10. Optimal Investment#

GPU

This lecture was built using hardware that has access to a GPU.

To run this lecture on Google Colab, click on the “play” icon top right, select Colab, and set the runtime environment to include a GPU.

To run this lecture on your own machine, you need to install Google JAX.

We require the following library to be installed.

!pip install --upgrade quantecon

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WARNING: Running pip as the 'root' user can result in broken permissions and conflicting behaviour with the system package manager. It is recommended to use a virtual environment instead: https://pip.pypa.io/warnings/venv

We study a monopolist who faces inverse demand curve

\[ P_t = a_0 - a_1 Y_t + Z_t, \]

where

\(P_t\) is price,
\(Y_t\) is output and
\(Z_t\) is a demand shock.

We assume that \(Z_t\) is a discretized AR(1) process, specified below.

Current profits are

\[ P_t Y_t - c Y_t - \gamma (Y_{t+1} - Y_t)^2 \]

Combining with the demand curve and writing \(y, y'\) for \(Y_t, Y_{t+1}\), this becomes

\[ r(y, z, y') := (a_0 - a_1 y + z - c) y - γ (y' - y)^2 \]

The firm maximizes present value of expected discounted profits. The Bellman equation is

\[ v(y, z) = \max_{y'} \left\{ r(y, z, y') + β \sum_{z'} v(y', z') Q(z, z') \right\}. \]

We discretize \(y\) to a finite grid y_grid.

In essence, the firm tries to choose output close to the monopolist profit maximizer, given \(Z_t\), but is constrained by adjustment costs.

Let’s begin with the following imports

import quantecon as qe
import jax
import jax.numpy as jnp
import matplotlib.pyplot as plt

Let’s check the GPU we are running

!nvidia-smi

Fri Sep 22 00:39:42 2023       
+-----------------------------------------------------------------------------+
| NVIDIA-SMI 470.182.03   Driver Version: 470.182.03   CUDA Version: 12.1     |
|-------------------------------+----------------------+----------------------+
| GPU  Name        Persistence-M| Bus-Id        Disp.A | Volatile Uncorr. ECC |
| Fan  Temp  Perf  Pwr:Usage/Cap|         Memory-Usage | GPU-Util  Compute M. |
|                               |                      |               MIG M. |
|===============================+======================+======================|

|   0  Tesla V100-SXM2...  Off  | 00000000:00:1E.0 Off |                    0 |
| N/A   29C    P0    38W / 300W |      0MiB / 16160MiB |      2%      Default |
|                               |                      |                  N/A |
+-------------------------------+----------------------+----------------------+
                                                                               
+-----------------------------------------------------------------------------+
| Processes:                                                                  |
|  GPU   GI   CI        PID   Type   Process name                  GPU Memory |
|        ID   ID                                                   Usage      |
|=============================================================================|
|  No running processes found                                                 |
+-----------------------------------------------------------------------------+

We will use 64 bit floats with JAX in order to increase the precision.

jax.config.update("jax_enable_x64", True)

We need the following successive approximation function.

def successive_approx(T,                     # Operator (callable)
                      x_0,                   # Initial condition
                      tolerance=1e-6,        # Error tolerance
                      max_iter=10_000,       # Max iteration bound
                      print_step=25,         # Print at multiples
                      verbose=False):
    x = x_0
    error = tolerance + 1
    k = 1
    while error > tolerance and k <= max_iter:
        x_new = T(x)
        error = jnp.max(jnp.abs(x_new - x))
        if verbose and k % print_step == 0:
            print(f"Completed iteration {k} with error {error}.")
        x = x_new
        k += 1
    if error > tolerance:
        print(f"Warning: Iteration hit upper bound {max_iter}.")
    elif verbose:
        print(f"Terminated successfully in {k} iterations.")
    return x

Let’s define a function to create an investment model using the given parameters.

def create_investment_model(
        r=0.01,                              # Interest rate
        a_0=10.0, a_1=1.0,                   # Demand parameters
        γ=25.0, c=1.0,                       # Adjustment and unit cost
        y_min=0.0, y_max=20.0, y_size=100,   # Grid for output
        ρ=0.9, ν=1.0,                        # AR(1) parameters
        z_size=150):                         # Grid size for shock
    """
    A function that takes in parameters and returns an instance of Model that
    contains data for the investment problem.
    """
    β = 1 / (1 + r)
    y_grid = jnp.linspace(y_min, y_max, y_size)
    mc = qe.tauchen(z_size, ρ, ν)
    z_grid, Q = mc.state_values, mc.P

