## 2.1    Supply

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### Model setup — production and prices
- Economy: S sectors; mass N_s of firms in each sector s ∈ S.
- Firm production function (Eq. (1)):
  - y_is = z_is f_s(k_is, A_s n_is, m_is)
  - Definitions preserved: y_is = gross output; k_is = fixed factor; n_is = labor input; m_is = other variable inputs; A_s = sector-specific labor-augmenting productivity; z_is = firm-specific productivity.
  - Assumption: f_s(., 0, .) = f_s(., ., 0) = 0.
- Prices and factor costs:
  - p_is = price of output of firm i in sector s
  - w_s = wage rate per effective unit of labor
  - r_s = user cost for fixed factors
  - p^m_s = price of other variable inputs
  - Factor prices vary only at sector level; prices assumed constant in the short run.

### Demand structure (Eqs. (2)–(8)
- Aggregation: nested CES final and intermediate demand (Eq. (2)):
  - D = [∑_s N_s ξ_s D_s^(η−1)/η]^(η/(η−1)), with ξ_s sectoral demand shifter and η elasticity of substitution between sectors.
  - Symmetry pre-COVID: N_s ξ_s = 1, ∀s. During COVID define ξ'_s and ˆξ_s ≡ ξ'_s / ξ_s.
- Sectoral and firm demand (Eqs. (3)–(5)):
  - D_s = (1/N_s ∫_0^{N_s} d_is^(ρ_s−1)/ρ_s di)^(ρ_s/(ρ_s−1))
  - d_is = ξ_s^η (p_is / P_s)^{−ρ_s} (P_s / P)^{−η} D
  - P_s = (1/N_s ∫_0^{N_s} p_is^{1−ρ_s} di)^{1/(1−ρ_s)}; P = (∑_s ξ_s^η N_s P_s^{1−η})^{1/(1−η)}
- Exact-hat relative demand (Eq. (6)) and symmetric pre-COVID simplification (Eq. (7)):
  - ˆd_is = ˆξ_s^η ˆP^{η−1} ˆPD
  - Under symmetry: ˆd_is = ˆξ_s^η / (∑_σ ˆξ_σ^η / S) ˆPD
  - Define ̃ξ_s^η ≡ ˆξ_s^η / (∑_σ ˆξ_σ^η / S) with ∑_s ̃ξ_s^η / S = 1 so (Eq. (8)) ˆd_s = ̃ξ_s^η ˆPD
- Interpretation:
  - Sectoral demand changes driven by relative pattern ˆξ_s (reallocation) and aggregate expenditure change ˆPD.
  - Elasticity η mediates sensitivity: high η → small ˆξ shocks produce large demand responses; low η → more uniform responses.

### Modeling the COVID-19 shock (Section 2.3)
- COVID-19 shock = combination of supply, productivity, and demand shocks at sectoral and aggregate levels.
- Supply-side assumptions:
  - Fixed factors immobile.
  - Labor supply constraint (Eq. (9)): n'_is ≤ x'_s n_is, where x'_s varies by sector (essential sectors may have x'_s = ∞; non-essential typically x'_s ≤ 1).
- Productivity: allow A'_s; expect A'_s / A_s ≤ 1.
- Demand-side: sector-specific ξ'_s (can be zero); aggregate nominal gross expenditures ˆPD taken as exogenous.
- Timing and reallocation:
  - Shock temporary; prices sticky at that horizon.
  - Workers cannot reallocate across sectors short run; laid-off workers not on firm payroll and generate no cashflow drain.
  - Firms produce demanded output given sticky prices.
  - Extension: firms may optimally ‘mothball’ (temporary shutdown) if cash flows lower under production.

### Firm cost minimization and production specification (Section 2.4)
- Cobb-Douglas production (Eq. (10)):
  - y = z k^α (A n)^β m^γ, with α + β + γ = 1 and β + γ < 1 (k fixed).
- Cost-minimization (Eq. (11)):
  - min_{m', n'} w n' + p^m m' subject to z k^α (A' n')^β m'^γ ≥ d' and n' ≤ x' n
  - Two solution cases depending on whether labor constraint binds.

### Case 1 — Labor not constrained (Section 2.4.1)
- First-order conditions (Eq. (12)):
  - ˆm = ˆn = ˆd^{1/(β+γ)} ˆA^{−β/(β+γ)} = (̃ξ^η ˆPD)^{1/(β+γ)} ˆA^{−β/(β+γ)} ≡ ˆx_c
  - Valid when ˆx_c < ˆx (unconstrained optimal labor demand).
  - Identity: ˆx_c^(β+γ) ˆA^β = ˆξ^η ˆPD
- Variable profits (Eq. (13)):
  - π' = p d' − w n' − p^m m' = p d (̃ξ^η ˆPD − (s_n + s_m) ˆx_c)
  - s_n = w n / p y and s_m = p^m m / p y are pre-COVID wage and material bills.

### Case 2 — Labor constrained (Section 2.4.2)
- When ˆx < ˆx_c (Eq. (14)):
  - ˆn = ˆx
  - ˆm = (̃ξ^η ˆPD)^{1/γ} (ˆA ˆx)^{−β/γ} = ˆx^{−β/γ} ˆx_c^{(β+γ)/γ}
  - Binding labor constraint reduces labor and increases intermediate use; effect stronger when γ is low.
- Variable profits when constrained (Eq. (15)):
  - π' = p d (̃ξ^η ˆPD − ˆx_c [ s_n (ˆx / ˆx_c) + s_m (ˆx / ˆx_c)^{−β/γ} ])
  - Net effect: variable costs must increase when firm is constrained; increase in material costs larger for low γ and high β.

### Business failures — liquidity-based closure rule (Section 2.5)
- Operating cash flow (Eq. (16)):
  - CF_is = p_is d_is − w n_is − p^m_s m_is − F_is − T_is = π_is − F_is − T_is
  - F_is = fixed factor costs (including r_s k_i,s); T_is = business taxes.
- Change in cashflows (Cases 1 & 2) (Eqs. (17)–(18)):
  - Case 1 (ˆx > ˆx_c): CF'_is − CF_is = p_is d_is [̃ξ^η_s ˆPD − 1 + (s_{n,is} + s_{m,is}) (1 − ˆx_c^s ) ]
  - Case 2 (ˆx < ˆx_c): CF'_is − CF_is = p_is d_is [̃ξ^η_s ˆPD − 1 + s_{n,is} (1 − ˆx_s ) + s_{m,is} (1 − ˆx_c(β_s + γ_s)/γ_s s ˆx^{−β_s/γ_s}_s ) ]
- Liquidity-based closure rule (Eq. (19) and (20)):
  - Firm closes if Z_is + CF'_is − ιL_is < 0
  - Equivalent: CF'_is − CF_is < ιL_is − Z_is − CF_is
  - Right-hand side observable in firm-level data; left-hand side constructed from Eqs. (17) and (18).
- Interpretation and caveats:
  - Static rule assumes tight borrowing constraint (no access to credit/restructuring); baseline estimates weekly bankruptcy rates under this assumption.
  - Section 6 evaluates end-of-year failure condition with longer smoothing.
  - Rationale: SME data limitations on equity and credit access motivate liquidity-focused criterion.

