82. Robust Consumption Smoothing and Precautionary Savings#

82.1. Overview#

This lecture studies a robust version of the LQ permanent income model due to Hansen et al. [1999] and Hansen and Sargent [2008].

It is the third of three lectures on the LQ permanent income model.

It builds on The LQ Permanent Income Model, which develops the standard model, and Consumption Smoothing with Incomplete and Complete Markets, which studies its cross-section and market-structure implications.

A consumer who distrusts his specification of the labor income process engages in a form of precautionary savings.

Our description of the model with concerns about robustness includes

  • how (for quantities) a concern for robustness is observationally equivalent to an increase in impatience

  • how the worst-case model that the consumer uses to shape his decision rule distorts the baseline model’s endowment process toward greater persistence

  • a frequency-domain representation of the effects of concerns about misspecification of the endowment process

  • a detection-error-probability characterization of the amount of model uncertainty

The lecture concludes by combining the Bewley economy of Consumption Smoothing with Incomplete and Complete Markets with the robustness machinery.

Using tools from Hansen and Sargent [2008], we show:

  • how a continuum of consumers \(i\) who use identical decision rules can nevertheless differ in their robustness parameters \(\sigma_i \leq 0\) and their discount factors \(\beta_i\), provided that the pair \((\sigma_i, \beta_i)\) lies on an observational-equivalence locus derived below

  • how every such consumer chooses the same consumption-saving rule as a baseline plain-vanilla \((\sigma = 0, \beta)\) agent with no concerns about misspecification of the endowment process

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

  • how distinct \((\sigma_i, \beta_i)\) agents act as if they have different subjective models of their non-financial income process

We first present the HST model in its general form, which includes physical capital and investment \(i_t\).

When we return to the Bewley economy of Consumption Smoothing with Incomplete and Complete Markets, we specialise to a pure endowment economy with no capital, so investment plays no role there.

Let’s begin with some imports.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm

82.2. A brief review#

We recall the essentials from The LQ Permanent Income Model and Consumption Smoothing with Incomplete and Complete Markets.

A consumer with quadratic utility and discount factor \(\beta\) faces the endowment process

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

The optimal decision rule has a state-space representation in which the state is current consumption \(c_t\) and the exogenous endowment state \(z_t\):

(82.2)#\[\begin{split} \begin{aligned} c_{t+1} &= c_t + (1-\beta)\,\check{G}(I-\beta\check{A})^{-1}\check{C}\, w_{t+1} \\ b_t &= \check{G}(I-\beta\check{A})^{-1} z_t - \frac{1}{1-\beta}\,c_t \\ y_t &= \check{G}\, z_t \\ z_{t+1} &= \check{A}\, z_t + \check{C}\, w_{t+1} \end{aligned} \end{split}\]

We again use the two-factor endowment \(y_t = z_{1t} + z_{2t}\),

(82.3)#\[\begin{split} \begin{pmatrix}z_{1,t+1}\\z_{2,t+1}\end{pmatrix} = \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix} \begin{pmatrix}z_{1t}\\z_{2t}\end{pmatrix} + \begin{pmatrix}\sigma_1 & 0\\0 & \sigma_2\end{pmatrix} \begin{pmatrix}w_{1,t+1}\\w_{2,t+1}\end{pmatrix} \end{split}\]

with \(z_{1t}\) a permanent component and \(z_{2t}\) a purely transitory component.

The following cell fixes the calibration used below.

# Parameters (as in the preceding lectures)
β = 0.95       # discount factor
σ1 = 0.15      # std of permanent shock
σ2 = 0.30      # std of transitory shock

82.3. A robust permanent income model#

82.3.1. Robustness and precautionary savings#

We now study a consumer who distrusts his specification of the stochastic process governing his labor income.

The model is due to Hansen et al. [1999] (HST), who estimated it on US quarterly consumption and investment data.

For a fuller treatment of the HST model and its asset-pricing implications, see Robust Permanent Income and Pricing.

A consumer who fears model misspecification engages in a form of precautionary savings that is distinct from the usual precautionary motive (which requires a convex marginal utility).

Here, the precautionary motive arises because the consumer wants to protect against misspecification of the conditional means of income shocks, and it operates even with quadratic preferences.

HST showed an important observational equivalence result: for quantities \((c_t, i_t)\) alone, a concern for robustness is indistinguishable from an increase in impatience (a decrease in \(\beta\)).

We develop this result carefully below.

82.3.2. The HST model#

HST’s model features a planner with preferences over consumption streams \(\{c_t\}\), mediated through service streams \(\{s_t\}\).

Let \(b\) be a preference shifter (utility bliss point).

The Bellman equation for the robust planner is

(82.4)#\[ -x^\top P x - p = \sup_c \inf_w \Bigl\{-(s-b)^2 + \beta\bigl(\theta (w^*)^\top w^* - \mathbb{E}\,(x^*)^\top P x^* - p\bigr)\Bigr\} \]

subject to the household technology, capital accumulation, endowment dynamics, and the state law:

(82.5)#\[\begin{split} \begin{aligned} s &= (1+\lambda)c - \lambda h \\ h^* &= \delta_h h + (1-\delta_h) c \\ k^* &= \delta_k k + i \\ c + i &= \gamma k + d \\ \begin{pmatrix}d\\b\end{pmatrix} &= U z \\ z^* &= A_{22} z + C_2(\epsilon^* + w^*) \end{aligned} \end{split}\]

Here \(^*\) denotes the next-period value; \(c\) is consumption; \(s\) is the scalar service measure; \(h\) is a habit stock; \(k\) is the capital stock; \(i\) is investment; \(d\) is an endowment/technology shock; \(b\) is a preference shock (bliss-point shifter, distinct from the bond/debt variable \(b_t\) used above); \(\epsilon^* \sim N(0,I)\) is the baseline shock; and \(w^*\) is a distortion to the conditional mean of \(\epsilon^*\) chosen by a minimizing agent.

