## IMF WORKING PAPERS — Fiscal Multipliers and Informality

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### Overview
- Paper objective: examine why fiscal multipliers are smaller in developing countries and identify a key driver: the interaction between the strength of the public expenditure multiplier and the size of informality.
- Sample: 141 developed and developing economies.
- Main empirical approach: local projection method of Jordà (2005) to estimate cumulative dynamic response of GDP to government spending shocks.
- Fiscal shocks identification: forecast errors for government spending from IMF World Economic Outlook (WEO) forecasts (October of the same year), following Auerbach and Gorodnichenko (2013) and related extensions.

### Key empirical findings (summary)
- High informality is associated with a marked reduction in the size of the fiscal multiplier.
- Result robust to controls including trade openness, capital mobility, exchange rate regime, debt-to-GDP ratio, economic development, and institutional quality.
- Irrespective of level of development or institutional quality, economies with low informality display higher cumulative fiscal multipliers than economies with high informality.
- Higher fiscal multipliers are not found in developed economies when informality is high.

### Fiscal shock identification and validation
- Two-step identification:
  - FE_{i,t|t-1} = ΔlnG_{i,t} − ΔlnG_{i,t|t-1}; then purge predictable components by projecting on lags of output, government spending, government revenue, real exchange rate, inflation; residual is fiscal shock.
- Rationale for WEO October forecasts:
  - WEO forecasts of fiscal variables are usually very accurate (An et al. 2018).
  - October forecasts reduce likelihood that forecast errors reflect endogenous contemporaneous responses.
  - Methodology addresses “fiscal foresight” by aligning econometrician and agent information sets.
- Exogeneity tests (per Ramey (2016)):
  - Regressions of fiscal shocks on lagged output gap, lagged public debt-to-GDP, contemporaneous change in revenue show none significantly correlated with fiscal shocks (Table A2).
  - Correlations between fiscal shocks and monetary policy shocks (Furceri et al. 2017) and World Uncertainty Index are close to zero (Table A3).
  - Shocks represent unanticipated movements by construction.

### Data and measures of the shadow economy
- GDP frequency: yearly basis from IMF WEO database.
- Two informality measures:
  - Medina and Schneider (2018) — MIMIC.
  - Elgin and Oztunali (2012); Elgin et al. (2019) — DGE two-sector model.
- Both estimate informal output in percent of official GDP.
- Estimation sample: unbalanced panel of 141 countries over 1995–2015.
- Note: other approaches (modified total electricity; currency demand) typically require base-year shadow economy estimates that are difficult to obtain.

### Empirical methodology (specification and interaction)
- Baseline: unconditional cumulative fiscal multiplier via local projections (Jordà 2005); government expenditures to GDP instrumented with identified fiscal shocks.
- Regression features:
  - Dependent variable: cumulative sum of log GDP from t to t+k for country i.
  - Main regressor: cumulative sum of government spending from t to t+k for country i.
  - Controls X_{i,t}: four lags of dependent variable and four lags of government expenditure to GDP.
  - Country fixed effects (α_i) and time fixed effects (γ_t) included.
- Nonlinear interaction with informality:
  - Smooth transition F(z_i) = exp(−γ z_i) / (1 + exp(−γ z_i)), z_i = (s_i − s̅_i)/sd(s_i).
  - Baseline γ = 5; robustness γ ∈ {3.6, 6.5}.
  - Coefficients m_hL (low informality, z → −∞) and m_hH (high informality, z → ∞).
  - Instruments: [shock_i,t * F(z_i)] and [shock_i,t * (1 − F(z_i))].
- Alternative specifications: linear interaction, median-dummy, varied lag structures, additional fixed effects, country-specific trends.

### Baseline empirical magnitudes and robustness (exact values preserved)
- Baseline unconditional cumulative fiscal multiplier:
  - about 0.25 one year after the shock.
  - about 0.6 four years after the shock.
- Heterogeneity by informality (thresholds and implied multipliers):
  - Very low informality (share of shadow economy below 17% of GDP):
    - Positive and statistically significant output response at all horizons.
    - Implied cumulative medium-term fiscal multiplier of about 2.
  - Very high informality (share of shadow economy above 44% of GDP):
    - Effect of government spending shocks smaller and not statistically significantly different from zero.
- Result consistency:
  - Holds for both DGE and MIMIC measures.
  - Difference in estimated multipliers:
    - Statistically significantly different from zero at all horizons for the DGE measure.
    - Statistically significantly different up to the first two horizons for the MIMIC measure (Table A6).
- Robustness checks:
  - Stable to alternative lag structures (Figure A4).
  - Similar across specifications: (i) only country fixed effects; (ii) country fixed effects plus country-specific time trends; (iii) both country-specific time trends and time fixed effects (Figure A5).
  - Changing γ from 3.6 to 6.5 yields effects similar to baseline (Table A7).
  - Linear interaction and median-dummy specifications produce robust results (Tables A8 and A9).
- Measurement error mitigation:
  - Robustness to two markedly different informality measures reduces concerns that measurement error drives findings.

### Distributional properties of computed government spending shocks
- Average (median) of the shock: -0.1 (0.02) percent.
- Bulk of shocks (between 1st and 99th percentile) ranges between -64 and 72 percent.
- Selected country descriptive statistics (Table A1; exact values preserved in source):
  - Afghanistan: Mean -7.68; Standard deviation 26.25; Min -34.45; Max 36.78
  - Albania: Mean 2.06; Standard deviation 25.75; Min -57.00; Max 37.00
  - Angola: Mean -18.35; Standard deviation 29.33; Min -52.98; Max 55.99
  - Australia: Mean 0.98; Standard deviation 8.61; Min -11.81; Max 15.47
  - Brazil: Mean -15.67; Standard deviation 14.58; Min -46.82; Max 15.19
  - China: Mean -1.31; Standard deviation 5.97; Min -16.16; Max 3.39
  - Germany: Mean 3.81; Standard deviation 3.31; Min -1.98; Max 9.06
  - India: Mean -2.11; Standard deviation 9.44; Min -25.96; Max 11.71
  - Japan: Mean 2.14; Standard deviation 3.68; Min -3.53; Max 7.84
  - United States: Mean -0.02; Standard deviation 1.80; Min -2.73; Max 2.27
  - Venezuela: Mean -5.31; Standard deviation 32.77; Min -53.26; Max 82.17
  - Zambia: Mean 1.29; Standard deviation 25.36; Min -50.92; Max 44.05

### Is informality a proxy for other factors?
- Extended specification includes interactions of government spending with country characteristics u:
  - u includes trade and financial openness (KOF), exchange rate regime (Ilzetzki et al. 2013), debt-to-GDP (WEO), log GDP per capita (WEO), Law and Order component of ICRG.
- Instrument for interaction: shock_i,t * (u̅_i − u̅); variables demeaned so baseline corresponds to average country.
- Results (Table 1, Table A10): very similar to baseline; confirm informality as an independent driver of fiscal multiplier magnitude.

### Interaction with development and institutional quality (triple-interaction findings)
- Triple-interaction uses smooth transitions for informality (z) and for development/institutional quality (x); γ-calibrated.
- Findings (Figures 3 and 4; Table A11, Table A12):
  - Regardless of development level, low-informality economies display relatively higher cumulative fiscal multipliers.
  - In less developed countries, difference between low and high informality is statistically significant at 90% for all periods considered.
  - In developed economies, difference is statistically significant only in the short term.
  - Conditioning on informality, fiscal multipliers do not vary significantly between high- and low-income countries (Table A11).
  - Conditioning on informality, multipliers do not vary significantly by institutional quality (Table A12).
- Conclusion: level of informality is a key element explaining differences in fiscal multipliers across countries, independent of income per capita and institutional quality.