### Taking the model to the data (Section 3) — required inputs
- Required empirical counterparts:
  - Sectoral and aggregate demand shocks: ̃ξ_s^η and ˆPD
  - Sectoral labor supply shocks: ˆx_s
  - Sectoral productivity shocks: ˆA_s
  - Firm-level pre-COVID factor shares: s_{n,is}, s_{m,is}
  - Firm-level sales: p_is d_is
  - Firm-level cash balances Z_is, financial expenses ιL_is, and pre-COVID cashflow CF_is
- With inputs construct CF'_is − CF_is via Eqs. (17) and (18) and apply failure condition Eq. (20) to predict business failures.

*Italic: Content summarized from the IMF working paper chapter "2.1    Supply" as provided.*

### 2.1    Supply

### 2.1    Supply

### Model setup — production and prices
- Economy: S sectors. In each sector s ∈ S there is a mass N_s of firms indexed by i. Mass of firms in each sector is taken as given.
- Firm production function (Eq. (1)):
  - y_is = z_is f_s(k_is, A_s n_is, m_is)
  - y_is = gross output; k_is = fixed factor (including capital, entrepreneurial talent); n_is = labor input; m_is = other variable inputs (materials / intermediate inputs); A_s = sector-specific labor-augmenting productivity so that A_s n_is is effective labor; z_is = firm-specific productivity.
  - Assumption: f_s(., 0, .) = f_s(., ., 0) = 0 (firms need both labor and intermediates).
- Prices:
  - p_is = price of output of firm i in sector s
  - w_s = wage rate per effective unit of labor
  - r_s = user cost for fixed factors
  - p^m_s = price of other variable inputs
  - Factor prices vary only at sector level. Prices of factors and output assumed constant in the short run.

### Demand structure (Eqs. (2)–(8))
- Aggregation: nested CES demand for final and intermediate uses (Eq. (2)):
  - D = [∑_s N_s ξ_s D_s^(η−1)/η]^(η/(η−1))
  - D = aggregate demand; D_s = sectoral demand; ξ_s = sectoral demand shifter; η = elasticity of substitution between sectors.
  - Symmetry pre-COVID: N_s ξ_s = 1, ∀s. Denote ξ'_s sectoral demand shifter during COVID-19 and ˆξ_s ≡ ξ'_s / ξ_s.
- Sectoral aggregation across varieties (Eq. (3)):
  - D_s = (1/N_s ∫_0^{N_s} d_is^(ρ_s−1)/ρ_s di)^(ρ_s/(ρ_s−1)), where ρ_s is sector-specific elasticity of substitution between varieties.
- Demand for variety i in sector s (Eq. (4)):
  - d_is = ξ_s^η (p_is / P_s)^{−ρ_s} (P_s / P)^{−η} D
- Sectoral and aggregate price indices (Eq. (5)):
  - P_s = (1/N_s ∫_0^{N_s} p_is^{1−ρ_s} di)^{1/(1−ρ_s)}
  - P = (∑_s ξ_s^η N_s P_s^{1−η})^{1/(1−η)}
  - With p_is and N_s constant pre-COVID, P_s constant; aggregate P can change because of ξ_s.
- Exact-hat expression for relative change in demand (Eq. (6)):
  - ˆd_is = ˆξ_s^η ˆP^{η−1} ˆPD
- Under symmetric pre-COVID equilibrium (Eq. (7)):
  - ˆd_is = ˆξ_s^η / (∑_σ ˆξ_σ^η / S) ˆPD
- Define ̃ξ_s^η ≡ ˆξ_s^η / (∑_σ ˆξ_σ^η / S), with ∑_s ̃ξ_s^η / S = 1, so (Eq. (8)):
  - ˆd_s = ̃ξ_s^η ˆPD
- Interpretation:
  - Sectoral demand changes driven by (i) relative pattern of sector shocks ˆξ_s (reallocation across sectors) and (ii) aggregate expenditure change ˆPD.
  - Elasticity η mediates sensitivity: high η → small ˆξ shocks produce large demand responses; low η → more uniform responses.

### Modeling the COVID-19 shock (Section 2.3)
- COVID-19 shock modeled as combination of supply, productivity, and demand shocks at sectoral and aggregate levels.
- Supply-side assumptions:
  - Fixed factors immobile.
  - Fraction of workers allowed to work in each sector: firm i with pre-COVID employment n_is can employ at most x'_s n_is during COVID (labor supply constraint Eq. (9)):
    - n'_is ≤ x'_s n_is, where n'_is is employment chosen during COVID.
  - x'_s varies by sector: essential sectors may have x'_s = ∞ (constraint never binds); non-essential sectors typically x'_s ≤ 1. Example interpretations: x'_s ≈ 1 for universities shifting online; x'_s < 1 for construction.
- Productivity changes:
  - Allow A'_s (labor productivity during COVID). Expect A'_s / A_s ≤ 1 (productivity may decline due to remote work, adjustment costs, spatial constraints).
- Demand-side shocks:
  - Sector-specific demand shifters ξ'_s (can be zero for sectors shut by policy e.g. restaurants under shelter-in-place).
  - Aggregate nominal gross expenditures shift ˆPD taken as exogenous in this paper.
- Timing and reallocation:
  - COVID-19 shock assumed temporary. Prices of goods and factors sticky at that horizon.
  - Workers cannot reallocate across sectors in short run; laid-off workers not on firm payroll and generate no cashflow drain.
  - Firms produce the demanded level of output because prices are sticky.
  - Extension considered later: firms can optimally ‘mothball’ (temporary shutdown) if cash flows lower under production.

### Firm cost minimization and production specification (Section 2.4)
- Production function specialized to Cobb-Douglas (Eq. (10)):
  - y = z k^α (A n)^β m^γ, with α + β + γ = 1 (sector-specific exponents).
  - Because k is fixed, relevant assumption: β + γ < 1 (decreasing returns to labor and intermediates jointly).
- Cost-minimization problem (Eq. (11)):
  - min_{m', n'} w n' + p^m m' subject to:
    - z k^α (A' n')^β m'^γ ≥ d'
    - n' ≤ x' n
  - d' is demand from Eq. (8); second line: if firm produces it must meet demand; third line: labor supply constraint.
- Two solution cases depending on whether labor constraint binds.

### Case 1 — Labor not constrained (Section 2.4.1)
- First-order conditions yield (Eq. (12)):
  - ˆm = ˆn = ˆd^{1/(β+γ)} ˆA^{−β/(β+γ)} = (̃ξ^η ˆPD)^{1/(β+γ)} ˆA^{−β/(β+γ)} ≡ ˆx_c
  - Interpretation: intermediate input and labor demand increase with output demand (̃ξ^η ˆPD) and decrease with productivity ˆA.
  - Solution valid as long as ˆn < ˆx, i.e. ˆx_c < ˆx. ˆx_c = unconstrained demand for labor driven by sectoral/aggregate demand and productivity changes.
  - Rearranged identity:
    - ˆx_c^(β+γ) ˆA^β = ˆξ^η ˆPD
    - Left-hand side = supply side (labor supply shock and productivity change) with exponent β + γ; right-hand side = demand side (sectoral/aggregate demand reduction).
- Variable profits for unconstrained firm (Eq. (13)):
  - π' = p d' − w n' − p^m m' = p d (̃ξ^η ˆPD − (s_n + s_m) ˆx_c)
  - s_n = w n / p y and s_m = p^m m / p y denote pre-COVID wage and material bills respectively.

