In addition to what’s in Anaconda, this lecture will deploy quantecon:
!pip install --upgrade quantecon
Let’s start with some standard imports:
import quantecon as qe
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
This lecture uses the method of Markov jump linear quadratic dynamic programming that is described in lecture Markov Jump LQ dynamic programming to extend the [Bar79] model of optimal tax-smoothing and government debt in a particular direction.
This lecture has two sequels that offer further extensions of the Barro model
The extensions are modified versions of his 1979 model later suggested by Barro (1999 [Bar99], 2003 [BM03]).
Barro’s original 1979 [Bar79] model is about a government that borrows and lends in order to minimize an intertemporal measure of distortions caused by taxes.
Technical tractability induced Barro [Bar79] to assume that
By using Markov jump linear quadratic dynamic programming we can allow interest rates to move over time in empirically interesting ways.
Also, by expanding the dimension of the state, we can add a maturity composition decision to the government’s problem.
It is by doing these two things that we extend Barro’s 1979 [Bar79] model along lines he suggested in Barro (1999 [Bar99], 2003 [BM03]).
Barro (1979) [Bar79] assumed
Barro’s model can be mapped into a discounted linear quadratic dynamic programming problem.
Partly inspired by Barro (1999) [Bar99] and Barro (2003) [BM03], our generalizations of Barro’s (1979) [Bar79] model assume
Our generalizations are designed to fit within a generalization of an ordinary linear quadratic dynamic programming problem in which matrices that define the quadratic objective function and the state transition function are time-varying and stochastic.
This generalization, known as a Markov jump linear quadratic dynamic program, combines
We want the stochastic time variation in the matrices defining the dynamic programming problem to represent variation over time in
As described in Markov Jump LQ dynamic programming, the idea underlying Markov jump linear quadratic dynamic programming is to replace the constant matrices defining a linear quadratic dynamic programming problem with matrices that are fixed functions of an $ N $ state Markov chain.
For infinite horizon problems, this leads to $ N $ interrelated matrix Riccati equations that pin down $ N $ value functions and $ N $ linear decision rules, applying to the $ N $ Markov states.
Barro’s 1979 [Bar79] model is designed to answer questions such as
Barro’s 1999 [Bar99] and 2003 [BM03] models are designed to answer more fine-grained questions such as
Thus, both the simple and the more fine-grained versions of Barro’s models are ways of precisely formulating the classic issue of How to pay for a war.
This lecture describes:
A sequel to this lecture describes applies Markov LQ control to settings in which a government issues risk-free debt of different maturities.
We begin by solving a version of the Barro (1979) [Bar79] model by mapping it into the original LQ framework.
As mentioned in this lecture, the Barro model is mathematically isomorphic with the LQ permanent income model.
Let $ T_t $ denote tax collections, $ \beta $ a discount factor, $ b_{t,t+1} $ time $ t+1 $ goods that the government promises to pay at $ t $, $ G_t $ government purchases, $ p_{t,t+1} $ the number of time $ t $ goods received per time $ t+1 $ goods promised.
Evidently, $ p_{t, t+1} $ is inversely related to appropriate corresponding gross interest rates on government debt.
In the spirit of Barro (1979) [Bar79], the stochastic process of government expenditures is exogenous.
The government’s problem is to choose a plan for taxation and borrowing $ \{b_{t+1}, T_t\}_{t=0}^\infty $ to minimize
$$ E_0 \sum_{t=0}^\infty \beta^t T_t^2 $$subject to the constraints
$$ T_t + p_{t,t+1} b_{t,t+1} = G_t + b_{t-1,t} $$$$ G_t = U_{g} z_t $$$$ z_{t+1} = A_{22} z_t + C_{2} w_{t+1} $$where $ w_{t+1} \sim {\cal N}(0,I) $
The variables $ T_t, b_{t, t+1} $ are control variables chosen at $ t $, while $ b_{t-1,t} $ is an endogenous state variable inherited from the past at time $ t $ and $ p_{t,t+1} $ is an exogenous state variable at time $ t $.