### Theoretical model and identified transmission channel
- Model: medium-size two-agent new-Keynesian (TANK) model with official and informal sectors.
- Key model assumptions and elements:
  - Formal and informal goods are sectoral bundles that are imperfect substitutes in household preferences; sector-specific formal and informal firms produce such goods.
  - Model abstracts from other structural asymmetries to focus on size of informal sector.
  - Main mechanism: informality reduces size of public expenditure multiplier via relative-price channel — high informality associated with stronger appreciation of relative price of official goods, triggering reallocation of private spending towards unofficial goods and weakening the official multiplier.
- Model extensions:
  - Non-separability between public and private consumption introduced.
  - Non-separability findings:
    - Useful only if low (high) informality is associated with complementarity (substitutability) between public and private goods.
    - A degree of complementarity close to values typically assumed in US DSGE models replicates estimates for low-informality case.
    - A relatively strong degree of substitutability contributes to even lower multipliers in economies with high informality.
- Contribution:
  - Theoretical novelty: identifies relative price of informal goods as key transmission channel, operating independently of institutional quality and development level.

### Model environment — households (structure and calibration highlights)
- Preferences and labor supply:
  - U_t = E_t ∑_{k=0}^{∞} β^k { ln(C̃_{t+k}^i) − ( χ (l_{t+k}^i)^{1+φ} / (1+φ) ) } (equation (21)).
  - Households supply continuum of differentiated labor types; wage-setting by unions with adjustment costs.
- Household types:
  - Ricardian fraction: (1−ω); intertemporal optimization with bonds and sectoral capital choices (equations (23)–(27)).
  - Non-Ricardian fraction: ω; consume entire current disposable income each period: c^{Rot}_t = (P^O_t / P^C_t) w^O_t l^O_t (1−τ_w) + (P^s_t / P^w_t) w^s_t l^s_t (equation (28)). Non-Ricardians do not pay lump-sum taxes.
- Wage-setting:
  - Symmetric-equilibrium wage-setting equation (equation (36)) embeds wage adjustment cost ξ_w and wage elasticity η_w.
  - Aggregate marginal utility weight λ_t = (1−ω) λ^{Opt}_t + ω λ^{Rot}_t (equation (37)).
- Key calibration parameters (exact values preserved):
  - β = 0.99.
  - Frisch elasticity 1/φ = 1.
  - η_w = 6, ξ_w = 50.
  - Official sector price adjustment cost φ_O = 50.
  - Official price-elasticity parameter σ_O = 6.
  - Informal sector nominal rigidities: φ_s = φ_O and σ_s = σ_O.
  - ε = 20 (substitution between official and informal consumption bundles).
  - Official capital income share α_O = 0.34.
  - Informal capital income share α_s = 0.24.
  - Depreciation rate δ = 0.02.
  - Steady-state relative capital-labor ratio k_O/l_O ÷ k_s/l_s = 2.15.
  - γ_I = 4 (investment adjustment cost).
  - Monetary policy Taylor rule parameter φ_π = 1.5.
  - Uniform tax rates τ_k = τ_w = 0.20.
  - Public-consumption-to-GDP ratio = 21%.
  - Public-debt-to-GDP ratio = 60%.
  - Autoregressive parameter for public consumption shocks ρ_G = 0.93.
  - ω = 0.25 (non-Ricardian share).
  - Complementarity case: γ_c = 0.7, e = 0.4.
  - Substitutability illustrative case: e = 1.5.
- Calibration of informality parameter z = y_s / y_O:
  - High-informality: z ≈ 45% described as [1−F(z)] > 0.9.
  - Low-informality: z ≈ 15% described as [F(z)] > 0.9.
  - Sample mean: z = 30%.
- Table A13 reports full parameter set (referenced in source).

### Model steady state and aggregation (selected steady-state relations)
- Steady-state normalizations and conditions:
  - R = 1/β.
  - 1 = π_O; 1 = π_s; π = 1.
  - Markups: m_cO = (σ_O − 1)/σ_O; m_cs = (σ_s − 1)/σ_s.
  - Real rental rates: r_k,O = [1/β − (1 − δ(1 − τ_k))] (1 − τ_k); r_k,s = [1/β − (1 − δ)].
  - Wage formulas and sectoral capital from marginal conditions preserved as in source.
- Consumption and investment aggregators (exact functional forms preserved in source).
- Labor normalization: steady-state labor l = 1 via parameter χ calibration.

### Model implications (as presented)
- Theoretical multipliers computed from IRFs of benchmark model replicate empirical pattern: larger fiscal multipliers when informality is low; smaller (or near-zero) multipliers when informality is high (Figures 6–12 and related notes).
- Non-separable utility cases can amplify differences depending on sign and magnitude of e and γ_c.

*Source: wpiea2022082-print-pdf*

### 1. Introduction ........................................................................................................

### 1. Introduction

### Overview
- The paper examines why fiscal multipliers are smaller in developing countries and identifies a key driver: the interaction between the strength of the public expenditure multiplier and the size of informality.
- Sample: a large sample of 141 developed and developing economies.
- Main empirical approach: local projection method of Jordà (2005) to estimate cumulative dynamic response of GDP to government spending shocks.
- Fiscal shocks identification: forecast errors for government spending from IMF World Economic Outlook (WEO) forecasts (October of the same year), following Auerbach and Gorodnichenko (2013) and extended as in Auerbach and Gorodnichenko (2012) and Furceri and Zdzienicka (2020).

### Key empirical findings
- High informality is associated with a marked reduction in the size of the fiscal multiplier.
- Finding robust to controls for country characteristics identified in the literature as determinants of cross-country differences in fiscal multipliers, including trade openness, capital mobility, exchange rate regime, debt-to-GDP ratio, economic development, and measures of institutional quality.
- Irrespective of the level of development (and the degree of institutional quality), economies with a low degree of informality display a higher cumulative fiscal multiplier than countries with high degree of informality.
- The converse does not hold: higher fiscal multipliers are not found in developed economies when informality is high.
- Distributional properties of computed government spending shocks for the entire sample (Figure A1 and Table A1 in Appendix):
  - Average (median) of the shock: -0.1 (0.02) percent.
  - Bulk of shocks (between the 1st and the 99th percentile) ranges between -64 and 72 percent.

### Fiscal shock identification and validation
- Two-step identification:
  1. Compute t-period forecast error for public spending for country i: FE_{i,t|t-1} = ΔlnG_{i,t} − ΔlnG_{i,t|t-1}, where ΔlnG_{i,t} is actual government spending growth and ΔlnG_{i,t|t-1} is the t−1 IMF forecast for ΔlnG_{i,t}.
  2. Purge the forecast error of predictable components by projecting it on lags of macro variables (output, government spending, government revenue, real exchange rate, inflation) and taking the residual as the fiscal shock.
- Rationale for using WEO October forecasts:
  - WEO forecasts of fiscal variables are usually very accurate (An et al. 2018) due to continuous IMF–finance ministry information flow.
  - Using forecasts made in October of the same year reduces the likelihood that forecast errors reflect endogenous responses to contemporaneous state of the economy.
  - Methodology solves the “fiscal foresight” problem by aligning econometrician and agent information sets.
- Exogeneity tests (per Ramey (2016)):
  - Regressions of fiscal shocks on endogenous macro variables (lagged output gap, lagged public debt-to-GDP ratios, contemporaneous change in revenue) show none are significantly correlated with the fiscal shocks (Table A2).
  - Correlations between fiscal shocks and other demand shocks (monetary policy shocks from Furceri et al. (2017) and the World Uncertainty Index by Ahir, Bloom and Furceri (2022)) are close to zero (Table A3).
  - Shocks represent unanticipated movements by construction.

### Theoretical model and identified transmission channel
- A medium-size two-agent new-Keynesian (TANK) model is developed that accounts for both official and informal economy sectors.
- Key modeling assumptions:
  - Formal and informal goods are treated as sectoral bundles that are imperfect substitutes in households’ preferences; sector-specific formal and informal firms produce such goods.
  - The model abstracts from other structural asymmetries (openness, institutional differences) to focus exclusively on size of informal sector.
- Main theoretical result:
  - Informality reduces the size of the public expenditure multiplier.
  - Main mechanism: high informality is associated with a stronger appreciation of the relative price of official goods, which triggers reallocation of private spending towards unofficial goods and weakens the official multiplier.
- Model extensions:
  - Incorporation of non-separability between public and private consumption in households’ preferences.
  - Findings on non-separability:
    - Non-separability is useful only if low (high) informality is associated with complementarity (substitutability) between public and private goods.
    - A degree of complementarity close to values typically assumed in US DSGE models replicates estimates for the low-informality case.
    - A relatively strong degree of substitutability contributes to even lower multipliers in economies with high informality.

