## wpiea2021055-print-pdf

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### Model setup and benchmark
- State vector: xt = (bt; Bt; yt−1; εt), with bt household debt, Bt aggregate debt, yt−1 = (yTt−1; yNt−1), εt = (εTt; εNt).
- Control vector: wt = (cTt; cNt; bt+1).
- Budget constraint (Lemma 1):
  - b t+1 + c T t + p N t c N t = b t (1 + r) + y T t + p N t y N t  (equation (1))
- Collateral constraint:
  - b t+1 ≤ (ϕ N p N t y N t + ϕ T y T t)  (equation (2))
- Aggregate bond law of motion and prices:
  - B t+1 = Ξ(B t; y t)  (equation (3))
  - p N t given ℱ t  (equation (4))
- Endowment process:
  - log(y t) = ρ log(y t−1) + ε t; ε t+1 ~ N(0; V); i.i.d.  (equation (5))
- Compact dynamics:
  - x t+1 = g(x t; w t; ε t+1); ε t+1 ~ N(0; V)  (equations (6)–(7))
- Given policy functions cTt(xt), cNt(xt), bt+1(xt) the benchmark transition density is f(xt+1 | xt); misspecified dynamics denoted ~f(xt+1 | xt).

### Robustness framework and distance measure
- Distance between benchmark and misspecified transition densities:
  - I(f; ~f) = E~f [ log( ~f(xt+1 | xt) / f(xt+1 | xt) ) | xt ]  (equation (8))
- Robust decision criterion:
  - Household solves a max–min problem choosing policies while an “evil” agent chooses ~f to minimize expected utility subject to a distance constraint.

### Robust optimization (minimax) and penalty representation
- Robust household problem (Proposition 1):
  - max_{cNt, cTt, bt+1} min_{~f} E0,~f [ Σ_{t=0}^∞ β^t u(c_t) ]  (equation (9))
  - c_t = [ ω (cTt)^{−θ} + (1−ω) (cNt)^{−θ} ]^{−1/θ}; θ > −1  (equation (10))
  - Dynamics: (b t+1; B t+1; y t+1) = g((b t; B t; y t); (c N t; c T t; b t+1); ε t+1)  (equation (11))
  - ε t+1 i.i.d ~ N(0; V)  (equation (12))
  - Distortion budget:
    - E0,~f [ Σ_{t=0}^∞ β^{t+1} I(f; ~f) ] ≤ θ̄  (equation (13))
    - 0 ≤ θ̄ ≤ Θ̄  (equation (15))
    - I(f; ~f) defined in (14)
- Multiplier (penalty) representation:
  - max_{cNt,cTt,bt+1} min_{~f} E0,~f [ Σ_{t=0}^∞ β^t u(c_t) ] + ξ E0,~f [ Σ_{t=0}^∞ β^{t+1} I(~f; f) ]  (equation (16))
  - ξ ∈ (ξ_L; +∞); lower ξ → more robustness (larger distortions); direct mapping between ξ and θ̄.

### Martingale representation and recursive Bellman form
- Define martingale M t = ~f(xt) / f(xt) with E f [ M t+1 | x t ] = M t and increment m t+1 = M t+1 / M t with E f [ m t+1 | x t ] = 1.
- Expectations under ~f via f:
  - E~f[y t | x0] = E f [ M t y t | x0 ]  (equation (17))
  - E~f[y t+1 | x t] = E f [ m t+1 y t+1 | x t ]  (equation (18))
- Recursive Bellman for robust household (Proposition 2):
  - V(b t; B t; y t) = max_{cNt,cTt,bt+1} c_t^{1−γ} / (1−γ) − β ξ log( E_{y t+1 | y t} [ e^{−V(b t+1; B t+1; y t+1)/ξ} | b t; B t; y t ] )  (equation (19))
  - c_t as in (20); ξ ∈ (ξ_L; +∞)  (equations (20), (24))
- Non-robust Bellman for comparison:
  - V(B t; y t; b t) = max_{cTt,cNt,bt+1} c(c_t)^{1−γ} / (1−γ) + β E_{y t+1 | y t} [ V(b t+1; B t+1; y t+1) | b t; B t; y t ]  (equation (25))

### Recursive competitive equilibrium (household) and markets
- Equilibrium objects: pN(Bt; yt), Ξ(B t; y t), decision rules b(B t; y t; b t), cT(...), cN(...), V(...).
- Market clearing:
  - y N t = c N(B t; y t; b t)  (equation (26))
  - Ξ(B t; y t) + c T(B t; y t; b t) = y T t + B t (1 + r)  (equation (27))
- Perceived law of motion equals actual:
  - B t+1 = Ξ(B t; y t) = b(B t; y t; B t)  (equation (28))
- Price of non-tradables:
  - p N(B t; y t) = ( (1−ω)/ω * ( y T t + B t (1 + r) − Ξ(B t; y t) ) / y N t )^{θ+1}  (equation (29))

### Planner problem and constrained-efficient equilibrium
- Planner chooses debt B t+1 and tradable consumption cTt; non-tradables consumed competitively with c N = y N t.
- Planner’s robust Bellman:
  - V(B t; y t) = max_{b t+1, cTt} c(cTt; yN(B t; y t))^{1−γ} / (1−γ) − β ξ log( E_{y t+1 | y t} [ e^{−V(b t+1; y t+1)/ξ} | b t; B t; y t ] )  (equation (30))
  - c(cTt; yN t) = [ ω (cTt)^{−θ} + (1−ω) (y N t)^{−θ} ]^{−1/θ}; θ > −1  (equation (31))
- Resource and collateral constraints:
  - B(B t; y t) + c T(B t; y t) = B t (1 + r) + y T t  (equation (32))
  - B(B t; y t) ≤ (ϕ N p N(B t; y t) y N t + ϕ T y T t)  (equation (33))
  - c t ≥ 0  (equation (34))
- Price internalization:
  - p N(B t; y t) = ( (1−ω)/ω * ( y T t + B t (1 + r) − B t+1 ) / y N t )^{θ+1}  (equation (35))
- Non-robust planner Bellman:
  - V(B t; y t) = max_{B t+1, cTt} c^{1−γ}_t / (1−γ) + β E_{y t+1 | y t} [ V(B t+1; y t+1) ]  (equation (36))

### Key mechanism: pecuniary externality and amplification
- Pecuniary externality: household does not internalize effect of debt on pN via tradable consumption; planner internalizes it when collateral depends on debt.
- Amplification with occasionally binding collateral constraint:
  - Negative shock → lower yTt and/or yNt → collateral binds → borrowing falls → tradable consumption falls → pN falls → constraint tightens → borrowing falls further.
  - When constraint binds: borrowing decreases in bad times (capital inflows decrease) and current account increases.
  - When constraint does not bind: borrowing increases in bad times (capital inflows increase) and current account decreases (consumption smoothing).
- Policy implication:
  - Macroprudential policies can reduce probability and magnitude of sudden stops by affecting incidence and severity of constraint binding.