    # Break up parameters into static and nonstatic components
    constants = β, a_0, a_1, γ, c
    sizes = y_size, z_size
    arrays = y_grid, z_grid, Q

    # Shift arrays to the device (e.g., GPU)
    arrays = tuple(map(jax.device_put, arrays))
    return constants, sizes, arrays

Let’s re-write the vectorized version of the right-hand side of the Bellman equation (before maximization), which is a 3D array representing

\[ B(y, z, y') = r(y, z, y') + \beta \sum_{z'} v(y', z') Q(z, z') \]

for all \((y, z, y')\).

def B(v, constants, sizes, arrays):
    """
    A vectorized version of the right-hand side of the Bellman equation
    (before maximization)
    """

    # Unpack
    β, a_0, a_1, γ, c = constants
    y_size, z_size = sizes
    y_grid, z_grid, Q = arrays

    # Compute current rewards r(y, z, yp) as array r[i, j, ip]
    y  = jnp.reshape(y_grid, (y_size, 1, 1))    # y[i]   ->  y[i, j, ip]
    z  = jnp.reshape(z_grid, (1, z_size, 1))    # z[j]   ->  z[i, j, ip]
    yp = jnp.reshape(y_grid, (1, 1, y_size))    # yp[ip] -> yp[i, j, ip]
    r = (a_0 - a_1 * y + z - c) * y - γ * (yp - y)**2

    # Calculate continuation rewards at all combinations of (y, z, yp)
    v = jnp.reshape(v, (1, 1, y_size, z_size))  # v[ip, jp] -> v[i, j, ip, jp]
    Q = jnp.reshape(Q, (1, z_size, 1, z_size))  # Q[j, jp]  -> Q[i, j, ip, jp]
    EV = jnp.sum(v * Q, axis=3)                 # sum over last index jp

    # Compute the right-hand side of the Bellman equation
    return r + β * EV

# Create a jitted function
B = jax.jit(B, static_argnums=(2,))

We define a function to compute the current rewards \(r_\sigma\) given policy \(\sigma\), which is defined as the vector

\[ r_\sigma(y, z) := r(y, z, \sigma(y, z)) \]

def compute_r_σ(σ, constants, sizes, arrays):
    """
    Compute the array r_σ[i, j] = r[i, j, σ[i, j]], which gives current
    rewards given policy σ.
    """

    # Unpack model
    β, a_0, a_1, γ, c = constants
    y_size, z_size = sizes
    y_grid, z_grid, Q = arrays

    # Compute r_σ[i, j]
    y = jnp.reshape(y_grid, (y_size, 1))
    z = jnp.reshape(z_grid, (1, z_size))
    yp = y_grid[σ]
    r_σ = (a_0 - a_1 * y + z - c) * y - γ * (yp - y)**2

    return r_σ


# Create the jitted function
compute_r_σ = jax.jit(compute_r_σ, static_argnums=(2,))

Define the Bellman operator.

def T(v, constants, sizes, arrays):
    """The Bellman operator."""
    return jnp.max(B(v, constants, sizes, arrays), axis=2)

T = jax.jit(T, static_argnums=(2,))

The following function computes a v-greedy policy.

def get_greedy(v, constants, sizes, arrays):
    "Computes a v-greedy policy, returned as a set of indices."
    return jnp.argmax(B(v, constants, sizes, arrays), axis=2)

get_greedy = jax.jit(get_greedy, static_argnums=(2,))

Define the \(\sigma\)-policy operator.

def T_σ(v, σ, constants, sizes, arrays):
    """The σ-policy operator."""