### Case 2 — Labor constrained (Section 2.4.2)
- When labor constraint binds (ˆx < ˆx_c), obtain (Eq. (14)):
  - ˆn = ˆx
  - ˆm = (̃ξ^η ˆPD)^{1/γ} (ˆA ˆx)^{−β/γ} = ˆx^{−β/γ} ˆx_c^{(β+γ)/γ}
  - Binding labor constraint reduces labor input and increases use of intermediate inputs; response stronger when γ (output elasticity of intermediates) is low.
- Variable profits when constrained (Eq. (15)):
  - π' = p d (̃ξ^η ˆPD − ˆx_c [ s_n (ˆx / ˆx_c) + s_m (ˆx / ˆx_c)^{−β/γ} ])
  - Comparison to unconstrained case:
    - Lower labor use tends to increase variable profits (term n ˆx / ˆx_c decreases since ˆx < ˆx_c).
    - Greater reliance on intermediates tends to lower profits (term m (ˆx / ˆx_c)^{−β/γ} increases).
    - Net effect: at unchanged demand, variable costs must increase when firm is constrained. Increase in material costs larger for low γ and high β.

### Business failures — liquidity-based closure rule (Section 2.5)
- Operating cash flow definition (Eq. (16)):
  - CF_is = p_is d_is − w n_is − p^m_s m_is − F_is − T_is = π_is − F_is − T_is
  - F_is = costs associated with fixed factors (rent, utilities, management compensation, including capital costs r_s k_i,s)
  - T_is = business taxes
- Difference in cash flows from observed to predicted under COVID used (assuming fixed costs and taxes not affected).
- Change in cashflow expressions:
  - Case 1 (labor supply does not bind, ˆx > ˆx_c) (Eq. (17)):
    - CF'_is − CF_is = p_is d_is [̃ξ^η_s ˆPD − 1 + (s_{n,is} + s_{m,is}) (1 − ˆx_c^s ) ]  (note: as given in source)
  - Case 2 (labor supply binds, ˆx < ˆx_c) (Eq. (18)):
    - CF'_is − CF_is = p_is d_is [̃ξ^η_s ˆPD − 1 + s_{n,is} (1 − ˆx_s ) + s_{m,is} (1 − ˆx_c(β_s + γ_s)/γ_s s ˆx^{−β_s/γ_s}_s ) ]  (note: expression as given in source)
- Closure rule (liquidity criterion) (Eq. (19)):
  - Firm closes if Z_is + CF'_is − ιL_is < 0
    - Z_is = cash balances; ιL_is = financial expenses (interest and principal repayments)
  - Reformulated (Eq. (20)):
    - CF'_is − CF_is < ιL_is − Z_is − CF_is
    - Right-hand side observable in firm-level data; left-hand side constructed from Eqs. (17) and (18).
- Interpretation and caveats:
  - Static rule assumes firms with temporary cashflow shortfall cannot access credit to smooth and thus illiquidity → insolvency.
  - Baseline estimates weekly bankruptcy rates (tight borrowing constraint); in Section 6 evaluate end-of-year failure condition with longer smoothing.
  - Ignores bankruptcy court restructuring; practical justification: SME bankruptcy regimes often do not preserve viable firms during a pandemic surge.
  - Data limitations: lack of reliable firm-level future profit forecasts, equity positions for SMEs, and direct information on continued access to credit. This motivates liquidity-based criterion focusing on SMEs.

### Taking the model to the data (Section 3)
- Required empirical counterparts to construct counterfactual cashflow changes and evaluate failures:
  - Sectoral and aggregate demand shocks: ̃ξ_s^η and ˆPD
  - Sectoral labor supply shocks: ˆx_s
  - Sectoral productivity shocks: ˆA_s
  - Firm-level pre-COVID factor shares: s_{n,is}, s_{m,is}
  - Firm-level sales: p_is d_is
  - Firm-level cash balances Z_is, financial expenses ιL_is, and pre-COVID cashflow CF_is
- With these inputs, construct CF'_is − CF_is via Eqs. (17) and (18) and apply failure condition Eq. (20) to determine predicted business failures.

*Italic: Content summarized from the IMF working paper chapter "2.1    Supply" as provided.*

### 3.1    Firm-Level Data

### 3.1    Firm-Level Data

### Data source and sample
- Data source: Orbis (BvD-Moody’s).
- Orbis coverage: more than 200 countries and over 200 million firms (private and publicly listed). Longitudinal dimension and representativeness vary by country.
- Analysis sample: seventeen countries using 2017 as base year (2018 data not yet available): Belgium, Czech Republic, Finland, France, Germany, Greece, Hungary, Italy, Japan, Korea, Poland, Portugal, Romania, Slovak Republic, Slovenia, Spain, and the United Kingdom.
- Focus: private, non-financial firms (NACE 1-digit sectors A, B, C, D, E, F, G, H, I, J, L, M, N, P, Q, R, S). Excluded: financial and insurance activities (K), public administration and defense (O), activities of households as employers (T), activities of extraterrestrial organizations and bodies (U), and sub-sectors 78 and 81 in Administration (N).

### Coverage metrics (Table 1: Orbis Coverage (2017))
- % of OECD Revenue — All / SMEs:
  - Belgium 60.45 / 52.1
  - Czech Republic 63.46 / 62.8
  - Finland 66.06 / 68.3
  - France 46.34 / 46.3
  - Germany 27.2 / 17.7
  - Greece 48.0 / 48.1
  - Hungary 63.9 / 48.7
  - Italy 63.57 / 58.0
  - Japan 42.5 / .
  - Korea 61.93 / 34.0
  - Poland 47.5 / 44.5
  - Portugal 63.3 / 72.9
  - Romania 60.6 / 40.0
  - Slovak Republic 52.0 / 73.2
  - Slovenia 49.3 / 61.0
  - Spain 58.4 / 69.9
  - United Kingdom 49.2 / 41.4
- Notes on coverage:
  - Coverage computed by summing Orbis revenue by 1-digit NACE sector and dividing by OECD SDBS revenue for matching sectors.
  - Japan: OECD-equivalent revenue data obtained from the Economic Census in 2015 (Orbis data are also for 2015); SME revenue data for Japan not available to evaluate coverage.
  - Countries highlighted in grey (in original) indicate coverage under 40% in analysis data.

