## Informality and Shock Propagation in an Open Economy — Working Paper No. WP/2025/190

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### Households: types, labor supply, and preferences
- Household types:
  - Two representative household types indexed by i = 1, 2.
    - i = 1: has access to credit markets and owns physical capital; adjusts physical capital to smooth consumption.
    - i = 2: no access to credit markets; consumes each period according to disposable income.
  - Each representative household consists of a continuum of workers indexed by labor variety j; labor varieties are uniformly distributed across household types.
- Labor markets and aggregation:
  - Workers allocate time between formal labor (differentiated: N^F_{i,j,t}) and informal labor (homogeneous in informal market: N^I_{i,j,t} = N^I_{i,t}).
  - Payroll taxes apply to formal sector labor; informal sector wages W^I_t are determined competitively.
  - Representative household formal wage: W^F_t = ∫ W^F_{j,t} dj.
  - Total labor effort for household i is CES:
    - N_{i,t} = [ ι_N (N^F_{i,t})^{(ρ_N−1)/ρ_N} + (1−ι_N) (N^I_{i,t})^{(ρ_N−1)/ρ_N} ]^{ρ_N/(ρ_N−1)}.
- Preferences and shocks:
  - Period utility (common to all households):
    - u(c_t, N_t, M_t/P_t) = z^u_{i,t} [ ln(c_t − ς c_{t−1}) − γ_N (1/(1+σ_N)) N_t^{1+σ_N} + γ_M (1/(1−σ_M)) (M_t/P_t)^{1−σ_M} ].
  - Parameters and meanings:
    - ς: habits in consumption
    - σ_N: inverse of the Frisch elasticity
    - σ_M: elasticity of money demand
    - γ_N and γ_M: weights on labor and real money balances
  - z^u_{i,t}: preference shock following an ARMA process.

### Households: budget constraints, investment, and capital
- Type i = 1 (credit-accessing, capital-owning) nominal budget constraint:
  - (1 + τ^c_t) P_t c_{1,t} + P^I_t I_t + M_{1,t} + B^h_t + S_t B^{h,*}_t =
    [ (1 − τ^k_t) R^k_t + τ^k_t δ Q_{t−1} ] K_{t−1}
    + (1 − τ^w_t) ∫ Z W^F_{j,t} N^F_{1,j,t} dj + W^I_t N^I_{1,t}
    + γ_O^{1−ν} S_t ̄O_t P^O_t + R_{t−1} B^h_{t−1} + S_t R^*_ {t−1} B^{h,*}_{t−1}
    + T_{1,t} + ξ_t + M_{1,t−1}.
  - Notation highlights (as used in the model):
    - P^I_t: nominal price of investment goods; I_t: investment.
    - B^h_t: nominal domestic government bond paying nominal interest R_t.
    - B^{h,*}_t: nominal foreign-currency bond paying R^*_t; S_t: nominal exchange rate.
    - R^k_t: nominal return on physical capital; Q_t: nominal price of installed capital; K_t: physical capital.
    - P^O_t and ̄O_t: external nominal price and flow of commodity exports.
    - τ^c_t, τ^w_t, τ^k_t: consumption, payroll, and capital income taxes.
    - T_{i,t}: government transfers; ξ_t: firms’ profits (nominal); δ: depreciation parameter.
    - γ_O^{1−ν}: per-capita share of commodity export revenues for type i = 1.
- Investment adjustment costs and capital law of motion:
  - Adjustment cost function: f(I_t / I_{t−1}) = (a/2) ( I_t / I_{t−1} − 1 )^2.
  - Capital accumulation with investment-specific shock z^k_t:
    - K_t = (1 − δ) K_{t−1} + z^k_t I_t [ 1 − f(I_t / I_{t−1}) ].
  - z^k_t: investment-specific exogenous shock following an ARMA process.
- Type i = 2 (no credit access) nominal budget constraint:
  - (1 + τ^c_t) P_t c_{2,t} + M_{2,t} = (1 − τ^w_t) ∫ Z W^F_{j,t} N^F_{2,j,t} dj + W^I_t N^I_{2,t} + T_{2,t} + M_{2,t−1}.