### Interpretation of robustness and planner types
- Two planner types when views differ:
  - Paternalistic planner: maximizes welfare according to her view (may fear misspecification).
  - Non-paternalistic planner: implements allocation from household’s perspective.
- Welfare measurement depends on which model is used (private agents’ worst-case ~f or benchmark f).
- Competing effects when planner and household disagree:
  - Planner may reduce welfare by using a different model.
  - Planner may improve welfare by correcting overborrowing externality.
- Welfare evaluated via compensating variation µ:
  - µ defined as constant proportional consumption subsidy making agents indifferent between DE and SP.
  - Equations governing µ differ depending on whether benchmark or worst-case probabilities are used (equations (40), (41)); homothetic simplification when no misspecification (equation (42)) does not apply under robustness.

### Macroprudential tax implementation conditions
- Taxes must satisfy household’s and planner’s Euler equations when collateral constraint not binding.
- Non-robust Euler form (Bianchi (2011)): equations (44) and (45).
- Robust Euler equations incorporate worst-case martingale increments m_{t+1} (equations (46)–(48)).
- Four cases for decentralizing planner allocation with tax ϕ:
  1. Neither household nor planner robust:
     - ϕ = E_{y_{t+1}|y_t}[µ_{t+1} Δ_{t+1}/Δ_{t+1}] / E_{y_{t+1}|y_t}[u_{T,t+1}]  (equation (49))
  2. Planner robust, household not:
     - 1 + ϕ = E_{y_{t+1}|y_t}[m_{t+1}(u_{T,t+1} + µ_{t+1} Δ_{t+1}/Δ_{t+1})] / E_{y_{t+1}|y_t}[u_{T,t+1}]  (equation (50))
  3. Both planner and household robust:
     - ϕ = E_{y_{t+1}|y_t}[m_{t+1} µ_{t+1} Δ_{t+1}/Δ_{t+1}] / E_{y_{t+1}|y_t}[m_{t+1} u_{T,t+1}]  (equation (51))
  4. Household robust, planner not:
     - 1 + ϕ = E_{y_{t+1}|y_t}[u_{T,t+1} + µ_{t+1} Δ_{t+1}/Δ_{t+1}] / E_{y_{t+1}|y_t}[m_{t+1} u_{T,t+1}]  (equation (52))
- Note: If household robust but planner not, subsidy (ϕ negative) may be required; decentralization requires accounting for distribution agents use in expectations.

### Calibration highlights (Section 4)
- Calibration targets:
  - Sudden stop probability: 13%.
  - Average debt-to-GDP ratio: 30%.
- Parameters:
  - Risk aversion γ = 2 baseline; alternative γ = 4 considered.
  - Robustness parameter θ = 0.35 baseline; alternative θ = 0.15 considered.
  - Baseline θ = 0.35 targets detection error probability of 20%.
  - Discount factor β = 0.8 (to match data moments; Bianchi (2011) used 0.91).
  - Intratemporal elasticity of substitution 1/(1 + ξ) = 0.83.
  - Share of tradables in CES χ = 0.31.
  - World risk-free rate r = 0.04.
  - Share of collateralized tradable ϕ_T = 0.31.
- Shock discretization:
  - Tradable and non-tradable shocks discretized with four grid points each (16 states), using Bianchi (2011) grids and transition probabilities as the “benchmark” distribution f.

### Debt policy quantitative findings (Figure 2 summary)
- Debt policies for state one standard deviation below mean in both endowments:
  - Both household and planner show U-shaped debt policy.
  - Binding region smaller for robust agents.
  - In non-binding region, planner accumulates larger bond holdings than household as precaution.
  - Robust agents save more than non-robust counterparts across non-binding region.
  - A robust household eventually saves more than social planner far from binding region.
  - Closer to binding area, a non-robust planner still saves more than household to internalize externality.
- Tax regions:
  - Green area: region where tax needed to implement robust planner’s allocation when households robust (θ = 0.35).
  - Grey area: tax region when both planner and household non-robust.
  - Robust planner taxes robust household over slightly narrower debt region (green) than non-robust planner taxes non-robust household (grey).
  - Cross-case (robust planner vs non-robust household) differences imply potentially much larger taxes and larger tax region.

### Simulation design (Section 4.2)
- Simulations:
  - 80,000 period allocations per experiment.
  - Baseline shocks drawn from benchmark distribution; worst-case probabilities used when analyzing welfare from household perspective.
  - Policies simulated: non-robust household, non-robust planner, robust household, robust planner.
  - Robust policies use θ = 0.15 and θ = 0.35.
  - Calibration set to produce 13% crisis probability under non-robust household and benchmark shocks.
- Sudden stop definition:
  - Constraint binds and increase in net capital outflows exceeds one standard deviation of net capital outflows in long-run distribution of bond holdings under decentralized non-robust allocation.
- Crisis severity:
  - Median crisis identified by median current account reversal in decentralized equilibrium; debt two periods before median crisis used as initial condition for three-period response comparisons.

### Simulation key findings: occurrence and severity of crises
- Planner internalization reduces sudden-stop occurrence:
  - From 5% to 0.3% under Bianchi’s calibration.
  - From 13% to 1.5% for the preferred calibration.
- Severity reductions (preferred calibration, comparisons relative to non-robust household baseline examples):
  - Reduction in consumption in median crisis: from -8.810 to -3.717 (Table 4 entries).
  - Depreciation of real exchange rate: from -20.055 to -7.930 (Table 4 examples).
  - Current account reversal: from 7.940 to 2.246 (Table 4 examples).
- Robust households:
  - Can reduce probability of sudden stop to 9% in some parameterizations.
  - Halve reduction in consumption and RER depreciation in median crises relative to non-robust counterparts.
- Robust planners further reduce probability and severity (e.g., crisis probability to 0.7% in some parameterizations).