The penalty parameter \(\theta > 0\) governs the consumer’s concern about robustness.

We use the transformation

\[ \sigma = -\theta^{-1} \leq 0 \]

so \(\sigma = 0\) corresponds to no robustness concern and \(\sigma < 0\) to an increasing concern.

When \(\lambda > 0\) and \(\delta_h \in (0,1)\), the technology (82.5) accommodates habit persistence (positive \(\lambda\)) or durability.

The stock \(h_t\) is a geometric weighted average of current and past consumption.

Equation \(c_t + k_t = Rk_{t-1} + d_t\) with \(R = \delta_k + \gamma\) combines capital accumulation with a linear production technology.

\(R\) is the physical gross return on capital.

Let \(x_t^\top = [h_{t-1},\, k_{t-1},\, z_t^\top]\).

The state transition equations are:

(82.6)#\[ x_{t+1} = A\, x_t + B\, u_t + C(\epsilon_{t+1} + w_{t+1}) \]

where \(u_t = c_t\) and \(w_{t+1}\) is the distortion to the conditional mean of \(\epsilon_{t+1}\).

HST estimated the model on U.S. quarterly data (1970Q1-1996Q3) using nondurables plus services for consumption and durable consumption plus gross private investment for investment.

Key estimates are summarised in the following table (reported in Appendix A of HST):

Parameter

Habit

No Habit

Risk-free rate

0.025

0.025

\(\beta\)

0.997

0.997

\(\delta_h\)

0.682

—

\(\lambda\)

2.443

0

\(\alpha_1\)

0.813

0.900

\(\alpha_2\)

0.189

0.241

\(\phi_1\)

0.998

0.995

\(\phi_2\)

0.704

0.450

\(2 \times \log L\)

779.05

762.55

HST imposed \(\beta R = 1\) and \(\delta_k = 0.975\), so \(\gamma\) is pinned down once \(\beta\) is estimated.

An annual real interest rate of 2.5% corresponds to \(\beta = 0.997\).

82.3.3. Solution when \(\sigma = 0\)#

When \(\sigma = 0\) the objective reduces to

(82.7)#\[ \mathbb{E}_0\sum_{t=0}^{\infty}\beta^t\bigl\{-(s_t - b_t)^2\bigr\} \]

Formulating a Lagrangian and deriving first-order conditions yields:

(82.8)#\[\begin{split} \begin{aligned} \mu_{st} &= b_t - s_t \\ \mu_{ct} &= (1+\lambda)\mu_{st} + (1-\delta_h)\mu_{ht} \\ \mu_{ht} &= \beta \mathbb{E}_t[\delta_h \mu_{h,t+1} - \lambda \mu_{s,t+1}] \\ \mu_{ct} &= \beta R\, \mathbb{E}_t\mu_{c,t+1} \end{aligned} \end{split}\]

Here \(\mu_{st}\) is the marginal valuation of consumption services, which summarises the endogenous state variables \(h_{t-1}\) and \(k_{t-1}\).

Equation (82.8) (last line) implies \(\mathbb{E}_t\mu_{c,t+1} = (\beta R)^{-1}\mu_{ct}\), so \(\mu_{st}\) is a martingale when \(\beta R = 1\):

(82.9)#\[ \mu_{st} = \mu_{s,t-1} + \nu^\top \epsilon_t \]

for some vector \(\nu\).

Solving forward and substituting gives

(82.10)#\[ \mu_{st} = \Psi_1 k_{t-1} + \Psi_2 h_{t-1} + \Psi_3\sum_{j=0}^{\infty} R^{-j} \mathbb{E}_t b_{t+j} + \Psi_4\sum_{j=0}^{\infty} R^{-j} \mathbb{E}_t d_{t+j} \]

where

(82.11)#\[ \Psi_1 = -(1+\lambda)R(1-R^{-2}\beta^{-1})\!\left[\frac{1-R^{-1}\tilde\delta_h}{1-R^{-1}\tilde\delta_h+\lambda(1-\tilde\delta_h)}\right], \quad \Psi_4 = R^{-1}\Psi_1 \]

and \(\tilde\delta_h = (\delta_h + \lambda)/(1+\lambda)\).

In the widely-studied special case \(\lambda = \delta_h = 0\), so \(s_t = c_t\) and \(\mu_{st} = b_t - c_t\), the marginal propensity to consume out of non-human wealth \(Rk_{t-1}\) equals that out of human wealth \(\sum_{j=0}^{\infty}R^{-j}\mathbb{E}_t d_{t+j}\), a well-known feature of the LQ model.

The formula for \(\mu_{st}\) can be written as \(\mu_{st} = M_s x_t\) where \(x_t\) follows (82.6).

