## 2. Model

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---

### Overview
- Framework: Overlapping generations model building on Erosa (2001) with imperfect financial markets and progressive personal income taxation to finance wasteful public expenditures.
- Demographics and lifecycle:
  - Each generation lives for 퐽<∞ periods.
  - Mandatory retirement at age 퐽_R<퐽.
  - Each individual endowed with one unit of time until retirement.
  - Heterogeneity: individual-specific managerial ability 푒∈Θ drawn at birth from invariant distribution Φ(푒); idiosyncratic wealth profiles endogenous via life-cycle saving.
- Occupational choice:
  - During working life, individuals choose between being workers or entrepreneurs based on managerial ability and financial wealth.
  - Workers supply labor inelastically and receive gross wage 푤.
  - Entrepreneurs produce using managerial ability, capital and labor with decreasing returns to scale technology.

### Households (Section 2.1)
- Preferences:
  - Lifetime utility for a newly born individual: ∑_{푗=0}^{퐽} 훽^{푗} u(푐_{푗}) with u(푐_{푗}) = c_{푗}^{1−휎}/(1−휎).
  - Discount factor 훽∈(0,1); relative risk aversion 휎>0.
- Endowments at birth:
  - Zero assets and managerial ability 푒 drawn from Φ(푒).
  - Managerial ability remains constant over life-cycle.

### Firms (Section 2.2)
- Small business (entrepreneurs) technology:
  - Output: e^{휂} (k^{훼} n^{(1−훼)})^{(1−휂)} where 훼, 휂 ∈ (0,1).
  - Decreasing returns to scale capturing span of managerial control.
- Formal vs informal entrepreneurs:
  - Formal entrepreneurs register, access credit, rent formal capital k^{f} and hire formal labor n^{f}; production observable to government.
  - Informal entrepreneurs do not register, operate in financial autarky, hire informal workers n^{i}, self-finance capital k^{i}, and may conceal production from tax authorities.
  - Detection of informal activity occurs only after monitoring.
- Corporate sector:
  - Aggregate production: A^{휂}(K_{c}^{훼} N_{c}^{(1−훼)})^{(1−휂)} − φ_{f}, with A>0 and fixed operational cost φ_{f}.
  - Large corporations not subject to collateral constraints.
- Corporate sector first-order conditions (interior maximum):
  - (1−휂)(1−훼) Y_{c}/L_{c} = w (equation (5))
  - (1−휂)훼 Y_{c}/K_{c} = r + δ (equation (6))

### Financial intermediaries (Section 2.3)
- Intermediaries:
  - Receive deposits at risk-free rate r, rent capital at rental rate r^{k}.
  - Zero-profit condition: r^{k} = r + δ (equation (2)) where δ∈(0,1) is capital depreciation rate.
- Collateral constraints:
  - Formal entrepreneur with wealth a faces k^{f} ≤ λ a (equation (3)), with λ≥1; λ=∞ → perfect capital markets; λ=1 → financial autarky.
  - Informal entrepreneurs (financial autarky): k^{i} ≤ a (equation (4)).
  - Large corporations assumed not subject to collateral constraints.

### Government (Section 2.4)
- Taxation:
  - Progressive tax function on personal income y: T(y) = y − λ_{y} y^{1−τ}, with τ∈(0,1) degree of progressivity and λ_{y}∈(0,1) determining average taxation.
- Tax enforcement and evasion:
  - Periodic audits; convicted informal entrepreneurs pay taxes due augmented by surcharge factor s>1.
  - Probability of audit in a period p(k) is increasing in capital rented k.

### Agents' decision problems (Section 2.5)
- State variables: managerial ability e and assets a.
- Working-age (j = 0,...,J_r−1) decision:
  - Value function: V(a,e) = max{V^{W}(a,e), V^{fE}(a,e), V^{iE}(a,e)}.
  - Worker V^{W}(a,e):
    - Max over c, a′: u(c) + β V(a′,e)
    - Budget: c + a′ = y_{w} + a − T(y_{w}) with y_{w} = w + r a
    - Borrowing constraint: a′ ≥ 0 (equation (7))
  - Formal entrepreneur V^{fE}(a,e):
    - Max over k^{f}, n^{f}, c, a′: u(c) + β V(a′,e)
    - Subject to k^{f} ≤ λ a and a′ ≥ 0
    - Earnings y^{E} = e^{η} (k^{f α} n^{f (1−α)})^{1−η} − w n^{f} − (r+δ) k^{f} + r a
    - Budget: c + a′ = y^{E} + a − T(y^{E})
    - Declared income y^{E} equals actual earnings.
  - Informal entrepreneur V^{iE}(a,e):
    - Chooses k^{i}, n^{i} to maximize p(k^{i}) V^{dE}(a,e) + (1−p(k^{i})) V^{ndE}(a,e)
    - Subject to k^{i} ≤ a.
    - If not detected V^{ndE}(a,e):
      - y^{E} = r a
      - c + a′ = y^{E} + π + a − T(y^{E}) where π = e^{η} (k^{i α} n^{i (1−α)})^{1−η} − w n^{i} − (r+δ) k^{i} (equation (8))
    - If detected V^{dE}(a,e):
      - y^{E} = r a
      - c + a′ = y^{E} + π + a − (1+s) [ T(y^{E} + π) − T(y^{E}) ]
- Retired agents (j = J_R,...,J):
  - Max over c, a′: u(c) + β V(a′,e)
  - Budget: c + a′ = y_{R} + a − T(y_{R}) with y_{R} = r a
  - Constraint: a′ ≥ 0 (equation (7))