### Distributional implications of planner vs private allocations
- Planner’s ergodic distribution assigns higher probability to lower debt levels (higher bond holdings) compared to private agents.
- On average both planner and private agents tend to hold the same level of debt: 30%.
- Planner reduces exposure to very high debt levels that risk triggering collateral constraint under negative shocks.

### Macroprudential tax simulation results (Section 4.3)
- Average taxes (mean over 80,000 periods, baseline shocks):
  - Bianchi (2011) replication (non-robust HH v non-robust SP): 7.274
  - Higher prob crisis (θ = 0.35) cases:
    - non-robust HH v non-robust SP: 15.971
    - non-robust HH v robust SP: 20.113
    - robust HH v robust SP: 16.999
    - robust HH v non-robust SP: -1.637
  - Higher prob crisis (θ = 0.15) cases:
    - non-robust HH v non-robust SP: 15.971
    - non-robust HH v robust SP: 22.260
    - robust HH v robust SP: 15.719
    - robust HH v non-robust SP: -6.247
- Probability tax is implemented (percent of periods):
  - Bianchi (2011) replication (non-robust HH v non-robust SP): 80.111
  - Higher prob crisis (θ = 0.35) examples:
    - non-robust HH v non-robust SP: 77.705
    - non-robust HH v robust SP: 87.911
    - robust HH v robust SP: 84.816
    - robust HH v non-robust SP: 26.461
  - Higher prob crisis (θ = 0.15) examples:
    - non-robust HH v non-robust SP: 77.705
    - non-robust HH v robust SP: 93.010
    - robust HH v robust SP: 90.689
    - robust HH v non-robust SP: 43.191
- Interpretation:
  - Calibrating to 13% crisis probability more than doubles average required macroprudential taxes relative to Bianchi (2011): 16% versus 7%.
  - Robust planner induces additional increase of almost 5 percentage points in average required taxes (up to 20% in some parameterizations).
  - Probability that a tax is needed increases (e.g., 88% vs 78% in comparative cases).
  - Taxes remain fairly stable when both agents fear misspecification, though implementation probability increases.
- If household robust but planner not, subsidy (negative ϕ) may be required; robust household underborrows relative to planner.

### Welfare gains and losses (Section 4.4)
- Welfare measured via compensating variation µ; value functions computed for all allocation-belief combinations.
- Average welfare gains (Table 7) under benchmark probabilities:
  - Bianchi replication:
    - non-robust HH v non-robust SP: 0.137
    - non-robust HH v robust SP: 0.129
    - robust HH v robust SP: 0.030
    - robust HH v non-robust SP: 0.028
  - Higher prob crisis (θ = 0.35 and θ = 0.15):
    - non-robust HH v non-robust SP: 0.270, 0.270
    - non-robust HH v robust SP: 0.241, 0.196
    - robust HH v robust SP: 0.090, -0.022
    - robust HH v non-robust SP: 0.096, 0.035
- Worst-case probabilities:
  - robust HH v robust SP: 0.108, 0.310, 0.339 (columns correspond to Bianchi, θ = 0.35, θ = 0.15)
  - robust HH v non-robust SP: 0.085, 0.227, 0.092
- Key interpretations:
  - For calibration yielding 13% sudden-stop probability, welfare gains from planner allocation are 0.27% of permanent consumption versus 0.137% in Bianchi replication.
  - Robust planner (θ = 0.35) increases welfare under benchmark shocks by 0.241% compared to 0.270% for non-robust planner; greater robustness (θ = 0.15) lowers gains to 0.196%.
  - When both agents fear misspecification and shocks follow benchmark f, gains are substantially reduced; with θ = 0.15 planner allocation can decrease welfare (robust planner “overshoots”).
  - Under worst-case probabilities, gains from robust planner relative to robust household can be larger.

### Comparison with higher risk aversion (Section 4.5)
- Robustness vs high risk aversion:
  - Similar decisions when debt high and collateral near binding.
  - Away from binding region:
    - Highly risk averse agent remains conservative.
    - Robust agent may accumulate larger debt because worst-case probabilities depend on state; far from binding region less concern about negative shocks.
- Figure/table comparisons (Table 9 columns order: HH θ = 0.35; SP θ = 0.35; HH θ = 0.15; SP θ = 0.15; HH γ = 4; SP γ = 4)
  - Long-run prob of crisis (%): 8.577 0.653 5.900 0.172 8.738 0.776
  - Long-run prob of constraint binding (%): 11.857 1.725 6.259 0.594 11.634 2.094
  - Average debt as % of GDP: 29.035 28.075 28.555 27.663 28.884 28.060
  - Decrease in consumption in median crisis: -4.912 -2.764 -2.574 -2.110 -13.898 -7.904
  - Depreciation RER in median crisis: -10.714 -5.280 -4.619 -3.400 -19.893 -4.504
  - CA reversal in median crisis: 7.940 1.193 7.940 0.458 7.054 0.119
  - Std dev of consumption: 0.065 0.054 0.059 0.053 0.063 0.054
  - Std dev of RER: 0.304 0.151 0.228 0.113 0.296 0.155
  - Std dev of CA (as % of GDP): 0.036 0.010 0.023 0.005 0.035 0.012
  - Corr consumption with GDP: 0.757 0.576 0.649 0.518 0.842 0.644
  - Corr RER with GDP: 0.792 0.632 0.674 0.560 0.860 0.636
  - Corr CA (as % of GDP) with GDP: -0.772 -0.608 -0.633 -0.506 -0.845 -0.562
- Policy and welfare implications:
  - Without model uncertainty, welfare gains arise from internalizing borrowing effects on collateral.
  - Planner robustness targets worst-case scenarios; desirability depends on welfare measurement distribution.
  - Robust planner can deliver welfare gains close to non-robust planner under benchmark shocks; if households are robust, planner gains are reduced.
  - Highly robust planner can overshoot and cause welfare losses.
  - Taxes required to implement robust planner’s allocation are about 5 percentage points higher on average and needed more frequently.
  - Robust planner can reduce crisis probability from 1.5% to 0.7% and reduce episode severity.

