisfies these equations into equation (3), we obtain the same sequence $ \vec \theta $ generated by system (9)

Thus, our construction of a Ramsey plan guarantees that promised inflation equals actual inflation

The inflation rate $ \theta_t $ that appears in system (9) and equation (3) plays three roles simultaneously:

  • In equation (3), $ \theta_t $ is the actual rate of inflation between $ t $ and $ t+1 $
  • In equation (2) and (3), $ \theta_t $ is also the public’s expected rate of inflation between $ t $ and $ t+1 $
  • In system (9), $ \theta_t $ is a promised rate of inflation chosen by the Ramsey planner at time $ 0 $

Time Inconsistency

As discussed in Stackelberg problems and Optimal taxation with state-contingent debt, a continuation Ramsey plan is not a Ramsey plan

This is a concise way of characterizing the time inconsistency of a Ramsey plan

The time inconsistency of a Ramsey plan has motivated other models of government decision making that alter either

  • the timing protocol and/or
  • assumptions about how government decision makers think their decisions affect private agents’ beliefs about future government decisions

A Constrained-to-a-Constant-$ \mu $ Ramsey Government

We now consider the following peculiar model of optimal government behavior

We have created this model in order to highlight an aspect of an optimal government policy associated with its time inconsistency, namely, the feature that optimal settings of the policy instrument vary over time

Instead of allowing the government to choose different settings of its instrument at different moments, we now assume that at time $ 0 $, a government chooses a constant sequence $ \mu_t = \check \mu $ for all $ t \geq 0 $ to maximize

$$ U(-\alpha \check \mu) - \frac{c}{2} \check \mu^2 $$

Here we have imposed the perfect foresight outcome that $ \theta_t = \check \mu $ when the government chooses a constant $ \mu $ for all $ t \geq 0 $

With the quadratic form (5) for the utility function $ U $, the maximizing $ \bar \mu $ is

$$ \check \mu = - \frac{\alpha a_1}{\alpha^2 a_2 + c } $$

Summary: We have introduced the constrained-to-a-constant $ \mu $ government in order to highlight the role of time-variation of $ \mu_t $ in generating time inconsistency of a Ramsey plan

Markov Perfect Governments

We now change the timing protocol by considering a government that chooses $ \mu_t $ and expects all future governments to set $ \mu_{t+j} = \bar \mu $

This assumption mirrors an assumption made in a different setting Markov Perfect Equilibrium

Further, the government at $ t $ believes that $ \bar \mu $ is unafffected by its choice of $ \mu_t $

The time $ t $ rate of inflation is then:

$$ \theta_t = \frac{\alpha}{1+\alpha} \bar \mu + \frac{1}{1+\alpha} \mu_t $$

The time $ t $ government policy maker then chooses $ \mu_t $ to maximize:

$$ W = U(-\alpha \theta_t) - \frac{c}{2} \mu_t^2 + \beta V(\bar \mu) $$

where $ V(\bar \mu) $ is the time $ 0 $ value $ v_0 $ of recursion (8) under a money supply growth rate that is forever constant at $ \bar \mu $

Substituting for $ U $ and $ \theta_t $ gives:

$$ W = a_0 + a_1(-\frac{\alpha^2}{1+\alpha} \bar \mu - \frac{\alpha}{1+\alpha} \mu_t) - \frac{a_2}{2}((-\frac{\alpha^2}{1+\alpha} \bar \mu - \frac{\alpha}{1+\alpha} \mu_t)^2 - \frac{c}{2} \mu_t^2 + \beta V(\bar \mu) $$

The first-order necessary condition for $ \mu_t $ is then:

$$ - \frac{\alpha}{1+\alpha} a_1 - a_2(-\frac{\alpha^2}{1+\alpha} \bar \mu - \frac{\alpha}{1+\alpha} \mu_t)(- \frac{\alpha}{1+\alpha}) - c \mu_t = 0 $$

Rearranging we get:

$$ \mu_t = \frac{- a_1}{\frac{1+\alpha}{\alpha}c + \frac{\alpha}{1+\alpha}a_2} - \frac{\alpha^2 a_2}{\left[ \frac{1+\alpha}{\alpha}c + \frac{\alpha}{1+\alpha} a_2 \right] (1+\alpha)}\bar \mu $$