### Monetary policy regimes and central bank operations
- Monetary policy regimes:
  - Two regimes allowed: inflation targeting with flexible exchange rates, and a peg regime.
  - Inflation targeting rule (central bank controls short-term nominal interest rate):
    - R_t / R = [ (R_{t−1} / R) ]^{ρ_R} [ (π^c_t / ̄π^c) ]^{φ_π}^{1−ρ_R} exp(z^m_t).
    - ρ_R: smoothing parameter; φ_π: sensitivity to deviations of inflation from target; z^m_t: monetary policy shock (exogenous, ARMA).
  - Peg regime:
    - Nominal devaluation rate constant: d_t = S_t / S_{t−1} = ̄d.
- Central bank balance sheet and flow of funds:
  - Balance sheet identity:
    - M_t = B^{cb}_t + S_t B^{cb,*}_t.
  - Flow of funds:
    - M_t − M_{t−1} + R_{t−1} B^{cb}_{t−1} + R^*_{t−1} S_t B^{cb,*}_{t−1} = B^{cb}_t + S_t B^{cb,*}_t + P^c_t qfb_t.
  - Quasi-fiscal balance qfb_t is a function of:
    - the return on external and domestic assets, the domestic inflation rate, and the real exchange rate.

### Labor market, unions, and wage/price rigidities (first-order conditions)
- Households and informal labor FOCs:
  - New first-order condition for informal labor supply (real terms) for type 1:
    - γ_N N^σ_N_{1,t} ∂N_{1,t}/∂N^I_{1,t} = λ_{1,t} w^I_t − ̃Υ_{1,t}.
    - ∂N_{1,t}/∂N^I_{1,t} = [ (1−ι_N) (N^I_{1,t}/N_{1,t})^{1/ρ_N} − 1/ρ_N ].
    - ̃Υ is the Lagrange multiplier of the labor force constraint (labor supply ≤ time endowment).
  - Households without credit markets obey analogous FOCs with subscript 2.
- Unions and wage setting:
  - Unions choose wage rate W^F_{j,t} subject to constraints; wages may be non-optimally fixed for some periods with stickiness and indexation:
    - W^F_{j,t} = W^F_{j,t−1} g_z π^{χ_w}_{t−1} ̄π^{(1−χ_w)} (when unions cannot optimally adjust).
    - Real wage adjustment: w^F_{j,t} = w^F_{j,t−1} g_z π^{χ_w}_{t−1} ̄π^{(1−χ_w)} / π^c_{t−1}.
  - Unions set wages above worker-first-order-condition levels:
    - w^F_{j,t+s} > w^F,foc_{j,t+s}.
  - Wage dispersion v^w_t > 1 implies:
    - N^F_t = v^w_t N^F,d_t; when wages are flexible, v^w_t = 1.
- Firms, factor pricing, and price setting:
  - First-order conditions for factor choices relate real marginal costs to factor prices:
    - w^F_t = mc_{a,l}^{F,t} (1−α_{a,F}) Y_{a,l}^{F,t} / N_{a,l}^{F,t}.
    - r^k_{F,a,t} = mc_{a,l}^{F,t} α_{a,F} Y_{a,l}^{F,t} / K_{a,l}^{F,t−1}.
  - Capital is sector-specific; composite capital is CES across formal and informal capital with parameters ζ, ς, θ_F, θ_I and shares χ.
  - Price stickiness: fraction (1−θ_H) re-optimize prices each period; others index to past inflation π^lH and long-run π.
  - Optimal reset price condition (representative form in text):
    - P^{*,F}_{a,l,t} / P_{a,l,t} = (ε_{a,F}/(ε_{a,F}−1)) × ratio of discounted expected marginal revenues and marginal costs.
  - Price dispersion ν_{a,ii,p,t} creates welfare costs appearing in aggregate supply through ν_{a,ii,p,t} Y_{a,ii,d,t}.