### Contribution to literature and novelty
- Empirical: first systematic analysis of the role of informality in shaping fiscal policy effectiveness across 141 economies.
- Theoretical: first contribution identifying the relative price of informal goods as a key transmission channel for fiscal shocks, independent from institutional quality and level of development.
- Distinction from prior literature:
  - Prior models often assume homogeneous goods and emphasize labor reallocation and tax enforcement/corruption channels related to institutional quality.
  - This paper’s channel—relative price adjustments between formal and informal goods—operates largely independently of social infrastructure measures.

### Paper structure
- Section 2: data and methodology.
- Section 3: empirical results.
- Section 4: model description and implications.
- Section 5: conclusions and policy implications.
- Appendix: additional empirical results, robustness checks, and formal derivation of the model steady state.

*IMF WORKING PAPERS — Fiscal Multipliers and Informality (1. Introduction).*

### 2.2 Shadow economy and other macroeconomic data

### 2.2 Shadow economy and other macroeconomic data

### Data and measures of the shadow economy
- GDP data: yearly basis from the IMF World Economic Outlook database.
- Two alternative informality measures used:
  - Medina and Schneider (2018) — Multiple Indicators Multiple Causes model (MIMIC).
  - Elgin and Oztunali (2012); Elgin et al. (2019) — deterministic dynamic general equilibrium (DGE) two-sector model (official and shadow economies).
- Both approaches estimate informal output in percent of official GDP.
- Estimation sample: an unbalanced panel of 141 countries over the period 1995–2015.
- Note: Two alternative methods (modified total electricity approach as in Colombo et al. (2016), and the currency demand approach) typically require base year estimates of the shadow economy size, which are difficult to obtain.

### Methodology
- Baseline estimation of the unconditional cumulative fiscal multiplier using the local projection method (Jordà 2005).
- Government expenditures to GDP are instrumented with identified fiscal policy shocks (as in Ramey and Zubairy 2018).
- Regression specification details:
  - Dependent variable: cumulative sum of log GDP from t to t+k for country i.
  - Main regressor: cumulative sum of government spending from t to t+k for country i.
  - Controls X_i,t include four lags of the dependent variable and four lags of government expenditure to GDP.
  - Country fixed effects (α_i) and time fixed effects (γ_t) included.
- Nonlinear interaction with informality:
  - Smooth transition function F(z_i) = exp(−γ z_i) / (1 + exp(−γ z_i)), where z_i = (s_i − s̅_i)/sd(s_i) and s_i is the country average informality.
  - Parameter γ calibrated in baseline to γ = 5; robustness checks vary γ between 3.6 and 6.5.
  - Coefficients m_hL (low informality, z → −∞) and m_hH (high informality, z → ∞).
  - Instruments: [shock_i,t * F(z_i)] and [shock_i,t * (1 − F(z_i))] for respective interactions.
- Alternative specifications tested:
  - Linear interaction between government spending and degree of informality.
  - Dummy approach: share of shadow economy above or below the median.
  - Robustness to lag-structure choices and to inclusion of different fixed effects and country-specific time trends.

### Key empirical findings (Baseline and robustness)
- Baseline unconditional cumulative fiscal multiplier:
  - Implied cumulative fiscal multiplier is about 0.25 one year after the shock.
  - Implied cumulative fiscal multiplier is about 0.6 four years after the shock.
- Heterogeneity by informality:
  - Countries with very low informality (share of shadow economy below 17% of GDP):
    - Output response to government spending shocks is positive and statistically significant at all horizons.
    - Implied cumulative medium-term fiscal multiplier of about 2.
  - Countries with very high informality (share of shadow economy above 44% of GDP):
    - Effect of government spending shocks is smaller and not statistically significantly different from zero.
  - Result holds for both DGE and MIMIC measures of informality.
  - Difference in estimated multipliers:
    - Statistically significantly different from zero at all time horizons for the DGE measure.
    - Statistically significantly different up to the first two horizons for the MIMIC measure (see Table A6).
- Robustness checks:
  - Results stable to alternative lag structures for log GDP and government expenditures to GDP (Figure A4).
  - Results similar across specifications: (i) only country fixed effects; (ii) country fixed effects plus country-specific time trends; (iii) both country-specific time trends and time fixed effects (Figure A5).
  - Changing γ from 3.6 to 6.5 yields effects similar to baseline (Table A7).
  - Linear interaction and median-dummy specifications produce robust results (Tables A8 and A9).
- Measurement error consideration:
  - Results are robust to two markedly different measures of informality, reducing concerns that measurement error drives findings.

### Is informality a proxy for other factors?
- Potential confounders considered: level of income, trade and financial openness, exchange rate regime, initial public debt, institutional quality.
- Extended specification (equation (5)) includes interactions of government spending with country characteristics u (trade and financial openness from KOF; exchange rate regime from Ilzetzki et al. 2013; debt-to-GDP from WEO; log GDP per capita from WEO; Law and Order component of ICRG).
- Instrument: shock_i,t * (u̅_i − u̅) used for interaction term ∑g * (u̅_i − u̅); variables demeaned so baseline response corresponds to average country.
- Results (Table 1 and Table A10):
  - Very similar to Figure 2.
  - Confirm informality as an independent driver of fiscal multiplier magnitude.

### Interaction with development and institutional quality
- Triple-interaction approach (equation (6)) uses smooth transition functions for informality (z) and for level of development or institutional quality (x), with γ-calibrated transition functions.
- Findings (Figures 3 and 4):
  - Regardless of level of development, economies with low informality display relatively higher cumulative fiscal multipliers.
  - In less developed countries, difference between low and high informality is statistically significant at 90% for all periods considered.
  - In developed economies, difference is statistically significant only in the short term.
  - Conditioning on informality, fiscal multipliers do not vary significantly between high- and low-income countries (Table A11).
  - Similar results when conditioning on institutional quality (Law and Order, ICRG): low-informality countries display larger cumulative multipliers irrespective of institutional quality; conditioning on informality, multipliers do not vary significantly by institutional quality (Table A12).
- Conclusion: level of informality is a key element explaining differences in fiscal multipliers across countries, independent of income per capita and institutional quality.

### Theoretical model overview (motivation and structure)
- Purpose: rationalize empirical result via an amplification mechanism determined by informal sector size, distinct from income or institutional quality.
- Model choice: standard two-sector DSGE (official and informal sectors); differences limited to:
  - i) capital intensity of informal firms,
  - ii) taxation of factor incomes (only official sector factor incomes taxed),
  - iii) steady-state size of informal sector,
  - iv) sign of non-separability between private and public goods.
- Production and demand structure:
  - Monopolistically competitive firms produce sectoral goods in official (o) and informal (s) sectors.
  - Informal sector firms have relatively more labor-intensive production technology.
  - Only factor incomes in the official sector are taxed.
  - Households’ consumption aggregator over sectoral goods:
    - c_t = [ (1 − α_c)^(1/ε) (c_t^o)^(ε−1)/ε + (α_c)^(1/ε) (c_t^s)^(ε−1)/ε ]^(ε/(ε−1)).
  - Consumption price index P_t^C defined accordingly.
  - Investment goods are a bundle of sectoral goods with analogous aggregator (α_I, ε_I).
  - Market clearing for investment goods: I_t = I_t^o + I_t^s.
- Household heterogeneity and fiscal transmission:
  - Limited Asset Market Participation (LAMP) / TANK setup: fraction of Ricardian households smooth consumption; remaining Non-Ricardian households consume current disposable income.
  - Non-separability between private and public goods introduced via bundle:
    - C̃_t^i = [ γ_c^(1/e) (c_t^i)^(e−1)/e + (1 − γ_c)^(1/e) (G_t)^(e−1)/e ]^(e/(e−1)).
  - Marginal utility of consumption λ_t^i determined by e and γ_c; calibration of e can generate positive or negative comovement between fiscal shock and marginal utility, affecting household demand beyond wealth effects.
- Modeling simplifications:
  - No labor matching frictions or non-trivial financial institutions included to keep model parsimonious and focused on informality amplification mechanism.