A Markov Perfect Equilibrium (MPE) outcome sets $ \mu_t = \bar \mu $:

$$ \mu_t = \bar \mu = \frac{-a_1}{\frac{1+\alpha}{\alpha}c + \frac{\alpha}{1+\alpha} a_2 + \frac{\alpha^2}{1+\alpha} a_2} $$

In light of results presented in the previous section, this can be simplified to:

$$ \bar \mu = - \frac{\alpha a_1}{\alpha^2 a_2 + (1+\alpha)c} $$

Equilibrium Outcomes for Three Models of Government Policy Making

Below we compute sequences $ \{ \theta_t,\mu_t \} $ under a Ramsey plan and compare these with the constant levels of $ \theta $ and $ \mu $ in a) a Markov Perfect Equilibrium, and b) a Ramsey plan in which the planner is restricted to choose $ \mu_t = \check\mu $ for all $ t \geq 0 $

We denote the Ramsey sequence as $ \theta^R, \mu^R $ and the MPE values as $ \theta^{MPE}, \mu^{MPE} $

The bliss level of inflation is denoted by $ \theta^* $

First we will create a class ChangLQ that solves the models and stores their values

In [1]:
import numpy as np
from quantecon import LQ
import matplotlib.pyplot as plt
%matplotlib inline


class ChangLQ:
    """
    Class to solve LQ Chang model
    """
    def __init__(self, α, α0, α1, α2, c, T=1000, θ_n=200):

        # Record parameters
        self.α, self.α0, self.α1 = α, α0, α1
        self.α2, self.c, self.T, self.θ_n = α2, c, T, θ_n

        # Create β using "Poor Man's Friedman Rule"
        self.β = np.exp(-α1 / (α * α2))

        # Solve the Ramsey Problem #

        # LQ Matrices
        R = -np.array([[α0,            -α1 * α / 2],
                       [-α1 * α/2, -α2 * α**2 / 2]])
        Q = -np.array([[-c / 2]])
        A = np.array([[1, 0], [0, (1 + α) / α]])
        B = np.array([[0], [-1 / α]])

        # Solve LQ Problem (Subproblem 1)
        lq = LQ(Q, R, A, B, beta=self.β)
        self.P, self.F, self.d = lq.stationary_values()

        # Solve Subproblem 2
        self.θ_R = -self.P[0, 1] / self.P[1, 1]

        # Find bliss level of θ
        self.θ_B = np.log(self.β)

        # Solve the Markov Perfect Equilibrium
        self.μ_MPE = -α1 / ((1 + α) / α * c + α / (1 + α)
                      * α2 + α**2 / (1 + α) * α2)
        self.θ_MPE = self.μ_MPE
        self.μ_check = -α * α1 / (α2 * α**2 + c)

        # Calculate value under MPE and Check economy
        self.J_MPE  = (α0 + α1 * (-α * self.μ_MPE) - α2 / 2 *
                      (-α * self.μ_MPE)**2 - c/2 * self.μ_MPE**2) / (1 - self.β)
        self.J_check = (α0 + α1 * (-α * self.μ_check) - α2/2 *
                       (-α * self.μ_check)**2 - c / 2 * self.μ_check**2) / (1 - self.β)

        # Siμlate Ramsey plan for large number of periods
        θ_series = np.vstack((np.ones((1, T)), np.zeros((1, T))))
        μ_series = np.zeros(T)
        J_series = np.zeros(T)
        θ_series[1, 0] = self.θ_R
        μ_series[0] = -self.F.dot(θ_series[:, 0])
        J_series[0] = -θ_series[:, 0] @ self.P @ θ_series[:, 0].T
        for i in range(1, T):
            θ_series[:, i] = (A - B @ self.F) @ θ_series[:, i-1]
            μ_series[i] = -self.F @ θ_series[:, i]
            J_series[i] = -θ_series[:, i] @ self.P @ θ_series[:, i].T

        self.J_series = J_series
        self.μ_series = μ_series
        self.θ_series = θ_series