It follows that

(82.12)#\[ \nu^\top = M_s C, \qquad \alpha = \sqrt{\nu^\top \nu} = \sqrt{M_s C C^\top M_s^\top} \]

The scalar \(\alpha\) plays a central role in the observational equivalence result below.

82.3.4. Observational equivalence#

HST state an observational-equivalence theorem.

Theorem 82.1 (Observational Equivalence, I)

Fix all parameters except \((\sigma, \beta)\) and suppose \(\beta R = 1\) when \(\sigma = 0\).

There exists \(\underline\sigma < 0\) such that for any \(\sigma \in (\underline\sigma, 0)\), the optimal consumption-investment plan for \((0,\beta)\) is also chosen by a robust decision maker with parameters \((\sigma, \hat\beta(\sigma))\), where

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

and \(\hat\beta(\sigma) < \beta\).

Since \(R > 1\) and \(\alpha^2 > 0\), a more negative \(\sigma\) (stronger robustness concern) lowers \(\hat\beta\).

A robust consumer wants to save more because his alter ego, a utility-minimizing agent, makes future income look worse than the approximating model predicts.

A lower discount factor makes a consumer less patient and therefore reduces saving.

When these two forces are balanced according to (82.13), consumption plans are identical across \((\sigma, \hat\beta(\sigma))\) pairs.

Proof. When \(\beta R = 1\) and \(\sigma = 0\), the marginal utility \(\mu_{st}\) obeys the martingale

(82.14)#\[ \mu_{st} = \mu_{s,t-1} + \alpha\,\tilde\epsilon_t \]

where \(\tilde\epsilon_t\) is scalar IID with mean zero and unit variance.

Activating a concern about robustness (\(\sigma < 0\)) implies the utility minimizing alter ego sets

\[ \tilde w_t = K(\sigma,\hat\beta)\,\mu_{s,t-1} \]

making the worst-case model for \(\mu_{st}\):

(82.15)#\[ \mu_{st} = (1 + \alpha\,K(\sigma,\hat\beta))\,\mu_{s,t-1} + \alpha\,\tilde\epsilon_t \]

For the allocation to remain the same, we require the robust Euler equation \(\hat\beta R\,\hat{\mathbb{E}}_t\mu_{s,t+1} = \mu_{st}\) to hold under the worst-case model, which gives

(82.16)#\[ (\hat\beta R)^{-1} = 1 + \alpha\, K(\sigma,\hat\beta) \]

The minimizing agent’s Bellman equation, a pure forecasting problem, yields

(82.17)#\[ \hat\zeta(\hat\beta) \equiv 1 + \alpha K(\sigma,\hat\beta) = \frac{1}{1 - \sigma\alpha^2 P(\hat\beta)} \]

where \(P(\hat\beta)\) solves the scalar Bellman equation:

(82.18)#\[ -P(\hat\beta) = \frac{\hat\beta - 1 + \sigma\alpha^2 + \sqrt{(\hat\beta-1+\sigma\alpha^2)^2 + 4\sigma\alpha^2}}{-2\sigma\alpha^2} \]

Solving (82.16)-(82.18) for \(\hat\beta\) gives exactly (82.13).

Equation (82.13) is the useful numerical object because it gives a straight-line map from the robustness parameter to the observationally equivalent discount factor.

82.3.5. Precautionary savings interpretation#

The consumer’s concern about model misspecification activates the precautionary savings motive that underlies the observational-equivalence theorem.

A concern about robustness makes the consumer save more.

Decreasing \(\beta\) makes the consumer save less.

The observational-equivalence theorem says that these two forces can be made to offset each other exactly.

In the special case \(\lambda = \delta_h = 0\), \(s_t = c_t\) and the consumption rule is

(82.19)#\[ c_t = (1 - R^{-2}\beta^{-1})\!\left[Rk_{t-1} + \mathbb{E}_t\sum_{j=0}^{\infty}R^{-j}d_{t+j}\right] + \left(\frac{(R\beta)^{-1}-1}{R-1}\right)\!b \]

The marginal propensity to consume out of non-human wealth \(Rk_{t-1}\) equals that out of human wealth \(\mathbb{E}_t\sum R^{-j}d_{t+j}\).

This equal-propensity property is a hallmark of the LQ model and persists when a concern for robustness is present, in contrast to usual precautionary-savings models with convex marginal utility.

Theorem 82.1 says that with \(\sigma < 0\), the observationally equivalent \(\hat\beta\) satisfies \(\hat\beta < \beta\).

If the starting point has \(\beta R = 1\), then \(\hat\beta R < 1\).

For a non-robust consumer with discount factor \(\hat\beta\) at the same interest rate, the Euler equation implies \(\mathbb{E}_t c_{t+1} < c_t\): expected consumption declines over time.

This downward drift is the impatience offset in Theorem 82.1.

It cancels the robust consumer’s precautionary-savings motive, leaving the consumption and investment quantities unchanged.

The upward-drift comparison appears in Theorem 82.2, which asks the reverse observational-equivalence question.

The classical precautionary motive arises because:

\[ u'''(c) > 0 \;\Rightarrow\; \mathbb{E}_t u'(c_{t+1}) > u'(\mathbb{E}_t c_{t+1}) \;\Rightarrow\; \mathbb{E}_t c_{t+1} > c_t \]

This channel requires convexity of marginal utility and is absent with quadratic preferences.