### Equilibrium (Section 2.6)
- Stationary equilibrium defined by price vector {r, w}, allocations {c(ω), a(ω)}, occupational choices b(e,a), formal and informal workers {n^{f}(ω), n^{i}(ω)}, investments {k^{f}(ω), k^{i}(ω)}, corporate sector {L_{c}, K_{c}}, tax function T(·), surcharge s, detection probability p(·), and invariant distribution ξ(ω) such that:
  - Policy functions solve agents' decision problems.
  - Corporate sector {L_{c}, K_{c}} satisfy (5) and (6).
  - Market clearing:
    - ∫(k^{f}(ω)+k^{i}(ω)) dξ(ω) + K_{c} = ∫ a(ω) dξ(ω)
    - ∫(n^{f}(ω)+n^{i}(ω)) dξ(ω) + L_{c} = ∫ 1_{W} dξ(ω)
  - Government budget constraint balanced:
    - G = ∫[ T(y(ω)) + 1_{D} s ( T(y(ω)+π(ω)) − T(y(ω)) ) ] dξ(ω)
  - Financial intermediaries zero profit condition (equation (2)).
  - ξ(ω) is the invariant distribution.

### Calibration (Section 3)
- Target economy: Brazil.
- Ability distribution:
  - Managerial ability drawn from generalized Pareto distribution with CDF Φ(e) = 1 − (1 + ν (e − μ) / κ)^{-1/ν}.
  - Parameters μ, κ, ν determine location, scale, shape; distribution discretized; first grid point e_{min} set to one with probability mass Φ(e_{min}).
- Detection probability p(k):
  - p(k) = 1 / (1 + p_{1} exp(− p_{2} k)).
- Two-step calibration:
  - Step 1: assign {β, σ, δ, α, φ_{f}, λ_{y}} mainly from external sources:
    - Period = year.
    - δ = 0.05.
    - β = 0.95 (text statement).
    - σ = 1.5.
    - α = 0.40.
    - φ_{f} determined so corporate firms earn zero profits in equilibrium.
    - λ_{y} = 0.70 (statutory income tax without progressivity = 0.30).
  - Step 2: jointly calibrate {λ, η, μ, κ, ν, Φ(e_{min}), τ, s, p_{1}, p_{2}, A} to match 11 data moments.
- Selected calibration targets (data moments):
  - Share of capital in total market output: 2.10.
  - Credit-to-GDP ratio: 0.42.
  - Size of informal economy relative to total formal output: 0.376.
  - Ratio of informal workers to formal workers: 0.354.
  - Share of informal firms in total number of firms: 0.698.
  - Size distribution targets for firm sizes from Ulyssea (2018).
- Table 1: Calibration Results — Parameter Values:
  - 훽: 0.951
  - 휎: 1.500
  - 휹: 0.050
  - 휶: 0.400
  - 휙_{f}: 0.506
  - 휆: 1.367
  - 휂: 0.217
  - 휇: 2.354
  - 휅: 0.410
  - 휈: 0.989
  - Φ(e_{min}): 0.719
  - 휏: 0.144
  - 휆_{y}: 0.700
  - 푠: 0.257
  - 푝_{1}: 1664
  - 푝_{2}: 1.790
  - 퐴: 2.176

### Numerical solution (computational appendix summary)
- Method: combination of the endogenous grid method (EGM) with value function iteration (VFI) following Fella (2014) to handle non-concavities from discrete occupational choices.
- State space ω = {e, a, b}; assets grid: 250 points; ability grid: 40 points.
- Iteration: guess (w_0, r_0) → solve corporate sector → solve household policy functions → simulate → check convergence → update prices until convergence.
- Life-cycle solved by backward induction from oldest generation to first.

### Key modeling implications
- Collateral constraints (k^{f} ≤ λ a) bind initially for all agents (assets at birth = 0), shaping occupational dynamics.
- Three ability zones arise determining worker, informal entrepreneur, and formal entrepreneur careers.
- Informal firms are small-scale due to financial autarky (k^{i} ≤ a) and increasing audit probability p(k).
- Detection risk creates trade-off between concealment gains and surcharge payments when audited.