*Source: wpiea2022082-print-pdf - 2.2 Shadow economy and other macroeconomic data.*

### 4.1 Households

### 4.1 Households

### Preferences and labor supply
- Representative household preferences: 
  - U_t = E_t ∑_{k=0}^{∞} β^k { ln(C̃_{t+k}^i) − ( χ (l_{t+k}^i)^{1+φ} / (1+φ) ) } (equation (21)).
- Labor market setup:
  - Each household i supplies a continuum of differentiated labor types l^i_h (h ∈ [0,1]).
  - Fully competitive labor packers aggregate labor types into the bundle l^d_t. Demand for generic labor type h:
    - l_{t}^{h,i} = (W_{t}^{h} / W_{t})^{-η_w} l^d_t (equation (22)).
  - Aggregate wage index:
    - W_t = ( ∫_0^1 (W_t^h)^{1−η_w} dh )^{1/(1−η_w)}.
  - Wage-setting delegated to type-specific unions; fraction of Ricardian and non-Ricardian households is uniformly distributed across unions; aggregate demand for each labor type uniformly distributed across households (as in Galí, López-Salido, and Vallés (2007)).

### Ricardian households (fraction (1−ω))
- Flow budget constraint in consumption units (equation (23)):
  - c^O_t + (P^I_t / P^C_t) (I^O_t + I^s_t) + (B_t / R_t P^C_t) =
    (P^O_t / P^C_t)[ (1−τ_k)(r^{k,O}_t − δ) k^{O}_{t−1} + (1−τ_w) w^O_t (P^O_t / P^l_t) l^O_t + Π^O_t − τ^{LS}_t ] + B_{t−1}/P^C_t + (P^s_t / P^C_t)[ (r^{k,s}_t − δ) k^{s}_{t−1} + w^s_t l^s_t + Π^s_t ].
- Intertemporal Euler conditions:
  - For government bonds: λ^{Opt}_t = β E_t λ^{Opt}_{t+1} / (π_{t+1} R_t) (equation (24)).
  - For sectoral capital j = {O, s} (equation (25)):
    - 1 = β E_t (λ^{Opt}_{t+1} / λ^{Opt}_t) [ q_{t+1}^j (1−δ) + (P_{t+1}^j / P^C_{t+1}) r^{k,j}_{t+1} (1−τ_{k,j}) ] / q_t^j.
- Capital accumulation (equation (26)):
  - k^j_t = (1−δ) k^j_{t−1} + [1 − S(I^j_t / I^j_{t−1}) ] I^j_t, where S( I^j_t / I^j_{t−1} ) = γ_I /2 ( I^j_t / I^j_{t−1} − 1 )^2.
- First-order condition for investment (equation (27)):
  - P^I_t / P^C_t = q^j_t { 1 − γ_I (I^j_t / I^j_{t−1} − 1 ) I^j_t / I^j_{t−1} − γ_I/2 ( I^j_t / I^j_{t−1} − 1 )^2 } + β (λ^{Opt}_{t+1} / λ^{Opt}_t) q^j_{t+1} γ_I ( I^j_{t+1} / I^j_t − 1 ) ( I^j_{t+1} / I^j_t )^2.

### Non-Ricardian households (fraction ω)
- Consume entire current disposable income each period (equation (28)):
  - c^{Rot}_t = (P^O_t / P^C_t) w^O_t l^O_t (1−τ_w) + (P^s_t / P^w_t) w^s_t l^s_t.
- Note: Non-Ricardian households do not pay lump-sum taxes (footnote 22).

### Wage setting and labor unions
- Labor union h maximizes:
  - L_{u,h} = E_t ∑_{k=0}^{∞} β^k { (1−ω) U^{Opt}(C̃^{Opt}_{t+k}) + ω U^{Rot}(C̃^{Rot}_{t+k}) − U(l_{t+k}) } subject to labor packer demand (22) and quadratic wage adjustment cost ξ_w/2 ( W^h_t / W^h_{t−1} − 1 )^2 l_t.
- Symmetric-equilibrium wage-setting equation (equation (36)):
  - w_t λ_t χ(l_t)^φ =
    η_w / (η_w − 1) { 1 − [ ξ_w / η_w ( w_t / w_{t−1} π_t / π_{t−1} χ_w^{−1} ) w_t / w_{t−1} π_t / π_{t−1} χ_w  − β λ_{t+1} ξ_w / η_w ( w_{t+1} / w_t π_{t+1} / π_t χ_w^{−1} ) ( l_{t+1} / l_t ) ( w_{t+1} / w_t ) ( π_{t+1} / π_t ) χ_w ] }.
- Aggregate marginal utility weight λ_t (equation (37)):
  - λ_t = (1−ω) λ^{Opt}_t + ω λ^{Rot}_t.
- Labor packers supply labor bundle at wage w_t; sectoral relations (equations (38)-(39)):
  - w_t = w^s_t P^s_t / P_t.
  - w_t = w^O_t P^O_t / P_t (1−τ_w).

### Calibration (parameters and steady states)
- Household and labor parameters:
  - β = 0.99.
  - Frisch elasticity 1/φ = 1.
  - Labor market: η_w = 6, ξ_w = 50.
- Price adjustment and markup parameters:
  - Official sector price adjustment cost φ_O = 50.
  - Official price-elasticity parameter σ_O = 6.
  - Informal sector nominal rigidities set equal to official: φ_s = φ_O and σ_s = σ_O.
- Substitution between official and informal consumption bundles:
  - ε = 20 (benchmark).
  - Footnote: Epstein and Finkelstein Shapiro (2017) set ε = 5; experiments yield marginal effects.
- Production shares and depreciation:
  - Official capital income share α_O = 0.34.
  - Informal capital income share α_s = 0.24.
  - Capital depreciation rate δ = 0.02.
  - Steady-state relative capital-labor ratio k_O/l_O ÷ k_s/l_s = 2.15.
- Investment adjustment cost parameter:
  - γ_I = 4.
- Monetary policy:
  - Taylor rule parameter φ_π = 1.5.
- Tax and fiscal steady-states:
  - Uniform tax rate on factor incomes τ_k = τ_w = 0.20.
  - Public-consumption-to-GDP ratio = 21%.
  - Public-debt-to-GDP ratio = 60%.
  - Autoregressive parameter for public consumption shocks ρ_G = 0.93.
- Non-Ricardian household share:
  - ω = 0.25.
- Complementarity/substitutability cases between public and private consumption:
  - Complementarity case: γ_c = 0.7, e = 0.4.
  - Substitutability illustrative case: e = 1.5.
- Calibration of informality parameter z = y_s / y_O to match empirical informality:
  - High-informality: z ≈ 45% described as [1−F(z)] > 0.9.
  - Low-informality: z ≈ 15% described as [F(z)] > 0.9.
  - Sample mean: z = 30%.
- Parameter χ set so steady-state labor l = 1 across different shares of informal sector.
- Table A13 (referenced) reports all parameter values.

Italicized source attribution:
*Source: wpiea2022082-print-pdf - 4.1 Households*

### References

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### Notes on presentation
- The References list appears in the context of the IMF Working Papers title: Fiscal Multipliers and Informality.