### Variables, cleaning, and transformations
- Required firm-level variables to evaluate bankruptcy rates: firm revenue, wage bill, material cost, number of employees, net income, depreciation, cash stock, and financial expenses.
- Cash flow calculation: cash flow = net income + depreciation − financial profits.
- Sectoral labor and material cost shares: revenue-weighted average of firm-level wage bill over revenue and material costs over revenue at the 2-digit NACE level.
- Winsorization: all level variables used for analysis winsorized at the 99.9th percentile.
- For Greece, Japan, Korea, and the United Kingdom (firms do not report labor and material costs separately): use costs of materials sold and divide between labor and materials using 2-digit industry cost shares derived from countries where labor and material costs are reported separately and country coverage of revenue exceeds 40% (Belgium, Czech Republic, Finland, France, Hungary, Italy, Poland, Portugal, Romania, Slovakia, Slovenia, and Spain).
- Elasticities (β_s and γ_s) estimation: estimated at 2-digit NACE level for each country as weighted average of firm revenue share of input expenditures (labor cost share of revenue and material cost share of revenue), with firm revenue as weights. Elasticities are revenue elasticities (lack of price data). For countries lacking separate labor/material reporting (Greece, Japan, Korea, United Kingdom) elasticities are averaged from the subset of countries with separate reporting and revenue coverage > 40%.

### Business failure rates (pre-COVID) — comparison with OECD (Table 2)
- OECD vs Orbis failure rates (pre-COVID):
  - Belgium OECD 3.0 / Orbis 8.3
  - Czech Republic 7.9 / 8.3
  - Finland 5.4 / 9.3
  - France 4.7 / 8.1
  - Germany 6.7 / 11.2
  - Greece 4.0 / 10.3
  - Hungary 8.7 / 8.9
  - Italy 6.7 / 9.7
  - Portugal 11.5 / 12.5
  - Romania 8.6 / 15.5
  - Slovak Republic 10.0 / 11.1
  - Slovenia 3.9 / 6.5
  - Spain 7.4 / 9.2
  - United Kingdom 13.9 / 11.4
- Notes:
  - OECD failure rates obtained from SDBS Business Demography Indicators for a subset of NACE sectors (B, C, D, E, F, G, H, I, J, L, M, N, P, Q, R, S).
  - Orbis failure rates calculate fraction of firms facing a liquidity shortfall in 2017 (cashflow + cash insufficient to cover financial expenses).
  - Orbis generally reports higher business failure rates than OECD for many high-coverage countries; a suspected reason is Orbis assumption that all illiquid firms fail (no access to credit or debt restructuring modeled).
  - Emphasis in analysis is on changes in business failure rates before and after COVID-19 rather than level comparisons.

### Role and vulnerability of SMEs
- SME definition: firms with less than 250 employees.
- Aggregate shares across "good coverage" European countries (Belgium, Czech Republic, Finland, France, Greece, Hungary, Italy, Poland, Portugal, Romania, Slovakia, Slovenia, Spain):
  - SMEs account for 53.4% of employment and 46.58% (large firms) / 53.42% (SME) employment share depicted.
  - Payroll: 48.94% SME share (51.06% large firms / 48.94% SME).
  - Revenue: 50.07% SME share (49.93% large firms / 50.07% SME).
  - Total assets: 45.61% SME share (54.39% large firms / 45.61% SME).
- Additional winsorized shares (when winsorized at the 99.9th percentile): SME employment share 63.2%, labor cost share 61.6%, revenue share 65.9%, total asset share 65.9%.
- Policy-relevant point: SMEs are particularly vulnerable to the COVID-19 shock because they tend to be bank-dependent and have limited ability to draw on credit lines, making them vulnerable to solvency problems following liquidity shortages.

### Shocks construction and calibration (overview drawing from sections 3.2–3.4)
- Classification of sectors into essential and non-essential sectors: based on U.S. Department of Homeland Security Guidance on the Essential Critical Infrastructure Workforce; essential examples include public health, public safety, food supply chain, energy infrastructure, transportation and logistics, critical manufacturing, hygiene products and services.
- Supply shock (x̂_s):
  - Follows Dingel and Neiman (2020) using O*NET “work context” and “generalized work activities” surveys to classify occupations into remote-capable versus not.
  - Uses BLS occupation prevalence by NAICS and NAICS-NACE crosswalk to compute fraction of employees who can/cannot work remotely at 4-digit NACE level.
  - Assumption for COVID-19: no workers in essential sectors lose their jobs; in non-essential sectors, all workers that cannot work remotely lose their jobs temporarily.
  - Aggregation to 1-digit NACE and cross-country aggregation uses gross value added sector shares. Figure 2 highlights that Accommodation & Food and Arts, Entertainment & Recreation are among most affected; Electricity and Water & Waste remain unaffected.
- Demand shock:
  - Uses O*NET to classify occupations by reliance on face-to-face interactions (external customers, physical proximity, assisting/caring for others, working with the public, selling).
  - Uses BLS and NAICS-NACE crosswalk to estimate interaction share at 4-digit NACE.
  - Assumption: demand unaffected in essential sectors; for non-essential industries demand = 1 − interaction share.
  - Interpret resulting estimate as ξ̂η_s. Sectoral demand shocks normalized to be consistent with aggregate demand equation by constructing ξ̃η_s = ξ̂η_s / (∑_σ ξ̂η_σ / S).
  - Figure 3 shows COVID-19 reallocates expenditure away from Arts, Entertainment & Recreation and Accommodation & Food toward essential sectors like Water & Waste and Electricity.
- Aggregate demand change (P̂D):
  - Measured using IMF projections of quarterly and annualized changes in GDP.
  - Total demand shock for sector s defined as d̂_s = ξ̃η_s P̂D.
- Productivity shock (Â_s):
  - Productivity modeled as combination of on-site (A_work_s) and remote (A_home_s) productivity weighted by fraction of on-site workers ω_s.
  - Before COVID: A_s = A_work_s ω_s + A_home_s (1 − ω_s).
  - During COVID: A′_s = A_work′_s ω′_s + A_home′_s (1 − ω′_s).
  - Under assumption A_work and A_home same before and during, Â_s = [ω′_s + (A_home_s / A_work_s)(1 − ω′_s)] / [ω_s + (A_home_s / A_work_s)(1 − ω_s)].
  - Under lockdown assumption for non-essential industries ω′_s = 0, expression collapses to Â_s = (A_home_s / A_work_s) / [ω_s + (A_home_s / A_work_s)(1 − ω_s)].
  - Calibration: use 2018 American Community Survey (ACS) for ω_s (share of remote workers by industry); calibrate A_remote / A_work = 0.8.
  - Implication stated: "This implies that ˆA=0.8 is the maximum decline in productivity possible for a sector where none of the workers worked remotely before COVID-19 to 100% during COVID-19."
- Production function parameters:
  - Labor and materials elasticities (β_s and γ_s) estimated at 2-digit NACE as described above.
  - Because price data unavailable, elasticities are revenue elasticities.