### Market clearing, aggregates, and external equilibrium
- Consumption aggregates:
  - C^X_t = ∫_j c^X_{j,1,t} dj + ∫_j c^X_{j,2,t} dj = c^X_{1,t} + c^X_{2,t}.
  - C^M_t and C^{S_i}_t defined analogously.
- Labor and capital aggregates:
  - N^F_t = ∫_j N^F_{j,1,t} dj + ∫_j N^F_{j,2,t} dj = N^F_{1,t} + N^F_{2,t}.
  - N^I_t = N^I_{1,t} + N^I_{2,t}.
  - Total capital demand K_{d,t−1} equals sum of sectoral and type-specific capital demands; equilibrium K_{t−1} = K_{d,t−1}.
- Goods and services market clearing:
  - Y^X_t = C^X_t + I^X_t + G^X_t + I^{g,X}_t + C^{*,X}_t.
  - Y^{S_i}_t = C^{S_i}_t + G^{S_i}_t + I^{g,S_i}_t.
- Balance of payments (external equilibrium):
  - s_t (nfa_t − nfa_{t−1}) = p^X_t C^{*,X}_t + s_t p^{O,∗}_t ̄O_t − s_t (C^M_t + I^M_t + G^M_t + I^{g,M}_t) + s_t (R^∗_{t−1}/π^∗_{t−1}) nfa_{t−1}.
- Financial and money markets:
  - Domestic bond market: B_t = B^h_t + B^{cb}_t.
  - Money market equilibrium: M^d_t = M_t.

### Appendix: empirical calibration, key parameters, and quantitative takeaways
- Bolivia empirical inputs and calibration:
  - World Bank Enterprise Surveys for Bolivia: 2006, 2010 and 2017 used to discipline model predictions.
  - International Labor Organization data point: 85 percent of employment in the informal economy.
  - Main empirical estimates for Bolivia:
    - 84 percent of firms operate informally, accounting for 68 percent of GDP.
    - Informality by sector:
      - Technical services: 34 percent of value added is informal.
      - Basic services: 79 percent of value added is informal.
      - Exportable goods: 75 percent of value added is informal.
    - Time path (Figure A3b):
      - Share of informal firms rose from around 80 percent in 2016 to 84 percent in 2023.
      - Share of informal employment rose from 76 percent in 2016 to 85 percent in 2023.
    - Drivers of increased informality since 2016: decline in export sectors, slower per capita income growth, higher unemployment, and a larger footprint of the state.
  - Data caveat: Micro data on informality is available infrequently; most recent World Bank Enterprise Survey for Bolivia: 2017.
- Key model calibration figures (Table A1):
  - External deficit: 5.0% of GDP.
  - Fiscal deficit: 10.3% of GDP.
  - Foreign reserves: 4.1% of GDP.
  - Share of informal employment: 85%.
  - Share of informal GDP: 68%.
  - Informal share of value added, technical services: 34%.
  - Informal share of value added, basic services: 79%.
  - Informal share of value added, exportable goods: 75%.
  - Elasticities of substitution:
    - Formal vs. informal inputs, exportable goods: ε_X,F = 5.
    - Formal vs. informal inputs, basic services: ε_basic,F = 5.
    - Formal vs. informal inputs, technical services: ε_technical,F = 2.
- Shocks analyzed (impulse responses reported in figures):
  - 1% increase in the Income Tax on Labor (Figure A1).
  - Responses to shocks in Figure A2:
    - (a) 1% Negative Productivity Shock.
    - (b) 1% Reduction in Government Consumption.
    - (c) 1% Increase in Foreign Demand.
    - (d) 30% Exchange Rate Devaluation.
- Structural and welfare implications:
  - High informality interacts with external deficits (5.0% of GDP) and fiscal deficits (10.3% of GDP) to shape shock propagation.
  - Wage and price rigidities create inefficiencies:
    - Wage dispersion factor v^w_t > 1 raises formal labor supply relative to firm demand.
    - Price dispersion ν_{a,ii,p,t} creates cross-firm output heterogeneity and welfare costs.