### Figures (captions and notes)
- Figure 1. Cumulative fiscal multiplier – unconditional
  - Note: The chart shows the impulse response functions and the associated 90 percent confidence bands; t = 0 is the year of shock. Estimates based on equation (1) using a sample of 141 countries over the period 1995-2015.
- Figure 2. Cumulative fiscal multiplier – the role of the shadow economy
  - Note: The charts show the impulse response functions and the associated 90 percent confidence bands; t = 0 is the year of shock. Estimates based on equation (2) using a sample of 141 countries over the period 1995-2015. Shadow economy estimates are from MIMIC and DGE models.
- Figure 3. The role of the shadow economy and level of economic development
  - Note: The chart shows the impulse response functions and the associated 90 percent confidence bands; t = 0 is the year of shock. Estimates based on equation (2) using a sample of 141 countries over the period 1995-2015. Shadow economy estimates are from MIMIC and DGE models.
- Figure 4. The role of the shadow economy and level of institutional quality
  - Note: The chart shows the impulse response functions and the associated 90 percent confidence bands; t = 0 is the year of shock. Estimates based on equation (2) using a sample of 141 countries over the period 1995-2015. Shadow economy estimates are from MIMIC and DGE models.
- Figure 5. The model economy
- Figure 6. Cumulative fiscal multiplier – unconditional
  - Note: The charts show the impulse response functions and the associated 90 percent confidence bands; t = 0 is the year of shock. Estimates based on equation (2) using a sample of 141 countries over the period 1995-2015. Theoretical multipliers are computed from IRFs of the benchmark model, separable utility and 30% share of the shadow economy, values are percentage deviations from the steady state.
- Figure 7. IRF shadow economy at 30% (circled line) and at 0% (black line). Separable utility - 1
  - Note: The charts show IRFs of the model with, separable utility 30% share of the shadow economy (circled purple line) and 0% share of the shadow economy (black line), values are percentage deviations from the steady state. On the x-axis numbers represent quarters. Aggregate variables denote the artificial economy obtained by aggregating official and shadow sectors.
- Figure 8. IRF shadow economy at 30% (circled line) and at 0% (black line). Separable utility - 2
  - Note: The charts show IRFs of the model with, separable utility 30% share of the shadow economy (circled purple line) and 0% share of the shadow economy (black line), values are percentage deviations from the steady state. On the x-axis numbers represent quarters.
- Figure 9. Cumulative fiscal multipliers at different degrees of informality
  - Note: The charts show IRFs of the model with separable utility at different shares of the shadow economy, values are percentage deviations from the steady state. On the x-axis numbers represent quarters.
- Figure 10. Cumulative fiscal multiplier. Separable and non separable utility
  - Note: The charts show the impulse response functions and the associated 90 percent confidence bands; t = 0 is the year of shock. Estimates based on equation (2) using a sample of 141 countries over the period 1995-2015. Shadow economy estimates are from MIMIC and DGE models. Theoretical multipliers are computed from IRFs of the benchmark model, separable and non-separable utility, 45% share of the shadow economy (high informality), and 15% share of the shadow economy (low informality); values are percentage deviations from the steady state.
- Figure 11. IRF High share of shadow economy (circled blue line) and Low share (starred red line). Non-separable utility – 1
  - Note: The charts show IRFs of the model with non-separable utility, 45% share of the shadow economy (high informality, circled blue line), and 15% share of the shadow economy (low informality, starred red line), values are percentage deviations from the steady state. On the x-axis numbers represent quarters.
- Figure 12. IRF High share of shadow economy (circled blue line) and Low share (starred red line). Non-separable utility – 2
  - Note: The charts show IRFs of the model with non-separable utility, 45% share of the shadow economy (high informality, circled blue line), and 15% share of the shadow economy (low informality, starred red line), values are percentage deviations from the steady state. On the x-axis numbers represent quarters.

### Tables (excerpt)
- Table 1. Robustness checks – Additional controls
  - Columns: All controls
  - Measure of informality | Horizon | Low Informality | High Informality | Significance level of difference
  - MIMIC
    - 0 | 1.169*** | 0.152 | 0.052
    - 1 | 1.792*** | 0.264 | 0.013
    - 2 | 2.429** | 0.635 | 0.027
    - 3 | 2.451 | 0.846 | 0.208
    - 4 | 3.130* | 0.820 | 0.137
  - DGE
    - 0 | 0.926** | 0.279 | 0.138
    - 1 | 1.434** | 0.437 | 0.066
    - 2 | 2.090* | 0.776 | 0.102
    - 3 | 2.268 | 0.974 | 0.326
    - 4 | 2.369 | 1.118 | 0.359
  - Note: Estimates are obtained using a sample of 141 countries over the period 1995-2015 and based on equation (4). *** p<0.01, ** p<0.05, * p<0.1.

### Appendix
- Appendix (section heading present; content not included in this excerpt)

*IMF Working Papers — Fiscal Multipliers and Informality (References, Figures, Tables, Appendix excerpts).*

### Appendix 1

### Appendix 1

### Figures — impulse responses and specification checks
- Figure A1: Distribution of Government Expenditure Shocks (chart).
- Figure A2: Fiscal multipliers – using forecasts of government expenditures made in April of the same year. Note: impulse response functions and associated 90 percent confidence bands; t = 0 is the year of shock. Estimates based on equation (1) using a sample of 141 countries over the period 1995-2015.
- Figure A3: Cumulative fiscal multiplier using forecasts of government expenditures made in April of the same year – the role of the shadow economy. Note: Estimates based on equation (2) using a sample of 141 countries over the period 1995-2015 and forecasts of government expenditures made in April of the same year. Shadow economy estimates are from MIMIC and DGE models.
- Figure A4: Alternative lag structures. Note: Estimates based on equation (1) using a sample of 141 countries over the period 1995-2015.
- Figure A5: Alternative specifications. Note: Estimates based on equation (1) using a sample of 141 countries over the period 1995-2015.

### Table A1 — Government Expenditure Shocks: descriptive statistics (selected entries)
- Each row: Country | Mean | Standard deviation | Min | Max.
- Selected countries (exact values as in the source):
  - Afghanistan: Mean -7.68; Standard deviation 26.25; Min -34.45; Max 36.78
  - Albania: Mean 2.06; Standard deviation 25.75; Min -57.00; Max 37.00
  - Angola: Mean -18.35; Standard deviation 29.33; Min -52.98; Max 55.99
  - Australia: Mean 0.98; Standard deviation 8.61; Min -11.81; Max 15.47
  - Brazil: Mean -15.67; Standard deviation 14.58; Min -46.82; Max 15.19
  - China: Mean -1.31; Standard deviation 5.97; Min -16.16; Max 3.39
  - Germany: Mean 3.81; Standard deviation 3.31; Min -1.98; Max 9.06
  - India: Mean -2.11; Standard deviation 9.44; Min -25.96; Max 11.71
  - Japan: Mean 2.14; Standard deviation 3.68; Min -3.53; Max 7.84
  - United States: Mean -0.02; Standard deviation 1.80; Min -2.73; Max 2.27
  - Venezuela: Mean -5.31; Standard deviation 32.77; Min -53.26; Max 82.17
  - Zambia: Mean 1.29; Standard deviation 25.36; Min -50.92; Max 44.05
- (Table continues for all sample countries listed in the source.)

### Table A2 — Regression of fiscal shocks on macroeconomic variables (coefficients and t-statistics)
- Dependent variable: Fiscal shock. Country and time fixed effects included but not reported. t-statistics based on robust standard errors in parentheses.
- Lagged output gap: coefficient -0.211 (t-statistic (-1.42))
- Lagged Public Debt to GDP: coefficient -0.001 (t-statistic (-1.40))
- Change in Revenue: coefficient 0.0346 (t-statistic (-0.42))

### Table A3 — Correlations between fiscal shocks, monetary policy shocks and WUI
- Fiscal shocks with:
  - Fiscal shocks: 1
  - Monetary Policy shocks (source: Furceri et al. 2017): -0.033
  - World Uncertainty Index (source: Ahir et al. 2021): -0.007

### Table A4 — Country sample and group classification
- Country group codes: AE = Advanced Economies; EM = Emerging Market economies; LIDC = Low Income and Developing Economies. Country group classification follows the World Bank definition.
- The table lists all countries in the analysis assigned to CG categories (AE, EM, LIDC) as in the source.