### Baseline scenario for bankruptcy rate estimation (Section 4 setup)
- Timing and duration:
  - COVID-19 shock assumed to hit in week 9 of the year (beginning of March).
  - Lockdown period duration: 8 weeks.
- Effects during lockdown:
  - 8 week lockdown lowers sectoral labor supply (x̂_s), demand (d̂_s = ξ̃η_s P̂D), and labor productivity (Â_s).
- Post-lockdown dynamics:
  - After lockdown ends, sectoral supply and productivity shocks return to pre-COVID levels.
  - Demand shocks remain active: aggregate demand component (P̂D) evolves according to IMF projections; sector-specific demand shocks (ξ̃η_s) evolve according to an AR(1) process with persistence 0.5 at quarterly frequency.
  - Persistence reflects continued subdued demand due to uncertainty and fear of infection after stay-at-home orders are lifted.
- Frequency and bankruptcy rule:
  - Bankruptcy rates evaluated at weekly frequency.
  - Rule: if at week’s end a firm is illiquid (cashflow + cash insufficient to cover financial expenses) it is assumed unable to access temporary credit and goes bankrupt.

*Source: wpiea2020207-print-pdf — 3.1    Firm-Level Data.*

### Section 6 relaxes this assumption.

### wpiea2020207-print-pdf - Section 6 relaxes this assumption.

### Baseline assumptions and interpretation
- Firms receive revenues throughout the year in equal weekly increments; labor and materials costs are paid throughout the year in equal weekly increments.
- Financial expenses are assumed to be paid monthly and taxes twice a year in June and December.
- Baseline bankruptcy rates abstract from any government or credit market interventions and are interpreted as an upper bound.
- Government policies (e.g., suspension of payroll taxes, direct assistance, loan guaranties) may have absorbed a large share of cashflow declines in practice, but these are not directly incorporated in the baseline; they are evaluated in Section 5 and partially reflected in IMF WEO projections.
- The preferred metric for business failures is the additional effect of COVID-19 on firm bankruptcies in 2020: COVID-19 bankruptcy rate minus non-COVID bankruptcy rate.

### Aggregate SME bankruptcy rates (Table 3)
- High coverage: Non-COVID 9.56, COVID-19 18.19, ∆ 8.63
- All: Non-COVID 9.43, COVID-19 18.17, ∆ 8.75
- Notes: Bankruptcy rates are first calculated at the 1-digit NACE level and aggregated to the country level using 2017 sector gross value added as weights (exceptions noted for UK, Korea, Japan). Aggregation across countries uses GDP weights. High coverage group includes Belgium, Czech Republic, Finland, France, Greece, Hungary, Italy, Poland, Portugal, Romania, Slovakia, Slovenia, and Spain. All countries group incorporates Germany, Japan, Korea, and the United Kingdom.

### Sector SME bankruptcy rates (Table 4)
- Agriculture: Non-COVID 9.44, COVID-19 13.52, ∆ 4.08
- Mining: Non-COVID 12.50, COVID-19 36.03, ∆ 23.54
- Manufacturing: Non-COVID 8.48, COVID-19 16.73, ∆ 8.25
- Electric, Gas & Air Con: Non-COVID 9.35, COVID-19 11.31, ∆ 1.96
- Water & Waste: Non-COVID 6.72, COVID-19 9.65, ∆ 2.93
- Construction: Non-COVID 7.97, COVID-19 10.19, ∆ 2.21
- Wholesale & Retail: Non-COVID 9.12, COVID-19 18.21, ∆ 9.10
- Transport & Storage: Non-COVID 7.64, COVID-19 13.28, ∆ 5.63
- Accom. & Food Service: Non-COVID 13.15, COVID-19 38.59, ∆ 25.44
- Info. & Comms: Non-COVID 10.00, COVID-19 15.92, ∆ 5.92
- Real Estate: Non-COVID 11.61, COVID-19 17.38, ∆ 5.76
- Prof., Sci., & Technical: Non-COVID 10.24, COVID-19 18.85, ∆ 8.60
- Administration: Non-COVID 8.32, COVID-19 19.39, ∆ 11.06
- Education: Non-COVID 10.86, COVID-19 30.04, ∆ 19.18
- Health & Social Work: Non-COVID 7.74, COVID-19 11.22, ∆ 3.48
- Arts, Ent., & Recreation: Non-COVID 12.95, COVID-19 36.55, ∆ 23.60
- Other Services: Non-COVID 12.80, COVID-19 31.42, ∆ 18.62
- Notes: Sector bankruptcy rates are calculated at the 1-digit NACE level for each country, then aggregated across countries using (country x sector) gross value added from the OECD as weights. Aggregation is over the high coverage group.

### Cross-sector heterogeneity — key mechanisms and findings
- Rise in bankruptcy rates under COVID-19 is driven by deterioration in firm cashflow, itself driven by total demand, sectoral supply, and productivity shocks.
- Service sectors with high customer orientation and limited remote work scope (Accommodation & Food Service, Arts, Entertainment & Recreation) see increases in bankruptcy rates exceeding 20 percentage points.
- Sectors with a high fraction of essential sub-sectors and no sectoral supply shocks (Agriculture, Health, Water & Waste) experience smaller increases (less than 5 percentage points).
- Sectors with fewer essential workers but lower total demand shocks or high scope for remote work (Manufacturing; Information & Communications; Professional, Scientific & Technical Activities) see moderate increases (under 10 percentage points).
- Example comparison — Arts, Entertainment & Recreation (sector R) vs Wholesale & Retail (sector G):
  - Sector R enters the crisis with lower average cash balances and higher non-COVID bankruptcy rates than sector G.
  - Sector-specific demand shock is substantially more severe in sector R, expanding the gap in bankruptcy rates during the lockdown.
  - Fraction of firms labor constrained: under 10% in sector R versus over 20% in sector G; in sector R demand collapse reduces optimal labor demand so constraint binds for fewer firms.
  - End-of-year bankruptcy rates: Arts, Entertainment & Recreation 36.5% vs Wholesale & Retail 18.2%.