*Source: wpiea2025190-source-pdf - Informality and Shock Propagation in an Open Economy — Working Paper No. WP/2025/190*

### 2.1    Households

### 2.1    Households

### Household types, labor supply, and market access
- Two representative household types indexed by i = 1, 2.
  - i = 1: has access to credit markets and owns physical capital; adjusts physical capital to smooth consumption.
  - i = 2: no access to credit markets; consumes each period according to disposable income.
- Each representative household consists of a continuum of workers indexed by labor variety j; labor varieties are uniformly distributed across household types.
- Workers choose time allocation between:
  - Formal labor market: labor variety remains differentiated (e.g., N^F_{i,j,t}).
  - Informal labor market: labor variety perceived as homogeneous in the informal market N^I_{i,j,t} = N^I_{i,t}.
- Payroll taxes apply to formal sector labor; informal sector wages W^I_t are determined competitively.
- Representative household formal wage W^F_t equals the average of differentiated wages: W^F_t = ∫ W^F_{j,t} dj.
- Total labor effort for household i is a CES aggregation of formal and informal labor:
  - N_{i,t} = [ ι_N (N^F_{i,t})^{(ρ_N−1)/ρ_N} + (1−ι_N) (N^I_{i,t})^{(ρ_N−1)/ρ_N} ]^{ρ_N/(ρ_N−1)}

### Preferences and shocks
- Period utility (common to all households):
  - u(c_t, N_t, M_t/P_t) = z^u_{i,t} [ ln(c_t − ς c_{t−1}) − γ_N (1/(1+σ_N)) N_t^{1+σ_N} + γ_M (1/(1−σ_M)) (M_t/P_t)^{1−σ_M} ]
- Parameters and interpretations:
  - ς: parameter controlling importance of habits in consumption
  - σ_N: inverse of the Frisch elasticity
  - σ_M: elasticity of money demand
  - γ_N and γ_M: weights on labor and real money balances in preferences
- z^u_{i,t}: preference shock following an ARMA process

### Budget constraint — household type i = 1 (credit-accessing, capital-owning)
- Nominal budget constraint:
  - (1 + τ^c_t) P_t c_{1,t} + P^I_t I_t + M_{1,t} + B^h_t + S_t B^{h,*}_t =
    [ (1 − τ^k_t) R^k_t + τ^k_t δ Q_{t−1} ] K_{t−1}
    + (1 − τ^w_t) ∫ Z W^F_{j,t} N^F_{1,j,t} dj + W^I_t N^I_{1,t}
    + γ_O^{1−ν} S_t ̄O_t P^O_t + R_{t−1} B^h_{t−1} + S_t R^*_ {t−1} B^{h,*}_{t−1}
    + T_{1,t} + ξ_t + M_{1,t−1}
- Notation and components:
  - P^I_t: nominal price of investment goods
  - I_t: investment in physical capital goods
  - B^h_t: nominal domestic government bond paying nominal interest R_t
  - B^{h,*}_t: nominal foreign-currency bond paying R^*_t
  - S_t: nominal exchange rate (domestic currency per unit of foreign currency)
  - R^k_t: nominal return on physical capital
  - Q_t: nominal price of a unit of installed capital
  - K_t: physical capital available at t
  - P^O_t: external nominal price of commodity goods
  - ̄O_t: flow of commodity exports
  - τ^c_t, τ^w_t, τ^k_t: consumption, payroll, and capital income taxes
  - T_{i,t}: government transfers (nominal)
  - ξ_t: firms’ profits (nominal)
  - δ: parameter determining depreciation rate of physical capital
  - γ_O: parameter governing share of commodity export revenues that households receive
  - γ_O^{1−ν}: per-capita share of commodity export revenues for type i = 1

### Investment adjustment costs and capital law of motion
- Investment adjustment cost function:
  - f(I_t / I_{t−1}) = (a/2) ( I_t / I_{t−1} − 1 )^2
- Capital accumulation with adjustment costs and investment-specific shock z^k_t:
  - K_t = (1 − δ) K_{t−1} + z^k_t I_t [ 1 − f(I_t / I_{t−1}) ]
- z^k_t: investment-specific exogenous shock following an ARMA process

