84. A Robust LQ Bewley Model#

84.1. Overview#

This lecture embeds the Bewley economy of Consumption Smoothing with Incomplete and Complete Markets in the robust permanent income framework of Robust Consumption Smoothing and Precautionary Savings.

It is the last of four lectures on the LQ permanent income model.

The result is a family of economies in which consumers disagree about the model generating their income, yet behave identically.

Using the observational-equivalence theorem of Hansen et al. [1999], we show

  • how a continuum of consumers \(i\) can differ in their robustness parameters \(\sigma_i \leq 0\) and their discount factors \(\beta_i\), provided each pair \((\sigma_i,\beta_i)\) lies on an observational-equivalence locus

  • how every such consumer chooses the same consumption-saving rule as a benchmark \((\sigma = 0, \beta)\) agent who fully trusts the endowment process

  • how the equilibrium interest rate \(R = \beta^{-1}\) and all aggregate and cross-section dynamics therefore coincide with those of the plain-vanilla Bewley model

  • how, despite all of this, distinct \((\sigma_i,\beta_i)\) agents hold genuinely different subjective models of their non-financial income

The economy is a pure endowment economy, so there is no physical capital and investment plays no role.

We read Robust Consumption Smoothing and Precautionary Savings as a prerequisite and carry over its notation.

Note

As in Robust Consumption Smoothing and Precautionary Savings, \(w_{t+1}\) is the baseline shock and \(v_{t+1}\) a distortion to its conditional mean, \(\sigma \le 0\) is the robustness parameter, \(\eta_1\) and \(\eta_2\) are the standard deviations of the permanent and transitory endowment shocks, and \(a_t\) denotes net assets.

Let’s begin with some imports.

import numpy as np
import matplotlib.pyplot as plt

84.2. Mapping the Bewley economy into the HST framework#

We specialise the robust model of Robust Consumption Smoothing and Precautionary Savings to \(\lambda = \delta_h = 0\), so there are no habits and no durable goods, and to a pure endowment economy with no physical capital, \(k_t = 0\).

In this case services equal consumption, \(s_t = c_t\).

The only traded security is a one-period risk-free bond, and \(a_t\) denotes the household’s net asset position, so that positive \(a_t\) is wealth.

The endowment process follows the state-space representation

(84.1)#\[\begin{split} \begin{aligned} z_{t+1} &= \check{A}\, z_t + \check{C}\, w_{t+1} \\ y_t &= \check{G}\, z_t \end{aligned} \end{split}\]

with the two-factor specification \(y_t = z_{1t}+z_{2t}\), \(\check A = \mathrm{diag}(1,0)\) and \(\check C = \mathrm{diag}(\eta_1,\eta_2)\).

The household’s augmented state vector is \(x_t = [a_t,\; z_t^\top]^\top\), and the law of motion of Robust Consumption Smoothing and Precautionary Savings specialises to

(84.2)#\[\begin{split} \begin{pmatrix} a_{t+1} \\ z_{t+1} \end{pmatrix} = \underbrace{\begin{pmatrix} R & R\check{G} \\ 0 & \check{A} \end{pmatrix}}_{A} \begin{pmatrix} a_t \\ z_t \end{pmatrix} + \underbrace{\begin{pmatrix} -R \\ 0 \end{pmatrix}}_{B} c_t + \underbrace{\begin{pmatrix} 0 \\ \check{C} \end{pmatrix}}_{C} (w_{t+1} + v_{t+1}) \end{split}\]

The objective is \(\mathbb{E}_0 \sum_{t=0}^\infty \beta^t\bigl[-(c_t - b)^2/2\bigr]\), which is the HST criterion with \(\sigma = 0\) and a constant bliss level \(b_t \equiv b\).

The robust Bellman equation with \(\sigma = 0\) therefore reduces exactly to the LQ problem of The LQ Permanent Income Model, confirming that the HST framework nests the Bewley model.

84.3. The robustness scalar for this economy#

Everything about the endowment process that matters for robustness is summarised by the scalar \(\alpha^2\) derived in Robust Consumption Smoothing and Precautionary Savings.

For the two-factor endowment it is

(84.3)#\[ \alpha^2 = \eta_1^2 + (1-\beta)^2\,\eta_2^2 \]

This is the variance of the consumption innovation \(h\,w_{t+1}\), where \(h = (1-\beta)\check G(I-\beta\check A)^{-1}\check C = \begin{pmatrix}\eta_1 & (1-\beta)\eta_2\end{pmatrix}\).

