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---

### Workers: preferences, constraints, and FOCs
- Preferences and utility:
  - Lifetime expected utility: E0 ∑_{t=0}^{∞} β^{t} U(c_{t} − G(h_{t})). (1)
  - U(·) is concave, twice continuously differentiable, satisfying the Inada conditions.
  - Composite commodity: c_{t} − G(h_{t}) (Greenwood, Hercowitz, and Huffman (1988)).
  - G(h_{t}) is convex, strictly increasing, continuously differentiable.
  - Functional forms:
    - U(c_{t} − G(h_{t})) = [(c_{t} − G(h_{t}))^{(1−σ)} − 1]/(1 − σ) with risk aversion coefficient σ.
    - G(h_{t}) = ψ h_{t}^{1+φ}/(1 + φ) with 1/φ the Frisch elasticity of labor supply.
- Hand-to-mouth assumption and motivation:
  - Workers cannot pledge labor income to borrow inter-temporally (hand-to-mouth).
  - Empirical motivation: Kaplan, Violante and Weidner (2014) document that one-third of all US households live hand-to-mouth.
  - Extension allowing worker access to credit discussed in Section 3.3.
- Budget constraint and first-order conditions:
  - Budget constraint: c_{t} ≤ w_{t} h_{t}. (2)
  - FOCs from maximizing (1) subject to (2):
    - Consumption: c_{t}: u_{c,t} = λ_{w t}. (3)
    - Labor: h_{t}: G_{h,t} = w_{t}. (4)
  - Notation:
    - Lagrange multipliers scaled by β^{t} (λ_{w t} = β^{t} \hat{λ}_{w t}).
    - X_{i,t} denotes derivative of X(·) w.r.t. i at t.

### Decentralized economy: entrepreneurs, taxes, and equilibrium linkages
- Entrepreneur budget constraint (as in source):
  - x_t + (1 + τ^b_{t-1}) b_t + p^v v_t + (1 + τ^l_t) w_t l_t + q_t k_{t+1} ≤ y_t + b_{t+1}/R + q_t k_t + T_t, with T_t = τ^b_{t-1} b_t + τ^l_t l_t.
- Taxes, timing, and rebates:
  - Tax revenues rebated back to private agents; tax payments and rebates settled at end of period after production.
- Private-agent FOCs in decentralized economy:
  - Euler for borrowing: U_{x,t} (1 − μ_t) = β R (1 + τ^b_t) E_t U_{x,t+1}. (33)
  - Entrepreneurs' labor demand: F_{l,t} = w_t (1 + τ^l_t + θ_l μ_t). (34)
- Interpretation and roles:
  - Borrowing tax/subsidy τ^b_t:
    - Levied on new borrowing b_{t+1} determined at t but applied at t+1 when debt repaid.
    - Internalizes pecuniary externalities via q_t and q_{t+1}.
  - Payroll tax/subsidy τ^l_t:
    - Targets wedge between competitive and social labor decisions.
    - Affects labor demand, wages, and working-capital-financed payroll share θ_l, with implications for collateral tightness and redistribution.
- Equilibrium linkage:
  - All other equilibrium conditions remain as in section 2.

### Optimal tax rates: formulas and interpretation
- Optimal borrowing tax (equation (35)):
  - τ^b_t = 1/(β R E_t U_{x,t+1}) [ μ^{SP}_t − U_{x,t} μ_t + ξ_t U_{xx,t} q_t − β R ξ_t E_t Ω_{t+1} ] − 1/(E_t U_{x,t+1}) E_t[ ξ_{t+1} U_{xx,t+1} q_{t+1} ].
  - When collateral constraint does not bind at t (μ_t = μ^{SP}_t = ξ_t = 0), reduces to:
    - τ^{MP}_t = − 1/(E_t U_{x,t+1}) E_t[ ξ_{t+1} U_{xx,t+1} q_{t+1} ] = − 1/(E_t U_{x,t+1}) E_t[ κ_{t+1} μ^{SP}_{t+1} q_{t+1} U_{xx,t+1} / U_{x,t+1} ]. (36)
    - This expression is always positive when the collateral constraint binds only in expectation, implying a tax on borrowing levied in good times to allow more borrowing in bad times (macroprudential interpretation).
  - When constraint binds at t, current q_t term can push for borrowing subsidy while future q_{t+1} term pushes for tax; planner balances these effects.
- Optimal payroll tax (equation (37) and decomposition (38)):
  - τ^l_t = ( θ_l μ_t + (λ^{SP,e}_t − λ^{SP,w}_t)/λ^{SP,e}_t ) (G_{ll,t}/G_{l,t}). (37)
  - Equivalently:
    - τ^l_t = ( θ_l μ_t + (U_{x,t} − ω U_{c,t})/(U_{x,t} − ξ_t U_{xx,t} q_t) − ξ_t U_{xx,t} q_t/(U_{x,t} − ξ_t U_{xx,t} q_t) ) (G_{ll,t}/l_t). (38)
  - Main components:
    1. Pecuniary externality via wage: θ_l μ_t G_{ll,t} l_t > 0 → desire to increase payroll tax (or decrease subsidy) to discourage labor demand and working-capital financing needs.
    2. Distributive externality: addresses gap between agents' marginal utilities of consumption:
      - If U_{x,t} > ω U_{c,t} → tighter payroll tax (or lower subsidy) to shift resources to entrepreneurs.
      - If U_{x,t} < ω U_{c,t} → lower payroll tax (or higher subsidy) to shift resources to workers.
  - Interaction term when constraint binds: −ξ_t U_{xx,t} q_t /(U_{x,t} − ξ_t U_{xx,t} q_t) > 0 can make payroll tax internalize asset-price pecuniary externality and distributive externality; raising payroll tax can mitigate Fisherian deflation ex post.
  - Payroll tax remains useful in two-agent economy even if θ_l = 0 due to distributive externality from heterogeneity.

### Additional tools: capital-holding tax/subsidy τ^k_t
- Implementability/pricing condition with τ^k_t (equation (39)):
  - (1 + τ^k_t) q_t U_{x,t} = β E_t U_{x,t+1} { F_{k,t+1} + q_{t+1} [ 1 + κ_{t+1}/θ_v ( F_{v,t+1}/p_v − 1 ) ] }.
- Without constraints on τ^k_t, planner could choose τ^k_t to implement any q_t and eliminate binding collateral constraint.
- Natural lower bound: τ^k_t ≥ τ, where τ < 0 (lower bound) imposed by transversality or funding/political constraints.
- Key takeaway: capital subsidies reduce but do not eliminate roles for borrowing and payroll taxes; borrowing taxes remain useful when capital tax hits lower bound. Quantitative analysis explores different τ values.

### Quantitative analysis: calibration and solution methods
- Solution methods:
  - Global, non-linear solution algorithm for planner (value function iteration with nested fixed point).
  - Competitive economy via Euler-equation iteration on first-order conditions.
  - Details in Section C of appendix.
- Time interpretation: each period = one year.
- Baseline calibration differences from Bianchi and Mendoza (2018):
  - R = 1.02 (vs. 1.01).
  - θ_v = θ_l = 0.09 (instead of 0.16) to target working-capital loan / GDP = 0.13.
  - Transition probability κ_h → κ_l set to P_{h,l} = 0.05 (5 percent) instead of 10 percent to match Sudden Stop probability = 4%.
  - Welfare weight on workers ω = 1 in baseline.
- Calibration parameters (as reported in source):
  - Risk aversion σ = 1
  - Labor disutility coefficient ψ = 0.352
  - Frisch elasticity of labor supply 1/φ = 2
  - Share of intermediate good in output α_v = 0.45
  - Share of labor in output α_l = 0.352
  - Share of assets in output α_k = 0.008
  - Interest rate R = 1.02
  - TFP process ρ_z = 0.78
  - σ_ε = 0.01
  - Discount factor β = 0.95
  - Working capital coefficient θ_v = θ_l = 0.09
  - Tight credit regime κ_l = 0.75
  - Normal credit regime κ_h = 0.90
  - Transition probability, κ_h to κ_l: P_{h,l} = 0.05
  - Transition probability, κ_l to κ_l: P_{l,l} = 0.00
  - Welfare weight on workers ω = 1
- Sudden Stop definition:
  - Event when drop in long-term borrowing (b) between two periods over output exceeds two standard deviations of its long-term distribution (consistent with Bianchi and Mendoza (2018)).
  - Sudden Stops are infrequent, accompanied by large current-account reversals, credit crunches, and a binding collateral constraint.