---

### 4. Results

### 4.1. The Benchmark Equilibrium
- Calibration moments (Data vs Model):
  - Capital-output ratio: 2.100 (Data) ; 2.178 (Model)
  - Credit-to-GDP ratio: 0.420 (Data) ; 0.420 (Model)
  - Ratio of informal to formal output: 0.376 (Data) ; 0.374 (Model)
  - Ratio of informal to formal workers: 0.354 (Data) ; 0.386 (Model)
  - Share of informal firms: 0.698 (Data) ; 0.745 (Model)
- Size distribution (selected):
  - Informal firms ≤ 2 workers: 0.957 (Data) ; 0.976 (Model)
  - Informal firms ≤ 5 workers: 0.998 (Data) ; 1.000 (Model)
  - Formal firms ≤ 5 workers: 0.701 (Data) ; 0.757 (Model)
  - Formal firms ≤ 6-10 workers: 0.141 (Data) ; 0.139 (Model)
  - Formal firms ≤ 11-20 workers: 0.083 (Data) ; 0.103 (Model)
  - Formal firms ≤ 21-50 workers: 0.048 (Data) ; 0.001 (Model)
- Lifecycle and mechanism highlights:
  - Age and wealth accumulation determine occupational choices; middle-ability agents become informal entrepreneurs first then formal once collateral constraints relax.
  - Informal capital for middle-ability agents tracks savings until a critical capital level; beyond that, savings rise while informal capital is held constant to avoid detection.
  - Financially constrained entrepreneurs face higher marginal product of capital than market interest rate, creating strong incentives to save and self-finance expansion; informal entrepreneurs amplify this via tax evasion when not audited.

### 4.2. The effects of a financial markets reform in Brazil
- Experiment: benchmark (휆 = 1.37) vs perfect capital markets (휆 = ∞), other parameters fixed.
- Selected results (Benchmark → No financial frictions):
  - Size informal economy: 37.43% → 22.97%
  - Ghost firms (% Total firms): 74.59% → 61.77%
  - ΔInformal production: −22.12% (percent deviation from benchmark)
  - ΔFormal production (Official GDP): 26.91%
  - ΔTotal production: 12.72%
  - ΔCapital: 18.65%
  - ΔTFP (Formal production): 3.68%
  - ΔTFP (Total production): 2.20%
  - ΔWage: 8.52%
  - Interest Rate: 0.28% → 9.61%
  - Total workers (% Population): 74.98% → 80.04%
  - Total informal entrepreneurs (% Population): 15.13% → 12.32%
  - Total formal entrepreneurs (% Population): 9.89% → 7.64%
  - ΔFiscal revenues: 15.44%
  - ΔTax evasion: −17.30%
  - Workers (% Total Wealth): 14.42% → 63.51%
  - Informal entrepreneurs (% Total Wealth): 25.46% → 25.98%
  - Formal entrepreneurs (% Total Wealth): 60.12% → 10.51%
  - Constrained formal entrepreneurs: 100% → 0%
  - Constrained informal entrepreneurs: 59.00% → 14.40%
- Mechanisms:
  - Removing collateral constraints disproportionately benefits formal entrepreneurs, raising formality for productive agents.
  - Higher wage (+8.52%) raises opportunity cost of entrepreneurship, reducing informal entry among lower-skilled workers.
  - Informality remains non-negligible under perfect credit markets: ghost firms 61.77% and informal production ≈ 23% of market output.

### 4.2.2. Official GDP and aggregate TFP: the amplification effect
- Counterfactual (no informality and no frictions) vs benchmark:
  - No informal production and removing frictions:
    - ΔGDP: 14.47%
    - ΔCapital: 23.52%
    - ΔWage: 7.87%
    - ΔTFP: 1.00%
  - Benchmark removing frictions:
    - ΔFormal production: 26.91%
    - ΔTFP (Formal production): 3.68%
- Interpretation:
  - Two channels raise official GDP and formal TFP when frictions removed:
    1. Higher production by high-ability entrepreneurs enabled by credit access.
    2. Reallocation from informal to formal sector (amplification).
  - Informality amplifies gains: increase in official GDP in the benchmark is almost twice that in the counterfactual without informality; TFP gains are more than three times higher when informality is present.

### 4.2.3. Wealth distribution
- Redistribution when removing financial frictions:
  - Workers’ share of total wealth: 14.42% → 63.51%.
  - Informal entrepreneurs’ share nearly unchanged: 25.46% → 25.98%.
  - Constrained informal entrepreneurs persist: 14.40% remain constrained even with perfect credit markets.
- Conclusion: informality dampens overall wealth redistribution from financial market reforms because informal entrepreneurs continue to self-finance and accumulate wealth.