        # Find range of θ in Ramsey plan
        θ_LB = min(θ_series[1, :])
        θ_LB = min(θ_LB, self.θ_B)
        θ_UB = max(θ_series[1, :])
        θ_UB = max(θ_UB, self.θ_MPE)
        θ_range = θ_UB - θ_LB
        self.θ_LB = θ_LB - 0.05 * θ_range
        self.θ_UB = θ_UB + 0.05 * θ_range
        self.θ_range = θ_range

        # Find value function and policy functions over range of θ
        θ_space = np.linspace(self.θ_LB, self.θ_UB, 200)
        J_space = np.zeros(200)
        check_space = np.zeros(200)
        μ_space = np.zeros(200)
        θ_prime = np.zeros(200)
        for i in range(200):
            J_space[i] = - np.array((1, θ_space[i])) @ self.P @ np.array((1, θ_space[i])).T
            μ_space[i] = - self.F @ np.array((1, θ_space[i]))
            x_prime = (A - B @ self.F) @ np.array((1, θ_space[i]))
            θ_prime[i] = x_prime[1]
            check_space[i] = (α0 + α1 * (-α * θ_space[i]) -
            α2/2 * (-α * θ_space[i])**2 - c/2 * θ_space[i]**2) / (1 - self.β)

        J_LB = min(J_space)
        J_UB = max(J_space)
        J_range = J_UB - J_LB
        self.J_LB = J_LB - 0.05 * J_range
        self.J_UB = J_UB + 0.05 * J_range
        self.J_range = J_range
        self.J_space = J_space
        self.θ_space = θ_space
        self.μ_space = μ_space
        self.θ_prime = θ_prime
        self.check_space = check_space

We will create an instance of ChangLQ with the following parameters

In [2]:
clq = ChangLQ(α=1, α0=1, α1=0.5, α2=3, c=2)
clq.β
Out[2]:
0.8464817248906141

The following code generates a figure that plots the value function from the Ramsey Planner’s problem, which is maximized at $ \theta^R_0 $

The figure also shows the limiting value $ \theta_\infty^R $ to which the inflation rate $ \theta_t $ converges under the Ramsey plan and compares it to the MPE value and the bliss value

In [3]:
def plot_value_function(clq):
    """
    Method to plot the value function over the relevant range of θ

    Here clq is an instance of ChangLQ

    """
    fig, ax = plt.subplots()

    ax.set_xlim([clq.θ_LB, clq.θ_UB])
    ax.set_ylim([clq.J_LB, clq.J_UB])

    # Plot value function
    ax.plot(clq.θ_space, clq.J_space, lw=2)
    plt.xlabel(r"$\theta$", fontsize=18)
    plt.ylabel(r"$J(\theta)$", fontsize=18)

    t1 = clq.θ_space[np.argmax(clq.J_space)]
    tR = clq.θ_series[1, -1]
    θ_points = [t1, tR, clq.θ_B, clq.θ_MPE]
    labels = [r"$\theta_0^R$", r"$\theta_\infty^R$",
              r"$\theta^*$", r"$\theta^{MPE}$"]

    # Add points for θs
    for θ, label in zip(θ_points, labels):
        ax.scatter(θ, clq.J_LB + 0.02 * clq.J_range, 60, 'black', 'v')
        ax.annotate(label,
                    xy=(θ, clq.J_LB + 0.01 * clq.J_range),
                    xytext=(θ - 0.01 * clq.θ_range,
                    clq.J_LB + 0.08 * clq.J_range),
                    fontsize=18)
    plt.tight_layout()
    plt.show()

plot_value_function(clq)

The next code generates a figure that plots the value function from the Ramsey Planner’s problem as well as that for a Ramsey planner that must choose a constant $ \mu $ (that in turn equals an implied constant $ \theta $)

In [4]:
def compare_ramsey_check(clq):
    """
    Method to compare values of Ramsey and Check