In the present setting \(\alpha^2\) has a second, equally concrete meaning that we met in Consumption Smoothing with Incomplete and Complete Markets.

Because individual consumption is a random walk with innovation variance \(\alpha^2\), the cross-section variance of consumption among agents of age \(t\) who started from a common initial condition is exactly \(t\,\alpha^2\).

So the same scalar that governs how much a consumer’s robustness concern bites also governs how fast the Bewley cross-section fans out.

84.4. The Bewley observational-equivalence locus#

Applying the observational-equivalence theorem Theorem 83.1 of Robust Consumption Smoothing and Precautionary Savings at the equilibrium interest rate \(R = \beta^{-1}\) gives the Bewley observational-equivalence locus

(84.4)#\[ \hat\beta(\sigma) = \beta + \frac{\sigma\,\alpha^2\,\beta}{1-\beta} \]

For \(\sigma < 0\) we have \(\hat\beta(\sigma) < \beta\).

An agent with the pair \((\sigma, \hat\beta(\sigma))\) is more concerned about model misspecification, because \(\sigma\) is lower, but also more impatient, because \(\hat\beta\) is lower.

The two forces cancel exactly, leaving the consumption decision rule unchanged.

The locus is admissible only down to the breakdown point of Robust Consumption Smoothing and Precautionary Savings,

(84.5)#\[ \underline\sigma = -\frac{(1-\beta)^2}{\alpha^2}, \qquad\text{at which}\qquad \hat\beta(\underline\sigma) = \beta^2 \]

Below \(\underline\sigma\) the individual robust control problem has no solution, so there is no economy to describe.

84.5. Equilibrium with heterogeneous types#

We can now populate the economy with a continuum of types that differ in their concern for robustness.

Proposition 84.1 (A robust Bewley equilibrium)

Let each agent \(i\) in the unit interval be indexed by a robustness parameter \(\sigma_i \in (\underline\sigma, 0]\), distributed according to any distribution \(\Phi\), and let agent \(i\) have discount factor

(84.6)#\[ \beta_i = \hat\beta(\sigma_i) = \beta + \frac{\sigma_i\,\alpha^2\,\beta}{1-\beta} \]

so that every pair \((\sigma_i,\beta_i)\) lies on the locus (84.4).

Then

  1. every agent’s optimal consumption plan is identical to that of the plain-vanilla \((\sigma = 0,\, \beta)\) agent,

  2. the equilibrium gross interest rate is \(R = \beta^{-1}\), independently of \(\Phi\), and

  3. the aggregate and cross-section dynamics coincide with those of the benchmark Bewley economy of Consumption Smoothing with Incomplete and Complete Markets.

Proof. By Theorem 83.1, an agent with parameters \((\sigma_i, \hat\beta(\sigma_i))\) facing gross interest rate \(R = \beta^{-1}\) chooses the same consumption-saving rule as the benchmark \((0,\beta)\) agent.

This holds agent by agent and does not require the \(\sigma_i\) to be equal, which gives part 1.

Since all individual rules coincide with the benchmark rule, the goods-market clearing condition \(\int c_t^i\, di = Y\) and the bond-market condition \(\int a_t^i\, di = 0\) are the benchmark conditions, so they are satisfied at \(R = \beta^{-1}\) for exactly the reason given in Consumption Smoothing with Incomplete and Complete Markets.

Because market clearing never refers to \(\Phi\), the equilibrium interest rate does not either, which gives part 2.

Part 3 follows because aggregate and cross-section objects are integrals of individual paths, and the individual paths are the benchmark paths.

The distribution \(\Phi\) of robustness types is therefore completely unidentified by quantity data.

An econometrician who observes \(\{c_t^i, a_t^i\}\) for every agent and every date cannot tell whether the economy is populated entirely by \(\sigma_i = 0\) agents, entirely by \(\sigma_i\) near \(\underline\sigma\) agents, or by any mixture.

84.6. Where the agents genuinely differ#

Agents on the locus are indistinguishable in what they do but not in what they believe.

An agent with \(\sigma_i < 0\) applies a worst-case distortion \(v_{t+1}^i = K(\sigma_i,\beta_i)\,\mu_{s,t}^i\) to its conditional expectations, while an agent with \(\sigma_i = 0\) takes the approximating model at face value.