### Sudden Stops, collateral constraints, and model statistics
- Binding collateral constraint vs Sudden Stop:
  - A binding collateral constraint is not sufficient for a Sudden Stop; multiplier can be very close to zero implying small credit corrections.
  - Probability that constraint binds is higher than probability of Sudden Stop in both representative- and two-agent economies.
  - Representative- and two-agent competitive equilibria produce similar probabilities of Sudden Stops and similar short-term / long-term borrowing ratios consistent with data.
  - Footnote: "99.6 percent and 100 percent of the Sudden Stop events in the representative-agent and two-agent economies, respectively, are characterized by changes in the current account as percentage of GDP that are high enough to be in the 95th percentile."
- Table (2) selected statistics (Panels A and B) — unconditional averages over simulation horizon:
  - PANEL A (Selected statistics)
    - Probability of Sudden Stop:
      - Repr. agent CE: 4.2
      - Repr. agent SP: 3.1
      - Two-agent CE: 4.0
      - Two-agent SP: 2.7
    - Probability of binding constraint:
      - Repr. agent CE: 13.8
      - Repr. agent SP: 15.7
      - Two-agent CE: 4.3
      - Two-agent SP: 4.7
    - Average Working capital loan / GDP:
      - Repr. agent CE: 13.1
      - Repr. agent SP: 13.1
      - Two-agent CE: 13.1
      - Two-agent SP: 13.7
    - Standard deviation (Working capital loan / GDP):
      - Repr. agent CE: 0.002
      - Repr. agent SP: 0.001
      - Two-agent CE: 0.002
      - Two-agent SP: 0.002
    - Average Credit-to-GDP (b_{t+1}) / GDP:
      - Repr. agent CE: 24.4
      - Repr. agent SP: 16.4
      - Two-agent CE: 21.1
      - Two-agent SP: 17.0
    - Standard deviation (Credit-to-GDP):
      - Repr. agent CE: 0.038
      - Repr. agent SP: 0.020
      - Two-agent CE: 0.033
      - Two-agent SP: 0.016
  - PANEL B (Selected aggregate moments)
    - Average aggregate consumption:
      - Repr. agent CE: 0.26
      - Repr. agent SP: 0.27
      - Two-agent CE: 0.26
      - Two-agent SP: 0.29
    - Standard deviation (aggregate consumption):
      - Repr. agent CE: 0.017
      - Repr. agent SP: 0.014
      - Two-agent CE: 0.015
      - Two-agent SP: 0.015
    - Average asset price:
      - Repr. agent CE: 0.11
      - Repr. agent SP: 0.09
      - Two-agent CE: 0.11
      - Two-agent SP: 0.10
    - Standard deviation (asset price):
      - Repr. agent CE: 0.008
      - Repr. agent SP: 0.005
      - Two-agent CE: 0.009
      - Two-agent SP: 0.004
    - Average wage:
      - Repr. agent CE: 0.28
      - Repr. agent SP: 0.28
      - Two-agent CE: 0.28
      - Two-agent SP: 0.29
    - Standard deviation (wage):
      - Repr. agent CE: 0.005
      - Repr. agent SP: 0.005
      - Two-agent CE: 0.006
      - Two-agent SP: 0.006
  - Data benchmark (Bianchi and Mendoza (2018)): short-term borrowing (working capital loans) as percent of GDP = 13.3%; long-term borrowing (NFA) as percent of GDP = 25% (2013 US data).
- Policy functions and optimal taxes (qualitative findings):
  - Planner chooses lower new borrowing b_{t+1} than private agents before borrowing constraint kink.
  - Planner’s borrowing constraint turns binding at lower debt levels than private agents because lower borrowing and consumption reduce collateral values.
  - Low productivity (z_t = z_l) and tight financial conditions (κ_t = κ_l):
    - Representative-agent: private and social borrowing dynamics very similar.
    - Two-agent: private equilibrium shows sharp decline in borrowing once constraint binds; planner’s decline is smoother due to redistribution from workers to entrepreneurs.

### Optimal borrowing and payroll tax dynamics (qualitative and numerical summaries)
- Borrowing tax dynamics:
  - High productivity (z_t = z_h) and favorable financial conditions (κ_t = κ_h):
    - Borrowing tax ~ zero at low borrowing, rises with borrowing, then decreases at collateral-constraint kink.
    - Decline sharper in representative-agent economy than in two-agent economy.
  - Representative-agent planner supports borrowing mainly by reducing borrowing tax when constraint binds; two-agent planner can also use payroll-tax transfers.
- Payroll tax dynamics and redistribution motives:
  - High productivity / favorable financial conditions:
    - Representative-agent: payroll tax zero when slack, slightly positive when constraint starts to bind.
    - Two-agent: payroll tax negative when slack (payroll subsidy), becomes less negative approaching kink; when constraint binds, planner decreases payroll subsidy further to address distributive externality.
  - Low productivity / unfavorable financial conditions:
    - Two-agent planner reduces payroll subsidy for higher indebtedness and ultimately levies higher positive payroll taxes to transfer resources from workers to entrepreneurs.
  - Ability to use payroll taxes (including subsidies) in two-agent framework provides an extra instrument to mitigate Sudden Stops and speed recovery.
- Table 3: tax and welfare numerical summaries (all numbers in percent; SS = Sudden Stop)
  - Representative vs Two agents:
    - Mean Payroll tax (unconditional): 0.1 (Representative) ; -9.9 (Two agents)
    - Mean Payroll tax (conditional on SS): 1.0 (Representative) ; -4.4 (Two agents)
    - Mean Tax on borrowing (unconditional): 1.4 (Representative) ; 1.8 (Two agents)
    - Mean Tax on borrowing (conditional on SS): -0.1 (Representative) ; -0.3 (Two agents)
    - Mean Welfare gains: 0.3 (Representative) ; 1.1 (Two agents)
- Interpretation:
  - Two-agent planner unconditionally prefers redistribution to workers (payroll subsidy = -9.9 percent).
  - Conditional on Sudden Stop, planner curtails payroll subsidy (to -4.4 percent) to address wage-related pecuniary externality and to redistribute toward entrepreneurs to support asset prices.
  - Borrowing tax near zero during Sudden Stops; planner balances current and future externalities, avoiding large borrowing subsidies.
  - Payroll tax is an important ex post tool complementing ex ante borrowing tax; planner sets slightly higher macroprudential borrowing tax in two-agent economy than representative-agent.

### Event analysis and welfare (simulation-based)
- Event construction and welfare metric:
  - Simulations: 100,000 periods for CE and SP.
  - Sudden Stop events identified separately for CE and SP; event windows are five years centered at event year.
  - Compensating consumption variation γ computed from:
    - E0 ∑_{t=0}^{∞} β^{t} [ U((1+γ)c^{CE}_{t} − G(l^{CE}_{t})) + U((1+γ)x^{CE}_{t}) ] = E0 ∑_{t=0}^{∞} β^{t} [ U(c^{SP}_{t} − G(l^{SP}_{t})) + U(x^{SP}_{t}) ]. (40)
  - Mean welfare gain = average γ with ergodic distribution.
- Dynamics during Sudden Stops (quantitative findings):
  - Planner mitigates Fisherian deflation; asset-price correction in planner’s economy is about 14 and 24 percentage points smaller in representative-agent and two-agent economies, respectively.
  - Corrections in credit-to-GDP and consumption (planner vs competitive):
    - Representative-agent: credit-to-GDP correction ~6 percentage points smaller; consumption correction ~8 percentage points smaller.
    - Two-agent: credit-to-GDP correction ~7 percentage points smaller; consumption correction ~21 percentage points smaller.
  - Cross-sectional consumption effects (competitive two-agent economy):
    - Entrepreneurs’ consumption decreases by about 30 percent during Sudden Stop vs long-run average.
    - Workers’ consumption decreases by about 10 percent during Sudden Stop vs long-run average.
    - Competitive representative-agent economy consumption drops by 19 percent.
  - Planner’s redistribution during Sudden Stops:
    - Drop in entrepreneurial consumption is about 21 percentage points smaller in planner’s equilibrium (two-agent).
    - Drop in workers’ consumption is about 3 percentage points larger in planner’s equilibrium (indicating redistribution from workers to entrepreneurs).
- Welfare implications:
  - Mean welfare gains (Table 3):
    - Representative-agent economy: 0.3 percent.
    - Two-agent economy: 1.1 percent.
  - Reasons higher gains in two-agent economy:
    - Tax system allows redistribution; planner reduces payroll subsidy during Sudden Stops to help entrepreneurs and support asset prices.
    - Probability of Sudden Stop somewhat lower in two-agent economy due to ex post interventions.
    - Payroll tax used in normal times to reduce consumption volatility (correlation evidence reported).
- Sensitivity to welfare weight on workers ω (Table 4 figures, all in percent; SS = Sudden Stop):
  - ω = 0.6 ; ω = 1 ; ω = 1.4
    - Mean Payroll tax (unconditional): 0.3 ; -9.9 ; -15.6
    - Mean Payroll tax (conditional on SS): 7.8 ; -4.4 ; -11.2
  - Interpretation: lower ω → payroll tax higher (smaller subsidy); higher ω → larger subsidy. In all ω cases, payroll tax is higher (subsidy smaller) conditional on a Sudden Stop.

