---

mystnb:

figure:

caption: The two observational-equivalence loci

caption: |

Two observational-equivalence experiments. Locus I (below $\beta$) holds the

*non-robust* agent fixed at $\beta R=1$ and reports the *robust* twin's

discount factor $\hat\beta(\sigma)$; locus II (above $\beta$) holds the

*robust* agent fixed at $\beta R=1$ and reports the *non-robust* twin's

discount factor $\tilde\beta(\hat\sigma)$.

name: fig-lqcs-oe-loci

---

β_bench = β # benchmark with β̂ R = 1

α2 = σ1**2 + (1 - β)**2 * σ2**2 # two-factor α² (see eq:bew_alpha2)

β_bench = β # benchmark with β R = 1

α2 = σ1**2 + (1 - β)**2 * σ2**2 # two-factor α^2 (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)

# Locus I (eq:obseq / eq:bew_locus): non-robust agent fixed at βR=1;

# report the robust twin's discount factor β̂(σ) < β

β_hat = β_bench + σ_hat_vals * α2 * β_bench / (1 - β_bench)

# Locus II (eq:obsequivn2): robust agent fixed at βR=1;

# report the non-robust twin's discount factor β̃(σ̂) > β

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)')

label=r'locus I: robust twin $\hat\beta(\sigma)<\beta$'

'\n(non-robust agent fixed at $\\beta R=1$)')

ax.plot(-σ_hat_vals, β_tilde, lw=2, color='C0',

label=r'locus II: non-robust twin $\tilde\beta(\hat\sigma)>\beta$'

'\n(robust agent fixed at $\\beta R=1$)')

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()

ax.set_xlabel(r'robustness concern $|\sigma|$')

ax.set_ylabel('discount factor of the equivalent agent')

ax.legend(fontsize=8.5)

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}")

```

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

Both loci pass through the benchmark $\beta$ at $\sigma = 0$ and separate as the robustness concern grows.

The key to reading the figure is that the two loci hold *different* agents fixed, so the discount factor plotted on the vertical axis refers to a different agent on each curve.

Locus I, from {prf:ref}`thm-lqcs-oe1`, holds the **non-robust** agent fixed at the benchmark $(\sigma = 0, \beta)$ with $\beta R = 1$ and reports the discount factor $\hat\beta(\sigma) < \beta$ of the **robust** agent that mimics it.

This is the sense in which HST call a concern for robustness observationally equivalent to a *lower* discount factor: because robustness already makes the agent save more, its discount factor must be lowered to hold the allocation at the benchmark.

Because the non-robust benchmark has $\beta R = 1$, its optimal consumption is a martingale, $\mathbb{E}_t c_{t+1} = c_t$.

The robust twin chooses the identical consumption process, so it too satisfies $\mathbb{E}_t c_{t+1} = c_t$.

The lower $\hat\beta$, which has $\hat\beta R < 1$, would on its own impart a downward drift, but the robust agent's precautionary saving offsets it exactly, leaving expected consumption flat.

Locus II, from {prf:ref}`thm-lqcs-oe2`, instead holds the **robust** agent fixed at $(\hat\sigma, \beta)$ with $\beta R = 1$ and reports the discount factor $\tilde\beta(\hat\sigma) > \beta$ of the **non-robust** agent that mimics it.

Here there is no impatience offset, so the common allocation inherits the robust agent's precautionary *upward* drift, which the non-robust twin reproduces through $\tilde\beta R > 1$.

Locus I, from {prf:ref}`thm-lqcs-oe1`, lies *below* $\beta$: activating robustness looks like an increase in impatience, which imparts a downward drift to expected consumption.

The two experiments encode the *same* economics: a concern for robustness adds precautionary saving that acts like extra patience.

Locus II, from {prf:ref}`thm-lqcs-oe2`, 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.

They differ only in which agent is anchored at $\beta R = 1$, and hence in whether the common saving motive shows up as an exactly-offsetting impatience adjustment (locus I, expected consumption flat) or as an upward drift in expected consumption (locus II).

### A robust LQ Bewley model

We now synthesise the lecture by embedding the Bewley economy of {doc}`lq_bewley_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 shall 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