In contrast, the robustness-based precautionary motive operates through distortions of conditional means of shocks, shifting the first moment of the innovation to non-financial income.

82.3.6. Observational equivalence and distorted expectations#

The observational-equivalence result can be interpreted using a Stackelberg multiplier game.

After the minimizing agent has committed to a distortion process \(\{w_{t+1}\}\), the maximizing consumer faces the following worst-case law of motion for the state \(X_t\):

(82.20)#\[\begin{split} \begin{aligned} X_{t+1} &= \bigl(A - BF(\sigma,\hat\beta) + CK(\sigma,\hat\beta)\bigr) X_t + C\tilde\epsilon_{t+1} \\ \begin{pmatrix}b_t\\d_t\end{pmatrix} &= S X_t \end{aligned} \end{split}\]

A robust consumer with concerns about possible misspecification of the approximating model’s stochastic process for non-financial income forms expectations of future income using the distorted transition matrix \(A - BF + CK\) rather than the approximating transition matrix \(A - BF\).

The distorted expectations operator \(\hat{\mathbb{E}}_t\) satisfies

\[ \hat{\mathbb{E}}_t X_{t+j} = (A - BF(\sigma,\hat\beta) + CK(\sigma,\hat\beta))^j X_t \]

Observational equivalence requires that the modified human-wealth formula

\[ \hat\Psi_4 \sum_{j=0}^{\infty} R^{-j}\hat{\mathbb{E}}_t d_{t+j} \]

equals its benchmark counterpart \(\Psi_4 \sum_{j=0}^{\infty} R^{-j} \mathbb{E}_t d_{t+j}\).

This is achieved by a mutual adjustment of the coefficients \(\hat\Psi_j\) through \(\hat\beta\) and the distorted expectation operator \(\hat{\mathbb{E}}_t\) through \(\sigma\).

The worst-case eigenvalue of \(A - BF + CK\) exceeds that of \(A - BF\) in modulus, so the worst-case distortions make the income process more persistent than under the approximating model.

This is the precautionary motive in state-space form: the minimizing agent makes future income look more risky by introducing low-frequency persistence.

82.3.7. Frequency domain interpretation#

The LQ permanent income framework has a natural frequency-domain interpretation.

The consumer’s concave utility makes him dislike high-frequency fluctuations in consumption, which he smooths by adjusting savings.

High-frequency fluctuations are easier to smooth, so the consumer is automatically robust to misspecification of high-frequency features of the income process.

Low-frequency fluctuations are harder to smooth because they are more persistent.

In the frequency-domain notation of HST, the transfer function from shocks \(\epsilon_t\) to the target \(s_t - b_t\) is \(G(\zeta)\), and the frequency decomposition of the \(H_2\) criterion is

\[ H_2 = -\frac{1}{2\pi}\int_{-\pi}^{\pi} \operatorname{trace}\!\bigl[G(\sqrt\beta\, e^{i\omega})^\top\,G(\sqrt\beta\, e^{i\omega})\bigr]\, d\omega \]

The integrand \(G^\top G\) is largest at low frequencies \(\omega \approx 0\), where the consumer’s welfare is most sensitive to income variability.

Recognizing this, the minimizing agent concentrates the worst-case distortions at low frequencies.

The distortion process has spectral density \(W(\zeta)^\top W(\zeta)\) that is concentrated near \(\omega = 0\).

The variance of the worst-case shocks grows as \(|\sigma|\) increases.

82.3.8. Detection error probabilities#

A natural way to discipline the choice of \(\sigma\) (or \(\theta\)) is to ask: how difficult would it be to statistically distinguish the approximating model from the worst-case model?

For a sample of length \(T\), one can use a log-likelihood ratio test to compare the two hypotheses.

The detection error probability (DEP) is the probability of making the wrong decision using the log-likelihood ratio statistic when one does not know which model generated the data.

Specifically:

\[ \text{DEP}(\sigma) = \frac{1}{2}\bigl[\mathbb{P}\{\text{prefer approx.} \mid \text{worst-case is true}\} + \mathbb{P}\{\text{prefer worst-case} \mid \text{approx. is true}\}\bigr] \]

When \(\sigma = 0\) the two models are identical and DEP \(= 0.5\).

As \(|\sigma|\) increases the models diverge and the DEP falls toward zero.

The full DEP calculation requires a specified approximating model, its worst-case counterpart, and the sample length used in the likelihood-ratio experiment.

We compute such a DEP for a robust Bewley model below.

Note

HST suggested that a DEP above 0.2 is “plausible”, meaning the models are still hard enough to distinguish statistically that a concern for robustness is warranted.

Values of \(\sigma\) corresponding to DEP \(\geq 0.2\) define a set of plausible worst-case models.

82.3.9. Robustness of decision rules#

To evaluate whether robust decision rules perform better than the non-robust rule when the data are generated by a distorted model, define the payoff when the decision rule is designed for robustness parameter \(\sigma_2\) and the data are generated by the distorted model associated with \(\sigma_1\):

(82.21)#\[ \pi(\sigma_1;\sigma_2) = -\mathbb{E}_{0,\sigma_1}\sum_{t=0}^{\infty}\beta^t\, x_t^\top H(\sigma_2)^\top H(\sigma_2)\, x_t \]

where the state evolves under decision rule \(F(\sigma_2)\) and worst-case shocks \(K(\sigma_1)\):

(82.22)#\[ x_{t+1} = \bigl(A - BF(\sigma_2) + CK(\sigma_1)\bigr)x_t + C\epsilon_{t+1} \]

For \(\sigma_1 = 0\) (approximating model generates data), the non-robust rule (\(\sigma_2 = 0\)) is optimal by construction.