### Labor shocks: extension, calibration, and implications
- Model extension:
  - Labor supply shock affects disutility of labor via ψ_t in G(h_t; ψ_t).
  - ψ_t two-state Markov: ψ_n and ψ_l.
  - Transition probabilities:
    - P(ψ_{t+1} = ψ_l | ψ_t = ψ_n) = 5%
    - P(ψ_{t+1} = ψ_l | ψ_t = ψ_l) = 0%
  - Calibration:
    - ψ_n = 0.352
    - ψ_l = 0.369 → calibrated so labor supply declines by 4% on average when shock materializes.
    - Probability and magnitude chosen to match 4 percent deviation events in OECD total hours worked (1980-2019), 5th percentile.
- Simulation results (two-agent economy with labor shocks) — Table (5) (figures in percent except credit-to-GDP in percentage points):
  - CE / SP
    - Probability of Sudden Stop: 4.1 / 2.4
    - Probability of Sudden Stop & Labor Shock: 0.4 / 0.2
    - Drop in credit-to-GDP in Sudden Stop: -11.2 / -4.6
    - Drop in credit-to-GDP in Sudden Stop & Labor Shock: -12.6 / -4.4
    - Drop in entr. consumption in Sudden Stop: -29.6 / -8.0
    - Drop in entr. consumption in Sudden Stop & Labor Shock: -36.9 / -10.3
    - Drop in asset price in Sudden Stop: -34.6 / -10.3
    - Drop in asset price in Sudden Stop & Labor Shock: -41.4 / -12.6
    - Payroll tax (unconditional): -10.1
    - Payroll tax (conditional on Sudden Stop): -5.1
    - Payroll tax (conditional on Sudden Stop & Labor Shock): -4.4
    - Borrowing tax (unconditional): -1.9
    - Borrowing tax (conditional on Sudden Stop): -0.13
    - Borrowing tax (conditional on Sudden Stop & Labor Shock): -0.03
    - Mean welfare gains: -1.2
- Optimal policy responses and asymmetries:
  - Representative-agent: labor shock tightens collateral constraint → somewhat higher payroll tax to tackle wage-related pecuniary externality.
  - Two-agent: planner uses payroll tax asymmetrically:
    - If constraint not binding (debt before kink): implement higher payroll subsidy (more negative payroll tax) to support workers.
    - If constraint binding (debt after kink): reduce payroll subsidy to avoid harming asset prices; transfers to entrepreneurs prioritized.
  - Borrowing tax:
    - Labor shock increases borrowing tax when constraint not binding to lean against future externalities; when constraint binds, planner cuts borrowing tax more when labor shock hits.
- Policy implications:
  - Both ex ante macroprudential borrowing tax and ex post payroll tax are required.
  - Payroll tax asymmetric use underscores redistribution channel importance when labor shocks interact with collateral constraints.

### Numerical algorithm and appendix highlights
- Collateral constraint derivation (Appendix A):
  - After repayment, total liabilities at start of t: b_{t+1}/R + θ_l w_t l_t + θ_v p_v v_t.
  - Lender outside option upon seizure: κ_t q_t k_t.
  - Incentive compatibility V_R ≤ V_NR yields:
    - b_{t+1}/R + θ_l w_t l_t + θ_v p_v v_t ≤ κ_t q_t k_t.
  - Derivation unchanged by payroll taxes for decentralized economy with payroll taxes.
- Representative-agent economy (Appendix B) key conditions:
  - Household problem: max E_0 ∑ β^{t} U(c_t − G(l_t)). (44) with c_t = w_t l_t + d_t. (45)
  - Firm collateral constraint: b_{t+1}/R + θ_v p_v v_t + θ_l w_t l_t ≤ κ_t q_t k_t. (50)
  - Firm FOCs include µ_{f,t} appearing in F_{v,t}, F_{l,t}, and Euler condition. (51)-(54)
  - Planner implementability/price condition (57) and multiplier relations (58)-(63).
  - Decentralized-tax expressions mirror two-agent counterparts (65)-(69).
- Numerical algorithm (Appendix C):
  - CE solver:
    - Euler-equation iteration across grid of 900 grid points = 150 debt values × 6 states.
    - Equations (70)-(77) solved for policy functions { ̃b, x, c, l, q, w, v, μ }.
    - Convergence tolerance ε = 10^{−3}.
  - SP solver:
    - Value function iteration with nested fixed-point; inner-loop fixed-grid optimization.
    - V(b,Ξ) representation (78) and constraints (79)-(83).
    - Inner-loop subgrid for ̃b: 5000 values.
    - Convergence tolerance ε = 10^{−3}.
  - Iteration steps and constraint-handling rules explicitly described in source.

*Source: wpiea2022147-print-pdf*

### 2.1  Workers

### 2.1  Workers

### Preferences and utility
- The economy is populated by a unit mass of identical workers with lifetime expected utility
  - E0 ∑_{t=0}^{∞} β^{t} U(c_{t} − G(h_{t})), (1)
- U(·) is standard concave, twice continuously differentiable, satisfying the Inada conditions.
- Utility depends on consumption c_{t} and labor supply h_{t} through the composite commodity c_{t} − G(h_{t}) (Greenwood, Hercowitz, and Huffman (1988)), where:
  - G(h_{t}) is convex, strictly increasing, continuously differentiable, measuring disutility of labor.
  - The composite form removes the wealth effect on labor supply to prevent counterfactual increases in labor during crisis.
- Functional forms specified:
  - U(c_{t} − G(h_{t})) = [(c_{t} − G(h_{t}))^{(1−σ)} − 1]/(1 − σ) with risk aversion coefficient σ.
  - G(h_{t}) = ψ h_{t}^{1+φ}/(1 + φ) with 1/φ being the Frisch elasticity of labor supply.

### Hand-to-mouth assumption and motivation
- Workers are hand-to-mouth agents who cannot pledge labor income to borrow inter-temporally; this represents a population segment without access to credit markets and consumption smoothing.
- Empirical motivation: Kaplan, Violante and Weidner (2014) document that one-third of all US households live hand-to-mouth.
- The model can be extended to allow worker access to credit at the cost of analytical complexity; implications of such an extension are discussed later in Section 3.3.

### Budget constraint and first-order conditions
- Workers’ budget constraint:
  - c_{t} ≤ w_{t} h_{t}, (2)
  - where w_{t} denotes the wage received for labor supplied.
- Maximization of (1) subject to (2) yields first-order optimality conditions:
  - Consumption: c_{t}: u_{c,t} = λ_{w t}, (3)
  - Labor: h_{t}: G_{h,t} = w_{t}, (4)
  - λ_{w t} denotes the Lagrange multiplier associated with the worker’s budget constraint (2).
- Notational conventions:
  - All Lagrange multipliers are scaled by β^{t} (e.g., λ_{w t} = β^{t} \hat{λ}_{w t} in the Lagrangean).
  - X_{i,t} denotes derivatives of function X(·) with respect to variable i at time t.