    Here clq is an instance of ChangLQ
    """
    fig, ax = plt.subplots()
    check_min = min(clq.check_space)
    check_max = max(clq.check_space)
    check_range = check_max - check_min
    check_LB = check_min - 0.05 * check_range
    check_UB = check_max + 0.05 * check_range
    ax.set_xlim([clq.θ_LB, clq.θ_UB])
    ax.set_ylim([check_LB, check_UB])
    ax.plot(clq.θ_space, clq.J_space, lw=2, label=r"$J(\theta)$")

    plt.xlabel(r"$\theta$", fontsize=18)
    ax.plot(clq.θ_space, clq.check_space,
            lw=2, label=r"$V^\check(\theta)$")
    plt.legend(fontsize=14, loc='upper left')

    θ_points = [clq.θ_space[np.argmax(clq.J_space)],
                    clq.μ_check]
    labels = [r"$\theta_0^R$", r"$\theta^\check$"]

    for θ, label in zip(θ_points, labels):
        ax.scatter(θ, check_LB + 0.02 * check_range, 60, 'k', 'v')
        ax.annotate(label,
                    xy=(θ, check_LB + 0.01 * check_range),
                    xytext=(θ - 0.02 * check_range, check_LB + 0.08 * check_range),
                    fontsize=18)
    plt.tight_layout()
    plt.show()

compare_ramsey_check(clq)

The next code generates figures that plot the policy functions for a continuation Ramsey planner

The left figure shows the choice of $ \theta' $ chosen by a continuation Ramsey planner who inherits $ \theta $

The right figure plots a continuation Ramsey planner’s choice of $ \mu $ as a function of an inherited $ \theta $

In [5]:
def plot_policy_functions(clq):
    """
    Method to plot the policy functions over the relevant range of θ

    Here clq is an instance of ChangLQ
    """
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))

    labels = [r"$\theta_0^R$", r"$\theta_\infty^R$"]

    ax = axes[0]
    ax.set_ylim([clq.θ_LB, clq.θ_UB])
    ax.plot(clq.θ_space, clq.θ_prime,
            label=r"$\theta'(\theta)$", lw=2)
    x = np.linspace(clq.θ_LB, clq.θ_UB, 5)
    ax.plot(x, x, 'k--', lw=2, alpha=0.7)
    ax.set_ylabel(r"$\theta'$", fontsize=18)

    θ_points = [clq.θ_space[np.argmax(clq.J_space)],
                    clq.θ_series[1, -1]]

    for θ, label in zip(θ_points, labels):
        ax.scatter(θ, clq.θ_LB + 0.02 * clq.θ_range, 60, 'k', 'v')
        ax.annotate(label,
                    xy=(θ, clq.θ_LB + 0.01 * clq.θ_range),
                    xytext=(θ - 0.02 * clq.θ_range,
                            clq.θ_LB + 0.08 * clq.θ_range),
                    fontsize=18)

    ax = axes[1]
    μ_min = min(clq.μ_space)
    μ_max = max(clq.μ_space)
    μ_range = μ_max - μ_min
    ax.set_ylim([μ_min - 0.05 * μ_range, μ_max + 0.05 * μ_range])
    ax.plot(clq.θ_space, clq.μ_space, lw=2)
    ax.set_ylabel(r"$\mu(\theta)$", fontsize=18)

    for ax in axes:
        ax.set_xlabel(r"$\theta$", fontsize=18)
        ax.set_xlim([clq.θ_LB, clq.θ_UB])

    for θ, label in zip(θ_points, labels):
        ax.scatter(θ, μ_min - 0.03 * μ_range, 60, 'black', 'v')
        ax.annotate(label, xy=(θ, μ_min - 0.03 * μ_range),
                    xytext=(θ - 0.02 * clq.θ_range,
                            μ_min + 0.03 * μ_range),
                    fontsize=18)
    plt.tight_layout()
    plt.show()

plot_policy_functions(clq)

The following code generates a figure that plots sequences of $ \mu $ and $ \theta $ in the Ramsey plan and compares these to the constant levels in a MPE and in a Ramsey plan with a government restricted to set $ \mu_t $ to a constant for all $ t $

In [6]:
def plot_ramsey_MPE(clq, T=15):
    """
    Method to plot Ramsey plan against Markov Perfect Equilibrium

