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# Uncertainty Traps¶

## Overview¶

In this lecture we study a simplified version of an uncertainty traps model of Fajgelbaum, Schaal and Taschereau-Dumouchel [FSTD15]

The model features self-reinforcing uncertainty that has big impacts on economic activity

In the model,

• Fundamentals vary stochastically and are not fully observable
• At any moment there are both active and inactive entrepreneurs; only active entrepreneurs produce
• Agents – active and inactive entrepreuneurs – have beliefs about the fundamentals expressed as probability distributions
• Greater uncertainty means greater dispersions of these distributions
• Entrepreneurs are risk averse and hence less inclined to be active when uncertainty is high
• The output of active entrepreneurs is observable, supplying a noisy signal that helps everyone inside the model infer fundamentals
• Entrepreneurs update their beliefs about fundamentals using Bayes’ Law, implemented via Kalman filtering

Uncertainty traps emerge because:

• High uncertainty discourages entrepreneurs from becoming active
• A low level of participation – i.e., a smaller number of active entrepreneurs – diminishes the flow of information about fundamentals
• Less information translates to higher uncertainty, further discouraging entrepreneurs from choosing to be active, and so on

Uncertainty traps stem from a positive externality: high aggregate economic activity levels generates valuable information

## The Model¶

The original model described in [FSTD15] has many interesting moving parts

Here we examine a simplified version that nonetheless captures many of the key ideas

### Fundamentals¶

The evolution of the fundamental process $$\{\theta_t\}$$ is given by

$\theta_{t+1} = \rho \theta_t + \sigma_{\theta} w_{t+1}$

where

• $$\sigma_\theta > 0$$ and $$0 < \rho < 1$$
• $$\{w_t\}$$ is IID and standard normal

The random variable $$\theta_t$$ is not observable at any time

### Output¶

There is a total $$\bar M$$ of risk averse entrepreneurs

Output of the $$m$$-th entrepreneur, conditional on being active in the market at time $$t$$, is equal to

(1)$x_m = \theta + \epsilon_m \quad \text{where} \quad \epsilon_m \sim N \left(0, \gamma_x^{-1} \right)$

Here the time subscript has been dropped to simplify notation

The inverse of the shock variance, $$\gamma_x$$, is called the shock’s precision

The higher is the precision, the more informative $$x_m$$ is about the fundamental

Output shocks are independent across time and firms

### Information and Beliefs¶

All entrepreneurs start with identical beliefs about $$\theta_0$$

Signals are publicly observable and hence all agents have identical beliefs always

Dropping time subscripts, beliefs for current $$\theta$$ are represented by the normal distribution $$N(\mu, \gamma^{-1})$$

Here $$\gamma$$ is the precision of beliefs; its inverse is the degree of uncertainty

These parameters are updated by Kalman filtering

Let

• $$\mathbb M \subset \{1, \ldots, \bar M\}$$ denote the set of currently active firms
• $$M := |\mathbb M|$$ denote the number of currently active firms
• $$X$$ be the average output $$\frac{1}{M} \sum_{m \in \mathbb M} x_m$$ of the active firms

With this notation and primes for next period values, we can write the updating of the mean and precision via

(2)$\mu' = \rho \frac{\gamma \mu + M \gamma_x X}{\gamma + M \gamma_x}$
(3)$\gamma' = \left( \frac{\rho^2}{\gamma + M \gamma_x} + \sigma_\theta^2 \right)^{-1}$

These are standard Kalman filtering results applied to the current setting

Exercise 1 provides more details on how (2) and (3) are derived, and then asks you to fill in remaining steps

The next figure plots the law of motion for the precision in (3) as a 45 degree diagram, with one curve for each $$M \in \{0, \ldots, 6\}$$

The other parameter values are $$\rho = 0.99, \gamma_x = 0.5, \sigma_\theta =0.5$$

Points where the curves hit the 45 degree lines are long run steady states for precision for different values of $$M$$

Thus, if one of these values for $$M$$ remains fixed, a corresponding steady state is the equilibrium level of precision

• high values of $$M$$ correspond to greater information about the fundamental, and hence more precision in steady state
• low values of $$M$$ correspond to less information and more uncertainty in steady state

In practice, as we’ll see, the number of active firms fluctuates stochastically

### Participation¶

Omitting time subscripts once more, entrepreneurs enter the market in the current period if

(4)$\mathbb E [ u(x_m - F_m) ] > c$

Here

• the mathematical expectation of $$x_m$$ is based on (1) and beliefs $$N(\mu, \gamma^{-1})$$ for $$\theta$$
• $$F_m$$ is a stochastic but previsible fixed cost, independent across time and firms
• $$c$$ is a constant reflecting opportunity costs

The statement that $$F_m$$ is previsible means that it is realized at the start of the period and treated as a constant in (4)

The utility function has the constant absolute risk aversion form

(5)$u(x) = \frac{1}{a} \left(1 - \exp(-a x) \right)$

where $$a$$ is a positive parameter

Combining (4) and (5), entrepreneur $$m$$ participates in the market (or is said to be active) when