### Shock decomposition and scenario comparison (Table 5)
- Scenarios (column headers):
  - (1) ̂PC : aggregate demand shock only
  - (2) ̂PC, ˆxs : aggregate demand and sectoral supply shocks
  - (3) ̂PC ̃ξηs : total demand shocks (aggregate + sector-specific demand)
  - (4) ̂PC ̃ξηs, ˆxs : total demand and sectoral supply shocks
  - (5) Baseline: ̂PC ̃ξηs, ˆxs, ˆAs (adds sectoral productivity shocks)
- Sectoral ∆ bankruptcy rates by scenario:
  - Agriculture: (1) 0.82, (2) 2.89, (3) 1.01, (4) 3.82, (5) 4.08
  - Mining: (1) 0.61, (2) 19.05, (3) 1.17, (4) 19.73, (5) 23.54
  - Manufacturing: (1) 1.00, (2) 6.11, (3) 0.95, (4) 6.56, (5) 8.25
  - Electric, Gas & Air Con: (1) 1.98, (2) 1.98, (3) 1.94, (4) 1.94, (5) 1.96
  - Water & Waste: (1) 3.33, (2) 3.33, (3) 2.73, (4) 2.73, (5) 2.93
  - Construction: (1) 1.48, (2) 1.77, (3) 2.00, (4) 2.00, (5) 2.21
  - Wholesale & Retail: (1) 1.65, (2) 4.83, (3) 4.83, (4) 4.83, (5) 9.10
  - Transport & Storage: (1) 6.83, (2) 7.00, (3) 5.13, (4) 5.16, (5) 5.63
  - Accom. & Food Service: (1) 0.07, (2) 75.04, (3) 9.20, (4) 20.04, (5) 25.44
  - Info. & Comms: (1) 2.12, (2) 3.56, (3) 4.99, (4) 4.99, (5) 5.92
  - Real Estate: (1) 1.60, (2) 2.18, (3) 5.76, (4) 5.71, (5) 5.76
  - Prof., Sci., & Technical: (1) 3.25, (2) 3.88, (3) 7.35, (4) 7.48, (5) 8.60
  - Administration: (1) 3.67, (2) 5.87, (3) 10.61, (4) 10.72, (5) 11.06
  - Education: (1) 2.01, (2) 49.45, (3) 18.60, (4) 18.60, (5) 19.18
  - Health & Social Work: (1) 1.85, (2) 12.16, (3) 3.14, (4) 3.14, (5) 3.48
  - Arts, Ent., & Recreation: (1) 1.92, (2) 51.04, (3) 18.92, (4) 21.27, (5) 23.60
  - Other Services: (1) 0.13, (2) 47.27, (3) 16.90, (4) 17.62, (5) 18.62
  - Average (GVA weighted): (1) 2.02, (2) 11.78, (3) 6.19, (4) 7.75, (5) 8.63
- Key interpretation:
  - Negative aggregate demand shocks alone have a limited effect in most sectors.
  - Including sectoral supply shocks sharply raises bankruptcy rates in labor-intensive sectors (e.g., Accommodation & Food Service).
  - Total demand shocks (aggregate + sector-specific demand) increase bankruptcy rates most in customer-oriented services (e.g., Arts, Entertainment & Recreation) and can reduce rates in sectors gaining reallocated demand (e.g., Water & Waste).
  - In some labor-intensive sectors, adding both total demand and sectoral supply shocks lowers bankruptcy rates relative to sectoral supply shocks alone because lower demand reduces firms’ optimal labor demand and thus the fraction of firms that are labor constrained.
  - Sectoral productivity shocks (baseline column) generally have little additional effect on bankruptcy rates in most sectors.

### Cross-country results (Table 6) and drivers
- Country-level baseline bankruptcy rates (Non-COVID, COVID-19, ∆):
  - Belgium: 7.75, 14.18, 6.42
  - Czech Republic: 8.24, 13.59, 5.35
  - Finland: 8.35, 16.91, 8.56
  - France: 9.03, 16.94, 7.91
  - Greece: 10.43, 16.37, 5.94
  - Hungary: 8.22, 14.01, 5.79
  - Italy: 9.91, 22.68, 12.77
  - Poland: 11.68, 20.45, 8.77
  - Portugal: 12.21, 19.65, 7.44
  - Romania: 15.77, 23.18, 7.41
  - Slovak Republic: 10.41, 16.05, 5.64
  - Slovenia: 7.25, 15.95, 8.71
  - Spain: 8.98, 15.50, 6.52
- Cross-country heterogeneity in bankruptcy rate changes ranges from 5.4 percentage points (Czech Republic) to 12.8 percentage points (Italy).
- Key drivers of cross-country differences:
  - Industrial composition of the economy.
  - Financial position of firms prior to the crisis (e.g., average cash balances).
  - Example: Italy vs France
    - Italy experiences a 12.8 percentage point increase in bankruptcy rates vs France’s 7.9 percentage points.
    - Italy faces slightly more severe initial total demand shock, slower recovery, larger sectoral supply shock, and Italian firms enter the crisis with substantially lower average cash balances than French firms, producing a much larger rate of SME failures.

*Italic: Source — wpiea2020207-print-pdf (section content provided).*

### 4.3    Financial Stability Implications of SME Bankruptcies

### 4.3    Financial Stability Implications of SME Bankruptcies

### Expected increase in non-performing loans (NPLs) due to SME bankruptcies
- Loans are qualified as non-performing for firms that fail, both under COVID-19 and in normal times.
- Table 7 country-level changes (COVID minus non-COVID) in NPL share (percentage points):
  - Belgium: 2.32
  - Czech Republic: 5.53
  - Finland: 7.83
  - France: 6.51
  - Greece: 5.34
  - Hungary: 4.02
  - Italy: 10.67
  - Poland: 5.91
  - Portugal: 7.69
  - Romania: 8.58
  - Slovak Republic: 6.72
  - Slovenia: 7.07
  - Spain: 6.53
  - Average (GDP-weighted): 7.24
- Range of increase in the share of NPLs: from 2.3 percentage points (Belgium) to almost 11 percentage points (Italy).

### Cross-country heterogeneity and interpretation
- Banking sector risk (increase in NPL share) is not perfectly correlated with increases in firm bankruptcy rates.
- Example: Poland, Slovenia, and Finland have similar changes in bankruptcy rates (~8.5 percentage points) but different changes in NPL share: Poland 5.9 percentage points, Finland 7.8 percentage points. Differences reflect heterogeneity in indebtedness among firms predicted to fail.

### Banking sector exposure from SME NPLs (Table 8)
- Change in SME NPLs under COVID-19 relative to non-COVID, reported as:
  - % Total Assets (column 1)
  - % Bank Tier-1 Capital (column 2)
  - CET1 ratio (risk-weighted) from ECB (column 3)
  - ∆CET1R: change in the CET1 capital ratio due to COVID-19 (column 4)
- Country-level values:
  - Belgium: 0.13% of total assets; 2.1% of Tier-1 Capital; CET1R 18.3%; ∆CET1R - 0.31%
  - Finland: 0.45% of total assets; 8.3% of Tier-1 Capital; CET1R 16.3%; ∆CET1R -1.15%
  - France: 0.30% of total assets; 6.2% of Tier-1 Capital; CET1R 14.3%; ∆CET1R -0.77%
  - Germany: 0.24% of total assets; 5.1% of Tier-1 Capital; CET1R 15.4%; ∆CET1R -0.68%
  - Greece: 1.12% of total assets; 12.5% of Tier-1 Capital; CET1R 14.9%; ∆CET1R -1.61%
  - Hungary: 0.26% of total assets; 2.7% of Tier-1 Capital; (CET1R unavailable)
  - Romania: 0.69% of total assets; 8.2% of Tier-1 Capital; (CET1R unavailable)
  - Spain: 0.44% of total assets; 8.5% of Tier-1 Capital; CET1R 11.8%; ∆CET1R -0.90%
  - Average (GDP-weighted): 0.30% of total assets; 5.83% of Tier-1 Capital; CET1R 14.69%; ∆CET1R -0.75%
- Interpretation:
  - Change in SME NPL share of total assets is a lower bound on banking-sector risk: averages 0.3%, ranges 0.13% (Belgium) to 1.12% (Greece).
  - Change in SME NPL share of Tier-1 Capital is an upper bound: averages 5.8%, ranges 2.1% (Belgium) to 12.5% (Greece).
  - Estimated decline in CET1 capital ratio averages -0.75 percentage points, ranging from -0.31 percentage points (Belgium) to -1.61 percentage points (Greece).
  - Given initial CET1 ratios (range: 11.8 percent (Spain) to 18.3 percent (Belgium)), the shock from SME failures due to COVID-19 is characterized as modest relative to historic stress scenarios (EBA’s 2018 adverse scenario implied ~4 percentage point decline in CET1).