### Budget constraint — household type i = 2 (no credit access)
- Nominal budget constraint:
  - (1 + τ^c_t) P_t c_{2,t} + M_{2,t} = (1 − τ^w_t) ∫ Z W^F_{j,t} N^F_{2,j,t} dj + W^I_t N^I_{2,t} + T_{2,t} + M_{2,t−1}

*Source: wpiea2025190-source-pdf - 2.1    Households*

### 2.5    Monetary Policy

### 2.5    Monetary Policy

### Monetary policy regimes
- The model allows for two monetary policy regimes: an inflation targeting regime with flexible exchange rates and a peg regime.
- Inflation targeting regime:
  - The central bank controls the short-term nominal interest rate and sets it following a rule that responds to deviations of inflation from the target.
  - The monetary policy rule is:
    - R_t / R = [ (R_{t−1} / R) ]^{ρ_R} [ (π^c_t / ̄π^c) ]^{φ_π}^{1−ρ_R} exp(z^m_t)
    - Where ρ_R is the smoothing parameter, φ_π measures the sensibility of the policy rule to deviations of inflation from the target, and z^m_t is the monetary policy shock.
    - The monetary policy shock z^m_t is exogenous and follows an ARMA model.
- Peg regime:
  - The nominal devaluation rate is constant:
    - d_t = S_t / S_{t−1} = ̄d

### Central bank balance sheet
- The central bank issues money M_t, holds foreign reserves B^{cb,*}_t, and net domestic assets comprising government and central bank bonds B^{cb}_t.
- The balance sheet identity:
  - M_t = B^{cb}_t + S_t B^{cb,*}_t

### Central bank flow of funds and quasi-fiscal balance
- The central bank flow of funds is:
  - M_t − M_{t−1} + R_{t−1} B^{cb}_{t−1} + R^*_{t−1} S_t B^{cb,*}_{t−1} = B^{cb}_t + S_t B^{cb,*}_t + P^c_t qfb_t
- The quasi-fiscal balance, qfb_t, is a function of:
  - the return on external and domestic assets,
  - the domestic inflation rate,
  - the real exchange rate.

*Source: wpiea2025190-source-pdf - 2.5    Monetary Policy*

### 2005. Ed. by Jeffrey A. Frenkel and Christopher Pissarides. MIT Press.

### Informality and Shock Propagation in an Open Economy — Working Paper No. WP/2025/190

### Appendix: Figures, Tables, and Key Parameters
- Figure A1: Impulse Response of Output and Inflation to 1% increase in the Income Tax on Labor.
- Figure A2: Impulse Response of consumption, investment, fiscal balance, current account, and real exchange rate to shocks:
  - (a) 1% Negative Productivity Shock
  - (b) 1% Reduction in Government Consumption
  - (c) 1% Increase in Foreign Demand
  - (d) 30% Exchange Rate Devaluation
- Table A1: Key Model Parameters
  - External deficit: 5.0% of GDP
  - Fiscal deficit: 10.3% of GDP
  - Foreign reserves: 4.1% of GDP
  - Share of informal employment: 85%
  - Share of informal GDP: 68%
  - Informal share of value added, technical services: 34%
  - Informal share of value added, basic services: 79%
  - Informal share of value added, exportable goods: 75%
  - Elasticities of substitution:
    - Formal vs. informal inputs, exportable goods: ε_X,F = 5
    - Formal vs. informal inputs, basic services: ε_basic,F = 5
    - Formal vs. informal inputs, technical services: ε_technical,F = 2