### 4.3. Development and financial frictions: discussion and policy implications
- Comparative steady-state responses (varying 휆; plotted vs credit-to-GDP):
  - Worsening collateral constraints to financial autarky reduces official GDP and formal TFP to 43.52% and 86.11% of their perfect-credit counterparts, respectively.
  - Size of informal economy declines monotonically with 휆: from maximum 120% of official GDP (pure self-financing) to minimum ≈ 23% (perfect credit markets).
  - Tax evasion falls with higher credit availability: from 20% of total output to ≈ 6%.
- Non-linear relationship and threshold:
  - Easing constraints significantly reduces informality when credit-to-GDP < 60% but has limited effect above this threshold.
  - Cross-country regression (equation (9)) interaction results (Table 5):
    - Dependent variable A (Size of informal sector):
      - constant: 25.97∗∗∗ (3.96)
      - credit: −0.06 (0.03)
      - credit * dummy: −0.19∗∗ (0.07)
      - dummy: 15.16∗∗∗ (4.41)
      - Adj. R^2: 0.41 ; Num. obs.: 155 ; RMSE: 9.26
    - Dependent variable B (GDP per capita, logs):
      - constant: 9.72∗∗∗ (0.38)
      - credit: 0.00 (0.00)
      - credit * dummy: 0.04∗∗∗ (0.01)
      - dummy: −2.21∗∗∗ (0.43)
      - Adj. R^2: 0.50 ; Num. obs.: 148 ; RMSE: 0.89
    - Significance levels: ∗∗∗ p<0.001, ∗∗ p<0.01, ∗ p<0.05.
  - Interpretation: higher credit-to-GDP associated with lower informality and higher GDP per capita mainly in less financially developed countries.
- Three informal firm types and policy trade-offs:
  - (i) High-ability but poor entrepreneurs (become formal with credit).
  - (ii) Parasite firms: productive enough to be formal but choose informality to evade taxes.
  - (iii) Low-skilled survival firms that cannot be formal.
  - Relaxing collateral constraints reduces informality via (i) and higher wages, but can increase parasite firms by raising gains from evasion; net effect depends on credit-to-GDP level.
- Policy comparisons: financial deepening vs flat tax reform
  - Financial deepening outperforms flat personal income tax (τ = 0) in economies with very tight borrowing constraints on:
    - Lower informality and tax evasion;
    - Higher total GDP, TFP, and fiscal revenues.
  - In financially developed economies, flat tax can outperform financial deepening.
  - Efficiency gains from flat tax larger the lower financial development because reduced progressivity raises formal-scale operation of top entrepreneurs and reduces gains from tax evasion among the richest.
- Policy implication summary:
  - Deepening financial markets effective to reduce informality and tax evasion in developing countries (credit-to-GDP < ~60%).
  - In financially developed countries, fiscal interventions (e.g., tax progressivity reforms) relatively more effective.
  - Perfect credit markets are a lower bound on informality absent fiscal policy change.

---

### 5. Conclusion

### Model and calibrated experiment (Brazil)
- Model features:
  - Occupational choice with progressive income taxation and informal production.
  - Informal entrepreneurs have no access to credit and face endogenous detection probability.
- Calibration and benchmark policy experiment (removing financial frictions):
  - Size of the informal economy decreases from 37% to 23%.
  - Official GDP increases by +27%.
  - Measured aggregate TFP increases by +4%.
  - Tax evasion falls by −17%.
  - Fiscal revenues increase by +15%.

### Financial development, non-linearities, and empirical testing
- Theoretical finding:
  - Non-linear relation between private credit-to-GDP and both size of informal economy and official GDP; beyond threshold further deepening has limited effect.
- Empirical strategy and findings:
  - Difference-in-difference analysis using 1991 and 2003 cross-country variations in Credit Controls Index.
  - Strong association between financial deepening and reduction in size of informal economy.
  - Splitting sample at 60% private credit-to-GDP: association stronger for least financially developed countries.
- Summary statistics (Table 6 highlights):
  - Inf share of GDP in 1991: N = 82; mean = 33.43; sd = 14.67
  - Inf share of GDP in 2003: N = 82; mean = 31.14; sd = 14.48
  - Private Credit-to-GDP in 1991: N = 62; mean = 62.08; sd = 55.62
  - Private Credit-to-GDP in 2003: N = 78; mean = 70.55; sd = 60.61
- Diff-and-diff estimates (Table 8):
  - Diff-and-diff Estimate:
    - All Countries: −2.479*** (0.712)
    - Least Financially Developed Countries: −2.817** (1.228)
    - Most Financially Developed Countries: −2.276*** (0.648)
  - R-squared: All = 0.505; Least = 0.586; Most = 0.664
  - Observations: All = 140; Least = 72; Most = 68

### Flat tax policy experiment
- Decreasing tax progressivity:
  - Decreases size of informal sector.
  - Reduces tax evasion.
  - Boosts total GDP and measured aggregate TFP.
- Effect amplified in presence of financial frictions.
- Comparative implication:
  - Financial market reforms preferred in developing countries (low financial development).
  - Flat tax reform more effective in medium- to high-financial-development countries.