$\frac{1}{a} \left\{ 1 - \mathbb E [ \exp \left( -a (\theta + \epsilon_m - F_m) \right) ] \right\} > c$

Using standard formulas for expectations of lognormal random variables, this is equivalent to the condition

(6)$\psi(\mu, \gamma, F_m) := \frac{1}{a} \left( 1 - \exp \left( -a \mu + a F_m + \frac{a^2 \left( \frac{1}{\gamma} + \frac{1}{\gamma_x} \right)}{2} \right) \right) - c > 0$

## Implementation¶

We want to simulate this economy

As a first step, let’s put together a class that bundles

• the parameters, the current value of $$\theta$$ and the current values of the two belief parameters $$\mu$$ and $$\gamma$$
• methods to update $$\theta$$, $$\mu$$ and $$\gamma$$, as well as to determine the number of active firms and their outputs

The updating methods follow the laws of motion for $$\theta$$, $$\mu$$ and $$\gamma$$ given above

The method to evaluate the number of active firms generates $$F_1, \ldots, F_{\bar M}$$ and tests condition (6) for each firm

The __init__ method encodes as default values the parameters we’ll use in the simulations below

Here’s the code, which can be obtained from GitHub

from __future__ import division
import numpy as np

class UncertaintyTrapEcon:

def __init__(self,
a=1.5,          # Risk aversion
gx=0.5,         # Production shock precision
rho=0.99,       # Correlation coefficient for theta
sig_theta=0.5,  # Std dev of theta shock
num_firms=100,  # Number of firms
sig_F=1.5,      # Std dev of fixed costs
c=-420,         # External opportunity cost
mu_init=0,      # Initial value for mu
gamma_init=4,   # Initial value for gamma
theta_init=0):  # Initial value for theta

# == Record values == #
self.a, self.gx, self.rho, self.sig_theta = a, gx, rho, sig_theta
self.num_firms, self.sig_F, self.c, = num_firms, sig_F, c
self.sd_x = np.sqrt(1/ gx)

# == Initialize states == #
self.gamma, self.mu, self.theta =  gamma_init, mu_init, theta_init

def psi(self, F):
temp1 = -self.a * (self.mu - F)
temp2 = self.a**2 * (1/self.gamma + 1/self.gx) / 2
return (1 / self.a) * (1 - np.exp(temp1 + temp2)) - self.c

def update_beliefs(self, X, M):
"""
Update beliefs (mu, gamma) based on aggregates X and M.
"""
# Simplify names
gx, rho, sig_theta = self.gx, self.rho, self.sig_theta
# Update mu
temp1 = rho * (self.gamma * self.mu + M * gx * X)
temp2 = self.gamma + M * gx
self.mu =  temp1 / temp2
# Update gamma
self.gamma = 1 / (rho**2 / (self.gamma + M * gx) + sig_theta**2)

def update_theta(self, w):
"""
Update the fundamental state theta given shock w.
"""
self.theta = self.rho * self.theta + self.sig_theta * w

def gen_aggregates(self):
"""
Generate aggregates based on current beliefs (mu, gamma).  This
is a simulation step that depends on the draws for F.
"""
F_vals = self.sig_F * np.random.randn(self.num_firms)
M = np.sum(self.psi(F_vals) > 0)  # Counts number of active firms
if M > 0:
x_vals = self.theta + self.sd_x * np.random.randn(M)
X = x_vals.mean()
else:
X = 0
return X, M


In the results below we use this code to simulate time series for the major variables

## Results¶

Let’s look first at the dynamics of $$\mu$$, which the agents use to track $$\theta$$

We see that $$\mu$$ tracks $$\theta$$ well when there are sufficient firms in the market

However, there are times when $$\mu$$ tracks $$\theta$$ poorly due to insufficient information

These are episodes where the uncertainty traps take hold

During these episodes

• precision is low and uncertainty is high
• few firms are in the market

To get a clearer idea of the dynamics, let’s look at all the main time series at once, for a given set of shocks

Notice how the traps only take hold after a sequence of bad draws for the fundamental

Thus, the model gives us a propagation mechanism that maps bad random draws into long downturns in economic activity

## Exercises¶

### Exercise 1¶

Fill in the details behind (2) and (3) based on the following standard result (see, e.g., p. 24 of [YS05])

Fact Let $$\mathbf x = (x_1, \ldots, x_M)$$ be a vector of IID draws from common distribution $$N(\theta, 1/\gamma_x)$$ and let $$\bar x$$ be the sample mean. If $$\gamma_x$$ is known and the prior for $$\theta$$ is $$N(\mu, 1/\gamma)$$, then the posterior distribution of $$\theta$$ given $$\mathbf x$$ is

$\pi(\theta \,|\, \mathbf x) = N(\mu_0, 1/\gamma_0)$

where

$\mu_0 = \frac{\mu \gamma + M \bar x \gamma_x}{\gamma + M \gamma_x} \quad \text{and} \quad \gamma_0 = \gamma + M \gamma_x$