### Informality in Bolivia: Empirical Estimates and Model Input (Appendix A.2)
- Data context and limitations:
  - Micro data on informality is available infrequently; most recent World Bank Enterprise Survey for Bolivia: 2017.
  - International Labor Organization data point: 85 percent of employment in the informal economy.
- Augmented factor model (based on Yao (2024)) design and use:
  - Model estimates factors linking causes and indicators of the informal economy and establishes predictive relationships with survey estimates across countries.
  - World Bank Enterprise Surveys (collected in Bolivia in 2006, 2010 and 2017) are used to discipline model predictions by setting the model average in those years to the average of survey data points.
- Main empirical estimates for Bolivia:
  - 84 percent of firms operate informally, accounting for 68 percent of GDP.
  - Informality by sector (Figure A3a):
    - Technical services: 34 percent of value added is informal.
    - Basic services: 79 percent of value added is informal.
    - Exportable goods: 75 percent of value added is informal.
  - Time path (Figure A3b):
    - Share of informal firms rose from around 80 percent in 2016 to 84 percent in 2023.
    - Share of informal employment rose from 76 percent in 2016 to 85 percent in 2023.
  - Drivers of increased informality since 2016: decline in export sectors, slower per capita income growth, higher unemployment, and a larger footprint of the state.
- Model note:
  - The MIMIC model is a special case of the augmented factor model; under strong assumptions it equals the first principal component, but Yao (2024) shows it is not a useful predictor of the degree of informality in survey data.

- Table A2: Causes and Indicators used in the augmented factor model
  - Causes and their data sources:
    - PPP GDP per capita, unemployment rate — World Development Indicators
    - Rule of law, control of corruption, government effectiveness, voice and accountability, regulatory quality, political stability and absence of violence/terrorism — Worldwide Governance Indicators
    - Trade openness, tax-to-GDP ratio, government consumption-to-GDP — World Economic Outlook Indicators
    - PPP GDP per capita — World Development Indicators (listed again)
    - Currency in circulation — International Financial Statistics
    - Labor participation rate (aged 15-64) — World Development Indicators
    - Electricity consumption — World Development Indicators

### Model Structure and Key First-Order Conditions (Appendix A.3)
- Households
  - Heterogeneous household types (with and without access to credit markets); informal labor supply chosen individually because informal labor markets are non-unionized.
  - Representative first order conditions include consumption Euler equations with habit ς, real pricing with Λ and λ multipliers, investment first order conditions featuring adjustment cost function f(·), and money demand expressed with γ_M and σ_M.
  - New first order condition for informal labor supply (real terms):
    - γ_N N^σ_N_{1,t} ∂N_{1,t}/∂N^I_{1,t} = λ_{1,t} w^I_t − ̃Υ_{1,t}
    - ∂N_{1,t}/∂N^I_{1,t} = [ (1−ι_N) (N^I_{1,t}/N_{1,t})^{1/ρ_N} − 1/ρ_N ]
    - ̃Υ is the Lagrange multiplier of the labor force constraint (labor supply ≤ time endowment).
  - Households without credit markets obey analogous FOCs with subscripts 2.

- Unions and Wage Setting
  - Optimization problem of unions choosing wage rate W^F_{j,t}, subject to households’ preferences, budget, capital dynamics, and wage adjustment constraints.
  - Wages may be non-optimally fixed for some periods; nominal wage stickiness and indexation specified:
    - W^F_{j,t} = W^F_{j,t−1} g_z π^{χ_w}_{t−1} ̄π^{(1−χ_w)} (for periods when unions cannot optimally adjust)
    - Real wage adjustment: w^F_{j,t} = w^F_{j,t−1} g_z π^{χ_w}_{t−1} ̄π^{(1−χ_w)} / π^c_{t−1}
  - Union FOCs yield complex recursive conditions equating expected discounted weighted marginal utility of labor supply to weighted marginal value of wages; unions set wages above worker-first-order-condition levels:
    - w^F_{j,t+s} > w^F,foc_{j,t+s}
  - Wage dispersion creates inefficiency measured by v^w_t > 1 such that:
    - N^F_t = v^w_t N^F,d_t
  - When wages are flexible, v^w_t = 1.

- Labor Market First-Order Conditions and Equilibrium
  - Union optimality conditions depend on expected sequences of marginal utilities and scaled wage distributions; markups between union MRS and real wage implied by ε_w:
    - Under flexible wages, representative condition: (̃λ_U ... ) w^F,∗_{j,t} = ε_w/(ε_w−1) > 1
  - Aggregate formal labor supply equals formal labor demand:
    - N^F_t = N^X,F_t + Σ_i N^{S_i,F}_t = N^F,d_t
  - Aggregate informal labor supply equals informal labor demand:
    - N^I_t = N^X,I_t + Σ_i N^{S_i,I}_t = N^I,d_t