To begin, we assume that $ p_{t,t+1} $ is constant (and equal to $ \beta $)
To map into the LQ framework, we use $ x_t = \begin{bmatrix} b_{t-1,t} \\ z_t \end{bmatrix} $ as the state vector, and $ u_t = b_{t,t+1} $ as the control variable.
Therefore, the $ (A, B, C) $ matrices are defined by the state-transition law:
$$ x_{t+1} = \begin{bmatrix} 0 & 0 \\ 0 & A_{22} \end{bmatrix} x_t + \begin{bmatrix} 1 \\ 0 \end{bmatrix} u_t + \begin{bmatrix} 0 \\ C_2 \end{bmatrix} w_{t+1} $$To find the appropriate $ (R, Q, W) $ matrices, we note that $ G_t $ and $ b_{t-1,t} $ can be written as appropriately defined functions of the current state:
$$ G_t = S_G x_t \hspace{2mm}, \hspace{2mm} b_{t-1,t} = S_1 x_t $$If we define $ M_t = - p_{t,t+1} $, and let $ S = S_G + S_1 $, then we can write taxation as a function of the states and control using the government’s budget constraint:
$$ T_t = S x_t + M_t u_t $$It follows that the $ (R, Q, W) $ matrices are implicitly defined by:
$$ T_t^2 = x_t'S'Sx_t + u_t'M_t'M_tu_t + 2 u_t'M_t'S x_t $$If we assume that $ p_{t,t+1} = \beta $, then $ M_t \equiv M = -\beta $.
In this case, none of the LQ matrices are time varying, and we can use the original LQ framework.
We will implement this constant interest-rate version first, assuming that $ G_t $ follows an AR(1) process:
$$ G_{t+1} = \bar G + \rho G_t + \sigma w_{t+1} $$To do this, we set $ z_t = \begin{bmatrix} 1 \\ G_t \end{bmatrix} $, and consequently:
$$ A_{22} = \begin{bmatrix} 1 & 0 \\ \bar G & \rho \end{bmatrix} \hspace{2mm} , \hspace{2mm} C_2 = \begin{bmatrix} 0 \\ \sigma \end{bmatrix} $$# Model parameters
β, Gbar, ρ, σ = 0.95, 5, 0.8, 1
# Basic model matrices
A22 = np.array([[1, 0],
[Gbar, ρ],])
C2 = np.array([[0],
[σ]])
Ug = np.array([[0, 1]])
# LQ framework matrices
A_t = np.zeros((1, 3))
A_b = np.hstack((np.zeros((2, 1)), A22))
A = np.vstack((A_t, A_b))
B = np.zeros((3, 1))
B[0, 0] = 1
C = np.vstack((np.zeros((1, 1)), C2))
Sg = np.hstack((np.zeros((1, 1)), Ug))
S1 = np.zeros((1, 3))
S1[0, 0] = 1
S = S1 + Sg
M = np.array([[-β]])
R = S.T @ S
Q = M.T @ M
W = M.T @ S
# Small penalty on the debt required to implement the no-Ponzi scheme
R[0, 0] = R[0, 0] + 1e-9
We can now create an instance of LQ
:
LQBarro = qe.LQ(Q, R, A, B, C=C, N=W, beta=β)
P, F, d = LQBarro.stationary_values()
x0 = np.array([[100, 1, 25]])
We can see the isomorphism by noting that consumption is a martingale in the permanent income model and that taxation is a martingale in Barro’s model.
We can check this using the $ F $ matrix of the LQ model.