### Exercise 2¶

Modulo randomness, replicate the simulation figures shown above

• Use the parameter values listed as defaults in the __init__ method of the UncertaintyTrapEcon class

## Solutions¶

import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
import itertools


### Exercise 1¶

This exercise asked you to validate the laws of motion for $$\gamma$$ and $$\mu$$ given in the lecture, based on the stated result about Bayesian updating in a scalar Gaussian setting. The stated result tells us that after observing average output $$X$$ of the $$M$$ firms, our posterior beliefs will be

$N(\mu_0, 1/\gamma_0)$

where

$\mu_0 = \frac{\mu \gamma + M X \gamma_x}{\gamma + M \gamma_x} \quad \text{and} \quad \gamma_0 = \gamma + M \gamma_x$

If we take a random variable $$\theta$$ with this distribution and then evaluate the distribution of $$\rho \theta + \sigma_\theta w$$ where $$w$$ is independent and standard normal, we get the expressions for $$\mu'$$ and $$\gamma'$$ given in the lecture.

### Exercise 2¶

First let’s replicate the plot that illustrates the law of motion for precision, which is

$\gamma_{t+1} = \left( \frac{\rho^2}{\gamma_t + M \gamma_x} + \sigma_\theta^2 \right)^{-1}$

Here $$M$$ is the number of active firms. The next figure plots $$\gamma_{t+1}$$ against $$\gamma_t$$ on a 45 degree diagram for different values of $$M$$

palette = itertools.cycle(sns.color_palette())
econ = UncertaintyTrapEcon()
rho, sig_theta, gx = econ.rho, econ.sig_theta, econ.gx # simplify names
g = np.linspace(1e-10, 3, 200)  # gamma grid
fig, ax = plt.subplots(figsize=(9, 9))
ax.plot(g, g, 'k-') # 45 degree line

for M in range(7):
g_next = 1 / (rho**2 / (g + M * gx) + sig_theta**2)
label_string = r"$M = {}$".format(M)
ax.plot(g, g_next, lw=2, label=label_string, color=next(palette))
ax.legend(loc='lower right', fontsize=14)
ax.set_xlabel(r'$\gamma$', fontsize=16)
ax.set_ylabel(r"$\gamma'$", fontsize=16)
ax.grid()
plt.show()


The points where the curves hit the 45 degree lines are the long run steady states corresponding to each $$M$$, if that value of $$M$$ was to remain fixed. As the number of firms falls, so does the long run steady state of precision.

Next let’s generate time series for beliefs and the aggregates – that is, the number of active firms and average output.

sim_length=2000

mu_vec = np.empty(sim_length)
theta_vec = np.empty(sim_length)
gamma_vec = np.empty(sim_length)
X_vec = np.empty(sim_length)
M_vec = np.empty(sim_length)

mu_vec[0] = econ.mu
gamma_vec[0] = econ.gamma
theta_vec[0] = 0

w_shocks = np.random.randn(sim_length)

for t in range(sim_length-1):
X, M = econ.gen_aggregates()
X_vec[t] = X
M_vec[t] = M

econ.update_beliefs(X, M)
econ.update_theta(w_shocks[t])

mu_vec[t+1] = econ.mu
gamma_vec[t+1] = econ.gamma
theta_vec[t+1] = econ.theta

# Record final values of aggregates
X, M = econ.gen_aggregates()
X_vec[-1] = X
M_vec[-1] = M


First let’s see how well $$\mu$$ tracks $$\theta$$ in these simulations

fig, ax = plt.subplots(figsize=(9, 6))
ax.plot(range(sim_length), theta_vec, alpha=0.6, lw=2, label=r"$\theta$")
ax.plot(range(sim_length), mu_vec, alpha=0.6, lw=2, label=r"$\mu$")
ax.legend(fontsize=16)
plt.show()


Now let’s plot the whole thing together

fig, axes = plt.subplots(4, 1, figsize=(12, 20))
# Add some spacing
fig.subplots_adjust(hspace=0.3)

series = (theta_vec, mu_vec, gamma_vec, M_vec)
names = r'$\theta$', r'$\mu$', r'$\gamma$', r'$M$'

for ax, vals, name in zip(axes, series, names):
# determine suitable y limits
s_max, s_min = max(vals), min(vals)
s_range = s_max - s_min
y_max = s_max + s_range * 0.1
y_min = s_min - s_range * 0.1
ax.set_ylim(y_min, y_max)
# Plot series
ax.plot(range(sim_length), vals, alpha=0.6, lw=2)
ax.set_title("time series for {}".format(name), fontsize=16)
ax.grid()

plt.show()


If you run the code above you’ll get different plots, of course. Try experimenting with different parameters to see the effects on the time series. (It would also be interesting to experiment with non-Gaussian distributions for the shocks, but this is a big exercise since it takes us outside the world of the standard Kalman filter.)

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