### Data and calculation notes for banking exposure measures
- Loans defined as sum of short-term and long-term loans.
- Four data sources used:
  1. Orbis: ∆SME NPL share (COVID vs non-COVID).
  2. European Banking Authority (EBA) 2018 country-level bank stress test: bank SME share of all loans and CET1 share of total assets.
  3. Eurostat Financial Balance Sheet: total loans and total assets of depository institutions.
  4. ECB Supervisory Banking Statistics (Q1 2019): CET1 capital ratio (risk-weighted) for subset of countries.
- Formulas reported:
  - Change in SME NPL value as fraction of total assets = [(total loans from Eurostat × bank SME share from EBA × ∆SME NPL share from Orbis)] / [total assets from Eurostat].
  - Change in SME NPL value as fraction of Tier-1 capital = [(total loans from Eurostat × bank SME share from EBA × ∆SME NPL share from Orbis)] / [(total assets from Eurostat) × (CET1 share from EBA)].
  - Change in CET1 capital ratio due to COVID: [(1 − SME NPLs % CET1) × CET1R] / [1 − (SME NPLs % CET1 × CET1R)].

### Implications for financial stability
- Aggregate impact on CET1 ratios is moderate (average decline -0.75 percentage points), suggesting banking-sector resilience to SME failures from COVID-19 is limited but not systemic under these estimates.
- Cross-country heterogeneity implies some countries (e.g., Greece) face proportionally larger capital ratio declines than others.

*Italic: Source — 4.3 Financial Stability Implications of SME Bankruptcies, wpiea2020207-print-pdf*

### 1.5 percent of GDP on survivor firms that don’t need it.   It also devotes a small amount of

### wpiea2020207-print-pdf - 1.5 percent of GDP on survivor firms that don’t need it.   It also devotes a small amount of

### Effects of 100% labor subsidies (SME firms during 8-week lockdown)
- Policy simulated: 100% labor cost subsidy to SME firms during the 8 week lockdown.
- Fiscal redistribution outcomes:
  - 1.5 percent of GDP spent on survivor firms that don’t need it.
  - 0.14 percent of GDP devoted to inefficiently saving 19 percent of ‘ghost firms’ (bankruptcy rates fall from 100% to 81.25%).
- Distributional outcomes:
  - Approximately 25% of the jobs saved and wages saved (and 20% of loans saved) from the labor subsidies can be attributed to retaining workers at ‘ghost firms’.
- Macroeconomic implication:
  - Major defect: wastes fiscal resources on surviving firms that don’t need support.

### Sectoral heterogeneity and measured impacts (Figure 7 summary)
- Bankruptcy rate reductions by 1-digit sector (top reduction to bottom): Accommodation & Food; Other Service; Education; Entertain & Recreation; Mining; Transport & Storage; Wholesale & Retail; Manufacturing; Administration; Prof., Sci. & Tech; Info & Comm; Health; Construction; Water & Waste; Real Estate; Agriculture; Electricity.
- Policy cost by 1-digit sector (GDP-weighted average across high coverage countries):
  - The sector receiving the bulk of policy spending is Manufacturing.
  - Accommodation & Food requires a fairly high amount of spending and shows the largest bankruptcy rate reductions.
  - Real Estate receives fairly high subsidies yet shows only modest reduction in bankruptcy rates.
- Wages and jobs saved by sector:
  - Bottom panels show wages saved relative to GDP and jobs saved relative to overall employment by sector.
  - Other than Accommodation & Food, sectors where most wages can be saved are not always the sectors with the highest proportion of firms saved.
  - Sectors such as Education and Other Services have many jobs that can be saved but not necessarily high wage jobs.
- Aggregation note:
  - Results aggregated across Belgium, Czech Republic, Finland, France, Greece, Hungary, Italy, Poland, Portugal, Romania, and Spain using total revenue of firms in Orbis as weights.

### Inefficiencies, reallocation, and fiscal design
- Reallocation concern:
  - Retaining workers at ‘ghost firms’ hinders proper reallocation of resources toward more productive uses.
  - ‘Ghost firms’ saved by subsidies are likely to fail once fiscal support ends.
- Cost structure:
  - The cost of bailing out these ‘ghost firms’ is small because there are few ghost firms to start with, but remains inefficient.
- Potential fiscal design to reduce burden:
  - Implement a mechanism to recoup some of the relief provided in future years if the firm makes enough profits — a tax on future excess profits could substantially lower the fiscal cost without impacting effectiveness.
  - Conceptually akin to the government taking an equity position: the subsidy in 2020 becomes a claim on future profits.
  - Footnote concept: "A negative tax now that will work as a direct transfer can be turned into a positive tax in later years conditional on excess profits if the policy scheme implemented through the tax system." 

### Targeting and cost-effectiveness
- Targeting support to specific sectors would likely make the labor subsidy more effective per dollar spent.
- Sectors identified as particularly effective targets:
  - Accommodation & Food, Education, Wholesale & Retail, and Manufacturing appear to save the majority of saveable jobs while keeping the share of saved jobs less than half the total policy cost in some cases.
- Trade-offs:
  - Targeting may mitigate risk of directing resources to firms that do not need support but requires sectoral identification of where subsidies yield the highest job and wage preservation per unit of fiscal cost.

*Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020207-print-pdf.pdf*

### 5.2    Policy Support Timing and Additional Lockdowns

### 5.2    Policy Support Timing and Additional Lockdowns

### Effects of timing and duration of labor subsidies (single lockdown)
- Baseline: cumulative bankruptcy rate week-by-week under the baseline COVID-19 scenario absent policy support is shown as the black dashed line.
- 100% labor subsidy during the 8 week lockdown beginning on the 1st week of March:
  - Shown as the blue solid line.
  - Lowers the bankruptcy rate considerably during the lockdown and produces a less steep rise after the lockdown, implying permanent effects in lowering the bankruptcy rate.
- Same total fiscal cost but slower payment: 50% of wages for 16 weeks (i.e., covering 50% of wages for 16 weeks vs 100% for 8 weeks):
  - Shown as the red line.
  - Does not change total subsidy given to each firm (provided the firm survives to 16 weeks after the lockdown) but delays payments.
  - Leads to a small but higher path for the bankruptcy rate over 2020 relative to immediate payments.
  - Implies timing matters, but less than whether support is received at all.
- Extending the original 100% labor subsidy for an additional 8 weeks (costing approximately twice as much as the 8-week subsidy):
  - Shown as the green line.
  - Lowers the bankruptcy rate profile even after support ends.
  - Overall, the bankruptcy rate decreases by approximately 2 percentage points.