*Source: wpiea2022147-print-pdf - 2.1  Workers*

### 3.2  Decentralized Economy

### 3.2  Decentralized Economy

### Decentralized budget and first-order conditions
- Entrepreneurs' budget constraint (equation numbering in source):  
  x_t + (1 + τ^b_{t-1}) b_t + p^v v_t + (1 + τ^l_t) w_t l_t + q_t k_{t+1} ≤ y_t + b_{t+1}/R + q_t k_t + T_t, with T_t = τ^b_{t-1} b_t + τ^l_t l_t. (Expression as in source.)
- Taxes and timing assumptions: tax revenues are rebated back to private agents; tax payments and rebates are settled at the end of the period after production.
- Private-agent first-order conditions in the decentralized economy:  
  - Euler for borrowing: U_{x,t} (1 − μ_t) = β R (1 + τ^b_t) E_t U_{x,t+1}. (equation (33))  
  - Entrepreneurs' labor demand: F_{l,t} = w_t (1 + τ^l_t + θ_l μ_t). (equation (34))
- Planner lacks direct lump-sum transfers and uses distortionary borrowing and payroll taxes/subsidies (Pigouvian in nature) both to address collateral-constraint-induced inefficiencies and to perform redistribution between workers and entrepreneurs.

### Interpretation and role of instruments
- Borrowing tax/subsidy τ^b_t:
  - Levied on new borrowing b_{t+1} determined at t but applied at t+1 when debt is repaid.
  - Aims to internalize pecuniary externalities operating via current and future prices of capital q_t and q_{t+1}.
- Payroll tax/subsidy τ^l_t:
  - Tackles the wedge between competitive and social optimal labor decisions.
  - Affects labor demand, wages, and thus the portion of payroll costs financed with working capital loans (θ_l), with implications for collateral tightness and redistribution.

### Equilibrium linkage
- All other equilibrium conditions in the decentralized economy remain the same as outlined in section 2 (as stated in source).

---

### 3.3  Optimal Tax Rates

- Optimal borrowing tax (from combining planner and private-agent Euler conditions):  
  τ^b_t = 1/(β R E_t U_{x,t+1}) [ μ^{SP}_t − U_{x,t} μ_t + ξ_t U_{xx,t} q_t − β R ξ_t E_t Ω_{t+1} ] − 1/(E_t U_{x,t+1}) E_t[ ξ_{t+1} U_{xx,t+1} q_{t+1} ]. (equation (35))
  - Functional form identical to Bianchi and Mendoza (2018).
  - Consists of two components corresponding to pecuniary externalities via q_t and q_{t+1}.
  - When the collateral constraint does not bind at t (μ_t = μ^{SP}_t = ξ_t = 0), the tax reduces to:  
    τ^{MP}_t = − 1/(E_t U_{x,t+1}) E_t[ ξ_{t+1} U_{xx,t+1} q_{t+1} ] = − 1/(E_t U_{x,t+1}) E_t[ κ_{t+1} μ^{SP}_{t+1} q_{t+1} U_{xx,t+1} / U_{x,t+1} ]. (equation (36))  
    - This expression can be shown to be always positive when the collateral constraint binds only in expectation, implying a tax on borrowing levied in good times to allow more borrowing in bad times (macroprudential interpretation).
  - When collateral constraint binds at t, the current-price-of-capital term is non-zero and can push for a subsidy on borrowing (to relax the collateral constraint via higher q_t), while the future-price term pushes for a tax (to avoid tighter future constraints). Planner balances these opposing effects.

- Optimal payroll tax (from planner labor condition and private-agent labor condition):  
  τ^l_t = ( θ_l μ_t + (λ^{SP,e}_t − λ^{SP,w}_t)/λ^{SP,e}_t ) (G_{ll,t}/G_{l,t}) l_t. (equation (37))
  - Decomposed using (21) and (22) into:  
    τ^l_t = ( θ_l μ_t + (U_{x,t} − ω U_{c,t})/(U_{x,t} − ξ_t U_{xx,t} q_t) − ξ_t U_{xx,t} q_t/(U_{x,t} − ξ_t U_{xx,t} q_t) ) (G_{ll,t}/l_t). (equation (38)) (expression as in source.)
  - Two main components:
    1. Pecuniary externality via the wage: θ_l μ_t G_{ll,t} l_t > 0 implies a desire to increase payroll tax (or decrease subsidy) to discourage labor demand and prefunding needs financed by loans.
    2. Distributive externality: planner addresses the gap between agents' marginal utilities of consumption. If U_{x,t} > ω U_{c,t} → tighter payroll tax (or lower subsidy) to shift resources from workers to entrepreneurs; if U_{x,t} < ω U_{c,t} → lower payroll tax (or higher subsidy) to shift resources from entrepreneurs to workers.
  - When collateral constraint binds and −ξ_t U_{xx,t} q_t /(U_{x,t} − ξ_t U_{xx,t} q_t) > 0, payroll tax also internalizes interaction between asset-price pecuniary externality and distributive externality; raising payroll tax can mitigate Fisherian deflation ex post during Sudden Stop episodes.
  - Payroll tax role differs from representative-agent economy: even if θ_l = 0 (no direct working-capital payroll channel), payroll tax remains useful in the two-agent economy to tackle distributive externality stemming from heterogeneity.

- Additional notes on payroll tax component:
  - The distributive-externality component pushing for a payroll tax is present in the model because workers do not participate in capital markets; it would persist if workers held capital but would reflect relative contributions to asset-price declines depending on capital holdings.

---

### 3.4  Additional Tools

- Capital-holding tax/subsidy τ^k_t:
  - If planner can levy τ^k_t on capital holdings k_{t+1} with proceeds rebated lump-sum to entrepreneurs, the implementability/pricing condition becomes:  
    (1 + τ^k_t) q_t U_{x,t} = β E_t U_{x,t+1} { F_{k,t+1} + q_{t+1} [ 1 + κ_{t+1}/θ_v ( F_{v,t+1}/p_v − 1 ) ] }. (equation (39))
  - Without constraints on τ^k_t, planner could choose τ^k_t to implement any q_t and effectively eliminate binding of the collateral constraint.
  - Natural lower bound: τ^k_t ≥ τ, where τ < 0 (lower bound), imposed by transversality condition or other funding/political constraints. The planner may face limits to subsidizing capital.
  - Key takeaway: even with capital subsidies, the planner would still find positive macroprudential borrowing taxes useful in states where capital tax hits its lower bound. Capital subsidies reduce but do not eliminate the role of borrowing and payroll taxes. Quantitative analysis in appendix considers different τ values.

---

### 4  Quantitative Analysis — Calibration and Summary Statistics

- Solution method: global, non-linear solution algorithm; competitive economy via iteration on first-order conditions; planner via value function iteration and nested fixed point algorithm. (Details in Section C of appendix as stated in source.)
- Time period interpretation: each period = one year.
- Deviations from Bianchi and Mendoza (2018) in baseline calibration:
  - Global interest rate set fixed at R = 1.02 (vs. long-term average 1.01 in Bianchi and Mendoza (2018)).
  - Set θ_v = θ_l = 0.09 (instead of 0.16) to obtain share of working-capital loan to GDP equal to 0.13 as in the data.
  - Transition probability κ_h → κ_l set to 5 percent (P_{h,l} = 0.05) instead of 10 percent to match a probability of a crisis (Sudden Stop) equal to 4%.
  - Welfare weight on workers ω = 1 in baseline (strictly utilitarian social welfare function); comparative statics performed with respect to ω in later sections.
- Calibration table (parameters and values as reported in source):
  - Risk aversion σ = 1
  - Labor disutility coefficient ψ = 0.352
  - Frisch elasticity of labor supply 1/φ = 2
  - Share of intermediate good in output α_v = 0.45
  - Share of labor in output α_l = 0.352
  - Share of assets in output α_k = 0.008
  - Interest rate R = 1.02
  - TFP process ρ_z = 0.78
  - σ_ε = 0.01 (noted as σ = 0.01 in source)
  - Discount factor β = 0.95
  - Working capital coefficient θ_v = θ_l = 0.09
  - Tight credit regime κ_l = 0.75
  - Normal credit regime κ_h = 0.90
  - Transition probability, κ_h to κ_l: P_{h,l} = 0.05
  - Transition probability, κ_l to κ_l: P_{l,l} = 0.00
  - Welfare weight on workers ω = 1
- Sudden Stop definition and measurement (as used in simulated statistics):
  - Sudden Stop event: period in which the drop in long-term borrowing (b) between two periods over output (current-account-to-GDP ratio) exceeds two standard deviations of its long-term distribution, consistent with Bianchi and Mendoza (2018).
  - Sudden Stops are infrequent, accompanied by large reversals in the current account and credit crunches, and by a binding collateral constraint.