As \(\sigma_1\) decreases (the data are generated by increasingly distorted models), the payoff of the \(\sigma_2 = 0\) rule deteriorates faster than that of robust rules.

Computing the payoff comparison requires solving the full HST matrix problem for \(F(\sigma_2)\) and \(K(\sigma_1)\).

82.3.10. Another observational equivalence result#

Theorem 82.2 (Observational Equivalence, II)

Fix all parameters except \((\sigma,\beta)\) and consider a consumption-investment allocation for \((\hat\sigma, \hat\beta)\) where \(\hat\beta R = 1\) and \(\hat\sigma < 0\).

Then there exists \(\tilde\beta > \hat\beta\) such that the \((\hat\sigma, \hat\beta)\) allocation also solves the \((0, \tilde\beta)\) problem.

Theorem 82.1 showed that starting from a benchmark with \(\beta R = 1\), activating robustness (\(\sigma < 0\)) is equivalent to reducing \(\beta\).

Theorem 82.2 goes in the opposite direction: it shows that the effects of activating a concern for robustness from a starting point with \(\beta R = 1\) are replicated by increasing \(\beta\) while setting \(\sigma = 0\).

In other words, when \(\beta R = 1\), a concern for robustness operates like an increase in the discount factor, pushing \(\beta R > 1\) and imparting an upward drift to the expected consumption profile.

Proof. With \(\hat\beta R = 1\) and \(\hat\sigma < 0\), the robust Euler equation implies

\[ \hat{\mathbb{E}}_t \mu_{c,t+1} = \mu_{ct} \]

One seeks \(\tilde\beta > \hat\beta\) and \(\sigma = 0\) such that the same allocation solves the non-robust problem with discount factor \(\tilde\beta\).

The key step is to observe that the worst-case distortion \(K(\hat\sigma, \hat\beta)\) introduces a drift in the marginal utility process that is equivalent to the drift produced by raising the discount factor above \(\hat\beta\).

Equating the two drifts and solving the scalar Bellman equation for \(K\) yields

(82.23)#\[ \tilde\beta(\hat\sigma) = \frac{\hat\beta(1+\hat\beta)}{2(1+\hat\sigma\alpha^2)} \left[1 + \sqrt{1 - 4\hat\beta\,\frac{1+\hat\sigma\alpha^2}{(1+\hat\beta)^2}}\right] \]

The solution satisfies \(\tilde\beta > \hat\beta\) when \(\hat\sigma < 0\).

The map (82.23) is a closed form, so we can plot it directly.

The next figure compares the two observational-equivalence loci for the two-factor calibration, using \(\alpha^2 = \sigma_1^2 + (1-\beta)^2\sigma_2^2\) (derived below in (82.27)).

We start from a benchmark with \(\hat\beta R = 1\), so \(\hat\beta = \beta\).

β_bench = β                                 # benchmark with β̂ R = 1
α2 = σ1**2 + (1 - β)**2 * σ2**2             # two-factor α² (see eq:bew_alpha2)

σ_hat_vals = np.linspace(0.0, -0.16, 60)

# Locus II (eq:obsequivn2): robustness ⟺ an *increase* in β (σ = 0)
disc = 1 - 4 * β_bench * (1 + σ_hat_vals * α2) / (1 + β_bench)**2
β_tilde = (β_bench * (1 + β_bench)) / (2 * (1 + σ_hat_vals * α2)) \
          * (1 + np.sqrt(disc))

# Locus I (eq:obseq / eq:bew_locus): robustness ⟺ a *decrease* in β (σ = 0)
β_hat = β_bench + σ_hat_vals * α2 * β_bench / (1 - β_bench)

fig, ax = plt.subplots()
ax.plot(-σ_hat_vals, β_tilde, lw=2, color='C0',
        label=r'locus II: $\tilde\beta(\hat\sigma)$ (upward drift)')
ax.plot(-σ_hat_vals, β_hat, lw=2, color='C3',
        label=r'locus I: $\hat\beta(\sigma)$ (downward drift)')
ax.axhline(β_bench, color='k', linestyle=':', lw=1,
           label=r'benchmark $\beta$ ($\beta R = 1$)')
ax.set_xlabel(r'robustness concern $-\hat\sigma$')
ax.set_ylabel('observationally equivalent discount factor')
ax.legend()
plt.show()

print(f"at σ̂ = {σ_hat_vals[-1]:.3f}:  β̃ = {β_tilde[-1]:.4f} > β = {β_bench}")
print(f"                       β̂ = {β_hat[-1]:.4f} < β = {β_bench}")
_images/9321247263c70d4ad0c89d4b4fd317f8acac9ac5190e117b5a214cc89ae074a0.png

Fig. 82.1 The two observational-equivalence loci#

at σ̂ = -0.160:  β̃ = 0.9905 > β = 0.95
                       β̂ = 0.8809 < β = 0.95

The two loci pass through the benchmark \(\beta\) at \(\hat\sigma = 0\) and separate as the robustness concern grows.

