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In addition to JAX and Anaconda, this lecture will need the following libraries:
:tags: [hide-output]
!pip install quantecon
This lecture describes Kesten processes, which are an important class of stochastic processes, and an application of firm dynamics.
The lecture draws on an earlier QuantEcon lecture, which uses Numba to accelerate the computations.
In that earlier lecture you can find a more detailed discussion of the concepts involved.
This lecture focuses on implementing the same computations in JAX.
Let's start with some imports:
import matplotlib.pyplot as plt
import quantecon as qe
import jax
import jax.numpy as jnp
from jax import random
from jax import lax
Let's check the GPU we are running
!nvidia-smi
A Kesten process is a stochastic process of the form
:label: kesproc
X_{t+1} = a_{t+1} X_t + \eta_{t+1}
where ${a_t}{t \geq 1}$ and ${\eta_t}{t \geq 1}$ are IID sequences.
We are interested in the dynamics of
We will focus on the nonnegative scalar case, where
In particular, we will assume that
- the initial condition
$X_0$ is nonnegative, -
${a_t}_{t \geq 1}$ is a nonnegative IID stochastic process and -
${\eta_t}_{t \geq 1}$ is another nonnegative IID stochastic process, independent of the first.
In this section we apply Kesten process theory to the study of firm dynamics.
It was postulated many years ago by Robert Gibrat that firm size evolves according to a simple rule whereby size next period is proportional to current size.
This is now known as Gibrat's law of proportional growth.
We can express this idea by stating that a suitably defined measure
:label: firm_dynam_gb
\frac{s_{t+1}}{s_t} = a_{t+1}
for some positive IID sequence
Subsequent empirical research has shown that this specification is not accurate, particularly for small firms.
However, we can get close to the data by modifying {eq}firm_dynam_gb to
:label: firm_dynam
s_{t+1} = a_{t+1} s_t + b_{t+1}
where
We now study the implications of this specification.
If the conditions of the Kesten--Goldie
Theorem
are satisfied, then {eq}firm_dynam implies that the firm size distribution
will have Pareto tails.
This matches empirical findings across many data sets.
But there is another unrealistic aspect of the firm dynamics specified in {eq}firm_dynam that we need to address: it ignores entry and exit.
In any given period and in any given market, we observe significant numbers of firms entering and exiting the market.
In this setting, firm dynamics can be expressed as
:label: firm_dynam_ee
s_{t+1} = e_{t+1} \mathbb{1}\{s_t < \bar s\} +
(a_{t+1} s_t + b_{t+1}) \mathbb{1}\{s_t \geq \bar s\}
The motivation behind and interpretation of can be found in our earlier Kesten process lecture.
What can we say about dynamics?
Although {eq}firm_dynam_ee is not a Kesten process, it does update in the
same way as a Kesten process when
So perhaps its stationary distribution still has Pareto tails?
We can investigate this question via simulation and rank-size plots.
The approach will be to
- generate
$M$ draws of$s_T$ when$M$ and$T$ are large and - plot the largest 1,000 of the resulting draws in a rank-size plot.
(The distribution of
In the simulation, we assume that each of
Here's code to update a cross-section of firms according to the dynamics in .
@jax.jit
def update_s(s, s_bar, a_random, b_random, e_random):
exp_a = jnp.exp(a_random)
exp_b = jnp.exp(b_random)
exp_e = jnp.exp(e_random)
s = jnp.where(s < s_bar,
exp_e,
exp_a * s + exp_b)
return s
Now we write a for loop that repeatedly calls this function, to push a cross-section of firms forward in time.
For sufficiently large T, the cross-section it returns (the cross-section at
time T) corresponds to firm size distribution in (approximate) equilibrium.
def generate_draws(M=1_000_000,
μ_a=-0.5,
σ_a=0.1,
μ_b=0.0,
σ_b=0.5,
μ_e=0.0,
σ_e=0.5,
s_bar=1.0,
T=500,
s_init=1.0,
seed=123):
key = random.PRNGKey(seed)
# Initialize the array of s values with the initial value
s = jnp.full((M, ), s_init)
# Perform updates on s for time t
for t in range(T):
keys = random.split(key, 3)
a_random = μ_a + σ_a * random.normal(keys[0], (M, ))
b_random = μ_b + σ_b * random.normal(keys[1], (M, ))
e_random = μ_e + σ_e * random.normal(keys[2], (M, ))
s = update_s(s, s_bar, a_random, b_random, e_random)
# Generate new key for the next iteration
key = random.fold_in(key, t)
return s
%time data = generate_draws().block_until_ready()
Running the above function again so we can see the speed with and without compile time.
%time data = generate_draws().block_until_ready()
Notice that we do not JIT-compile the for loops, since
- acceleration of the outer loop makes little difference terms of compute time and
- compiling the outer loop is often very slow.
Let's produce the rank-size plot and check the distribution:
fig, ax = plt.subplots()
rank_data, size_data = qe.rank_size(data, c=0.01)
ax.loglog(rank_data, size_data, 'o', markersize=3.0, alpha=0.5)
ax.set_xlabel("log rank")
ax.set_ylabel("log size")
plt.show()
The plot produces a straight line, consistent with a Pareto tail.
If the time horizon is not too large, we can try to further accelerate our code
by replacing the for loop with
lax.fori_loop.
Note, however, that
- as mentioned above, there is not much speed gain in accelerating outer loops,
lax.fori_loophas a more complicated syntax, and, most importantly,- the
lax.fori_loopimplementation consumes far more memory, as we need to have to store large matrices of random draws
Hence the code below will fail due to out-of-memory errors when T and M are large.
Here is the lax.fori_loop version:
@jax.jit
def generate_draws_lax(μ_a=-0.5,
σ_a=0.1,
μ_b=0.0,
σ_b=0.5,
μ_e=0.0,
σ_e=0.5,
s_bar=1.0,
T=500,
M=500_000,
s_init=1.0,
seed=123):
key = random.PRNGKey(seed)
keys = random.split(key, 3)
# Generate random draws and initial values
a_random = μ_a + σ_a * random.normal(keys[0], (T, M))
b_random = μ_b + σ_b * random.normal(keys[1], (T, M))
e_random = μ_e + σ_e * random.normal(keys[2], (T, M))
s = jnp.full((M, ), s_init)
# Define the function for each update
def update_s(i, s):
a, b, e = a_random[i], b_random[i], e_random[i]
s = jnp.where(s < s_bar,
jnp.exp(e),
jnp.exp(a) * s + jnp.exp(b))
return s
# Use lax.scan to perform the calculations on all states
s_final = lax.fori_loop(0, T, update_s, s)
return s_final
%time data = generate_draws_lax().block_until_ready()
In this case, M is small enough for the code to run and
we see some speed gain over the for loop implementation:
%time data = generate_draws_lax().block_until_ready()
Here we produce the same rank-size plot:
fig, ax = plt.subplots()
rank_data, size_data = qe.rank_size(data, c=0.01)
ax.loglog(rank_data, size_data, 'o', markersize=3.0, alpha=0.5)
ax.set_xlabel("log rank")
ax.set_ylabel("log size")
plt.show()
Let's rerun the for loop version on smaller M to compare the speed
%time generate_draws(M=500_000).block_until_ready()
Let's run it again to get rid of the compile time.
%time generate_draws(M=500_000).block_until_ready()
We see that the lax.fori_loop version is faster than the for loop version
when memory is not an issue.