Monetary Policy Model

A central bank must set nominal interest rate so as to minimize deviations of inflation rate and GDP gap from established targets.

A monetary authority wishes to control the nominal interest rate $x$ in order to minimize the variation of the inflation rate $s_1$ and the gross domestic product (GDP) gap $s_2$ around specified targets $s^∗_1$ and $s^∗_2$, respectively. Specifically, the authority wishes to minimize expected discounted stream of weighted squared deviations

\begin{equation} L(s) = \frac{1}{2}(s − s^∗)'\Omega(s − s^∗) \end{equation}

where $s$ is a $2\times 1$ vector containing the inflation rate and the GDP gap, $s^∗$ is a $2\times 1$ vector of targets, and $\Omega$ is a $2 \times 2$ constant positive definite matrix of preference weights. The inflation rate and the GDP gap are a joint controlled exogenous linear Markov process

\begin{equation} s_{t+1} = \alpha + \beta s_t + \gamma x_t + \epsilon_{t+1} \end{equation}

where $\alpha$ and $\gamma$ are $2 \times 1$ constant vectors, $\beta$ is a $2 \times 2$ constant matrix, and $\epsilon$ is a $2 \times 1$ random vector with mean zero. For institutional reasons, the nominal interest rate $x$ cannot be negative. What monetary policy minimizes the sum of current and expected future losses?

This is an infinite horizon, stochastic model with time $t$ measured in years. The state vector $s \in \mathbb{R}^2$ contains the inflation rate and the GDP gap. The action variable $x \in [0,\infty)$ is the nominal interest rate. The state transition function is $g(s, x, \epsilon) = \alpha + \beta s + \gamma x + \epsilon$

In order to formulate this problem as a maximization problem, one posits a reward function that equals the negative of the loss function $f(s,x) = −L(s)$

The sum of current and expected future rewards satisfies the Bellman equation

\begin{equation} V(s) = \max_{0\leq x}\left\{-L(s) + \delta + E_\epsilon V\left(g(s,x,\epsilon)\right)\right\} \end{equation}

Given the structure of the model, one cannot preclude the possibility that the nonnegativity constraint on the optimal nominal interest rate will be binding in certain states. Accordingly, the shadow-price function $\lambda(s)$ is characterized by the Euler conditions

\begin{align} \delta\gamma'E_\epsilon \lambda\left(g(s,x,\epsilon)\right) &= \mu \\ \lambda(s) &= -\Omega(s-s^*) + \delta\beta'E_\epsilon \lambda\left(g(s,x,\epsilon)\right) \end{align}

where the nominal interest rate $x$ and the long-run marginal reward $\mu$ from increasing the nominal interest rate must satisfy the complementarity condition \begin{equation} x \geq 0, \qquad \mu \leq 0, \qquad x > 0 \Rightarrow \mu = 0 \end{equation}

It follows that along the optimal path

\begin{align} \delta\gamma'E_\epsilon \lambda_{t+1} &= \mu_t \\ \lambda_t &= -\Omega(s_t-s^*) + \delta\beta'E_\epsilon \lambda_{t+1}\\ x \geq 0, \qquad \mu \leq 0, &\qquad x > 0 \Rightarrow \mu = 0 \end{align}

Thus, in any period, the nominal interest rate is reduced until either the long-run marginal

reward or the nominal interest rate is driven to zero.

In [1]:
%matplotlib inline
from warnings import simplefilter
from compecon import BasisChebyshev, DPmodel, BasisSpline
from compecon.quad import qnwnorm
from demos.setup import np, plt, demo
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm

Model Parameters

In [2]:
alpha   = np.array([[0.9, -0.1]]).T             # transition function constant coefficients
beta    = np.array([[-0.5, 0.2], [0.3, -0.4]])  # transition function state coefficients
gamma   = np.array([[-0.1, 0.0]]).T             # transition function action coefficients
omega   = np.identity(2)                        # central banker's preference weights
sbar    = np.array([[1, 0]]).T                  # equilibrium targets
sigma   = 0.08 * np.identity(2),                # shock covariance matrix
delta   = 0.9                                   # discount factor

State Space

In [3]:
n = 21          
smin = [-2, -3] 
smax = [ 2,  3] 

basis = BasisChebyshev(n, smin, smax, method='complete',
                       labels=['GDP gap', 'inflation'])

Action space

There is only one action variable x: the nominal interest rate, which must be nonnegative.