    Here clq is an instance of ChangLQ
    """
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))

    plots = [clq.θ_series[1, 0:T], clq.μ_series[0:T]]
    MPEs = [clq.θ_MPE, clq.μ_MPE]
    labels = [r"\theta", r"\mu"]

    axes[0].hlines(clq.θ_B, 0, T-1, 'r', label=r"$\theta^*$")

    for ax, plot, MPE, label in zip(axes, plots, MPEs, labels):
        ax.plot(plot, label=r"$" + label + "^R$")
        ax.hlines(MPE, 0, T-1, 'orange', label=r"$" + label + "^{MPE}$")
        ax.hlines(clq.μ_check, 0, T, 'g', label=r"$" + label + "^\check$")
        ax.set_xlabel(r"$t$", fontsize=16)
        ax.set_ylabel(r"$" + label + "_t$", fontsize=18)
        ax.legend(loc='upper right')

    plt.tight_layout()
    plt.show()

plot_ramsey_MPE(clq)

Time Inconsistency of Ramsey Plan

The variation over time in $ \vec \mu $ chosen by the Ramsey planner is a symptom of time inconsistency

  • The Ramsey planner reaps immediate benefits from promising lower inflation later to be achieved by costly distorting taxes
  • These benefits are intermediated by reductions in expected inflation that precede the reductions in money creation rates that rationalize them, as indicated by equation (3)
  • A government authority offered the opportunity to ignore effects on past utilities and to reoptimize at date $ t \geq 1 $ would, if allowed, want to deviate from a Ramsey plan

Note: A modified Ramsey plan constructed under the restriction that $ \mu_t $ must be constant over time is time consistent (see $ \check \mu $ and $ \check \theta $ in the above graphs)

Meaning of Time Inconsistency

In settings in which governments choose sequentially, many economists regard a time inconsistent plan implausible because of the incentives to deviate that occur along the plan

A way to summarize this defect in a Ramsey plan is to say that it is not credible because there endure incentives for policy makers to deviate from it

For that reason, the Markov perfect equilibrium concept attracts many economists

  • A Markov perfect equilibrium plan is constructed to insure that government policy makers who choose sequentially do not want to deviate from it

The no incentive to deviate from the plan property is what makes the Markov perfect equilibrium concept attractive

Ramsey Plan Strikes Back

Research by Abreu [Abr88], Chari and Kehoe [CK90] [Sto89], and [Sto91] discovered conditions under which a Ramsey plan can be rescued from the complaint that it is not credible

They accomplished this by expanding the description of a plan to include expectations about adverse consequences of deviating from it that can serve to deter deviations

We turn to such theories of sustainable plans next

A Fourth Model of Government Decision Making

This is a model in which

  • The government chooses $ \{\mu_t\}_{t=0}^\infty $ not once and for all at $ t=0 $ but sequentially
  • private agents’ forecasts of $ \{\mu_{t+j+1}, \theta_{t+j+1}\}_{j=0}^\infty $ respond to whether the government at $ t $ confirms or disappoints their forecasts of $ \mu_t $ brought into period $ t $ from period $ t-1 $
  • the government at each time $ t $ understands how private agents’ forecasts will respond to its choice of $ \mu_t $
  • at each $ t $, the government chooses $ \mu_t $ to maximize a continuation discounted utility of a representative household

A Theory of Government Decision Making

$ \vec \mu $ is chosen by a sequence of government decision makers, one for each $ t \geq 0 $

We assume the following within-period and between-period timing protocol for each $ t \geq 0 $:

  • at time $ t-1 $, private agents expect that the government will set $ \mu_t = \tilde \mu_t $, and more generally that it will set $ \mu_{t+j} = \tilde \mu_{t+j} $ for all $ j \geq 0 $
  • Those forecasts determine a $ \theta_t = \tilde \theta_t $ and an associated log of real balances $ m_t - p_t = -\alpha\tilde \theta_t $ at $ t $
  • Given those expectations and the associated $ \theta_t $, at $ t $ a government is free to set $ \mu_t \in {\bf R} $
  • If the government at $ t $ confirms private agents’ expectations by setting $ \mu_t = \tilde \mu_t $ at time $ t $, private agents expect the continuation government policy $ \{\tilde \mu_{t+j+1}\}_{j=0}^\infty $ and therefore bring expectation $ \tilde \theta_{t+1} $ into period $ t+1 $
  • If the government disappoints private agents by setting $ \mu_t \neq \tilde \mu_t $, private agents expect $ \{\mu^A_j\}_{j=0}^\infty $ as the continuation policy for $ t+1 $, i.e., $ \{\mu_{t+j+1}\} = \{\mu_j^A \}_{j=0}^\infty $ and therefore expect $ \theta_0^A $ for $ t+1 $. Here $ \vec \mu^A = \{\mu_j^A \}_{j=0}^\infty $ is an alternative government plan to be described below