_Source: wpiea2022147-print-pdf - 3.2  Decentralized Economy_

### 99.5 percent of the time.  However, the opposite is not true, i.e.  a binding collateral constraint is

### wpiea2022147-print-pdf - 99.5 percent of the time.  However, the opposite is not true, i.e.  a binding collateral constraint is

### Sudden Stops and Collateral Constraints
- A binding collateral constraint is not sufficient for the occurrence of a Sudden Stop; the constraint can bind with multipliers very close to zero implying small corrections in credit.
- The probability that the constraint binds is higher than the probability of a Sudden Stop event materializing for both the representative- and two-agent economies.
- Representative- and two-agent competitive equilibria produce similar probabilities of Sudden Stop episodes and similar ratios of short-term (working capital loan) and long-term borrowing over GDP, consistent with the data.
- Footnote finding: "99.6 percent and 100 percent of the Sudden Stop events in the representative-agent and two-agent economies, respectively, are characterized by changes in the current account as percentage of GDP that are high enough to be in the 95th percentile."

### Model Statistics and Key Moments (Table (2) — Panels A and B)
- Note: Panels A and B report numbers in percent and levels, respectively. Unconditional averages over the simulation horizon. Credit-to-GDP is the ratio of the long-term loan (b_{t+1}) to GDP. CE=Competitive Economy, SP=Social Planner.
- PANEL A (Selected statistics)
  - Probability of Sudden Stop:
    - Repr. agent CE: 4.2
    - Repr. agent SP: 3.1
    - Two-agent CE: 4.0
    - Two-agent SP: 2.7
  - Probability of binding constraint:
    - Repr. agent CE: 13.8
    - Repr. agent SP: 15.7
    - Two-agent CE: 4.3
    - Two-agent SP: 4.7
  - Average Working capital loan / GDP:
    - Repr. agent CE: 13.1
    - Repr. agent SP: 13.1
    - Two-agent CE: 13.1
    - Two-agent SP: 13.7
  - Standard deviation (Working capital loan / GDP):
    - Repr. agent CE: 0.002
    - Repr. agent SP: 0.001
    - Two-agent CE: 0.002
    - Two-agent SP: 0.002
  - Average Credit-to-GDP (b_{t+1}) / GDP:
    - Repr. agent CE: 24.4
    - Repr. agent SP: 16.4
    - Two-agent CE: 21.1
    - Two-agent SP: 17.0
  - Standard deviation (Credit-to-GDP):
    - Repr. agent CE: 0.038
    - Repr. agent SP: 0.020
    - Two-agent CE: 0.033
    - Two-agent SP: 0.016
- PANEL B (Selected aggregate moments)
  - Average aggregate consumption:
    - Repr. agent CE: 0.26
    - Repr. agent SP: 0.27
    - Two-agent CE: 0.26
    - Two-agent SP: 0.29
  - Standard deviation (aggregate consumption):
    - Repr. agent CE: 0.017
    - Repr. agent SP: 0.014
    - Two-agent CE: 0.015
    - Two-agent SP: 0.015
  - Average asset price:
    - Repr. agent CE: 0.11
    - Repr. agent SP: 0.09
    - Two-agent CE: 0.11
    - Two-agent SP: 0.10
  - Standard deviation (asset price):
    - Repr. agent CE: 0.008
    - Repr. agent SP: 0.005
    - Two-agent CE: 0.009
    - Two-agent SP: 0.004
  - Average wage:
    - Repr. agent CE: 0.28
    - Repr. agent SP: 0.28
    - Two-agent CE: 0.28
    - Two-agent SP: 0.29
  - Standard deviation (wage):
    - Repr. agent CE: 0.005
    - Repr. agent SP: 0.005
    - Two-agent CE: 0.006
    - Two-agent SP: 0.006
- Data benchmark note: In the data (Bianchi and Mendoza (2018)), short-term borrowing (working capital loans) as percent of GDP is 13.3%, and long-term borrowing (NFA) as percent of GDP is 25% (based on 2013 US data). The model targets the probability of Sudden Stops and working capital loan over GDP in the two-agent competitive economy and uses the same calibrated parameter for the representative-agent competitive equilibrium.

### Policy Functions Analysis and Optimal Taxes
- General findings on borrowing policy rules (Figure 1 description):
  - For both representative- and two-agent economies, the social planner (SP) chooses lower new borrowing b_{t+1} than private agents before the borrowing constraint turns binding (the kink point).
  - The planner’s borrowing constraint turns binding at lower levels of debt than for private agents because lower borrowing and consumption reduce collateral values, tightening the constraint.
  - When productivity is low (z_t = z_l) and financial conditions are tight (κ_t = κ_l):
    - Representative-agent: private and social equilibrium borrowing follow very similar dynamics.
    - Two-agent: private equilibrium shows a sharp decline in new borrowing once the borrowing constraint binds; social planner’s decline is smoother because the planner can redistribute resources from workers to entrepreneurs to speed recovery of entrepreneurs’ borrowing ability.
- Borrowing tax dynamics (Figure 2, top chart):
  - For high productivity (z_t = z_h) and favorable financial conditions (κ_t = κ_h):
    - In both economies, the borrowing tax rate is about zero for low levels of current period borrowing, rises as borrowing increases, then decreases at the point when the collateral constraint turns binding.
    - The decline of the tax rate is sharper in the representative-agent economy than in the two-agent economy (the two-agent decline is smoother).
  - Mechanism difference:
    - Representative-agent planner supports borrowing primarily by reducing the borrowing tax when the constraint binds.
    - Two-agent planner can also support borrowing by transferring resources from workers to entrepreneurs via the payroll tax.

### Payroll Tax and Redistribution Motives
- Payroll tax dynamics (Figure 2, bottom charts):
  - High productivity and favorable financial conditions (z_t = z_h, κ_t = κ_h):
    - Representative-agent: payroll tax is zero when constraint slack and turns slightly positive when the borrowing constraint starts to bind.
    - Two-agent: payroll tax is negative when constraint is slack (payroll subsidy), becomes less negative and approaches zero as the constraint tightens; once constraint binds, the planner decreases the payroll subsidy further to address distributive externality.
  - Low productivity and unfavorable financial conditions (z_t = z_l, κ_t = κ_l):
    - Planner in the two-agent economy reduces the payroll subsidy for higher indebtedness and ultimately levies increasingly higher positive payroll taxes to transfer resources from workers to entrepreneurs.
    - The positive payroll tax motive is twofold: address the pecuniary externality via wages and redistribute to constrained entrepreneurs to support asset prices and enable borrowing and expansion.
- Implication: The ability to use payroll taxes (including negative payroll taxes/subsidies) in the two-agent framework provides an additional policy instrument to mitigate Sudden Stops and speed recovery relative to representative-agent frameworks.

### Robustness and Extensions
- The model introduces a working capital loan component to payroll costs, tightening the collateral constraint relative to some prior models.
- Robustness checks: the authors solved for θ_l = 0 and θ_l = 0.09 using the same calibration otherwise; Section A of the Online Appendix reports results and detailed sensitivity analysis showing results continue to hold under alternative parameter choices.

*Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022147-print-pdf.pdf*

### 4.3  Event Analysis and Welfare

### 4.3 Event Analysis and Welfare

### Event construction and methodology
- Competitive and planner’s equilibria are simulated for 100,000 periods.
- Sudden Stop events are identified separately for the competitive and planner’s equilibria; the comparison of dynamics in Figure 3 is performed at the Sudden Stop episodes of the competitive equilibrium.
- Five-year event windows centered around the year when the event materializes are constructed and averages for each variable computed across the cross section of events at each date.
- Results are presented in terms of deviations from the long-term average.
- Equation used to compute the compensating consumption variation γ for the two-agent economy (similar for the representative-agent economy but only accounts for the consumption of the representative agent):
  E0 ∑_{t=0}^{∞} β^{t} [ U((1+γ)c^{CE}_{t} − G(l^{CE}_{t})) + U((1+γ)x^{CE}_{t}) ] = E0 ∑_{t=0}^{∞} β^{t} [ U(c^{SP}_{t} − G(l^{SP}_{t})) + U(x^{SP}_{t}) ].(40)
  - Superscripts CE and SP denote values in the competitive and social planner’s equilibria, respectively.
  - Mean welfare gain is the average γ computed with the ergodic distribution.

### Dynamics during Sudden Stops: quantitative findings
- When a Sudden Stop materializes, borrowing ability is curtailed in both equilibria, leading to drops in credit-to-GDP, consumption, and asset prices.
- The planner internalizes pecuniary externalities when the constraint binds and mitigates Fisherian deflation:
  - Asset-price correction (drop) in the planner’s economy is about 14 and 24 percentage points smaller in the representative-agent and two-agent economies, respectively.
  - Correction in credit-to-GDP (consumption) in the planner’s economy is about:
    - Representative-agent: 6 (credit-to-GDP) and 8 (consumption) percentage points smaller.
    - Two-agent: 7 (credit-to-GDP) and 21 (consumption) percentage points smaller.
- Cross-sectional consumption effects (competitive two-agent economy, calibrated):
  - Entrepreneurs’ consumption decreases by about 30 percent during a Sudden Stop compared to long-run average.
  - Workers’ consumption decreases by about 10 percent during a Sudden Stop compared to long-run average.
  - Competitive representative-agent economy consumption drops by 19 percent.
- Planner’s redistribution during Sudden Stops:
  - Drop in entrepreneurial consumption is about 21 percentage points smaller in the planner’s equilibrium (two-agent economy).
  - Drop in workers’ consumption is about 3 percentage points larger in the planner’s equilibrium (indicating redistribution from workers to entrepreneurs during Sudden Stops).

### Tax policy results and numerical summaries (Table 3: Decentralized Representative & Two-agent Economies)
- Note: All numbers are in percent. SS = Sudden Stop.
- Representative vs Two agents (columns):
  - Mean Payroll tax (unconditional): 0.1 (Representative) ; -9.9 (Two agents)
  - Mean Payroll tax (conditional on SS): 1.0 (Representative) ; -4.4 (Two agents)
  - Mean Tax on borrowing (unconditional): 1.4 (Representative) ; 1.8 (Two agents)
  - Mean Tax on borrowing (conditional on SS): -0.1 (Representative) ; -0.3 (Two agents)
  - Mean Welfare gains: 0.3 (Representative) ; 1.1 (Two agents)

### Interpretation of tax results and policy trade-offs
- Payroll tax:
  - Unconditionally, the planner would like to redistribute resources to workers (payroll subsidy) in the two-agent economy (mean unconditional payroll tax = -9.9 percent).
  - Conditional on a Sudden Stop, the planner substantially curtails the average payroll subsidy (mean conditional payroll tax = -4.4 percent in two-agent economy) for two reasons:
    - To address the amplified pecuniary externality operating via the wage by reducing the payroll subsidy (increasing the payroll tax).
    - To redistribute towards entrepreneurs during Sudden Stops because higher entrepreneurial consumption has positive effects on asset prices and tightness of the constraint.
  - In the representative-agent economy, the change in average payroll tax conditional on a Sudden Stop is relatively small because the redistributive motive is absent; payroll tax mainly tackles pecuniary externality via the wage.
- Borrowing tax:
  - Planner sets a borrowing tax close to zero in both economies during Sudden Stops to mitigate the pecuniary externality operating via the asset price.
  - Planner does not introduce large borrowing subsidies during Sudden Stops because optimal policy balances current and future externalities from binding constraints that push in opposite directions.
  - Unconditional average borrowing taxes: 1.4 percent (Representative) ; 1.8 percent (Two agents).
  - Conditional on Sudden Stop average borrowing taxes: -0.1 percent (Representative) ; -0.3 percent (Two agents).
- Complementarity of tools:
  - Payroll tax is an important ex post tool during Sudden Stops due to limits on borrowing subsidies.
  - Payroll tax is complementary to ex ante (macroprudential) borrowing tax; ex post intervention does not eliminate the need for ex ante measures because anticipated redistribution can weaken entrepreneurs’ precautionary motive and induce more borrowing in normal times.
  - Planner levies a slightly higher macroprudential borrowing tax in the two-agent economy than in the representative-agent one to discourage excessive borrowing.

### Welfare implications
- Sudden Stops reduce consumption and asset prices and tighten working-capital funding; reducing frequency and severity of Sudden Stops yields welfare gains by lowering consumption volatility and increasing production efficiency.
- Mean welfare gains (Table 3):
  - Representative-agent economy: 0.3 percent.
  - Two-agent economy: 1.1 percent.
- Reasons welfare gains are higher in the two-agent economy:
  - Tax system allows redistribution between agents; planner decreases payroll subsidy during Sudden Stops to help entrepreneurs and support asset prices.
  - Probability of Sudden Stop is somewhat lower in the two-agent economy, partly due to ability to intervene ex post and mitigate Sudden Stops.
  - Planner uses payroll tax for redistribution even in normal times, reducing consumption volatility:
    - Correlation of entrepreneurial consumption and payroll tax conditional on Sudden Stops is close to minus one in both economies.
    - Outside Sudden Stops: correlation is -0.05 in representative-agent economy and -0.39 in two-agent economy, indicating continued use of payroll tax in normal times in two-agent economy.
  - Redistribution from entrepreneurs to workers outside Sudden Stops can slightly raise probability of a binding collateral constraint in planner’s equilibrium, even though probability of Sudden Stop is considerably lower.

### Sensitivity to planner’s weight on workers (ω) and payroll tax policy
- Results shown for payroll tax only; rest of results qualitatively similar to ω = 1.
- Figure 4 (described):
  - Payroll tax is higher (subsidy is smaller) for lower weight ω on workers; payroll tax is lower (subsidy larger) for higher ω.
  - Panels show payroll tax as a function of outstanding debt b_t for high productivity/high financial conditions (z_t = z_h, κ_t = κ_h) and low productivity/low financial conditions (z_t = z_l, κ_t = κ_l) for ω = 0.6, ω = 1, ω = 1.2.
- Table 4: Payroll taxes in two-agent economy for different ω’s
  - Note: All figures are in percent. SS = Sudden Stop.
  - ω = 0.6 ; ω = 1 ; ω = 1.4
    - Mean Payroll tax (unconditional): 0.3 ; -9.9 ; -15.6
    - Mean Payroll tax (conditional on SS): 7.8 ; -4.4 ; -11.2
- Interpretation:
  - In all ω cases, average payroll tax is higher (subsidy is smaller) conditional on a Sudden Stop as planner reduces redistribution to workers to support entrepreneurs and mitigate Fisherian deflation.
  - For ω = 0.6 the payroll tax is positive (redistribution from workers to entrepreneurs); for ω = 1.4 the payroll tax is more negative (larger subsidy to workers).

*Source: IMF Working Paper — 4.3 Event Analysis and Welfare (wpiea2022147-print-pdf - 4.3 Event Analysis and Welfare).*

### 4.5  Labor shocks

### 4.5  Labor shocks

### Overview
- The model is extended to include a labor supply shock that affects the disutility from supplying labor and captures periods when workers are more or less willing to supply labor.
- The labor supply shock has two main equilibrium effects:
  - Directly affects workers’ welfare.
  - Through production and general equilibrium effects, affects entrepreneurs’ welfare and the asset price.
- The planner uses taxes on borrowing and payroll to alleviate adverse effects of a negative labor supply shock.
- The payroll tax exhibits an asymmetric response with respect to the tightness of the borrowing constraint in the two-agent economy; this asymmetry highlights the importance of the redistribution channel.