### Aggregate implications and policy message
- Financial markets interact with informality producing important aggregate consequences up to a credit-to-GDP threshold.
- Policy prioritization suggested:
  - In developing countries: prioritize financial market reforms to relax credit constraints.
  - In medium- to high-financial-development countries: consider tax progressivity reforms to reduce informality and raise measured output and TFP.

*Source content unit "2. Model" from wpiea2020211-print-pdf.*

### 2. Model

### 2. Model

### Overview
- Framework: Overlapping generations model building on Erosa (2001) with imperfect financial markets and progressive personal income taxation to finance wasteful public expenditures.
- Demographics and lifecycle:
  - Each generation lives for 퐽<∞ periods.
  - Mandatory retirement at age 퐽_R<퐽.
  - Each individual endowed with one unit of time until retirement.
  - Heterogeneity: individual-specific managerial ability 푒∈Θ drawn at birth from invariant distribution Φ(푒); idiosyncratic wealth profiles endogenous via life-cycle saving.
- Occupational choice:
  - During working life, individuals choose between being workers or entrepreneurs based on managerial ability and financial wealth.
  - Workers supply labor inelastically and receive gross wage 푤.
  - Entrepreneurs produce using managerial ability, capital and labor with decreasing returns to scale technology.

### Households (Section 2.1)
- Preferences:
  - Lifetime utility for a newly born individual: ∑_{푗=0}^{퐽} 훽^{푗} u(푐_{푗}) with u(푐_{푗}) = c_{푗}^{1−휎}/(1−휎).
  - Discount factor 훽∈(0,1); relative risk aversion 휎>0.
- Endowments at birth:
  - Zero assets and managerial ability 푒 drawn from Φ(푒).
  - Managerial ability remains constant over life-cycle.

### Firms (Section 2.2)
- Small business (entrepreneurs) technology (following Quadrini (2000)):
  - Output: e^{휂} (k^{훼} n^{(1−훼)})^{(1−휂)} where 훼, 휂 ∈ (0,1).
  - Decreasing returns to scale capturing span of managerial control (Lucas (1978)).
- Formal vs informal entrepreneurs:
  - Formal entrepreneurs register, access credit, rent formal capital k^{f} and hire formal labor n^{f}; production observable to government.
  - Informal entrepreneurs do not register, operate in financial autarky, hire informal workers n^{i}, self-finance capital k^{i}, and may conceal production from tax authorities.
  - Detection of informal activity occurs only after monitoring.
- Corporate sector:
  - Aggregate production: A^{휂}(K_{c}^{훼} N_{c}^{(1−훼)})^{(1−휂)} − φ_{f} (equation (1)), with A>0 and fixed operational cost φ_{f}.
  - Large corporations not subject to collateral constraints.

- Corporate sector first-order conditions (interior maximum):
  - (1−휂)(1−훼) Y_{c}/L_{c} = w (equation (5))
  - (1−휂)훼 Y_{c}/K_{c} = r + δ (equation (6))

### Financial intermediaries (Section 2.3)
- Intermediaries:
  - Receive deposits at risk-free rate r, rent capital at rental rate r^{k}.
  - Zero-profit condition: r^{k} = r + δ (equation (2)) where δ∈(0,1) is capital depreciation rate.
- Collateral constraints (as in Buera and Shin (2013)):
  - Formal entrepreneur with wealth a faces k^{f} ≤ λ a (equation (3)), with λ≥1; λ=∞ → perfect capital markets; λ=1 → financial autarky.
  - Informal entrepreneurs (financial autarky): k^{i} ≤ a (equation (4)).
- Large corporations assumed not subject to collateral constraints.

### Government (Section 2.4)
- Taxation:
  - Progressive tax function on personal income y: T(y) = y − λ_{y} y^{1−τ}, with τ∈(0,1) degree of progressivity and λ_{y}∈(0,1) determining average taxation.
- Tax enforcement and evasion:
  - Periodic audits; convicted informal entrepreneurs pay taxes due augmented by surcharge factor s>1.
  - Probability of audit in a period p(k) is increasing in capital rented k (larger establishments are more visible).