Temptation to Deviate from Plan

The government’s one-period return function $ s(\theta,\mu) $ described in equation (6) above has the property that for all $ \theta $

$$ s(\theta, 0 ) \geq s(\theta, \mu) \quad $$

This inequality implies that whenever the policy calls for the government to set $ \mu \neq 0 $, the government could raise its one-period return by setting $ \mu =0 $

Disappointing private sector expectations in that way would increase the government’s current payoff but would have adverse consequences for subsequent government payoffs because the private sector would alter its expectations about future settings of $ \mu $

The temporary gain constitutes the government’s temptation to deviate from a plan

If the government at $ t $ is to resist the temptation to raise its current payoff, it is only because it forecasts adverse consequences that its setting of $ \mu_t $ would bring for subsequent government payoffs via alterations in the private sector’s expectations

Sustainable or Credible plan

We call a plan $ \vec \mu $ sustainable or credible if at each $ t \geq 0 $ the government chooses to confirm private agents’ prior expectation of its setting for $ \mu_t $

The government will choose to confirm prior expectations if the long-term loss from disappointing private sector expectations – coming from the government’s understanding of the way the private sector adjusts its expectations in response to having its prior expectations at $ t $ disappointed – outweigh the short-term gain from disappointing those expectations

The theory of sustainable or credible plans assumes throughout that private sector expectations about what future governments will do are based on the assumption that governments at times $ t \geq 0 $ will act to maximize the continuation discounted utilities that describe those governments’ purposes

This aspect of the theory means that credible plans come in pairs:

  • a credible (continuation) plan to be followed if the government at $ t $ confirms private sector expectations
  • a credible plan to be followed if the government at $ t $ disappoints private sector expectations

That credible plans come in pairs seems to bring an explosion of plans to keep track of

  • each credible plan itself consists of two credible plans
  • therefore, the number of plans underlying one plan is unbounded

But Dilip Abreu showed how to render manageable the number of plans that must be kept track of

The key is an object called a self-enforcing plan

Abreu’s Self-Enforcing Plan

A plan $ \vec \mu^A $ is said to be self-enforcing if

  • the consequence of disappointing private agents’ expectations at time $ j $ is to restart the plan at time $ j+1 $
  • that consequence is sufficiently adverse that it deters all deviations from the plan

More precisely, a government plan $ \vec \mu^A $ is self-enforcing if

$$ \begin{aligned} v_j^A & = s(\theta^A_j, \mu^A_j) + \beta v_{j+1}^A \\ & \geq s(\theta^A_j, 0 ) + \beta v_0^A \equiv v_j^{A,D}, \quad j \geq 0 \end{aligned} \tag{10} $$

(Here it is useful to recall that setting $ \mu=0 $ is the maximizing choice for the government’s one-period return function)

The first line tells the consequences of confirming private agents’ expectations, while the second line tells the consequences of disappointing private agents’ expectations

A consequence of the definition is that a self-enforcing plan is credible

Self-enforcing plans can be used to construct other credible plans, ones with better values

A sufficient condition for a plan $ \vec \mu $ to be credible or sustainable is that

$$ \begin{aligned} \tilde v_j & = s(\tilde \theta_j, \mu_j) + \beta \tilde v_{j+1} \\ & \geq s(\tilde \theta_j, 0) + \beta v_0^A \quad \forall j \geq 0 \end{aligned} $$

A key step in constructing a credible plan is first constructing a self-enforcing plan that has a low time $ 0 $ value

The idea is to use the self-enforcing plan as a continuation plan when the government’s choice at time $ t $ fails to confirm private agents’ expectation

We shall use a construction featured in [Abr88] to construct a self-enforcing plan with low time $ 0 $ value