### Model and calibration
- Workers’ preferences with the labor supply shock:
  - E0 ∑∞_{t=0} β^t U(c_t − G(h_t; ψ_t))
  - ψ_t is a two-state, regime-switching Markov process taking values ψ_n and ψ_l for “normal times” and “crisis times,” respectively.
- Transition probabilities:
  - P(ψ_{t+1} = ψ_l | ψ_t = ψ_n) = 5%
  - P(ψ_{t+1} = ψ_l | ψ_t = ψ_l) = 0%
- Calibration:
  - ψ_n = 0.352 (same as benchmark)
  - ψ_l = 0.369, calibrated so that labor supply declines by 4% on average when the negative labor supply shock materializes.
  - Probability and magnitude chosen to match the probability of a 4 percent deviation from an HP trend in annual OECD total hours worked for 1980-2019; such events belong to the 5th percentile of the distribution of percent-deviations from an HP trend.

### Simulation results (two-agent economy with labor shocks)
- Qualitative results from the economy without labor shocks continue to hold, but the labor shock amplifies magnitudes:
  - Larger drops in credit as a percentage of GDP, entrepreneurial consumption, and asset prices.
  - The labor shock amplifies Fisherian deflation dynamics.
- The effects are similar in the representative-agent economy (not reported here).
- Table (5) — Statistics for two-agent economy with labor shocks (all figures in percent apart from credit-to-GDP expressed in percentage points as the ratio of the long-term loan to GDP). CE=Competitive Economy; SP=Social Planner.
  - CESP
  - Probability of Sudden Stop 4.1 2.4
  - Probability of Sudden Stop & Labor Shock 0.4 0.2
  - Drop in credit-to-GDP in Sudden Stop -11.2 -4.6
  - Drop in credit-to-GDP in Sudden Stop & Labor Shock -12.6 -4.4
  - Drop in entr. consumption in Sudden Stop -29.6 -8.0
  - Drop in entr. consumption in Sudden Stop & Labor Shock -36.9 -10.3
  - Drop in asset price in Sudden Stop -34.6 -10.3
  - Drop in asset price in Sudden Stop & Labor Shock -41.4 -12.6
  - Payroll tax (unconditional) --10.1
  - Payroll tax (conditional on Sudden Stop) --5.1
  - Payroll tax (conditional on Sudden Stop & Labor Shock) --4.4
  - Borrowing tax (unconditional) -1.9
  - Borrowing tax (conditional on Sudden Stop) -0.13
  - Borrowing tax (conditional on Sudden Stop & Labor Shock) --0.03
  - Mean welfare gains -1.2

### Optimal policy responses and asymmetries
- Representative-agent economy:
  - A labor shock increases the tightness of the borrowing constraint, requiring a somewhat higher payroll tax to tackle the pecuniary externality operating through wages.
- Two-agent economy:
  - The planner can use the payroll tax both to address the pecuniary externality through wages and to perform redistribution.
  - When the labor supply shock hits but the borrowing constraint is not binding (debt levels before the kink point), the planner implements a higher payroll subsidy (more negative payroll tax) to support workers directly affected by the shock.
  - When the borrowing constraint binds (debt levels after the kink), transferring resources from entrepreneurs to workers negatively affects asset prices; the planner reduces the payroll subsidy in this state.
  - Table 5 shows the average payroll tax conditional on a Sudden Stop and negative labor shock is the highest, highlighting the payroll tax’s role for redistributive purposes.
- Borrowing tax:
  - The labor shock does not introduce an asymmetry in the optimal borrowing tax between the two economies.
  - In both economies, the tax on borrowing is higher in the presence of a labor shock and a non-binding borrowing constraint compared to when labor shock events are absent (to lean against future pecuniary externalities).
  - When the constraint binds, the planner cuts the borrowing tax by more when the labor shock hits to help alleviate negative effects.

### Policy implications and interpretation
- The distributive externality operates alongside the pecuniary externality via collateral asset prices; their interaction matters for optimal policy during Sudden Stops.
- Optimal policy requires both:
  - An ex ante macroprudential tax on borrowing to limit borrowing when financing constraints are loose.
  - An ex post payroll tax to reallocate resources to entrepreneurs and support asset prices when financing constraints are tight.
- The payroll tax’s asymmetric use in the two-agent economy emphasizes redistribution as an important driver of policy response in the presence of labor supply shocks.

*Source: wpiea2022147-print-pdf - 4.5  Labor shocks*

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### Appendix

### A Derivation of the Collateral Constraint
- Setup:
  - At the beginning of period t, after previous period borrowing, b_t, has been repaid, total liabilities are b_{t+1}/R + θ_l w_t l_t + θ_v p_v v_t.
  - Lender’s outside option upon seizure/liquidation yields κ_t q_t k_t, where κ_t is the liquidation value of borrower’s assets.
  - Borrower has full negotiation power (following Jermann and Quadrini (2012)).
- Value from renegotiation:
  - V_R = b_{t+1}/R + θ_l w_t l_t + θ_v p_v v_t − κ_t q_t k_t + β E_t V(k_{t+1}, b_{t+1}). (Equation 42)
- Value from honoring debt:
  - V_NR = β E_t V(k_{t+1}, b_{t+1}). (Equation 43)
- Incentive compatibility and collateral constraint:
  - V_R ≤ V_NR implies b_{t+1}/R + θ_l w_t l_t + θ_v p_v v_t ≤ κ_t q_t k_t. (Resulting collateral constraint)
- Note on payroll taxes:
  - For decentralized economy with payroll taxes, funds borrower can divert do not include τ_{l,t} θ_l w_t l_t; derivation of collateral constraint remains unchanged and applies in both competitive and social planner equilibria.

### B Representative-agent Economy

B.1 Competitive Economy
- Household problem:
  - max_{c_t, l_t} E_0 ∑_{t=0}^{∞} β^{t} U(c_t − G(l_t)). (Equation 44)
  - Budget: c_t = w_t l_t + d_t. (Equation 45)
  - First-order conditions:
    - U_{c,t} = λ_{h,t}. (46)
    - G_{l,t} = w_t. (47)
- Firm problem (owned by households):
  - Objective: max_{b_{t+1}, k_{t+1}, l_t, v_t} E_0 ∑_{t=0}^{∞} β^{t} (U_{c,t+1}/U_{c,t}) d_t. (48)
  - Dividend definition: d_t ≡ b_{t+1}/R + q_t k_t + F(z_t, k_t, l_t, v_t) − b_t − p_v v_t − q_t k_{t+1} − w_t l_t. (49)
  - Collateral constraint: b_{t+1}/R + θ_v p_v v_t + θ_l w_t l_t ≤ κ_t q_t k_t. (50)
  - First-order conditions:
    - F_{v,t} = p_v (1 + θ_v μ_{f,t}). (51)
    - F_{l,t} = w_t (1 + θ_l μ_{f,t}). (52)
    - U_{c,t} (1 − μ_{f,t}) = β E_t U_{c,t+1}. (53)
    - q_t U_{c,t} = β E_t [U_{c,t+1} (q_{t+1} + F_{k,t+1}) + κ_{t+1} U_{c,t+1} μ_{f,t+1} q_{t+1}]. (54)