In [4]:
def bounds(s, i, j):
    lb  = np.zeros_like(s[0])
    ub  = np.full(lb.shape, np.inf)
    return lb, ub

Reward Function

In [5]:
def reward(s, x, i, j):
    s = s - sbar  #  todo make sure they broadcast (:,ones(1,n))'
    f = np.zeros_like(s[0])
    for ii in range(2):
        for jj in range(2):
            f -= 0.5 * omega[ii, jj] * s[ii] * s[jj]
    fx = np.zeros_like(x)
    fxx = np.zeros_like(x)
    return f, fx, fxx

State Transition Function

In [6]:
def transition(s, x, i, j, in_, e):
    g = alpha + beta @ s + gamma @ x + e
    gx = np.tile(gamma, (1, x.size))
    gxx = np.zeros_like(s)
    return g, gx, gxx

The continuous shock must be discretized. Here we use Gauss-Legendre quadrature to obtain nodes and weights defining a discrete distribution that matches the first 6 moments of the Normal distribution (this is achieved with m=3 nodes and weights) for each of the state variables.

In [7]:
m   = [3, 3]
mu  = [0, 0] 
[e,w] = qnwnorm(m,mu,sigma)

Model structure

In [8]:
bank = DPmodel(basis, reward, transition, bounds,
               x=['interest'], discount=delta, e=e, w=w)

Compute Unconstrained Deterministic Steady-State

In [9]:
sstar, xstar, pstar = bank.lqapprox(sbar, [0])

If Nonnegativity Constraint Violated, Re-Compute Deterministic Steady-State

In [10]:
if xstar < 0:
    I = np.identity(2)
    xstar = 0
    sstar = np.linalg.solve(np.identity(2) - beta, alpha)

frmt = '\t%-21s = %5.2f' 
print('Deterministic Steady-State')
print(frmt % ('GDP Gap', sstar[0]))
print(frmt % ('Inflation Rate', sstar[1]))
print(frmt % ('Nominal Interest Rate', xstar))
Deterministic Steady-State
	GDP Gap               =  0.61
	Inflation Rate        =  0.06
	Nominal Interest Rate =  0.00

Solve the model

We solve the model by calling the solve method in bank. On return, sol is a pandas dataframe with columns GDP gap, inflation, value, interest, and resid. We set a refined grid nr=5 for this output.

In [11]:
sol = bank.solve(nr=5)
Solving infinite-horizon model collocation equation by Newton's method
iter change       time    
   0       1.9e+00    0.5574
   1       5.9e-02    1.1994
   2       8.5e-05    1.7157
   3       3.3e-07    2.1045
   4       3.0e-09    2.5088
Elapsed Time =    2.51 Seconds
In [12]:
Index(['GDP gap', 'inflation', 'value', 'interest', 'resid'], dtype='object')

To make the 3D plots, we need to reshape the columns of sol.

In [13]:
gdp_gap, inflation, value, interest, resid = [x.reshape((5*n, 5*n)) for x in sol.values.T]

Optimal policy

In [14]:
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(gdp_gap, inflation, interest, cmap=cm.coolwarm)
plt.xlabel('GDP gap')
# plt.zlabel('Nomianal Interest Rate')  #zlabel not defined!
plt.title('Optimal Monetary Policy')
<matplotlib.text.Text at 0x12e88dfe358>

Value function

In [15]:
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(gdp_gap, inflation, value, cmap=cm.coolwarm)
plt.xlabel('GDP gap')
# plt.zlabel('Nomianal Interest Rate')  #zlabel not defined!
plt.title('Value Function')
<matplotlib.text.Text at 0x12e88f29438>


In [16]:
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(gdp_gap, inflation, resid, cmap=cm.coolwarm)
plt.xlabel('GDP gap')
# plt.zlabel('Nomianal Interest Rate')  #zlabel not defined!
plt.title('Bellman Equation Residual')
<matplotlib.text.Text at 0x12e88e961d0>