    # Unpack model
    β, a_0, a_1, γ, c = constants
    y_size, z_size = sizes
    y_grid, z_grid, Q = arrays

    r_σ = compute_r_σ(σ, constants, sizes, arrays)

    # Compute the array v[σ[i, j], jp]
    zp_idx = jnp.arange(z_size)
    zp_idx = jnp.reshape(zp_idx, (1, 1, z_size))
    σ = jnp.reshape(σ, (y_size, z_size, 1))
    V = v[σ, zp_idx]

    # Convert Q[j, jp] to Q[i, j, jp]
    Q = jnp.reshape(Q, (1, z_size, z_size))

    # Calculate the expected sum Σ_jp v[σ[i, j], jp] * Q[i, j, jp]
    Ev = jnp.sum(V * Q, axis=2)

    return r_σ + β * Ev

T_σ = jax.jit(T_σ, static_argnums=(3,))

Next, we want to computes the lifetime value of following policy \(\sigma\).

This lifetime value is a function \(v_\sigma\) that satisfies

\[ v_\sigma(y, z) = r_\sigma(y, z) + \beta \sum_{z'} v_\sigma(\sigma(y, z), z') Q(z, z') \]

We wish to solve this equation for \(v_\sigma\).

Suppose we define the linear operator \(L_\sigma\) by

\[ (L_\sigma v)(y, z) = v(y, z) - \beta \sum_{z'} v(\sigma(y, z), z') Q(z, z') \]

With this notation, the problem is to solve for \(v\) via

\[ (L_{\sigma} v)(y, z) = r_\sigma(y, z) \]

In vector for this is \(L_\sigma v = r_\sigma\), which tells us that the function we seek is

\[ v_\sigma = L_\sigma^{-1} r_\sigma \]

JAX allows us to solve linear systems defined in terms of operators; the first step is to define the function \(L_{\sigma}\).

def L_σ(v, σ, constants, sizes, arrays):

    β, a_0, a_1, γ, c = constants
    y_size, z_size = sizes
    y_grid, z_grid, Q = arrays

    # Set up the array v[σ[i, j], jp]
    zp_idx = jnp.arange(z_size)
    zp_idx = jnp.reshape(zp_idx, (1, 1, z_size))
    σ = jnp.reshape(σ, (y_size, z_size, 1))
    V = v[σ, zp_idx]

    # Expand Q[j, jp] to Q[i, j, jp]
    Q = jnp.reshape(Q, (1, z_size, z_size))

    # Compute and return v[i, j] - β Σ_jp v[σ[i, j], jp] * Q[j, jp]
    return v - β * jnp.sum(V * Q, axis=2)

L_σ = jax.jit(L_σ, static_argnums=(3,))

Now we can define a function to compute \(v_{\sigma}\)

def get_value(σ, constants, sizes, arrays):

    # Unpack
    β, a_0, a_1, γ, c = constants
    y_size, z_size = sizes
    y_grid, z_grid, Q = arrays

    r_σ = compute_r_σ(σ, constants, sizes, arrays)

    # Reduce L_σ to a function in v
    partial_L_σ = lambda v: L_σ(v, σ, constants, sizes, arrays)

    return jax.scipy.sparse.linalg.bicgstab(partial_L_σ, r_σ)[0]

get_value = jax.jit(get_value, static_argnums=(2,))

Finally, we introduce the solvers that implement VFI, HPI and OPI.

# Implements VFI-Value Function iteration

def value_iteration(model, tol=1e-5):
    constants, sizes, arrays = model
    _T = lambda v: T(v, constants, sizes, arrays)
    vz = jnp.zeros(sizes)

    v_star = successive_approx(_T, vz, tolerance=tol)
    return get_greedy(v_star, constants, sizes, arrays)

# Implements HPI-Howard policy iteration routine

def policy_iteration(model, maxiter=250):
    constants, sizes, arrays = model
    σ = jnp.zeros(sizes, dtype=int)
    i, error = 0, 1.0
    while error > 0 and i < maxiter:
        v_σ = get_value(σ, constants, sizes, arrays)
        σ_new = get_greedy(v_σ, constants, sizes, arrays)
        error = jnp.max(jnp.abs(σ_new - σ))
        σ = σ_new
        i = i + 1
        print(f"Concluded loop {i} with error {error}.")
    return σ

# Implements the OPI-Optimal policy Iteration routine

def optimistic_policy_iteration(model, tol=1e-5, m=10):
    constants, sizes, arrays = model
    v = jnp.zeros(sizes)
    error = tol + 1
    while error > tol:
        last_v = v
        σ = get_greedy(v, constants, sizes, arrays)
        for _ in range(m):
            v = T_σ(v, σ, constants, sizes, arrays)
        error = jnp.max(jnp.abs(v - last_v))
    return get_greedy(v, constants, sizes, arrays)

model = create_investment_model()
print("Starting HPI.")
qe.tic()
out = policy_iteration(model)
elapsed = qe.toc()
print(out)
print(f"HPI completed in {elapsed} seconds.")