### Table A5 — Descriptive statistics: Informality measures
- Informality: MIMIC — N 2223; mean 31.509; sd 12.566; min 6.390; max 70.570
- Informality: DGE — N 2286; mean 31.962; sd 11.908; min 7.909; max 65.994

### Table A6 — Cumulative fiscal multiplier – the role of the shadow economy (exact values)
- Estimates obtained using a sample of 141 countries over 1995-2015 and based on equation (2). Significance: *** p<0.01, ** p<0.05, * p<0.1.
- MIMIC:
  - Horizon 0: Low Informality 0.783***; High Informality -0.000; Significance level of difference 0.031
  - Horizon 1: Low Informality 1.183***; High Informality -0.010; Significance level of difference 0.018
  - Horizon 2: Low Informality 1.527**; High Informality -0.001; Significance level of difference 0.036
  - Horizon 3: Low Informality 1.648*; High Informality 0.114; Significance level of difference 0.139
  - Horizon 4: Low Informality 2.013*; High Informality 0.125; Significance level of difference 0.118
- DGE:
  - Horizon 0: Low Informality 0.604***; High Informality 0.035; Significance level of difference 0.027
  - Horizon 1: Low Informality 1.044***; High Informality 0.013; Significance level of difference 0.005
  - Horizon 2: Low Informality 1.239***; High Informality -0.001; Significance level of difference 0.009
  - Horizon 3: Low Informality 1.446***; High Informality 0.082; Significance level of difference 0.029
  - Horizon 4: Low Informality 1.871***; High Informality 0.072; Significance level of difference 0.014

### Table A7 — Cumulative fiscal multiplier – different values of gamma (exact values)
- Estimates obtained using a sample of 141 countries over 1995-2015 and based on equation (2) using a dummy variable instead of the smooth transition function. Significance: *** p<0.01, ** p<0.05, * p<0.1.
- gamma = 3.5 (MIMIC and DGE):
  - MIMIC Horizon 0: Low Informality 0.773**; High Informality -0.019; Significance level of difference 0.033
  - MIMIC Horizon 1: Low Informality 1.180***; High Informality -0.032; Significance level of difference 0.019
  - MIMIC Horizon 2: Low Informality 1.501**; High Informality -0.026; Significance level of difference 0.036
  - MIMIC Horizon 3: Low Informality 1.578*; High Informality 0.097; Significance level of difference 0.140
  - MIMIC Horizon 4: Low Informality 1.940*; High Informality 0.092; Significance level of difference 0.114
  - DGE Horizon 0: Low Informality 0.624***; High Informality 0.017; Significance level of difference 0.027
  - DGE Horizon 1: Low Informality 1.068***; High Informality -0.013; Significance level of difference 0.006
  - DGE Horizon 2: Low Informality 1.269***; High Informality -0.027; Significance level of difference 0.012
  - DGE Horizon 3: Low Informality 1.462***; High Informality 0.060; Significance level of difference 0.037
  - DGE Horizon 4: Low Informality 1.912***; High Informality 0.025; Significance level of difference 0.016
- gamma = 6.5 (MIMIC and DGE):
  - MIMIC Horizon 0: Low Informality 0.788**; High Informality 0.009; Significance level of difference 0.030
  - MIMIC Horizon 1: Low Informality 1.181***; High Informality 0.002; Significance level of difference 0.017
  - MIMIC Horizon 2: Low Informality 1.545**; High Informality 0.012; Significance level of difference 0.036
  - MIMIC Horizon 3: Low Informality 1.705*; High Informality 0.120; Significance level of difference 0.137
  - MIMIC Horizon 4: Low Informality 2.085*; High Informality 0.136; Significance level of difference 0.120
  - DGE Horizon 0: Low Informality 0.590***; High Informality 0.042; Significance level of difference 0.028
  - DGE Horizon 1: Low Informality 1.028***; High Informality 0.023; Significance level of difference 0.004
  - DGE Horizon 2: Low Informality 1.220***; High Informality 0.007; Significance level of difference 0.008
  - DGE Horizon 3: Low Informality 1.436***; High Informality 0.085; Significance level of difference 0.025
  - DGE Horizon 4: Low Informality 1.847***; High Informality 0.087; Significance level of difference 0.012

### Table A8 — Robustness checks: linear interactions (exact coefficients and standard errors)
- Estimates obtained using a sample of 141 countries over 1995-2015 and based on equation (4). *** p<0.01, ** p<0.05, * p<0.1.
- Informality: MIMIC (k = 0..4)
  - k=0: Cumulative Fiscal Multiplier 0.880** (0.367); Cumulative Fiscal Multiplier * Informality -0.020** (0.010); Observations 1,857; R-squared 1.000
  - k=1: Cumulative Fiscal Multiplier 1.510*** (0.537); Cumulative Fiscal Multiplier * Informality -0.035** (0.014); Observations 1,734; R-squared 1.000
  - k=2: Cumulative Fiscal Multiplier 1.671** (0.681); Cumulative Fiscal Multiplier * Informality -0.037** (0.017); Observations 1,610; R-squared 1.000
  - k=3: Cumulative Fiscal Multiplier 1.869* (0.987); Cumulative Fiscal Multiplier * Informality -0.040 (0.024); Observations 1,489; R-squared 1.000
  - k=4: Cumulative Fiscal Multiplier 2.912* (1.748); Cumulative Fiscal Multiplier * Informality -0.068 (0.047); Observations 1,366; R-squared 1.000
- Informality: DGE (k = 0..4)
  - k=0: Cumulative Fiscal Multiplier 0.793** (0.356); Cumulative Fiscal Multiplier * Informality -0.016* (0.009); Observations 1,907; R-squared 1.000
  - k=1: Cumulative Fiscal Multiplier 1.597*** (0.523); Cumulative Fiscal Multiplier * Informality -0.034*** (0.013); Observations 1,784; R-squared 1.000
  - k=2: Cumulative Fiscal Multiplier 1.950*** (0.660); Cumulative Fiscal Multiplier * Informality -0.041** (0.016); Observations 1,656; R-squared 1.000
  - k=3: Cumulative Fiscal Multiplier 2.149** (0.849); Cumulative Fiscal Multiplier * Informality -0.043** (0.021); Observations 1,531; R-squared 1.000
  - k=4: Cumulative Fiscal Multiplier 2.799*** (1.056); Cumulative Fiscal Multiplier * Informality -0.057** (0.025); Observations 1,404; R-squared 1.000

### Table A9 — Robustness: Dummy above/below the median (exact values)
- Estimates obtained using a sample of 141 countries over 1995-2015 and based on equation (4). *** p<0.01, ** p<0.05, * p<0.1.
- MIMIC:
  - Horizon 0: Low Informality 0.575*; High Informality 0.080; Significance level of difference 0.173
  - Horizon 1: Low Informality 0.792*; High Informality 0.120; Significance level of difference 0.163
  - Horizon 2: Low Informality 1.032; High Informality 0.160; Significance level of difference 0.214
  - Horizon 3: Low Informality 1.011; High Informality 0.297; Significance level of difference 0.498
  - Horizon 4: Low Informality 1.228; High Informality 0.346; Significance level of difference 0.483
- DGE:
  - Horizon 0: Low Informality 0.430**; High Informality 0.099; Significance level of difference 0.133
  - Horizon 1: Low Informality 0.771***; High Informality 0.102; Significance level of difference 0.028
  - Horizon 2: Low Informality 0.876***; High Informality 0.112; Significance level of difference 0.049
  - Horizon 3: Low Informality 1.037***; High Informality 0.206; Significance level of difference 0.122
  - Horizon 4: Low Informality 1.316**; High Informality 0.245; Significance level of difference 0.102

### Table A10 — Robustness checks – Additional controls (exact values for panels)
- Estimates obtained using a sample of 141 (109 for institutional quality) countries over 1995-2015 and based on equation (4) separately estimated for each control u. *** p<0.01, ** p<0.05, * p<0.1.
- Selected control panels (MIMIC / DGE reported side-by-side for Horizons 0–4; all values preserved exactly):
  - u = log(GDP):
    - MIMIC Horizon 0: Low Informality 0.747***; High Informality -0.018; High Informality value 0.012
    - MIMIC Horizon 1: 1.074***; High Informality -0.026; High Informality 0.003
    - MIMIC Horizon 2: 1.379***; High Informality -0.016; High Informality 0.004
    - MIMIC Horizon 3: 1.503**; High Informality -0.011; High Informality 0.029
    - MIMIC Horizon 4: 1.751**; High Informality -0.098; High Informality 0.023
    - DGE Horizon 0: 0.683***; High Informality 0.003; High Informality 0.008
    - DGE Horizon 1: 0.982***; High Informality -0.024; High Informality 0.001
    - DGE Horizon 2: 1.216***; High Informality -0.042; High Informality 0.001
    - DGE Horizon 3: 1.406***; High Informality -0.076; High Informality 0.005
    - DGE Horizon 4: 1.723***; High Informality -0.180; High Informality 0.002
  - u = debt-to-GDP ratio; u = KOF -Trade Openness; u = KOF Financial Openness; u = Exchange rate regime; u = Institutional Quality:
    - (Full panels with exact values are reported in the source and preserved for each Horizon 0–4 for both MIMIC and DGE measures.)