### Agents' decision problems (Section 2.5)
- State variables: managerial ability e and assets a.
- Working-age (j = 0,...,J_r−1) decision:
  - Value function: V(a,e) = max{V^{W}(a,e), V^{fE}(a,e), V^{iE}(a,e)}.
  - Worker value V^{W}(a,e):
    - Max over c, a′: u(c) + β V(a′,e)
    - Budget constraint: c + a′ = y_{w} + a − T(y_{w}) with y_{w} = w + r a
    - Borrowing constraint: a′ ≥ 0 (equation (7))
  - Formal entrepreneur V^{fE}(a,e):
    - Max over k^{f}, n^{f}, c, a′: u(c) + β V(a′,e)
    - Subject to collateral constraint (3), worker constraint (7), and
      - y^{E} = e^{η} (k^{f α} n^{f (1−α)})^{1−η} − w n^{f} − (r+δ) k^{f} + r a
      - c + a′ = y^{E} + a − T(y^{E})
    - Declared income y^{E} equals actual earnings.
  - Informal entrepreneur V^{iE}(a,e):
    - Chooses k^{i}, n^{i} to maximize p(k^{i}) V^{dE}(a,e) + (1−p(k^{i})) V^{ndE}(a,e)
    - Subject to financial autarky (equation (4)).
    - Value if not detected V^{ndE}(a,e):
      - Max over c, a′: u(c) + β V(a′,e)
      - y^{E} = r a
      - c + a′ = y^{E} + π + a − T(y^{E}) where π = e^{η} (k^{i α} n^{i (1−α)})^{1−η} − w n^{i} − (r+δ) k^{i} (equation (8))
      - Concealed profit π is hidden from tax authorities when not detected.
    - Value if detected V^{dE}(a,e):
      - Max over c, a′: u(c) + β V(a′,e)
      - y^{E} = r a
      - c + a′ = y^{E} + π + a − (1+s) [ T(y^{E} + π) − T(y^{E}) ]
      - Detection forces payment of unreported taxes scaled by surcharge factor s.
- Retired agents (j = J_R,...,J):
  - Value function: Max over c, a′: u(c) + β V(a′,e)
  - Budget: c + a′ = y_{R} + a − T(y_{R}) with y_{R} = r a
  - Constraint: a′ ≥ 0 (equation (7))

### Equilibrium (Section 2.6)
- A stationary equilibrium is given by price vector {r, w}, allocations {c(ω), a(ω)}, occupational choices b(e,a), formal and informal workers {n^{f}(ω), n^{i}(ω)}, investments {k^{f}(ω), k^{i}(ω)}, corporate sector {L_{c}, K_{c}}, tax function T(·), surcharge s, detection probability p(·), and invariant distribution ξ(ω) such that:
  - Policy functions solve agents' decision problems.
  - Corporate sector {L_{c}, K_{c}} satisfy optimality conditions (5) and (6).
  - Market clearing:
    - ∫(k^{f}(ω)+k^{i}(ω)) dξ(ω) + K_{c} = ∫ a(ω) dξ(ω)
    - ∫(n^{f}(ω)+n^{i}(ω)) dξ(ω) + L_{c} = ∫ 1_{W} dξ(ω) where 1_{W} is indicator for worker.
  - Government budget constraint balanced:
    - G = ∫[ T(y(ω)) + 1_{D} s ( T(y(ω)+π(ω)) − T(y(ω)) ) ] dξ(ω) where 1_{D} indicates audited informal entrepreneur.
  - Financial intermediaries zero profit condition (equation (2)).
  - ξ(ω) is the invariant distribution.

### Calibration (Section 3)
- Target economy: Brazil.
- Ability distribution:
  - Managerial ability drawn from generalized Pareto distribution with CDF Φ(e) = 1 − (1 + ν (e − μ) / κ)^{-1/ν}.
  - Parameters μ, κ, ν determine location, scale, shape.
  - Distribution discretized; first grid point e_{min} set to one with probability mass Φ(e_{min}).
- Detection probability p(k):
  - Parametrized as logistic: p(k) = 1 / (1 + p_{1} exp(− p_{2} k)).
  - Gives probability of detection increasing quickly with capital; parameters p_{1}, p_{2} control level and steepness.
- Two-step calibration:
  - Step 1: assign values to {β, σ, δ, α, φ_{f}, λ_{y}} mainly from external sources:
    - Period corresponds to a year.
    - Capital depreciation rate δ set to 0.05.
    - Subjective discount factor β is fixed to 0.95 (text statement).
    - Relative risk aversion σ set to 1.5.
    - Labor share α set to 0.40.
    - Operational fixed cost φ_{f} determined so corporate firms earn zero profits in equilibrium.
    - λ_{y} in taxation function set to 0.70, corresponding to a statutory income tax (without progressivity) of 0.30.
  - Step 2: jointly calibrate remaining parameters {λ, η, μ, κ, ν, Φ(e_{min}), τ, s, p_{1}, p_{2}, A} to match 11 data moments.