Locus I, from Theorem 82.1, lies below \(\beta\): activating robustness looks like an increase in impatience, which imparts a downward drift to expected consumption.

Locus II, from Theorem 82.2, lies above \(\beta\): the same robustness concern, viewed from a benchmark with \(\beta R = 1\), looks like an increase in patience, which imparts an upward drift.

82.3.11. A robust LQ Bewley model#

We now synthesise the lecture by embedding the Bewley economy of Consumption Smoothing with Incomplete and Complete Markets into the HST framework and applying the observational-equivalence theorem.

In this way, we construct a family of robust Bewley economies, parameterised by a robustness level \(\sigma \leq 0\), whose equilibrium quantities are identical to those of the plain vanilla Bewley model.

We first map the Bewley economy into HST notation, specialising the robust model to \(\lambda = \delta_h = 0\) (no habits, no durable goods) and to a pure endowment economy (no physical capital, \(k_t = 0\)).

In this case:

Services equal consumption: \(s_t = c_t\).

The only traded security is the one-period risk-free bond, and we write the household’s net asset position as \(a_t=-b_t\) so that positive \(a_t\) denotes wealth rather than debt.

The endowment process follows the state-space representation (82.1).

The household’s augmented state vector is \(x_t = [a_t,\; z_t^\top]^\top\), and the law of motion (82.6) specialises to

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

The objective is \(\mathbb{E}_0 \sum_{t=0}^\infty \beta^t [-(c_t - \gamma)^2/2]\), which is the HST criterion (82.7) with \(\sigma = 0\) and \(b_t \equiv \gamma\) (a fixed bliss level).

The robust Bellman equation (82.4) 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.

We next compute the robustness parameter \(\alpha^2\).

From the \((c_t,z_t)\) representation (82.2), the consumption innovation is

(82.25)#\[ c_{t+1} - c_t = h\, w_{t+1}, \qquad h = (1-\beta)\,\check{G}(I-\beta\check{A})^{-1}\check{C} \]

The vector \(h\) plays the role of \(\nu^\top = M_s C\) in the HST scalar \(\alpha\) formula (82.12).

Consequently,

(82.26)#\[ \alpha^2 = h h^\top = (1-\beta)^2\, \check{G}(I-\beta\check{A})^{-1}\check{C}\check{C}^\top(I-\beta\check{A}^\top)^{-1}\check{G}^\top \]

For the two-factor model (82.3) with \(\check{A} = \mathrm{diag}(1,0)\) and \(\check{C} = \mathrm{diag}(\sigma_1,\sigma_2)\) this simplifies to

(82.27)#\[ \alpha^2 = \sigma_1^2 + (1-\beta)^2\,\sigma_2^2 \]

The permanent shock variance \(\sigma_1^2\) enters with coefficient 1 because a unit permanent shock is fully capitalised into consumption.

The transitory shock variance \(\sigma_2^2\) enters with the small coefficient \((1-\beta)^2\) because only its annuity value is consumed.

Applying Theorem 82.1 (82.13) with equilibrium interest rate \(R = \beta_0^{-1}\) and \(\alpha^2\) from (82.27) gives the Bewley observational equivalence locus:

(82.28)#\[ \hat\beta(\sigma) = \beta_0 + \frac{\sigma\,\alpha^2\,\beta_0}{1-\beta_0} \]

For \(\sigma < 0\), we have \(\hat\beta(\sigma) < \beta_0\).

An agent with the pair \((\sigma, \hat\beta(\sigma))\) on this locus is more concerned about model misspecification (lower \(\sigma\)) but also more impatient (lower \(\hat\beta\)); the two forces cancel exactly, leaving the consumption decision rule unchanged.

These ingredients combine into a robust Bewley equilibrium.

Proposition 82.1

Suppose all agents in the Bewley economy share a common pair \((\sigma, \hat\beta(\sigma))\) lying on the locus (82.28), with \(R = \beta_0^{-1}\).

Then every agent’s optimal consumption plan is identical to that of the plain vanilla \((\sigma = 0,\, \beta_0)\) economy, and the equilibrium interest rate remains \(R = \beta_0^{-1}\).

Proof. By Theorem 82.1, each agent’s consumption-saving rule is identical to the benchmark.

The goods-market clearing condition \(\int c_t^i\, di = Y\) is therefore satisfied at \(R = \beta_0^{-1}\) for the same reason as in the benchmark Bewley economy.

82.3.11.1. Heterogeneous \((\beta_i, \sigma_i)\) preferences#

A richer extension populates the economy with a continuum of types, each indexed by a robustness parameter \(\sigma_i \in [\underline\sigma, 0]\), with discount factor

(82.29)#\[ \beta_i = \hat\beta(\sigma_i) = \beta_0 + \frac{\sigma_i\,\alpha^2\,\beta_0}{1-\beta_0} \]

Since all pairs \((\sigma_i, \beta_i)\) lie on (82.28), every agent adopts the same consumption rule as the benchmark.

Aggregate dynamics are unchanged because the cross-section mean of consumption equals \(Y\) and the cross-section variance grows at rate \(\alpha^2\) per period.

The equilibrium interest rate is unchanged: \(R = \beta_0^{-1}\).