Because $ u_t = -F x_t $, we have
$$ T_t = S x_t + M u_t = (S - MF) x_t $$and
$$ T_{t+1} = (S-MF)x_{t+1} = (S-MF)(Ax_t + B u_t + C w_{t+1}) = (S-MF)((A-BF)x_t + C w_{t+1}) $$Therefore, the mathematical expectation of $ T_{t+1} $ conditional on time $ t $ information is
$$ E_t T_{t+1} = (S-MF)(A-BF)x_t $$Consequently, taxation is a martingale ($ E_t T_{t+1} = T_t $) if
$$ (S-MF)(A-BF) = (S-MF) , $$which holds in this case:
S - M @ F, (S - M @ F) @ (A - B @ F)
This explains the fanning out of the conditional empirical distribution of taxation across time, computing by simulation the Barro model a large number of times:
T = 500
for i in range(250):
x, u, w = LQBarro.compute_sequence(x0, ts_length=T)
plt.plot(list(range(T+1)), ((S - M @ F) @ x)[0, :])
plt.xlabel('Time')
plt.ylabel('Taxation')
plt.show()
We can see a similar, but a smoother pattern, if we plot government debt over time.
T = 500
for i in range(250):
x, u, w = LQBarro.compute_sequence(x0, ts_length=T)
plt.plot(list(range(T+1)), x[0, :])
plt.xlabel('Time')
plt.ylabel('Govt Debt')
plt.show()
To implement the extension to the Barro model in which $ p_{t,t+1} $ varies over time, we must allow the M matrix to be time-varying.
Our $ Q $ and $ W $ matrices must also vary over time.
We can solve such a
model using the LQMarkov
class that solves Markov jump linear
quandratic control problems as described above.
The code for the class can be viewed here.
The class takes lists of matrices that corresponds to $ N $ Markov states.
The value and policy functions are then found by iterating on a coupled system of matrix Riccati difference equations.
Optimal $ P_s,F_s,d_s $ are stored as attributes.
The class also contains a “method” for simulating the model.
We can use the above class to implement a version of the Barro model with a time-varying interest rate. The simplest way to extend the model is to allow the interest rate to take two possible values. We set:
$$ p^1_{t,t+1} = \beta + 0.02 = 0.97 $$$$ p^2_{t,t+1} = \beta - 0.017 = 0.933 $$Thus, the first Markov state has a low interest rate, and the second Markov state has a high interest rate.
We also need to specify a transition matrix for the Markov state.
We use:
$$ \Pi = \begin{bmatrix} 0.8 & 0.2 \\ 0.2 & 0.8 \end{bmatrix} $$(so each Markov state is persistent, and there is an equal chance of moving from one state to the other)
The choice of parameters means that the unconditional expectation of $ p_{t,t+1} $ is 0.9515, higher than $ \beta (=0.95) $.
If we were to set $ p_{t,t+1} = 0.9515 $ in the version of the model with a constant interest rate, government debt would explode.
# Create list of matrices that corresponds to each Markov state
Π = np.array([[0.8, 0.2],
[0.2, 0.8]])
As = [A, A]
Bs = [B, B]
Cs = [C, C]
Rs = [R, R]
M1 = np.array([[-β - 0.02]])
M2 = np.array([[-β + 0.017]])
Q1 = M1.T @ M1
Q2 = M2.T @ M2
Qs = [Q1, Q2]
W1 = M1.T @ S
W2 = M2.T @ S
Ws = [W1, W2]
# create Markov Jump LQ DP problem instance
lqm = qe.LQMarkov(Π, Qs, Rs, As, Bs, Cs=Cs, Ns=Ws, beta=β)
lqm.stationary_values();
The decision rules are now dependent on the Markov state:
lqm.Fs[0]
lqm.Fs[1]
Simulating a large number of such economies over time reveals interesting dynamics.
Debt tends to stay low and stable but recurrently surges.
T = 2000
x0 = np.array([[1000, 1, 25]])
for i in range(250):
x, u, w, s = lqm.compute_sequence(x0, ts_length=T)
plt.plot(list(range(T+1)), x[0, :])
plt.xlabel('Time')
plt.ylabel('Govt Debt')
plt.show()