From Robust Consumption Smoothing and Precautionary Savings, the worst-case law for agent \(i\)’s marginal utility is

(84.7)#\[ \mu_{s,t+1}^i = \zeta_i\, \mu_{st}^i + \alpha\, w_{t+1}, \qquad \zeta_i = \frac{\beta}{\beta_i} = \left[1 + \frac{\sigma_i\alpha^2}{1-\beta}\right]^{-1} > 1 \]

With \(\lambda = \delta_h = 0\) and a constant bliss point we have \(\mu_{st} = b - c_t\), so agent \(i\)’s worst-case expected consumption path is

(84.8)#\[ \hat{\mathbb{E}}_t\, c_{t+h}^i = b - \zeta_i^{\,h}\,(b - c_t) \]

Under the approximating model, by contrast, consumption is a martingale, \(\mathbb{E}_t c_{t+h} = c_t\) for every \(h\).

Equation (84.8) says that a robust agent below its bliss point expects, under its worst-case model, that consumption will drift away from bliss at the geometric rate \(\zeta_i\).

The more robust the agent, the faster the drift it guards against, and the more precautionary saving it does.

That extra saving is exactly offset by the lower \(\beta_i\), which is why the realized path is the same for all types.

84.7. Computation#

We use the calibration of the preceding lectures.

β = 0.95        # benchmark discount factor, R = 1/β
η1 = 0.15       # std of permanent shock
η2 = 0.30       # std of transitory shock
b = 1.0         # bliss point

R = 1 / β
h = np.array([η1, (1 - β) * η2])       # consumption innovation loadings
α2 = h @ h
α = np.sqrt(α2)
σ_lo = -(1 - β)**2 / α2                # breakdown point, eq:rbew-breakdown

print(f"α^2 = {α2:.6f}   α = {α:.6f}")
print(f"breakdown σ̲ = {σ_lo:.6f},  where β̂ = {β + σ_lo * α2 * β / (1 - β):.6f}"
      f"  (β² = {β**2:.6f})")

α^2 = 0.022725   α = 0.150748
breakdown σ̲ = -0.110011,  where β̂ = 0.902500  (β² = 0.902500)

Next we build a set of types spread across the admissible range and record what distinguishes them.

We reuse the detection error probability of Robust Consumption Smoothing and Precautionary Savings to report how hard each type’s worst-case model would be to detect in a sample of \(T = 40\) quarters.

def worst_case_persistence(σ, β, α2):
    "Worst-case persistence ζ(σ) of marginal utility, eq:rbew-zeta."
    return 1 / (1 + σ * α2 / (1 - β))


def simulate_paths(ζ, α, T, n_paths, seed):
    "Simulate n_paths draws of μ_{t+1} = ζ μ_t + α w_{t+1} from μ_0 = 0."
    rng = np.random.default_rng(seed)
    paths = np.zeros((n_paths, T + 1))
    shocks = rng.standard_normal((n_paths, T))
    for t in range(T):
        paths[:, t + 1] = ζ * paths[:, t] + α * shocks[:, t]
    return paths


def log_likelihood_ratio(paths, ζ, α):
    "Return log p_worst(path) - log p_approx(path)."
    lag, lead = paths[:, :-1], paths[:, 1:]
    return 0.5 * (np.sum(((lead - lag) / α)**2, axis=1)
                  - np.sum(((lead - ζ * lag) / α)**2, axis=1))


def detection_error_probability(ζ, α, T=40, n_paths=10_000, seed=1234):
    "Finite-sample DEP for the approximating and worst-case scalar laws."
    if np.isclose(ζ, 1.0):
        return 0.5
    approx = simulate_paths(1.0, α, T, n_paths, seed)
    worst = simulate_paths(ζ, α, T, n_paths, seed + 1)
    return 0.5 * (np.mean(log_likelihood_ratio(worst, ζ, α) < 0)
                  + np.mean(log_likelihood_ratio(approx, ζ, α) > 0))

σ_types = np.array([0.0, 0.3, 0.6, 0.9]) * σ_lo
β_types = β + σ_types * α2 * β / (1 - β)
ζ_types = worst_case_persistence(σ_types, β, α2)
dep_types = np.array([detection_error_probability(ζ, α) for ζ in ζ_types])

print(f"{'σ_i':>10}{'β_i':>10}{'ζ_i':>10}{'DEP':>8}")
for σ_i, β_i, ζ_i, dep in zip(σ_types, β_types, ζ_types, dep_types):
    print(f"{σ_i:10.4f}{β_i:10.4f}{ζ_i:10.4f}{dep:8.3f}")

       σ_i       β_i       ζ_i     DEP
   -0.0000    0.9500    1.0000   0.500
   -0.0330    0.9357    1.0152   0.426
   -0.0660    0.9215    1.0309   0.333
   -0.0990    0.9072    1.0471   0.233

These four types differ substantially in patience and in the pessimism of their worst-case model.