*Source: wpiea2021055-print-pdf - 2.1 Modelling robustness; 2.2 Interpreting robustness; 3 Macroprudential taxes; 4 Quantitative results; 4.2 Simulations; 4.3 Macroprudential taxes; 4.4 Welfare gains and losses; 4.5 Comparing robustness with higher risk aversion*

### 2.1  Modelling robustness

### 2.1  Modelling robustness

### State variables, agent problem, and benchmark model
- State vector xt = (bt; Bt; yt−1; εt), where bt is the household’s level of debt, Bt is aggregate debt, yt−1 = (yTt−1; yNt−1) is the endowment of tradable and non-tradable goods, and εt = (εTt; εNt) are innovations.
- Control vector wt = (cTt; cNt; bt+1): consumption and next-period household debt.
- Household maximizes utility subject to budget and collateral constraints given the state.
- Lemma 1 — equations governing state evolution:
  - Budget constraint:
    b t+1 + c T t + p N t c N t = b t (1 + r) + y T t + p N t y N t  (1)
  - Collateral constraint:
    b t+1 ≤ (ϕ N p N t y N t + ϕ T y T t)  (2)
  - Evolution of aggregate bond holdings:
    B t+1 = Ξ(B t; y t)  (3)
  - Price of non-tradables:
    p N t given ℱ t  (4)
  - Endowment process y t = (y N t; y T t):
    log(y t) = ρ log(y t−1) + ε t
    ε t+1 ~ N(0; V); i.i.d.  (5)
  - Compact notation:
    x t+1 = g(x t; w t; ε t+1)  (6)
    ε t+1 ~ N(0; V); i.i.d.  (7)
- Given policy functions cTt(xt), cNt(xt), bt+1(xt), the benchmark model implies transition density f(xt+1 | xt). Under misspecification, ~f(xt+1 | xt) can be any conditional distribution not too far from f.

### Robustness and distance measure
- Conditional relative entropy as distance between benchmark and misspecified transition densities:
  I(f; ~f) = E~f [ log( ~f(xt+1 | xt) / f(xt+1 | xt) ) | xt ]  (8)
- Robust household chooses policies to maximize utility under a “worst-case” transition density chosen by an “evil” minimizing agent.

### Robust optimization problem (minimax) and constraint on distortions
- Proposition 1 — robust household solves:
  max_{cNt, cTt, bt+1} min_{~f} E0,~f [ Σ_{t=0}^∞ β^t u(c_t) ]  (9)
  where
  c_t = [ ω (cTt)^{−θ} + (1−ω) (cNt)^{−θ} ]^{−1/θ}; θ > −1  (10)
  (b0; B0; y0) given
  (b t+1; B t+1; y t+1) = g((b t; B t; y t); (c N t; c T t; b t+1); ε t+1)  (11)
  ε t+1 i.i.d ~ N(0; V)  (12)
  E0,~f [ Σ_{t=0}^∞ β^{t+1} I(f; ~f) ] ≤ θ̄  (13)
  I(f; ~f) = E~f [ log( ~f(b t+1; B t+1; y t+1 | b t; B t; y t ) / f(b t+1; B t+1; y t+1 | b t; B t; y t ) ) | b t; B t; y t ]  (14)
  0 ≤ θ̄ ≤ Θ̄  (15)
- Condition (13) restricts the minimizing agent to consider worst-case ~f within distance θ̄ from the benchmark.
- Multiplier representation (introducing penalty parameter ξ) of the minimax problem:
  max_{cNt,cTt,bt+1} min_{~f} E0,~f [ Σ_{t=0}^∞ β^t u(c_t) ] + ξ E0,~f [ Σ_{t=0}^∞ β^{t+1} I(~f; f) ]  (16)
  - ξ ∈ (ξ_L; +∞) regulates penalty for deviations; lower ξ → more robustness (larger allowed distortions).
  - There is a direct mapping between ξ and θ̄.

### Martingale representation and recursive formulation
- Define martingale M t = ~f(xt) / f(xt) with E f [ M t+1 | x t ] = M t and increment m t+1 = M t+1 / M t satisfying E f [ m t+1 | x t ] = 1.
- Expectations under ~f expressed via f:
  E~f[y t | x0] = E f [ M t y t | x0 ]  (17)
  E~f[y t+1 | x t] = E f [ m t+1 y t+1 | x t ]  (18)
- Proposition 2 — recursive Bellman form for robust household:
  V(b t; B t; y t) = max_{cNt,cTt,bt+1} c_t^{1−γ} / (1−γ) − β ξ log( E_{y t+1 | y t} [ e^{−V(b t+1; B t+1; y t+1)/ξ} | b t; B t; y t ] )  (19)
  with
  c_t = [ ω (cTt)^{−θ} + (1−ω) (cNt)^{−θ} ]^{−1/θ}; θ > −1  (20)
  b t; B t; y t given  (21)
  (b t+1; B t+1; y t+1) = g((b t; B t; y t); (c N t; c T t; b t+1); ε t+1)  (22)
  ε t+1 i.i.d ~ N(0; V)  (23)
  ξ ∈ (ξ_L; +∞)  (24)
- Non-robust agent’s Bellman equation for comparison:
  V(B t; y t; b t) = max_{cTt,cNt,bt+1} c(c_t)^{1−γ} / (1−γ) + β E_{y t+1 | y t} [ V(b t+1; B t+1; y t+1) | b t; B t; y t ]  (25)

### Recursive competitive equilibrium (household) and markets
- Recursive competitive equilibrium (Bianchi (2011) structure): price pN(Bt; yt), law of motion B t+1 = Ξ(B t; y t), individual decision rules b(B t; y t; b t), cT(...), cN(...), and value V(...) such that:
  - Decision rules and value function solve individual recursive problem (19).
  - Markets clear:
    y N t = c N(B t; y t; b t)  (26)
    Ξ(B t; y t) + c T(B t; y t; b t) = y T t + B t (1 + r)  (27)
  - Perceived law of motion equals actual:
    B t+1 = Ξ(B t; y t) = b(B t; y t; B t)  (28)
  - Price of non-tradables:
    p N(B t; y t) = ( (1−ω)/ω * ( y T t + B t (1 + r) − Ξ(B t; y t) ) / y N t )^{θ+1}  (29)