print("Starting VFI.")
qe.tic()
out = value_iteration(model)
elapsed = qe.toc()
print(out)
print(f"VFI completed in {elapsed} seconds.")

print("Starting OPI.")
qe.tic()
out = optimistic_policy_iteration(model, m=100)
elapsed = qe.toc()
print(out)
print(f"OPI completed in {elapsed} seconds.")

Here’s the plot of the Howard policy, as a function of \(y\) at the highest and lowest values of \(z\).

model = create_investment_model()
constants, sizes, arrays = model
β, a_0, a_1, γ, c = constants
y_size, z_size = sizes
y_grid, z_grid, Q = arrays

σ_star = policy_iteration(model)

fig, ax = plt.subplots(figsize=(9, 5))
ax.plot(y_grid, y_grid, "k--", label="45")
ax.plot(y_grid, y_grid[σ_star[:, 1]], label="$\\sigma^*(\cdot, z_1)$")
ax.plot(y_grid, y_grid[σ_star[:, -1]], label="$\\sigma^*(\cdot, z_N)$")
ax.legend(fontsize=12)
plt.show()

Concluded loop 1 with error 50.
Concluded loop 2 with error 26.
Concluded loop 3 with error 17.
Concluded loop 4 with error 10.
Concluded loop 5 with error 7.
Concluded loop 6 with error 4.
Concluded loop 7 with error 3.
Concluded loop 8 with error 1.
Concluded loop 9 with error 1.
Concluded loop 10 with error 1.
Concluded loop 11 with error 1.
Concluded loop 12 with error 0.

_images/c86c3446a7c20f5de97b1ad524c96a0d31550860af15c6ae47b8beafcc1f5fd8.png

Let’s plot the time taken by each of the solvers and compare them.

m_vals = range(5, 600, 40)

model = create_investment_model()
print("Running Howard policy iteration.")
qe.tic()
σ_pi = policy_iteration(model)
pi_time = qe.toc()

Running Howard policy iteration.
Concluded loop 1 with error 50.
Concluded loop 2 with error 26.
Concluded loop 3 with error 17.
Concluded loop 4 with error 10.
Concluded loop 5 with error 7.
Concluded loop 6 with error 4.
Concluded loop 7 with error 3.
Concluded loop 8 with error 1.
Concluded loop 9 with error 1.
Concluded loop 10 with error 1.
Concluded loop 11 with error 1.
Concluded loop 12 with error 0.
TOC: Elapsed: 0:00:0.04

print(f"PI completed in {pi_time} seconds.")
print("Running value function iteration.")
qe.tic()
σ_vfi = value_iteration(model, tol=1e-5)
vfi_time = qe.toc()
print(f"VFI completed in {vfi_time} seconds.")

PI completed in 0.04567694664001465 seconds.
Running value function iteration.

TOC: Elapsed: 0:00:1.38
VFI completed in 1.3857970237731934 seconds.

opi_times = []
for m in m_vals:
    print(f"Running optimistic policy iteration with m={m}.")
    qe.tic()
    σ_opi = optimistic_policy_iteration(model, m=m, tol=1e-5)
    opi_time = qe.toc()
    print(f"OPI with m={m} completed in {opi_time} seconds.")
    opi_times.append(opi_time)

fig, ax = plt.subplots(figsize=(9, 5))
ax.plot(m_vals, jnp.full(len(m_vals), pi_time),
        lw=2, label="Howard policy iteration")
ax.plot(m_vals, jnp.full(len(m_vals), vfi_time),
        lw=2, label="value function iteration")
ax.plot(m_vals, opi_times, lw=2, label="optimistic policy iteration")
ax.legend(fontsize=12, frameon=False)
ax.set_xlabel("$m$", fontsize=12)
ax.set_ylabel("time(s)", fontsize=12)
plt.show()

_images/6d58ebc931b9bc834ec117aff9d3b8da7b8e27b4865944cdcc85c5adebad9221.png