Agents are observationally indistinguishable to an outside econometrician because data on \((c_t^i, a_t^i)\) cannot reveal whether agent \(i\) has \(\sigma_i = 0\) or \(\sigma_i < 0\).

Agents differ in their internal model because an agent with \(\sigma_i < 0\) applies a worst-case distortion \(w_{t+1}^i = K(\sigma_i, \beta_i)\,\mu_{s,t}^i\) to her conditional expectations, while an agent with \(\sigma_i = 0\) takes the approximating model at face value.

This sets the stage for a Bewley model with heterogeneous ambiguity aversion: although every agent acts identically in terms of observable choices, they hold different subjective models of the income process and have different attitudes toward model uncertainty.

82.3.11.2. Computation#

# Bewley parameters
β0_bew = β       # 0.95
σ1_bew = σ1      # 0.15
σ2_bew = σ2      # 0.30
R_bew = 1.0 / β0_bew

# Two-factor Bewley α^2
α2_bew = σ1_bew**2 + (1 - β0_bew)**2 * σ2_bew**2

print(f"α^2  (Bewley, two-factor)        = {α2_bew:.6f}")
print(f"  permanent component  σ1^2     = {σ1_bew**2:.6f}  "
      f"({100*σ1_bew**2/α2_bew:.1f} % of α^2)")
print(f"  transitory component (1-β)^2σ2^2= {(1-β0_bew)**2*σ2_bew**2:.6f}  "
      f"({100*(1-β0_bew)**2*σ2_bew**2/α2_bew:.1f} % of α^2)")

α^2  (Bewley, two-factor)        = 0.022725
  permanent component  σ1^2     = 0.022500  (99.0 % of α^2)
  transitory component (1-β)^2σ2^2= 0.000225  (1.0 % of α^2)

The calculation shows why permanent shocks dominate \(\alpha^2\) in this calibration.

We now solve the scalar robust forecasting problem attached to this \(\alpha^2\).

The solution selects the Bellman-equation root that satisfies the observational-equivalence Euler equation.

def robust_scalar_solution(σ, β0, α2):
    """
    Solve the scalar robust marginal-utility problem on the
    observational-equivalence locus.
    """
    α = np.sqrt(α2)
    R = 1.0 / β0

    if np.isclose(σ, 0.0):
        return β0, np.nan, 1.0, 0.0

    β_hat = β0 + σ * α2 * β0 / (1 - β0)
    disc = (β_hat - 1 + σ * α2)**2 + 4 * σ * α2
    root_disc = np.sqrt(max(disc, 0.0))
    target_ζ = 1 / (β_hat * R)

    candidates = []
    for sign in (1.0, -1.0):
        P = (β_hat - 1 + σ * α2 + sign * root_disc) / (-2 * σ * α2)
        ζ = 1 / (1 - σ * α2 * P)
        K = (ζ - 1) / α
        candidates.append((abs(ζ - target_ζ), P, ζ, K))

    _, P, ζ, K = min(candidates, key=lambda x: x[0])
    return β_hat, P, ζ, K


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


def simulate_scalar_paths(ζ, α, T, n_paths, seed):
    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 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_paths = simulate_scalar_paths(1.0, α, T, n_paths, seed)
    worst_paths = simulate_scalar_paths(ζ, α, T, n_paths, seed + 1)

    llr_approx = log_likelihood_ratio(approx_paths, ζ, α)
    llr_worst = log_likelihood_ratio(worst_paths, ζ, α)

    return 0.5 * (np.mean(llr_worst < 0) + np.mean(llr_approx > 0))

The next figure reports worst-case dynamics and model-detection probabilities implied by this solved scalar problem.

α_bew = np.sqrt(α2_bew)
β_min = 0.88
σ_min = (β_min - β0_bew) * (1 - β0_bew) / (α2_bew * β0_bew)
σ_vals = np.linspace(0.0, σ_min, 31)

solutions = np.array([robust_scalar_solution(σ, β0_bew, α2_bew) for σ in σ_vals])
β_hat_vals = solutions[:, 0]
ζ_vals = solutions[:, 2]
K_vals = solutions[:, 3]
dep_vals = np.array([
    detection_error_probability(ζ, α_bew)
    for ζ in ζ_vals
])

fig, axes = plt.subplots(1, 2, figsize=(11.2, 4.1))

horizons = np.arange(31)
for σ in [0.0, σ_vals[10], σ_vals[20]]:
    β_hat, P, ζ, K = robust_scalar_solution(σ, β0_bew, α2_bew)
    label = rf'$\sigma={σ:.3f}$, $\zeta={ζ:.3f}$'
    axes[0].plot(horizons, ζ**horizons, lw=2, label=label)

axes[0].set_xlabel('horizon')
axes[0].set_ylabel(r'response of $\mu_{s,t+h}$')
axes[0].set_title('worst-case impulse response')
axes[0].legend(fontsize=8.5)

axes[1].plot(-σ_vals, dep_vals, lw=2, color='C0')
axes[1].axhline(0.2, color='C3', linestyle='--', lw=1.2,
                label='DEP = 0.2')
axes[1].set_xlabel(r'robustness concern $-\sigma$')
axes[1].set_ylabel('detection error probability')
axes[1].set_ylim(0.0, 0.52)
axes[1].set_title('finite-sample detectability')
axes[1].legend(fontsize=8.5)

fig.tight_layout()
plt.show()
_images/f370302dd3af573beaca9e5cd5b8a6c7c710902b340b66b177e9a773a9401a4e.png

Fig. 82.2 Solved robust scalar model#

The left panel shows that the solved worst-case law makes marginal utility more persistent as \(\sigma\) becomes more negative.