### Planner problem and constrained-efficient equilibrium
- Planner chooses debt, lets goods markets clear competitively; households receive proceeds lump-sum.
- Planner faces same collateral constraint and resource constraints and internalizes effect of debt on pN via tradable consumption.
- Recursive constrained-efficient equilibrium: pN(B t; y t), planner law B(B t; y t), value V(B t; y t), household decision rules cT(B t; y t), cN(B t; y t) (given planner’s debt), such that:
  - Planner’s robust Bellman:
    V(B t; y t) = max_{b t+1, cTt} c(cTt; yN(B t; y t))^{1−γ} / (1−γ) − β ξ log( E_{y t+1 | y t} [ e^{−V(b t+1; y t+1)/ξ} | b t; B t; y t ] )  (30)
    where
    c(cTt; yN t) = [ ω (cTt)^{−θ} + (1−ω) (y N t)^{−θ} ]^{−1/θ}; θ > −1  (31)
  - Resource and collateral constraints:
    B(B t; y t) + c T(B t; y t) = B t (1 + r) + y T t  (32)
    B(B t; y t) ≤ (ϕ N p N(B t; y t) y N t + ϕ T y T t)  (33)
    c t ≥ 0  (34)
  - Price:
    p N(B t; y t) = ( (1−ω)/ω * ( y T t + B t (1 + r) − B t+1 ) / y N t )^{θ+1}  (35)
  - Consumption of non-tradables: c N(B t; y t) = y N t.
- Non-robust planner’s Bellman for comparison:
  V(B t; y t) = max_{B t+1, cTt} c^{1−γ}_t / (1−γ) + β E_{y t+1 | y t} [ V(B t+1; y t+1) ]  (36)

### Key mechanism: pecuniary externality, amplification, and policy implications
- Planner internalizes how debt affects pN via tradable consumption; household does not, creating a pecuniary externality when collateral depends on debt.
- Amplification mechanism with occasionally binding collateral constraints:
  - Negative shock → lower yTt and/or yNt → collateral constraint may bind → borrowing falls → tradable consumption falls → pN falls → constraint tightens further → borrowing falls more. Relaxation of constraint can occur by increasing tradable consumption (raises pN).
  - Consequence: in bad times borrowing decreases (capital inflows decrease) and the current account increases when constraint binds. When constraint does not bind, typical consumption smoothing leads to borrowing increasing in bad times (capital inflows increase) and current account decreases.
  - Positive shock relaxes constraint further; in good times borrowing decreases (capital inflows decrease) and current account increases.
- Role for macroprudential policies: reduce probability and magnitude of sudden stops by affecting incidence and severity of constraint binding.

### Modelling robustness for household and/or planner
- Introducing model misspecification fears for household and/or planner adds an additional decision layer.
- Because household and planner are distinct agents with different objectives and information, modeling either or both as fearing misspecification yields different interpretations and equilibrium implications; these cases are discussed in the following section.

*Source: wpiea2021055-print-pdf - 2.1  Modelling robustness*

### 2.2  Interpreting robustness

### 2.2 Interpreting robustness

### Conceptual framework and planner types
- Introducing a friction (the collateral constraint) breaks the equivalence between the representative household’s allocation and the social planner’s allocation implied by the welfare theorems.
- The planner faces two dimensions of ambiguity:
  - The model that governs the evolution of the state (planner’s model).
  - The model that households believe governs the evolution of the state (household’s model).
- When planner and household views differ, two planner types arise:
  - Paternalistic planner: maximizes welfare according to her view of the world (may fear misspecification).
  - Non-paternalistic planner: implements the allocation from the household’s perspective.
- The planner’s robust problem (equation (37))—with expectations taken with respect to the misspecified model ~f—can be interpreted as either:
  - A paternalistic planner fearing misspecification, or
  - A non-paternalistic planner believing the private sector fears misspecification and sharing the same ambiguity.

### Welfare measurement scenarios (household perspective and benchmark perspective)
- Welfare measurement requires choosing which model (distribution of shocks) is used to compute expected utility; this choice matters for whether planner intervention improves welfare.
- Two measurement approaches are considered:

1) Welfare measured from the private agents’ perspective (Table 1). Key logical possibilities shown:
  - Superscripts indicate evaluation at decentralized competitive equilibrium (DE) or planner’s (SP) allocation.
  - Subscripts indicate whether welfare is computed under the benchmark distribution f or the worst-case distribution ~f (which coincides with the household’s view in Table 1).
  - Value functions follow either the traditional Bellman equation (equation (38)) or the modified Bellman equation (equation (39)), depending on probability beliefs. Where fi ∈ {f, ~f}.

2) Welfare measured based on the benchmark f (“truth”) (Table 2).
  - Welfare gains are computed using the benchmark model regardless of agents’ or planner’s beliefs.
  - If the planner knows more than agents, she can improve welfare; if she knows less, she can cause harm.
  - Upper row of Table 2 is identical to Table 1; the bottom-left corner (planner robust and benchmark used to measure welfare) is the case where planner achieves maximum welfare.

- Summary implication:
  - Presence of externalities plus fear of misspecification implies planner’s allocation may not improve household welfare or gains may be subdued.
  - Planner’s allocation guarantees improvement only when the planner’s view is used to compute expected utility.
  - When planner and household disagree and welfare is measured from the household’s perspective, two opposing forces exist: planner may reduce welfare by using a different model, but may improve welfare by correcting overborrowing externality.

### Compensating variation (welfare metric)
- Welfare gains/losses computed using a compensating variation approach (as in Lucas (1987)):
  - Compute the constant proportional increase in consumption (the percent subsidy to consumption) that makes agents indifferent between the decentralized equilibrium (DE) and the planner’s allocation (SP); denote this subsidy by µ (previously notationally µ(B; y) but reduced to µ when fixed over time).
  - If welfare computed using benchmark probabilities, µ satisfies equation (40).
  - If worst-case probabilities govern shocks, µ satisfies equation (41).
  - When no misspecification (homothetic utility), µ can be simplified using equation (42):
    - (1 + µ(B; y))^(1-γ) V(DE; f_DE)_f(B; y; B) = V(SP; f_SP)_f(B; y; B)  (equation (42))
  - Homotheticity cannot be used when fear of misspecification is present because the modified Bellman equation is nonlinear; welfare gains are computed numerically in that case.