- Calibration targets and data moments (selected):
  - Share of capital in total market output: 2.10 (from Allub and Erosa (2018)).
  - Credit-to-GDP ratio: 0.42 (from Allub and Erosa (2018)).
  - Size of informal economy relative to total formal output: 0.376 (Medina and Schneider (2018) average for Brazil 1991-2015).
  - Ratio of informal workers to formal workers: 0.354 (Ulyssea (2018)).
  - Share of informal firms in total number of firms: 0.698 (Ulyssea (2018)).
  - Size distribution targets: shares of formal firms with up to 5, 5 to 10, 11 to 20, and 21 to 50 employees; shares of informal firms with less than 2 and less than 5 workers (targets taken from Ulyssea (2018)).
- Estimation:
  - Parameter values jointly pinned down by minimizing a loss function computing distance between targeted moments and model counterparts.

- Table 1: Calibration Results — Parameter Values (as reported)
  - 훽 Subjective discount factor discount factor: 0.951
  - 휎 Relative risk aversion coefficient: 1.500
  - 휹 Capital depreciation rate: 0.050
  - 휶 Production function parameter: 0.400
  - 휙_{f} Operational fixed cost: 0.506
  - 휆 Tightness of financial frictions: 1.367
  - 휂 Span of control: 0.217
  - 휇 Location Pareto distribution: 2.354
  - 휅 Scale in Pareto distribution: 0.410
  - 휈 Shape in Pareto distribution: 0.989
  - Φ(e_{min}) Probability mass (Pareto): 0.719
  - 휏 Degree in taxes progressivity: 0.144
  - 휆_{y} Measure of the average taxation: 0.700
  - 푠 Surcharge factor: 0.257
  - 푝_{1} Probability of detection parameter: 1664
  - 푝_{2} Probability of detection parameter: 1.790
  - 퐴 TFP in the corporate sector: 2.176

*Italic: Source content unit "2. Model" from wpiea2020211-print-pdf.*

### 4. Results

### 4. Results

### 4.1. The Benchmark Equilibrium

- Calibration targets (selected moments; Data vs Model):
  - Capital-output ratio: 2.100 (Data) ; 2.178 (Model)
  - Credit-to-GDP ratio: 0.420 (Data) ; 0.420 (Model)
  - Ratio of informal to formal output: 0.376 (Data) ; 0.374 (Model)
  - Ratio of informal to formal workers: 0.354 (Data) ; 0.386 (Model)
  - Share of informal firms: 0.698 (Data) ; 0.745 (Model)
  - Size distribution: informal firms
    - ≤ 2 workers: 0.957 (Data) ; 0.976 (Model)
    - ≤ 5 workers: 0.998 (Data) ; 1.000 (Model)
  - Size distribution: formal firms
    - ≤ 5 workers: 0.701 (Data) ; 0.757 (Model)
    - ≤ 6-10 workers: 0.141 (Data) ; 0.139 (Model)
    - ≤ 11-20 workers: 0.083 (Data) ; 0.103 (Model)
    - ≤ 21-50 workers: 0.048 (Data) ; 0.001 (Model)

- Key qualitative mechanisms and lifecycle patterns:
  - Age and wealth accumulation determine occupational choices because collateral constraints (equation (3)) bind initially for all agents (assets at birth = 0), forcing initial work.
  - Three ability zones (lower, middle, upper) emerge from two thresholds on the ability distribution; they determine whether agents remain workers, become informal entrepreneurs, or formal entrepreneurs.
  - Low-ability agents remain workers for their entire working life; their ability is too low for viable entrepreneurship even with minimal detection risk from tax audits.
  - Middle- and high-ability agents switch occupations as soon as they accumulate enough assets to self-finance informal businesses; formal entry occurs later once assets relax collateral constraints.
  - Informal firms are small-scale relative to formal firms due to:
    - financial autarky limiting capital to own savings (equation (4));
    - audit probability p(k) increasing with capital, giving incentive to produce at small scale.
  - Informal capital for middle-ability agents initially tracks savings until a critical capital level; beyond that point savings continue to rise while informal capital stays constant to avoid detection.
  - Financially constrained entrepreneurs face higher marginal product of capital than the market interest rate, creating strong incentives to save and self-finance expansion; this is amplified for informal entrepreneurs who evade taxes when not audited.
  - Once informal entrepreneurs accumulate sufficient assets they optimally become formal entrepreneurs; switch ages differ by ability.

### 4.2. The effects of a financial markets reform in Brazil

- Experiment: compare benchmark (휆 = 1.37) with perfect capital markets (휆 = ∞), holding other calibrated parameters constant.