- Firms: Cost Minimization and Price/Quantity Setting
  - Intermediate formal and informal firms choose labor and capital per standard first order conditions relating marginal costs to factor prices:
    - w^F_t = mc_{a,l}^{F,t} (1−α_{a,F}) Y_{a,l}^{F,t} / N_{a,l}^{F,t}
    - r^k_{F,a,t} = mc_{a,l}^{F,t} α_{a,F} Y_{a,l}^{F,t} / K_{a,l}^{F,t−1}
    - Analogous conditions for informal firms and sectors.
  - Real marginal cost mc_{a,l,ii,t} includes factor prices and capital adjustment costs.
  - Capital is sector-specific and composite capital is CES across formal and informal capital with parameters ζ, ς, θ_F, θ_I and shares χ.
  - Firms face price stickiness: fraction (1−θ_H) optimally re-optimize prices each period; others index to past inflation π^lH and long-run π.
  - Firm pricing first order condition yields optimal reset price formula:
    - P^{*,F}_{a,l,t} / P_{a,l,t} = (ε_{a,F}/(ε_{a,F}−1)) × ratio of discounted expected marginal revenues and marginal costs (full recursive expression provided in text).
  - Price dispersion ν_{a,ii,p,t} defined; welfare cost of price dispersion enters aggregate supply:
    - Aggregate intermediate input supply: ∫_0^1 Y_{a,l,ii,t} dl = ν_{a,ii,p,t} Y_{a,ii,d,t}

- Market Clearing and Aggregate Definitions (Model Equilibrium)
  - Consumption aggregates:
    - C^X_t = ∫_j c^X_{j,1,t} dj + ∫_j c^X_{j,2,t} dj = c^X_{1,t} + c^X_{2,t}
    - C^M_t and C^{S_i}_t defined analogously.
  - Aggregate labor aggregates:
    - N^F_t = ∫_j N^F_{j,1,t} dj + ∫_j N^F_{j,2,t} dj = N^F_{1,t} + N^F_{2,t}
    - N^I_t = N^I_{1,t} + N^I_{2,t}
  - Total capital demand K_{d,t−1} equals sum of sectoral and type-specific capital demands; equilibrium K_{t−1} = K_{d,t−1}.
  - Goods and services market clearing:
    - Y^X_t = C^X_t + I^X_t + G^X_t + I^{g,X}_t + C^{*,X}_t
    - Y^{S_i}_t = C^{S_i}_t + G^{S_i}_t + I^{g,S_i}_t
  - Balance of payments (external equilibrium):
    - s_t (nfa_t − nfa_{t−1}) = p^X_t C^{*,X}_t + s_t p^{O,∗}_t ̄O_t − s_t (C^M_t + I^M_t + G^M_t + I^{g,M}_t) + s_t (R^∗_{t−1}/π^∗_{t−1}) nfa_{t−1}
  - Domestic bond market: B_t = B^h_t + B^{cb}_t
  - Money market equilibrium: M^d_t = M_t

### Key Quantitative and Structural Takeaways (from Appendix material)
- High informality in Bolivia used as calibration/input:
  - 85% of employment in informal economy (ILO data).
  - Model estimate: 84% of firms operate informally; informal share of GDP: 68%.
  - Sectoral informal shares: technical services 34%; basic services 79%; exportable goods 75%.
- Model highlights mechanisms through which informality interacts with:
  - External deficits (calibrated at 5.0% of GDP) and fiscal deficits (10.3% of GDP).
  - Shocks analyzed: 1% income tax increase on labor, 1% negative productivity shock, 1% reduction in government consumption, 1% increase in foreign demand, and a 30% exchange rate devaluation (impulse responses reported in figures).
- Wage and price rigidities create inefficiencies:
  - Wage dispersion factor v^w_t > 1 raises formal labor supply relative to firm demand.
  - Price dispersion ν_{a,ii,p,t} creates cross-firm output heterogeneity and welfare costs.

*Source: Informality and Shock Propagation in an Open Economy — Working Paper No. WP/2025/190*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025190-source-pdf.pdf_