The right panel computes the DEP from the exact likelihood ratio between the approximating scalar law \(\mu_{t+1}=\mu_t+\alpha\epsilon_{t+1}\) and the solved worst-case law \(\mu_{t+1}=\zeta(\sigma)\mu_t+\alpha\epsilon_{t+1}\).

82.3.12. Concluding remarks#

We close with a summary of the key messages from all three lectures.

The LQ permanent income model, a rational-expectations version of Friedman’s permanent income hypothesis, has two complementary state-space representations:

  1. \((b_t, z_t)\) representation: emphasises that the consumer’s optimal borrowing is history dependent and cointegrated with consumption.

  2. \((c_t, z_t)\) representation: emphasises that consumption is a martingale (random walk) and that assets \(b_t\) are encoded in consumption, so the impulse response function of consumption is “box-shaped”: a permanent shift in the level.

We embedded this single-agent model in a Bewley equilibrium with a continuum of ex-post heterogeneous consumers.

The equilibrium gross interest rate \(R = \beta^{-1}\) is supported by constant average consumption, though the cross-section variance of consumption grows linearly with age.

A complete-markets version of the same model achieves full risk sharing and a time-invariant consumption distribution at the cost of more complex financial arrangements (Arrow securities).

A concern for model misspecification, parameterised by \(\sigma = -\theta^{-1} \leq 0\), alters the permanent income model.

A concern for robustness generates a precautionary savings motive even under quadratic preferences by distorting the conditional means of income shocks.

The distorted worst-case model makes the income process more persistent, shifting power toward low frequencies where the permanent income consumer is most vulnerable.

The observational equivalence theorem Theorem 82.1 shows that for quantities \((c_t, i_t)\) alone, a concern for robustness is indistinguishable from a reduction in \(\beta\).

The reverse theorem Theorem 82.2 shows that, starting from \(\beta R = 1\), robustness is observationally equivalent to an increase in \(\beta\), which imparts an upward drift to expected consumption.

Detection error probabilities provide a principled way to calibrate \(\sigma\): choose \(|\sigma|\) small enough that the approximating and worst-case models remain difficult to distinguish statistically.

The observationally equivalent \((\sigma, \hat\beta)\) pairs do have different implications for asset prices, a point explored further by HST in the asset-pricing context.

The robust Bewley economy shows how agents can have the same consumption decision rule and support the same equilibrium interest rate \(R = \beta_0^{-1}\) while differing in their worst-case subjective income dynamics.

82.4. Exercises#

Exercise 82.1

We translate from the benchmark Bewley economy to HST notation.

Specialise the robust-control setup to the no-habit, no-capital LQ Bewley environment (\(\lambda = \delta_h = 0\), \(k_t = 0\)), and let the endowment process be the two-factor model in (82.3).

  1. Write the household state as \(x_t = [a_t, z_t^\top]^\top\), where \(a_t=-b_t\) is net assets, and derive matrices \((A, B, C)\) for the law of motion (82.6).

  2. Show that when \(\sigma = 0\), the Bellman problem coincides with the LQ permanent-income problem.

  3. Derive \(\alpha^2\) and verify

\[ \alpha^2 = \sigma_1^2 + (1-\beta)^2\sigma_2^2. \]

Interpret economically why the permanent and transitory components enter with different weights.

Exercise 82.2

This exercise studies a continuum of robust but observationally equivalent Bewley consumers.

Fix a benchmark pair \((\beta_0, \sigma = 0)\) with \(R = \beta_0^{-1}\) and define

\[ \beta(\sigma) = \beta_0 + \frac{\sigma\alpha^2\beta_0}{1-\beta_0}, \qquad \sigma \in [-\bar\sigma, 0]. \]

Suppose a unit interval of consumers is indexed by \(i\) with type \(\sigma_i \in [-\bar\sigma, 0]\) and discount factor \(\beta_i = \beta(\sigma_i)\).

  1. Use Theorem 82.1 to show that each type has the same consumption rule as the benchmark \((\beta_0, 0)\) agent.

  2. Prove that aggregate consumption and bond-market clearing imply the same equilibrium interest rate \(R = \beta_0^{-1}\) as in the plain-vanilla Bewley model.

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

Exercise 82.3

This exercise separates quantities from beliefs without introducing an additional calibration.

Consider two agents \(a\) and \(b\) in the robust Bewley economy with \(\sigma^a < \sigma^b \leq 0\) and \(\beta^j = \beta_0 + \sigma^j\alpha^2\beta_0/(1-\beta_0)\) for \(j \in \{a,b\}\).

  1. Use (82.25) and (82.28) to show that the two agents have the same consumption innovation \(h\epsilon_{t+1}\).

  2. Show that if the two agents start from the same \((a_t,z_t)\) and observe the same shock \(\epsilon_{t+1}\), then their next-period choices of consumption and assets coincide.

  3. Explain why the two agents can nevertheless disagree about the worst-case conditional mean of \(\epsilon_{t+1}\).

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