### Key equations referenced
- Planner’s robust problem: equation (37).
- Traditional Bellman: equation (38).
- Modified Bellman (robust): equation (39).
- Compensating variation conditions: equations (40) and (41).
- Simplified homothetic expression: equation (42).
- DE value function under benchmark: equation (43).

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### 3 Macroprudential taxes (implementation conditions)
- For periods when the collateral constraint is not binding, taxes on debt that implement the planner’s allocation must satisfy both household’s and planner’s Euler equations evaluated at the planner’s allocation.
- Non-robust Euler equations (Bianchi (2011)): equations (44) and (45), where:
  - u_T,t is marginal utility with respect to tradables,
  - ϕ_t is the tax on debt,
  - µ_t is the shadow value of relaxing the collateral constraint,
  - Δ_t captures sensitivity of collateral value to tradable consumption.
- Robust Euler equations (planner and/or household fear misspecification): equations (46)–(48), where expectations are under worst-case martingale increments m_{t+1} defined in equation (48).
- Decentralization requires accounting for the distribution of shocks that households use when forming expectations, regardless of the true distribution.

- Four cases for taxes or subsidies implementing the planner’s allocation:
  1. Neither household nor planner robust (Bianchi (2011)):
     - ϕ = E_{y_{t+1}|y_t}[µ_{t+1} Δ_{t+1}/Δ_{t+1}] / E_{y_{t+1}|y_t}[u_{T,t+1}]  (equation (49))
  2. Planner robust, household not:
     - 1 + ϕ = E_{y_{t+1}|y_t}[m_{t+1}(u_{T,t+1} + µ_{t+1} Δ_{t+1}/Δ_{t+1})] / E_{y_{t+1}|y_t}[u_{T,t+1}]  (equation (50))
  3. Both planner and household robust:
     - ϕ = E_{y_{t+1}|y_t}[m_{t+1} µ_{t+1} Δ_{t+1}/Δ_{t+1}] / E_{y_{t+1}|y_t}[m_{t+1} u_{T,t+1}]  (equation (51))
  4. Household robust, planner not:
     - 1 + ϕ = E_{y_{t+1}|y_t}[u_{T,t+1} + µ_{t+1} Δ_{t+1}/Δ_{t+1}] / E_{y_{t+1}|y_t}[m_{t+1} u_{T,t+1}]  (equation (52))
- The last case (household robust, planner not) is the only case where the planner’s constraint can bind for some current debt values while the household’s does not; in that case a subsidy (ϕ negative) may be required.

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### 4 Quantitative results: calibration and debt policy
- Calibration approach:
  - Follows Bianchi (2011) closely but re-calibrates discount factor and share of collateralized tradable to match target moments:
    - Target probability of sudden stop: 13%.
    - Target average debt-to-GDP ratio: 30%.
  - Rationale: Bianchi targets a probability of 5%, but empirical evidence (Jeanne and Ranciere (2011); Furceri et al (2011)) suggests sudden stops are about twice as frequent (around 10% and as high as 20%); hence focus on a higher probability justifying higher macroprudential taxes.
  - Also present Bianchi’s exact calibration for comparability.

- Risk aversion and misspecification parameters:
  - Baseline risk aversion γ = 2 (standard).
  - Also consider γ = 4 for sensitivity to higher risk aversion.
  - Degree of misspecification θ (notation ϕ in text is γ; robustness parameter denoted by θ):
    - Baseline θ = 0.35 chosen by targeting detection error probabilities (agents would not “easily” distinguish between worst-case and benchmark).
    - Also compare with θ = 0.15.
  - Detection error probabilities used to discipline choice of θ; baseline θ = 0.35 targets an error detection probability of 20%.

- Selected calibration parameters (Table 3):
  - Risk aversion γ = 2 (Standard, Bianchi (2011))
  - Intratemporal elasticity of substitution 1/(1 + ξ) = 0.83 (Bianchi (2011), conservative)
  - Discount factor β = 0.8 (To match data moments - Bianchi (2011): 0.91)
  - Share of tradables in CES χ = 0.31 (Bianchi (2011), to match data moments)
  - World risk-free interest rate r = 0.04 (Standard, Bianchi (2011))
  - Share of collateralized tradable ϕ_T = 0.31 (To match data moments - Bianchi (2011): 0.32)
  - Degree of misspecification θ = 0.35 (To target an error detection probability of 20%)

- Shock process discretization:
  - Following Bianchi (2011): discretize tradable and non-tradable shocks with four grid points each (total 16 states).
  - Use same bivariate autoregressive first-order calibration and same grid values and transition probabilities as Bianchi (2011); refer to these transition probabilities as the “benchmark” distribution f.

- Debt policy: household vs planner (Figure 2 summary)
  - Debt policies plotted for the state where tradable and non-tradable endowments are about one standard deviation below the mean.
  - Dashed lines: household and planner without misspecification (replicate Bianchi (2011) with calibration changes).
  - Solid lines: household and planner under robustness (θ = 0.35).
  - Observed patterns:
    - Both household and planner show a U-shaped debt policy.
    - Explanation: absent the constraint, optimal bond holdings increase monotonically with current debt for consumption smoothing; when collateral constraint binds, higher current debt tightens future collateral via lower tradable consumption and price effects, producing an upward slope in the binding region.
    - The binding region is smaller for robust agents.
    - In non-binding regions, planner accumulates larger bond holdings than household as precautionary savings against future constraint binding.
    - Robust agents save more than non-robust counterparts across the non-binding region.
    - A robust household eventually saves more than the social planner when current debt is far from the binding region.
    - Closer to the binding area, a non-robust planner still saves more than the household to internalize collateral externalities.
  - Tax regions:
    - Green area: region of current debt holdings where a tax is needed to implement a robust planner’s allocation when households are robust (θ = 0.35).
    - Grey area: corresponding tax region when both planner and household are non-robust.
    - A robust planner taxes a robust household over a slightly narrower debt region (green) than a non-robust planner taxes a non-robust household (grey).
    - The difference between a robust planner and a non-robust household in debt choices is significantly larger, implying potentially much larger taxes and a larger tax region would be needed to implement the socially optimal allocation in that cross-case.

- Calibration targets and rationale summarized:
  - Sudden stop probability targeted: 13% (calibration choice).
  - Average debt-to-GDP targeted: 30%.
  - Baseline robustness parameter θ = 0.35; alternative θ = 0.15 considered.
  - Risk aversion baseline γ = 2; alternative γ = 4 considered.