We now confirm that they nonetheless behave identically.

Each type faces the same shocks and starts from the same initial consumption, and each consumes according to the benchmark rule \(c_{t+1} = c_t + h\,w_{t+1}\).

T = 60
rng = np.random.default_rng(42)
shocks = rng.standard_normal((T, 2))          # common shocks for all types

c_paths = np.zeros((len(σ_types), T + 1))
for i in range(len(σ_types)):
    for t in range(T):
        c_paths[i, t + 1] = c_paths[i, t] + h @ shocks[t]

print("max absolute difference across types:"
      f" {np.abs(c_paths - c_paths[0]).max():.2e}")

max absolute difference across types: 0.00e+00

The paths agree exactly, which is Proposition 84.1 part 1 in action.

The next figure contrasts what the types do with what they believe.

fig, axes = plt.subplots(1, 2, figsize=(11, 4))

for i, σ_i in enumerate(σ_types):
    axes[0].plot(c_paths[i], lw=3 - 0.6 * i, alpha=0.9,
                 label=rf'$\sigma_i={σ_i:.3f}$')
axes[0].set_xlabel('$t$')
axes[0].set_ylabel('$c_t$')
axes[0].set_title('realized consumption')
axes[0].legend()

horizons = np.arange(41)
c_now = c_paths[0, 20]
axes[1].axhline(c_now, color='k', linestyle=':', lw=1.2,
                label='approximating model')
for σ_i, ζ_i in zip(σ_types, ζ_types):
    if σ_i == 0.0:
        continue
    axes[1].plot(horizons, b - ζ_i**horizons * (b - c_now), lw=2,
                 label=rf'worst case, $\sigma_i={σ_i:.3f}$')
axes[1].set_xlabel('horizon $h$')
axes[1].set_ylabel(r'$\hat{\mathbb{E}}_t\,c_{t+h}$')
axes[1].set_title('expected consumption under each belief')
axes[1].legend()

fig.tight_layout()
plt.show()
_images/c3e002aaf89b59e29c95af912ad8112d361d386e1bd00e6aa691d32b6d6b60de.png

Fig. 84.1 Same actions, different beliefs. Left: realized consumption paths for four robustness types facing common shocks; the curves lie exactly on top of one another. Right: each type’s worst-case expected consumption path from a common date, against the flat martingale forecast of the approximating model.#

The left panel of Fig. 84.1 shows four curves drawn on top of one another.

The right panel shows that the same four agents expect very different futures.

The \(\sigma_i = 0\) agent expects consumption to stay where it is.

Every robust agent guards against a future in which consumption drifts away from the bliss point, and the drift is faster the more robust the agent.

Finally we check part 3 of Proposition 84.1, that the cross-section behaves as in the benchmark Bewley economy.

We simulate a large population in which each agent draws its own type and its own shocks, and compare the cross-section variance of consumption to the benchmark prediction \(t\,\alpha^2\).

n_agents, T = 20_000, 40
rng = np.random.default_rng(1234)

# each agent draws a robustness type; types do not affect behaviour
σ_i = rng.uniform(σ_lo, 0.0, size=n_agents)

shocks = rng.standard_normal((n_agents, T, 2))
c = np.zeros((n_agents, T + 1))
c[:, 1:] = np.cumsum(shocks @ h, axis=1)

print(f"{'t':>5}{'cross-section var':>20}{'t·α²':>12}")
for t in [10, 20, 30, 40]:
    print(f"{t:5d}{c[:, t].var():20.5f}{t * α2:12.5f}")

    t   cross-section var        t·α²
   10             0.22744     0.22725
   20             0.45652     0.45450
   30             0.67723     0.68175
   40             0.89604     0.90900

The cross-section variance grows linearly at rate \(\alpha^2\), exactly as in Consumption Smoothing with Incomplete and Complete Markets, and the distribution of robustness types leaves no trace in the data.