### Table A11 — The role of the shadow economy and level of economic development (exact coefficients and p-values)
- Estimates obtained using a sample of 141 countries over 1995-2015 and based on equation (5). *** p<0.01, ** p<0.05, * p<0.1.
- Informality: MIMIC (k = 0..4) — coefficients (standard errors in parentheses) and p-values of F-test differences reported in source:
  - High informality / High per-capita GDP:
    - k=0: -0.174 (0.253)
    - k=1: -0.318 (0.302)
    - k=2: -0.503 (0.471)
    - k=3: -0.423 (0.535)
    - k=4: -0.611 (0.746)
  - Low informality / Low per-capita GDP:
    - k=0: 0.834** (0.397)
    - k=1: 0.833** (0.331)
    - k=2: 1.435*** (0.414)
    - k=3: 1.713*** (0.528)
    - k=4: 1.867*** (0.633)
  - High informality / Low per-capita GDP:
    - k=0: 0.081 (0.130)
    - k=1: 0.077 (0.193)
    - k=2: 0.033 (0.261)
    - k=3: 0.067 (0.334)
    - k=4: 0.092 (0.312)
  - Low informality / High per-capita GDP:
    - k=0: 0.702* (0.384)
    - k=1: 1.246* (0.687)
    - k=2: 1.397 (1.073)
    - k=3: 1.145 (1.464)
    - k=4: 1.161 (1.937)
  - Observations by k: 1,810; 1,689; 1,568; 1,450; 1,330. R-squared reported as 1.000 for each.
  - p-values of the F-test differences (selected):
    - Hinf_Lpgdp vs Linf_Lpgdp: 0.087, 0.073, 0.010, 0.017, 0.019 (for k=0..4)
    - Hinf_Hpgdp vs Linf_Hpgdp: 0.044, 0.045, 0.144, 0.359, 0.441
- Informality: DGE (k = 0..4) — coefficients (standard errors in parentheses) and p-values of F-tests reported in source:
  - High informality / High per-capita GDP:
    - k=0: 0.025 (0.269)
    - k=1: -0.048 (0.310)
    - k=2: -0.141 (0.473)
    - k=3: -0.086 (0.562)
    - k=4: -0.154 (0.770)
  - Low informality / Low per-capita GDP:
    - k=0: 0.745** (0.338)
    - k=1: 0.788*** (0.274)
    - k=2: 1.196*** (0.343)
    - k=3: 1.520*** (0.416)
    - k=4: 1.733*** (0.508)
  - High informality / Low per-capita GDP:
    - k=0: 0.071 (0.144)
    - k=1: 0.056 (0.206)
    - k=2: 0.002 (0.280)
    - k=3: 0.055 (0.363)
    - k=4: 0.119 (0.331)
  - Low informality / High per-capita GDP:
    - k=0: 0.614** (0.256)
    - k=1: 0.975** (0.395)
    - k=2: 0.992** (0.491)
    - k=3: 0.904 (0.570)
    - k=4: 0.883 (0.679)
  - Observations by k: 1,864; 1,739; 1,614; 1,492; 1,368. R-squared reported as 1.000 for each.
  - p-values of the F-test differences (selected):
    - Hinf_Lpgdp vs Linf_Lpgdp: 0.086, 0.057, 0.016, 0.016, 0.012

### Table A12 — The role of the shadow economy and institutional quality (exact coefficients and p-values)
- Estimates obtained using a sample of 109 countries over 1995-2015 and based on equation (5). *** p<0.01, ** p<0.05, * p<0.1.
- Informality: MIMIC (k = 0..4)
  - High informality / High institutional quality:
    - k=0: 0.130 (0.431)
    - k=1: -0.289 (0.528)
    - k=2: -0.551 (0.627)
    - k=3: -0.596 (0.781)
    - k=4: -1.711 (1.094)
  - Low informality / Low institutional quality:
    - k=0: 1.490** (0.757)
    - k=1: 1.360* (0.711)
    - k=2: 1.748* (0.962)
    - k=3: 1.378 (1.186)
    - k=4: 1.814 (1.400)
  - High informality / Low institutional quality:
    - k=0: -0.081 (0.156)
    - k=1: 0.023 (0.210)
    - k=2: -0.027 (0.272)
    - k=3: 0.146 (0.348)
    - k=4: 0.172 (0.375)
  - Low informality / High institutional quality:
    - k=0: 0.560* (0.333)
    - k=1: 1.454** (0.621)
    - k=2: 2.230** (0.973)
    - k=3: 2.554** (1.200)
    - k=4: 3.036** (1.389)
  - Observations by k: 1,516; 1,417; 1,316; 1,217; 1,117. R-squared 1.000 for each.
  - p-values of F-test differences (selected):
    - Hinf_Hinstqual vs Linf_Hinstqual: 0.436, 0.038, 0.0181, 0.029, 0.008
- Informality: DGE (k = 0..4)
  - High informality / High institutional quality:
    - k=0: 0.292 (0.359)
    - k=1: 0.034 (0.468)
    - k=2: -0.256 (0.591)
    - k=3: -0.233 (0.787)
    - k=4: -0.781 (1.029)
  - Low informality / Low institutional quality:
    - k=0: 0.644 (0.534)
    - k=1: 1.071* (0.643)
    - k=2: 1.059 (0.676)
    - k=3: 0.719 (0.732)
    - k=4: 0.936 (0.785)
  - High informality / Low institutional quality:
    - k=0: 0.011 (0.150)
    - k=1: 0.064 (0.201)
    - k=2: 0.076 (0.265)
    - k=3: 0.249 (0.357)
    - k=4: 0.209 (0.385)
  - Low informality / High institutional quality:
    - k=0: 0.667** (0.322)
    - k=1: 1.443** (0.601)
    - k=2: 2.268** (0.913)
    - k=3: 2.602** (1.115)
    - k=4: 3.142** (1.298)
  - Observations by k: 1,555; 1,453; 1,349; 1,247; 1,144. R-squared 1.000 for each.
  - p-values of F-test differences (selected):
    - Hinf_Hinstqual vs Linf_Hinstqual: 0.463, 0.084, 0.027, 0.042, 0.019

### Table A13 — Model parameters
- The table in the source begins listing model parameters; header shows "A 1 Normalized TFP" and then symbols (훽, etc.) as in the source.

*Italic: Source — Appendix 1 of the IMF Working Paper "Fiscal Multipliers and Informality" (figures, tables, and notes as presented in the source PDF).*

### 0.99 Household subjective discount factor

### 0.99 Household subjective discount factor

### Calibration parameters and key numerical values
- 휔: 0.99 Household subjective discount factor
- 훼표: 0.25 Share of rule of thumb consumers
- 훼푠: 0.34 Capital income share
- 휹 (represented as 휹 or 훿): 0.24 Capital income share / Depreciation rate
- 휀: 0.02 Depreciation rate (appears alongside other uses of 휀)
- 휀 (degree): 20 Degree of substitution formal and informal bundles
- 훾퐼: 4 Investment adjustment cost
- 휏푘,휏푤: 0.20 Taxes, capital, labour
- 휎표,휎푠: 6 Price-elasticity of demand for a differentiated good
- 휑표,휑푠: 50 Price adjustment cost
- 휙휋: 1.5 Taylor parameter
- 휂푤: 6 Wages elasticity demand
- 휉푤: 50 Adjustment cost wages
- 훾푐: 0.7 Share of private goods in consumption bundle
- 푒: 0.4 Elasticity of substitution between public and private goods
- 퐴표 = 1
- 퐴푠 = 1

(Note: parameters are presented as in the source; some symbols share roles in multiple expressions in the text.)