_Italic: Source: wpiea2021055-print-pdf - 2.2 Interpreting robustness (PDF chapter/section)._

### 4.2  Simulations

### 4.2  Simulations

### Simulation design and calibration
- 80,000 period allocations are obtained given some history of realized shocks for the endowment process.
- Shocks for the baseline simulations are realizations from the benchmark distribution; when analyzing welfare gains, simulations using worst-case probabilities (from the household perspective) are also reported.
- Simulated series are produced for allocations following the optimal choices of:
  - a non-robust household,
  - a non-robust planner,
  - a robust household,
  - a robust planner.
- Robust policies use values for  of 0.15 (high degree of misspecification) and 0.35 (low degree of misspecification).
- Calibration is set to render a probability of crisis of 13% under the household’s non-robust allocation and the benchmark distribution of shocks (results that replicate Bianchi (2011) are also reported for comparability).

### Definition of sudden stop and simulation procedure
- A sudden stop occurs when the collateral constraint binds and this leads to an increase in net capital outflows (i.e., an increase in bond holdings) that exceeds one standard deviation of net capital outflows according to the long-run distribution of bond holdings of the decentralized economy (under the non-robust household’s allocation).
- For crisis severity, the median financial crisis in the decentralized equilibrium (under the non-robust household’s allocation) is identified by the median current account reversal; the level of debt two periods before this median crisis is backtracked and used as the initial condition for three-period simulations comparing responses by robust/non-robust households and planners.

### Key simulation findings (comparison across allocations)
- A social planner that internalizes the effect of borrowing on collateral substantially reduces the occurrence of sudden stops:
  - From 5% to 0.3% for Bianchi’s calibration.
  - From 13% to 1.5% for the preferred calibration.
- Severity reductions under planner intervention (preferred calibration, comparisons relative to non-robust household baseline):
  - Reduction in consumption in median crisis: from -8.810 (non-robust HH v non-robust SP baseline in Table 4) to -3.717 (non-robust HH v robust SP) and other listed entries — see Table 4 for full matrix.
  - Lower depreciation of the real exchange rate: from -20.055 to -7.930 (examples from Table 4).
  - Smaller current account reversal: from 7.940 to 2.246 (examples from Table 4).
- Robust households save more for precautionary motives and:
  - Decrease probability of a sudden stop to 9% in some parameterizations.
  - Halve the reduction in consumption and depreciation of the real exchange rate in median crises relative to their non-robust counterparts (no significant improvements in median current account reversal in some cases).
- Robust planners further reduce both probability and severity of crises (e.g., probability of crisis to 0.7% in some parameterizations).

### Distributional implications
- Long-run (ergodic) distribution of bond holdings for the planner assigns a higher probability to lower levels of debt (high bond holdings) compared to private agents.
- On average both planner and private agents—whether robust or not—tend to hold the same level of debt (30%).
- The planner reduces exposure to very high levels of debt (low bond holdings) that risk triggering collateral-constraint binding events under negative output shocks.

---

### 4.3  Macroprudential taxes

### Taxes required to decentralize planner allocations
- The difference in optimal bond holdings between a non-robust household and a robust planner widens over the non-binding region; this implies larger macroprudential taxes are required to decentralize a robust planner’s allocation.
- For all states and across the entire bond grid, macroprudential taxes required to decentralize a robust planner (= 0:35) are larger and apply over the entire set of possible current bond holdings such that the constraint is not yet binding.
- If the household is robust but the planner is not, a subsidy is required to implement the planner’s allocation because the robust household underborrows relative to the planner.

### Simulation averages and probabilities (Tables 5 and 6)
- Average taxes (mean over 80,000 periods; shocks from baseline distribution):
  - Bianchi (2011) replication (non-robust HH v non-robust SP): 7.274
  - Higher prob crisis (= 0:35) cases:
    - non-robust HH v non-robust SP: 15.971
    - non-robust HH v robust SP: 20.113
    - robust HH v robust SP: 16.999
    - robust HH v non-robust SP: -1.637
  - Higher prob crisis (= 0:15) cases:
    - non-robust HH v non-robust SP: 15.971
    - non-robust HH v robust SP: 22.260
    - robust HH v robust SP: 15.719
    - robust HH v non-robust SP: -6.247
- Probability that a tax is implemented (percent of simulation periods):
  - Bianchi (2011) replication (non-robust HH v non-robust SP): 80.111
  - Higher prob crisis (= 0:35) examples:
    - non-robust HH v non-robust SP: 77.705
    - non-robust HH v robust SP: 87.911
    - robust HH v robust SP: 84.816
    - robust HH v non-robust SP: 26.461
  - Higher prob crisis (= 0:15) examples:
    - non-robust HH v non-robust SP: 77.705
    - non-robust HH v robust SP: 93.010
    - robust HH v robust SP: 90.689
    - robust HH v non-robust SP: 43.191

### Interpretation
- Changing calibration to achieve a 13% probability of crisis more than doubles average required macroprudential taxes relative to Bianchi (2011): 16% versus 7%.
- A robust planner induces an additional increase (almost 5 percentage points) in average required taxes (e.g., up to 20% in some parameterizations).
- Probability that a tax is needed to implement the planner’s allocation increases (e.g., 88% compared to 78% in comparative cases).
- Taxes remain fairly stable when both agents fear misspecification, though the probability of implementation increases.

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### 4.4  Welfare gains and losses

### Method and value functions
- Welfare comparisons use the compensating variation approach (Section 2.2).
- Value functions for all combinations of allocations and probability assessments (benchmark or worst-case densities) are derived; Figures 14–17 in the Appendix depict these value functions.
- Welfare measure 
(B;y) is state dependent; Figure 6 shows gains for a state with a negative one-standard deviation shock to tradables and non-tradables under a non-robust planner/non-robust household with benchmark probabilities.
- For average welfare gains, means are computed over simulation periods in which a macroprudential tax is required to implement the planner’s allocation.
- When computing average gains comparing robust planner vs robust household from the household perspective, simulations use shocks from the household’s worst-case distribution.