84.8. Concluding remarks#

We embedded the single-agent robust permanent income model in a Bewley equilibrium with a continuum of agents who differ in how much they distrust their income process.

Provided each agent’s pair \((\sigma_i,\beta_i)\) lies on the observational-equivalence locus (84.4), every agent chooses the benchmark consumption rule.

The equilibrium interest rate \(R = \beta^{-1}\), the aggregate dynamics, and the linear growth of the cross-section variance of consumption are all inherited unchanged from the plain-vanilla Bewley model of Consumption Smoothing with Incomplete and Complete Markets.

This is a strong non-identification result.

Quantity data pin down the decision rule, but they cannot decompose it into a degree of impatience and a degree of concern about misspecification.

What the agents do not share is their view of the world: each robust type acts as if its consumption were about to drift away from bliss at its own rate \(\zeta_i\).

Two routes lead out of this observational equivalence.

One is to look at asset prices, which do distinguish the \((\sigma,\hat\beta)\) pairs, and which are studied in Robust Permanent Income and Pricing.

The other is to bound the plausible range of \(\sigma\) statistically, using the detection error probabilities of Robust Consumption Smoothing and Precautionary Savings.

84.9. Exercises#

Exercise 84.1

This exercise asks you to carry out the translation from the benchmark Bewley economy into HST notation.

Specialise the robust-control setup to the no-habit, no-capital LQ Bewley environment, so \(\lambda = \delta_h = 0\) and \(k_t = 0\), and let the endowment follow the two-factor model.

  1. Write the household state as \(x_t = [a_t, z_t^\top]^\top\), where \(a_t\) is net assets, and derive the matrices \((A, B, C)\) in (84.2).

  2. Show that when \(\sigma = 0\) the Bellman problem coincides with the LQ permanent-income problem of The LQ Permanent Income Model.

  3. HST define \(\alpha^2 = \nu^\top\nu\) with \(\nu^\top = M_s C\), where \(\mu_{st} = M_s x_t\). Compute \(M_s\) for this economy and verify that this route delivers the same \(\alpha^2\) as (84.3).

Exercise 84.2

This exercise works through the equilibrium logic of Proposition 84.1.

Fix a benchmark pair \((\beta, \sigma = 0)\) with \(R = \beta^{-1}\) and let a unit interval of consumers be indexed by \(i\), with type \(\sigma_i \in (\underline\sigma, 0]\) and discount factor \(\beta_i = \hat\beta(\sigma_i)\) from (84.4).

  1. Use Theorem 83.1 of Robust Consumption Smoothing and Precautionary Savings to show that each type has the same consumption rule as the benchmark \((\beta, 0)\) agent.

  2. Show that goods- and bond-market clearing imply the same equilibrium interest rate \(R = \beta^{-1}\) as in the plain-vanilla Bewley model, whatever the distribution of types.

  3. Explain why agents can be observationally equivalent in quantities while holding different worst-case subjective models.

  4. Suppose instead that agents share a common \(\beta\) but differ in \(\sigma_i\), so that they are not on the locus. Explain why \(R = \beta^{-1}\) would then generally fail to clear the bond market.

Exercise 84.3

This exercise separates quantities from beliefs for two individual agents.

Consider agents \(a\) and \(b\) with \(\sigma^a < \sigma^b \leq 0\), both on the locus (84.4).

  1. Show that the two agents have the same consumption innovation \(h\,w_{t+1}\).

  2. Show that if they start from the same \((a_t, z_t)\) and observe the same shock \(w_{t+1}\), their next-period consumption and assets coincide.

  3. Using (84.8), compute the ratio of the two agents’ worst-case forecasts of \(b - c_{t+h}\) and show that it grows geometrically in \(h\).

  4. Summarise what is and is not identified by data on quantities alone.

Exercise 84.4

This exercise asks how much belief heterogeneity is statistically plausible.

Restrict attention to types whose worst-case model has a detection error probability of at least \(0.2\) in a sample of \(T = 40\).

  1. Find the most robust admissible type \(\sigma^{\min}\) by bisection.

  2. For that type, report \(\beta_i\), \(\zeta_i\), and the horizon \(h\) at which its worst-case forecast of \(b - c_{t+h}\) is twice the approximating model’s forecast.

  3. Repeat part 1 with \(T = 160\) and comment on what a longer sample does to the plausible amount of belief heterogeneity.