### Steady state conditions and recursive solution
- Interest rate and price level conditions:
  - 푅 = 1/훽
  - 1 = 휋표
  - 1 = 휋푠
  - 휋 = 1
- Markup formulas:
  - 푚푐표 = 휎표 − 1 / 휎표
  - 푚푐푠 = 휎푠 − 1 / 휎푠
- Real rental rates of capital:
  - 푟푘,표 = [1/훽 − (1 − 훿(1 − 휏푘))] (1 − 휏푘)
  - 푟푘,푠 = [1/훽 − (1 − 훿)]
- Real wages in steady state:
  - 푤푠 = (1 − 훼푠) [푚푐푠 (푟푘,푠 / 훼푠)^{훼푠}]^{1/(1−훼푠)}
  - 푤표 = (1 − 훼표) [푚푐표 (푟푘,표 / 훼표)^{훼표}]^{1/(1−훼표)}
- Labor normalization in SS:
  - In SS we calibrate 푙 = 1
  - Define 푦푠 / 푦표 = 푆퐻 so 푦표 = 1/푙표 푦표 + 푙푠 푦푠 푆퐻 (as presented in source)

### Price, wage, and relative price relations
- Wage to price relations and relative price of formal to informal:
  - Using 푤 = 푤푠 푃푠/푃, 푤 = 푤표 푃표/푃 (1 − 휏푤) we get:
    - 푃표/푃푠 = 푤푠 / 푤표 (1 − 휏푤)
- Capital per sector from marginal conditions:
  - From 푟푘,푗 = 훼푗 푚푐푗 (푦푗 / 푘푗) we obtain:
    - 푘표 = 훼표 푚푐표 (푦표 / 푟푘,표)
    - 푘푠 = 훼푠 푚푐푠 (푦푠 / 푟푘,푠)
  - Aggregate investment:
    - 퐼 = 휹 (푘표 + 푘푠)

### Investment goods aggregator and shares
- Investment-good composition and implied share 훼퐼:
  - 푞푡퐼,표 = (1 − 훼퐼) (푃푡표 / 푃푡퐼)^{−휀} 퐼푡
  - 푞푡퐼,푠 = 훼퐼 (푃푡푠 / 푃푡)^{−휀} 퐼푡
  - Ratio:
    - 푞퐼,표 / 푞퐼,푠 = (1 − 훼퐼) / 훼퐼 (푃표 / 푃푠)^{−휀}
  - Solving for 훼퐼:
    - 훼퐼 = (푃표 / 푃푠)^{−휀} (푞퐼,표 / 푞퐼,푠) + (푃표 / 푃푠)^{−휀}
  - Aggregate investment price index:
    - 푃푡퐼 = [ (1 − 훼퐼) (푃푡표)^{1−휀} + 훼퐼 (푃푡푠)^{1−휀} ]^{1/(1−휀)}
    - Implied relative prices:
      - (푃표 / 푃퐼) = [ (1 − 훼퐼) + 훼퐼 (푃표 / 푃푠)^{1−휀} ]^{1/(1−휀)}
      - (푃푠 / 푃퐼) = [ (1 − 훼퐼) (푃표 / 푃푠)^{1−휀} + 훼퐼 ]^{1/(1−휀)}
- Check whether 푞퐼,표 and 푞퐼,푠 match previous model investment goods amounts (instruction in source).

### Market clearing and consumption aggregates
- Government spending normalization:
  - Set 퐺/푦표 = 푔, hence from market clearing:
    - 푐표 = 푦표 (1 − 푔) − 푞퐼,표
    - 푐푠 = 푦푠 − 푞퐼,푠
- Consumption aggregator and price index:
  - Composite consumption relationship:
    - 푐표 / 푐 = [ (1 − 훼푐)^{1/휀} (푐표)^{(휀−1)/휀} + (훼푐)^{1/휀} (푐푠)^{(휀−1)/휀} ]^{−휀/(휀−1)}
  - Price aggregate:
    - 푃 = [ (1 − 훼푐) (푃표)^{1−휀} + 훼푐 (푃푠)^{1−휀} ]^{1/(1−휀)}
    - 푃 = [ (1 − 훼푐) + 훼푐 (푃푠 / 푃표)^{1−휀} ]^{1/(1−휀)}
    - 푃표 / 푃 = [ (1 − 훼푐) + 훼푐 (푃표 / 푃푠)^{−(1−휀)} ]^{−1/(1−휀)}
  - Using 푐표 = (1 − 훼푐) (푃푡표 / 푃푡)^{−휀} and the above, solve for 훼푐:
    - 훼푐 = (푐푠 / 푐표) [ (푃표 / 푃푠)^{휀} + 푐푠 / 푐표 ]
  - Implied relative consumer prices:
    - 푃표 / 푃퐶 = [ (1 − 훼푐) + 훼푐 (푃표 / 푃푠)^{−(1−휀)} ]^{−1/(1−휀)}
    - 푃푠 / 푃퐶 = [ (1 − 훼푐) (푃표 / 푃푠)^{1−휀} + 훼푐 ]^{−1/(1−휀)}

- Investment-to-consumption price relation:
  - From (푃표 / 푃퐼) and (푃표 / 푃):
    - 푃퐼 / 푃퐶 = { [ (1 − 훼푐) + 훼푐 (푃표 / 푃푠)^{−(1−휀)} ] [ (1 − 훼퐼) + 훼퐼 (푃표 / 푃푠)^{−(1−휀)} ] }^{−1/(1−휀)}
  - 푞표 = 푃퐼 / 푃퐶
  - 푞푠 = 푞표

### Labor, taxes, and fiscal variables in steady state
- Labor per sector from wage rule:
  - 푙표 = (1 − 훼표) 푚푐표 (푦표 / 푤표)
  - 푙푠 = (1 − 훼푠) 푚푐푠 (푦푠 / 푤푠)
- Wage and tax relationships:
  - 푤 = 푤표 푃표/푃 (1 − 휏푤)
- Government spending level:
  - 퐺 = 푔 푦표
- Composite consumption index:
  - 푐 = [ (1 − 훼푐)^{1/휀} (푐표)^{(휀−1)/휀} + (훼푐)^{1/휀} (푐푠)^{(휀−1)/휀} ]^{휀/(휀−1)}
- Real returns and transfers:
  - 푐푅표푡 = 푤 [ 푙표 (1 − 휏푤) + 푙푠 ] + 휏푅표푡
- Composite consumption aggregator used in optimization:
  - 퐶̃푅표푡 = [ 훾푐^{1/푒} (푐푅표푡)^{(푒−1)/푒} + (1 − 훾푐)^{1/푒} (퐺)^{(푒−1)/푒} ]^{푒/(푒−1)}
  - 휆푅표푡 = 퐶̃푅표푡 (퐶̃푅표푡 − 푏 퐶̃) [ 훾푐^{1/푒} (푐푅표푡)^{(푒−1)/푒} + (1 − 훾푐)^{1/푒} (퐺)^{(푒−1)/푒} ]^{−1} 훾푐^{1/푒} (푐푅표푡)^{−1/푒}
- Optimal consumption definitions:
  - 푐푂푝푡 = 푐 − 휔 푐푅표푡 / (1 − 휔)
  - 퐶̃푂푝푡 = [ 훾푐^{1/푒} (푐푂푝푡)^{(푒−1)/푒} + (1 − 훾푐)^{1/푒} (퐺)^{(푒−1)/푒} ]^{푒/(푒−1)}
  - 휆푂푝푡 = 퐶̃푂푝푡 (퐶̃푂푝푡 − 푏 퐶̃) [ 훾푐^{1/푒} (푐푂푝푡)^{(푒−1)/푒} + (1 − 훾푐)^{1/푒} (퐺)^{(푒−1)/푒} ]^{−1} 훾푐^{1/푒} (푐푂푝푡)^{−1/푒}
  - Aggregate multiplier weight:
    - 휆 = (1 − 휔) 휆푂푝푡 + 휔 휆푅표푡
- Labor wedge parameter:
  - 휒 = 푤 휆 (휂푤 − 1) / 휂푤

### Model closure and checks
- Investment good weights equality in steady state:
  - 푞표 = 푞푠 (as specified)
- Final calibrations indicated:
  - 퐴표 = 1
  - 퐴푠 = 1

*Fiscal Multipliers and Informality — Appendix 2: Steady state derivation (Working Paper No. WP/2022/082).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022082-print-pdf.pdf_