Abreu Carrot-Stick Plan

[Abr88] invented a way to create a self-enforcing plan with low initial value

Imitating his idea, we can construct a self-enforcing plan $ \vec \mu $ with a low time $ 0 $ value to the government by insisting that the government set $ \mu_t $ to a value yielding low one-period utilities to the household for a long time, after which the government forever does things yielding high one-period utilities

  • low one-period utilities early are a stick
  • high one-period utilities later are a carrot

Consider a plan $ \vec \mu^A $ in which the government sets $ \mu_t^A = \bar \mu $ (a high positive number) for $ T_A $ periods, and then reverts to the Ramsey plan

Denote this sequence by $ \{\mu_t^A\}_{t=0}^\infty $

The sequence of inflation rates implied by this plan, $ \{\theta_t^A\}_{t=0}^\infty $, can be calculated using:

$$ \theta_t^A = \frac{1}{1+\alpha} \sum_{j=0}^{\infty} \left(\frac{\alpha}{1+\alpha}\right)^j \mu^A_{t+j} $$

The value of $ \{\theta_t^A,\mu_t^A \}_{t=0}^\infty $ is

$$ v^A_0 = \sum_{t=0}^{T_A-1} \beta^t s(\theta_t^A,\mu_t^A) +\beta^{T_A} J(\theta^R_0) $$

Example of Self-Enforcing Plan

The following example implements Abreu’s idea

The government sets $ \mu_t^A = 0.1 $ for $ t=0, 1, \ldots, 9 $ and then starts the Ramsey plan

We have computed outcomes for this plan

For this plan, we plot the $ \theta^A $, $ \mu^A $ sequences as well as the implied $ v^A $ sequence

Notice that because the government sets money supply growth high for 10 periods, inflation starts high

Inflation gradually slowly declines immediately because people immediately expect the government to lower the money growth rate after period $ 10 $

From the 10th period onwards, the inflation rate $ \theta^A_t $ associated with this Abreu plan starts the Ramsey plan from its beginning, i.e., $ \theta^A_{t+10} =\theta^R_t \ \ \forall t \geq 0 $

In [7]:
def abreu_plan(clq, T=1000, T_A=10, μ_bar=0.1, T_Plot=20):

    # Append Ramsey μ series to stick μ series
    clq.μ_A = np.append(np.ones(T_A) * μ_bar, clq.μ_series[:-T_A])

    # Calculate implied stick θ series
    clq.θ_A = np.zeros(T)
    discount = np.zeros(T)
    for t in range(T):
        discount[t] = (clq.α / (1 + clq.α))**t
    for t in range(T):
        length = clq.μ_A[t:].shape[0]
        clq.θ_A[t] = 1 / (clq.α + 1) * sum(clq.μ_A[t:] * discount[0:length])

    # Calculate utility of stick plan
    U_A = np.zeros(T)
    for t in range(T):
        U_A[t] = clq.β**t * (clq.α0 + clq.α1 * (-clq.θ_A[t]) -
                 clq.α2 / 2 * (-clq.θ_A[t])**2 - clq.c * clq.μ_A[t]**2)

    clq.V_A = np.zeros(T)
    for t in range(T):
        clq.V_A[t] = sum(U_A[t:] / clq.β**t)

    # Make sure Abreu plan is self-enforcing
    clq.V_dev = np.zeros(T_Plot)
    for t in range(T_Plot):
        clq.V_dev[t] = (clq.α0 + clq.α1 * (-clq.θ_A[t]) -
                            clq.α2 / 2 * (-clq.θ_A[t])**2) + \
                            clq.β * clq.V_A[0]

    fig, axes = plt.subplots(3, 1, figsize=(8, 12))

    axes[2].plot(clq.V_dev[0:T_Plot], label="$V^{A, D}_t$", c="orange")

    plots = [clq.θ_A, clq.μ_A, clq.V_A]
    labels = [r"$\theta_t^A$", r"$\mu_t^A$", r"$V^A_t$"]

    for plot, ax, label in zip(plots, axes, labels):
        ax.plot(plot[0:T_Plot], label=label)
        ax.set(xlabel="$t$", ylabel=label)
        ax.legend()

    plt.tight_layout()
    plt.show()

abreu_plan(clq)