### Effects of a second lockdown and policy support (two lockdowns)
- Second lockdown modeling assumptions:
  - Second lockdown is 6 weeks instead of 8.
  - Starts in week 32 of the year (mid-August).
  - Black dotted line: weekly evolution with only 1 lockdown (baseline).
  - Black dashed line: effect of a second lockdown without policy support.
- Without policy support the second lockdown raises bankruptcy rates by between 2-3% – much less than the marginal effect of the first lockdown, for three reasons:
  1. The second lockdown is shorter.
  2. Many vulnerable businesses were already forced into bankruptcy by the first lockdown; remaining businesses have considerably stronger cash positions.
  3. Sectoral demand is assumed to recover only gradually, so the net fall in sectoral demand during the second lockdown is smaller than in the first.
- 100% labor subsidy applied:
  - Blue line: subsidy only in the first lockdown.
  - Red line: subsidy in both lockdowns (additional 6-week 100% labor subsidy during the second lockdown).
  - Imposing a second lockdown without providing policy support raises bankruptcy rates by around 2-3 percentage points relative to providing policy support during both lockdowns.
  - Providing policy support in both lockdowns leads to an end-of-year bankruptcy rate level almost the same as the end-of-year bankruptcy rate from the single lockdown 8-week 100% labor subsidy scenario.
  - Implication: provided the government has the fiscal capacity to provide the needed policy support, additional lockdowns may be imposed without necessarily requiring additional rises in firm bankruptcies.

### Policy tools, targeting, and sectoral/job implications
- Typical policy tools such as covering firms’ financial expenses or subsidizing payroll:
  - Shown to be either too small or very imprecisely targeted relative to infeasible targeted benchmarks.
- Generous policies:
  - Costlier fiscally but have noticeable effects on bankruptcy rates and jobs lost, particularly in Accommodation & Food, Manufacturing, Education and Wholesale & Trade sectors.
- Timing optimization:
  - Considerably less important than providing adequate support.
- Summary quantitative trade-off (from conclusions):
  - Baseline estimate: absent intervention and with impaired access to credit markets, the rate of business failures for SMEs would almost double, increasing by 8.8 percentage points in 2020.
  - Jobs at risk: about 3.1 percent of employment.
  - Financial sector impact: decline of the CET1 capital ratio of 0.75 percentage points on average.
  - A subsidy corresponding to 15% of the firm’s annual wage bill in a normal year would:
    - Reduce business failures by 5.6 percentage points.
    - Save 2.96 percent of employment.
    - Cost 1.8 percent of GDP.
  - Such policies risk misallocation: bulk of support goes to firms that don’t need it; a smaller fraction goes to firms that would fail anyway.

### Extensions: mothballing and annual bankruptcy assessment (smoothing cash)
- Two modifications to the baseline to allow firms more coping mechanisms:
  1. Mothballing: firms may temporarily shut down when workplace restrictions make meeting demand prohibitively expensive; they still pay rent and fixed costs but incur no variable costs while closed and may re-open later.
  2. Annual bankruptcy assessment: instead of assessing liquidity weekly and bankrupting firms when end-of-week balances turn negative, bankruptcy is determined only at the end of the calendar year (conceptually similar to access to zero-interest loans repayable by 31st December 2020).
- Aggregate impacts (Table 12 summaries):
  - Mothballing leads to a reduction in both non-COVID and COVID-19 bankruptcy rates and reduces the impact of COVID-19 (∆) by about one percentage point.
  - Combining mothballing and annual bankruptcy assessment reduces the impact of COVID-19 on bankruptcy rates (∆) by 3 to 4 percentage points depending on country sample.
  - Interpretation: allowing firms mechanisms to smooth cash balances throughout the year could significantly lower resulting bankruptcy rates.

### Sectoral impacts of extensions (selected figures from Table 13 and narrative)
- Introducing mothballing (effect on change in bankruptcy rates, COVID - non-COVID):
  - Wholesale & Retail: change falls from 9.10 to 9.03 (a reduction of 0.07).
  - Mining: reduction of 6.97 (from 23.54 to 16.57 under mothballing).
  - Accommodation & Food Service: reduction of 5.35 (from 25.44 to 20.09 under mothballing).
- Combining mothballing and annual bankruptcy calculation:
  - Decline in bankruptcy rate due to COVID-19 across sectors ranges from 0.7 percentage points in Real Estate to 18.4 percentage points in Mining.
- Drivers:
  - Largest impacts occur in sectors with large labor supply shocks and a high fraction of firms that are labor constrained during COVID.
  - Mining is uniquely affected: high demand with strong workplace restrictions makes mothballing particularly helpful; assessing liquidity at year-end particularly advantages mining as demand remains above pre-COVID after restrictions ease.

### Cross-country impacts of extensions (selected figures from Table 14 and narrative)
- Impact of mothballing on the COVID – non-COVID bankruptcy rate change (relative to baseline) varies across countries:
  - From -0.58 percentage points in Italy to -1.95 percentage points in Romania.
- Impact of incorporating annual bankruptcy rate calculations varies across countries:
  - Ranges from a 1.7 percentage point decline in Greece to a 2.9 percentage point decline in Finland.
- Despite these changes, the ordering of most-to-least affected countries remains similar to the baseline.

*Source: wpiea2020207-print-pdf - 5.2    Policy Support Timing and Additional Lockdowns*

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### Appendix A — Mothballing: mechanisms and conditions
- Context: Firms may choose to “mothball” (temporarily shut down production) during the COVID-19 period if production costs are excessive, yielding higher cash-flow relative to operating under constrained inputs.
- Mothballing implementation: For a mothballed firm set n_is = n_is' = m_is = m_is' = 0 and variable profits π' = 0.
- Economic intuition:
  - Mothballing is particularly relevant for firms facing severe labor constraints that would otherwise need to substitute with intermediate inputs at excessively high cost (see Bresnahan and Raff (1991)).
  - Conditional on meeting demand, firms minimize costs by re-optimizing over labor n_is' (subject to labor supply constrained Eq. (9)) and other flexible input m_is'.
  - Firms can incur negative variable profits when attempting to meet demand d'—especially labor-constrained firms with low material output elasticity—making mothballing preferable.
- Mothballing condition for non-constrained firms (Equation (A.1)):
  - ˆA^β ≤ ( ̃ξ η ̂PD )^{1−β−γ} (s_n + s_m)^{β+γ} . (A.1)
- Mothballing condition for labor constrained firms (Equation (A.2)):
  - ˆA^β ≤ ( ̃ξ η ̂PD )^{1−β−γ} ( s_n (ˆx/ˆx_c) + s_m (ˆx/ˆx_c)^{−β/γ} )^{β+γ} . (A.2)
- Interpretation of the conditions:
  - Mothballing is more likely when firms experience larger productivity shocks (a lower ˆA).
  - The condition is relaxed for labor-constrained firms and highlights greater mothballing propensity for those with low material elasticity.
  - Because cost shares s_n and s_m are measured at the firm level, the mothballing conditions apply at the individual firm level according to Eqs. (A.1) and (A.2).
- Note: The authors mention a theoretical possibility that unconstrained firms could prefer to mothball if they experience a large increase in sectoral demand ( ̃ξ η ), but state this case is not empirically relevant in their estimation.

*IMF Working Paper — References and Appendix A (Mothballing).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020207-print-pdf.pdf_