B.2 Planner’s Economy
- Planner maximization:
  - max_{c_t, b_{t+1}, l_t, v_t, q_t} E_0 ∑_{t=0}^{∞} β^{t} U(c_t − G(l_t)), subject to resource, collateral, and implementability constraints.
  - Resource constraint: c_t = b_{t+1}/R + F(z_t,1,l_t,v_t) − b_t − p_v v_t. (λ_{SP,r,t}) (55)
  - Collateral constraint: b_{t+1}/R + θ_v p_v v_t + θ_l G_{l,t} l_t ≤ κ_t q_t. (μ_{SP,r,t}) (56)
  - Implementability/price condition:
    - U_{c,t} q_t = β E_t U_{c,t+1} { F_{k,t+1} + q_{t+1} [1 + κ_{t+1} θ_v (F_{v,t+1}/p_v − 1)] }. (ξ_{r,t}) (57)
- First-order conditions and multiplier relations:
  - λ_{SP,r,t} = U_{c,t} − ξ_{r,t} U_{cc,t} q_t. (58)
  - λ_{SP,r,t} = β R (λ_{SP,r,t+1} − ξ_{r,t} Ω_{t+1}) + μ_{SP,r,t}. (59)
  - U_{c,t} G_{l,t} = λ_{SP,r,t} F_{l,t} − θ_l μ_{SP,r,t} (G_{ll,t} l_t + G_{l,t}) + ξ_{r,t} U_{cc,t} G_{l,t} q_t. (60)
  - μ_{SP,r,t} = λ_{SP,r,t} θ_v (F_{v,t}/p_v − 1). (61)
  - ξ_{r,t} = κ_t μ_{SP,r,t} U_{c,t}. (62)
- Optimal labor condition (combining relations):
  - F_{l,t} − G_{l,t} (1 + θ_l μ_{f,t}) = θ_l μ_{f,t} G_{ll,t} l_t. (63)

B.3 Decentralized Economy and Optimal Tax Rates
- Firm budget with policy instruments:
  - d_t ≡ b_{t+1}/R + q_t k_t + F(z_t,k_t,l_t,v_t) − (1 + τ_{b,r,t−1}) b_t − p_v v_t − q_t k_t − (1 + τ_{l,r,t}) w_t l_t + T_{b,r,t} + T_{l,r,t}. (64)
  - Rebates: T_{r,t} = τ_{b,r,t−1} b_t + τ_{l,r} l_t.
- Euler and labor demand with taxes:
  - U_{c,t} (1 − μ_{f,t}) = β R (1 + τ_{b,r,t}) E_t U_{c,t+1}. (65)
  - F_{l,t} = w_t (1 + τ_{l,r,t} + θ_l μ_{f,t}). (66)
- Payroll tax expression:
  - τ_{l,r,t} = θ_l μ_{f,t} G_{ll,t} / (G_{l,t} l_t). (67)
- Tax on borrowing and macroprudential tax:
  - τ_{b,r,t} = 1/(β R E_t U_{c,t+1}) [ μ_{SP,r,t} − U_{c,t} μ_{f,t} + ξ_{r,t} U_{cc,t} q_t − β R ξ_{r,t} E_t Ω_{t+1} ] − 1/(E_t U_{c,t+1}) E_t ξ_{r,t+1} U_{cc,t+1} q_{t+1}. (68)
  - τ_{MP,r,t} = − 1/(E_t U_{c,t+1}) E_t ξ_{r,t+1} U_{cc,t+1} q_{t+1}. (69)

### C Numerical Algorithm

Competitive equilibrium (CE) solver
- Method: Euler-equation iteration algorithm solving system (70)-(77) across a grid.
- Grid specification:
  - Total grid points: 900 grid points = 150 values of debt (b) × 6 states (3 productivity states × 2 pledgeable fraction states).
- Equations to solve for policy functions { ̃b(b, Ξ), x(b, Ξ), c(b, Ξ), l(b, Ξ), q(b, Ξ), w(b, Ξ), v(b, Ξ), μ(b, Ξ) } subject to:
  - (70) x(b,Ξ) + b + p_v v(b,Ξ) + w(b,Ξ) l(b,Ξ) = F(z,1,v(b,Ξ),l(b,Ξ)) + ̃b(b,Ξ)/R.
  - (71) ̃b(b,Ξ)/R + θ_v p_v v(b,Ξ) + θ_l w(b,Ξ) l(b,Ξ) ≤ κ(b,Ξ) q(b,Ξ).
  - (72) μ(b,Ξ) = 1 − β R E_{Ξ'|Ξ} U_x(x(b',Ξ')) / U_x(x(b,Ξ)).
  - (73) q(b,Ξ) U_x(x(b,Ξ)) = β E_{Ξ'|Ξ} [ U_x(x(b',Ξ')) × ( q(b',Ξ') + F_k(z',1,v(b',Ξ'),l(b,Ξ)) + κ(b',Ξ') μ(b',Ξ') q(b',Ξ') ) ].
  - (74) c(b,Ξ) = w(b,Ξ) l(b,Ξ).
  - (75) F_l(z,1,v(b,Ξ),l(b,Ξ)) = w(b,Ξ) (1 + θ μ(b,Ξ)).
  - (76) G_h(l(b,Ξ)) = w(b,Ξ).
  - (77) F_v(z,1,v(b,Ξ),l(b,Ξ)) = p_v (1 + θ μ(b,Ξ)).
- Iteration steps:
  1. Conjecture future policy functions for each grid point in b; use initial guess for first iteration.
  2. For each b, solve (70)-(77) for current policies. Distinguish whether collateral constraint (71) binds:
     - i. If constraint binds, solve and check μ(b,Ξ) > 0 via (72); if true proceed.
     - ii. If constraint does not bind, set μ(b,Ξ) = 0 and solve.
  3. Update conjectured future policy functions using current solutions.
  4. Convergence criterion: stop when sup_{b,Ξ} || y^i(b,Ξ) − y^{i−1}(b,Ξ) || < ε, where y = {̃b, c} and ε = 10^{−3}. Results checked for robustness to stricter ε.

Social planner (SP) solver
- Method: Value function iteration with nested fixed-point (inner loop fixed-grid optimization), delivering time-consistent policies.
- Value function representation:
  - V(b,Ξ) = max_{̃b,c,x,w,v,l,q,μ} ( ω U(c(b,Ξ) − G(l(b,Ξ))) + U(x(b,Ξ)) + β E_{Ξ'|Ξ} [ V(b',Ξ') ] ). (78)
- Constraints: (79)-(83) analogous to CE but in planner context.
- Additional policy function relations:
  - λ_{SP,w}(b,Ξ) = ω U_c(c(b,Ξ) − G(l(b,Ξ))). (84)
  - λ_{SP,e}(b,Ξ) = U_x(x(b,Ξ)) / [ 1 + κ(b,Ξ) (F_v(z,1,v(b,Ξ),l(b,Ξ))/p_v − 1)/θ ] × U_{xx}(x(b,Ξ))/U_x(x(b,Ξ)) q(b,Ξ). (85)
- Iteration steps:
  1. Outer loop: define future policies V(b',Ξ'), ̃b(b',Ξ'), x(b',Ξ'), c(b',Ξ'), q(b',Ξ'), w(b',Ξ'), l(b',Ξ'), v(b',Ξ') using previous-iteration solutions; initialize with CE solution for first iteration. Compute initial conjectures for l(b,Ξ), w(b,Ξ), λ_{SP,w}, λ_{SP,e}.
  2. Inner loop: for each b, solve for V(b,Ξ), ̃b(b,Ξ), x(b,Ξ), c(b,Ξ), v(b,Ξ) given future policies and outer-loop conjectures. Distinguish two cases regarding collateral constraint (80):
     - i. If collateral constraint does not bind, set μ(b,Ξ) = 0, solve for v(b,Ξ), then for each candidate ̃b on a subgrid of 5000 values compute x, c using (79) and (83), choose ̃b maximizing V(b,Ξ). If maximizing ̃b < sup_{̃b} (computed assuming constraint binds), proceed to step 3; else go to ii.
     - ii. If collateral constraint binds, for same 5000-point subgrid compute v, x, c from (80), (79), (83) and choose ̃b that maximizes current utility plus continuation value. Compute μ(b,Ξ) = (F_v(z,1,v(b,Ξ),l(b,Ξ))/p_v − 1) and restrict μ ≥ 0.
  3. Derive new l(b,Ξ) via optimality condition (24) using λ_{SP,w} and λ_{SP,e} and update c, x, w, q.
  4. Convergence criterion: stop when sup_{b,Ξ} || V^i(b,Ξ) − V^{i−1}(b,Ξ) || < ε, with ε = 10^{−3}.

- Numerical parameters and grid details explicitly stated:
  - CE grid: 900 grid points = 150 debt values × 6 states (3×2).
  - Inner-loop subgrid for ̃b: 5000 values.
  - Convergence tolerance: ε = 10^{−3}.

*Sudden Stops and Optimal Policy in a Two-agent Economy — Working Paper No. WP/2022/147*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022147-print-pdf.pdf_