To confirm that the plan $ \vec \mu^A $ is self-enforcing, the right panel plots an object that we call $ V_t^{A,D} $, defined in the second line of equation (10) above

$ V_t^{A,D} $ is the value at $ t $ of deviating from the self-enforcing plan $ \vec \mu^A $ by setting $ \mu_t = 0 $ and then restarting the plan at $ v^A_0 $ at $ t+1 $

Notice that $ v_t^A > v_t^{A,D} $

This confirms that $ \vec \mu^A $ is a self-enforcing plan

We can also verify the inequalities required for $ \vec \mu^A $ to be self-confirming numerically as follows

In [8]:
np.all(clq.V_A[0:20] > clq.V_dev[0:20])
Out[8]:
True

Given that plan $ \vec \mu^A $ is self-enforcing, we can check that the Ramsey plan $ \vec \mu^R $ is sustainable by verifying that:

$$ v^R_t \geq s(\theta^R_t,0) + \beta v^A_0 , \quad \forall t \geq 0 $$
In [9]:
def check_ramsey(clq, T=1000):
    # Make sure Ramsey plan is sustainable
    R_dev = np.zeros(T)
    for t in range(T):
        R_dev[t] = (clq.α0 + clq.α1 * (-clq.θ_series[1, t]) -
                    clq.α2 / 2 * (-clq.θ_series[1, t])**2) + clq.β * clq.V_A[0]

    return np.all(clq.J_series > R_dev)

check_ramsey(clq)
Out[9]:
True

Recursive Representation of a Sustainable Plan

We can represent a sustainable plan recursively by taking the continuation value $ v_t $ as a state variable

We form the following 3-tuple of functions:

$$ \begin{aligned} \hat \mu_t & = \nu_\mu(v_t) \\ \theta_t & = \nu_\theta(v_t) \\ v_{t+1} & = \nu_v(v_t, \mu_t ) \end{aligned} \tag{11} $$

In addition to these equations, we need an initial value $ v_0 $ to characterize a sustainable plan

The first equation of (11) tells the recommended value of $ \hat \mu_t $ as a function of the promised value $ v_t $

The second equation of (11) tells the inflation rate as a function of $ v_t $

The third equation of (11) updates the continuation value in a way that depends on whether the government at $ t $ confirms private agents’ expectations by setting $ \mu_t $ equal to the recommended value $ \hat \mu_t $, or whether it disappoints those expectations

Comparison of Equilibrium Values

We have computed plans for

  • an ordinary (unrestricted) Ramsey planner who chooses a sequence $ \{\mu_t\}_{t=0}^\infty $ at time $ 0 $
  • a Ramsey planner restricted to choose a constant $ \mu $ for all $ t \geq 0 $
  • a Markov perfect sequence of governments

Below we compare equilibrium time zero values for these three

We confirm that the value delivered by the unrestricted Ramsey planner exceeds the value delivered by the restricted Ramsey planner which in turn exceeds the value delivered by the Markov perfect sequence of governments

In [10]:
clq.J_series[0]
Out[10]:
6.67918822960449
In [11]:
clq.J_check
Out[11]:
6.676729524674898
In [12]:
clq.J_MPE
Out[12]:
6.663435886995107

We have also computed sustainable plans for a government or sequence of governments that chooses sequentially

These include

  • a self-enforcing plan that gives a low initial value $ v_0 $
  • a better plan – possibly one that attains values associated with Ramsey plan – that is not self-enforcing

Note on Dynamic Programming Squared

The theory deployed in this lecture is an application of what we nickname dynamic programming squared

The nickname refers to the fact that a value satisfying one Bellman equation is itself an argument in a second Bellman equation

Thus, our models have involved two Bellman equations:

  • equation (1) expresses how $ \theta_t $ depends on $ \mu_t $ and $ \theta_{t+1} $
  • equation (4) expresses how value $ v_t $ depends on $ (\mu_t, \theta_t) $ and $ v_{t+1} $

A value $ \theta $ from one Bellman equation appears as an argument of a second Bellman equation for another value $ v $