### Average welfare gains (Table 7)
- Benchmark probabilities; Bianchi (2011) replication:
  - 1) non-robust HH v non-robust SP: 0.137
  - 2) non-robust HH v robust SP: 0.129
  - 3) robust HH v robust SP: 0.030
  - 4) robust HH v non-robust SP: 0.028
- Benchmark probabilities; higher prob crisis (= 0:35 and = 0:15):
  - non-robust HH v non-robust SP: 0.270, 0.270
  - non-robust HH v robust SP: 0.241, 0.196
  - robust HH v robust SP: 0.090, -0.022
  - robust HH v non-robust SP: 0.096, 0.035
- Worst-case probabilities:
  - 5) robust HH v robust SP: 0.108, 0.310, 0.339 (columns correspond to Bianchi (2011), = 0:35, = 0:15)
  - 6) robust HH v non-robust SP: 0.085, 0.227, 0.092

### Key welfare interpretations
- For the calibration yielding a 13% sudden-stop probability in the decentralized economy, welfare gains from implementing the planner’s allocation are twice as large as in Bianchi (2011): 0.27% of permanent consumption versus 0.137% (Bianchi replication).
- A robust planner’s allocation increases welfare even when welfare is evaluated assuming benchmark shock realizations:
  - Robust planner with = 0:35 achieves a welfare gain of 0.24% of permanent consumption under benchmark probabilities, compared to 0.27% for a non-robust planner.
  - Higher degree of robustness (= 0:15) lowers welfare gain to 0.20% in comparable cases.
- If both agents fear misspecification but shocks follow the benchmark distribution, gains from implementing the planner’s allocation are substantially reduced:
  - Example: average gain of 0.096% when planner is robust and household is robust (= 0:35, benchmark probabilities).
  - If fear of misspecification is sufficiently high (e.g., = 0:15), a planner’s allocation can decrease welfare (robust planner “overshoots” in saving).
- Under worst-case probabilities, gains from a robust planner relative to a robust household can be larger (see worst-case block in Table 7).

### Summary table of welfare gains and losses (conceptual)
- Benchmark probabilities:
  - Non-robust HH v Non-robust SP: Average gain 0:270%
  - Non-robust HH v Robust SP: Average gain 0:241%
  - Robust HH v Robust SP: Average gain 0:096% (or 0:090% depending on ordering)
- Worst-case probabilities:
  - Robust HH v Robust SP: Average gain 0:227% or 0:31% depending on ordering
  - Robust HH v Non-robust SP: Average gain 0:085% or 0:339% in alternate comparisons
(Note: See Table 8 for the detailed arrangement of comparisons and exact numeric entries.)

*Source: wpiea2021055-print-pdf — 4.2 Simulations / 4.3 Macroprudential taxes / 4.4 Welfare gains and losses*

### 4.5  Comparing robustness with higher risk aversion

### 4.5  Comparing robustness with higher risk aversion

### Key comparative insight
- Robust agents and highly risk averse agents tend to make very similar decisions when debt is high and the collateral constraint is close to binding.
- For levels of debt further away from the binding region:
  - A highly risk averse agent remains conservative in terms of savings.
  - A robust agent accumulates larger amounts of debt because the robust agent’s worst-case scenario of probabilities for the following period depends on the current state; if the economy is far from the binding region, there is less concern about negative future shocks.
- Figure7 parameters noted in the source:
  - For the robust agents, the degree of misspecification considered is characterized by = 0:35.
  - Non-robust agents in this plot have a high risk aversion of = 4.

### Simulation comparisons (Table 9)
- Columns correspond to:
  - HH = 0:35
  - SP = 0:35
  - HH = 0:15
  - SP = 0:15
  - HH = 4
  - SP = 4
- Results (each line lists the six column values in order):
  - Long-run prob of crisis* (%): 8.577 0.653 5.900 0.172 8.738 0.776
  - Long-run prob of constraint binding (%): 11.857 1.725 6.259 0.594 11.634 2.094
  - Average debt as % of GDP: 29.035 28.075 28.555 27.663 28.884 28.060
  - Decrease in consumption in median crisis: -4.912 -2.764 -2.574 -2.110 -13.898 -7.904
  - Depreciation RER in median crisis: -10.714 -5.280 -4.619 -3.400 -19.893 -4.504
  - CA reversal in median crisis: 7.940 1.193 7.940 0.458 7.054 0.119
  - Std dev of consumption: 0.065 0.054 0.059 0.053 0.063 0.054
  - Std dev of RER: 0.304 0.151 0.228 0.113 0.296 0.155
  - Std dev of CA (as % of GDP): 0.036 0.010 0.023 0.005 0.035 0.012
  - Corr consumption with GDP: 0.757 0.576 0.649 0.518 0.842 0.644
  - Corr RER with GDP: 0.792 0.632 0.674 0.560 0.860 0.636
  - Corr CA (as % of GDP) with GDP: -0.772 -0.608 -0.633 -0.506 -0.845 -0.562
- Notable summary points from these simulations:
  - Both a robust and a highly risk averse household and planner arrive at similar probabilities of a crisis (around 8.5% based on a household’s debt policy, and this is reduced to 0.7% for a planner).
  - Both types of planners halve the reduction in consumption in the median crisis relative to baseline, although the median crisis under highly risk averse agents is more severe than that under robustness for some metrics (e.g., consumption decrease and RER depreciation).

### Policy and welfare implications (from Conclusion and results)
- Without model uncertainty, welfare gains arise from internalizing how borrowing in good times affects collateral value during crises.
- When the planner fears model misspecification (robustness), policy desirability depends on the distribution of shocks used for welfare computation; robustness targets a “worst-case” scenario.
- Welfare findings:
  - A robust planner may achieve a welfare gain almost as large as a non-robust planner when shocks follow the benchmark distribution.
  - If households also make robust choices, welfare gains from planner robustness are significantly reduced because households save more as precaution, indirectly addressing overborrowing.
  - A highly-robust planner can “overshoot” savings and induce welfare losses.
  - If the robust agents’ feared worst-case scenario actually governs shocks, welfare gains are the largest possible.
- Macroprudential taxes and crisis outcomes:
  - The taxes required to implement a robust planner’s allocation are 5 percentage points higher, and the probability that they need to be implemented is also larger.
  - A robust planner can bring the probability of a crisis from 1.5% to 0.7% and reduce the severity of episodes.

*Source: wpiea2021055-print-pdf - 4.5  Comparing robustness with higher risk aversion*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021055-print-pdf.pdf_