- Table 3: Selected results — Benchmark vs No financial frictions (휆 1.37 → ∞)
  - Size informal economy: 37.43% (Benchmark) → 22.97% (No financial frictions)
  - Ghost firms (% Total firms): 74.59% → 61.77%
  - ΔInformal production: -22.12% (percent deviation from benchmark)
  - ΔFormal production (Official GDP): 26.91%
  - ΔTotal production: 12.72%
  - ΔCapital: 18.65%
  - ΔTFP (Formal production): 3.68%
  - ΔTFP (Total production): 2.20%
  - ΔWage: 8.52%
  - Interest Rate: 0.28% → 9.61%
  - Total workers (% Population): 74.98% → 80.04%
  - Total informal entrepreneurs (% Population): 15.13% → 12.32%
  - Total formal entrepreneurs (% Population): 9.89% → 7.64%
  - ΔFiscal revenues: 15.44%
  - ΔTax evasion: -17.30%
  - Workers (% Total Wealth): 14.42% → 63.51%
  - Informal entrepreneurs (% Total Wealth): 25.46% → 25.98%
  - Formal entrepreneurs (% Total Wealth): 60.12% → 10.51%
  - Constrained formal entrepreneurs: 100% → 0%
  - Constrained informal entrepreneurs: 59.00% → 14.40%

- Main mechanisms and intuition:
  - Removing collateral constraints asymmetrically benefits formal entrepreneurs (who gain full credit access) while informal entrepreneurs remain in financial autarky; this raises the attractiveness of formality, especially for the most productive agents.
  - Highly productive agents under perfect credit markets can operate relatively large formal businesses immediately; gains from scale can outweigh gains from tax evasion.
  - Higher wage (+8.52%) increases opportunity cost of entrepreneurship and reduces entry into informal entrepreneurship among lower-skilled workers.
  - Combined effects reduce informal entrepreneurs from 15.13% to 12.32% of working-age population and lower tax evasion by 17.30%, raising fiscal revenues by 15.44%.
  - Despite removing financial frictions, informality remains non-negligible: ghost firms remain high (61.77%) and informal production accounts for about 23% of market output — suggesting perfect credit markets are a lower bound for informality absent fiscal policy change.

### 4.2.2. Official GDP and aggregate TFP: the amplification effect

- Key quantitative outcomes (benchmark vs counterfactual with no informality — Table 4):
  - With no informal production and removing financial frictions:
    - ΔGDP: 14.47% (relative to collateral-constrained economy with no informal sector baseline)
    - ΔCapital: 23.52%
    - ΔWage: 7.87%
    - ΔTFP: 1.00%
  - In the benchmark with informality (Table 3), removing frictions yields:
    - ΔFormal production: 26.91%
    - ΔTFP (Formal production): 3.68%

- Interpretation:
  - Two channels raise official GDP and formal TFP when financial frictions are removed:
    1. Higher production by relatively high-ability entrepreneurs enabled by credit access.
    2. Reallocation of entrepreneurial ability, labor, and capital from informal to formal sector due to imperfect tax enforcement — an amplification channel driven by reduced informality.
  - Informality amplifies gains: increase in official GDP in the benchmark is almost twice that in the counterfactual without informality; TFP gains are more than three times higher when informality is present.

### 4.2.3. Wealth distribution

- Redistribution effects when removing financial frictions:
  - Workers’ share of total wealth: 14.42% (Benchmark) → 63.51% (No financial frictions)
  - Entrepreneurs’ share of total wealth under perfect capital markets: 36.49% (implied) — about 15 percentage points higher than in the model with complete tax enforcement (Table 4 shows Entrepreneurs (% Total Wealth) 84.34% → 21.33% when no informal sector is assumed).
  - Informal entrepreneurs’ share of total wealth remains nearly unchanged when removing collateral constraints (25.46% → 25.98%).
  - Constrained informal entrepreneurs persist: 14.40% remain constrained even under perfect credit markets (because informal entrepreneurs have no access to credit regardless of formal-sector collateral rules).
  - Conclusion: informality dampens the overall wealth redistribution effect of financial market reforms because informal entrepreneurs continue to self-finance and accumulate wealth.

### 4.3. Development and financial frictions: discussion and policy implications

- Comparative steady-state responses (varying 휆; results plotted against credit-to-GDP ratio):
  - Worsening collateral constraints to financial autarky reduces official GDP and formal TFP to 43.52% and 86.11% of their perfect-credit-markets counterparts, respectively.
  - Size of the informal economy monotonically declines with 휆 (credit availability): from a maximum of 120% of official GDP (pure self-financing) to a minimum of about 23% (perfect credit markets).
  - Tax evasion also falls with higher credit availability: from 20% of total output (self-financing) to about 6% (perfect credit markets).

- Non-linear relationship and threshold:
  - There is a non-linear effect: easing collateral constraints significantly reduces informality when credit-to-GDP ratio is below 60% but has limited effect above this threshold.
  - Empirical test (cross-country regression; equation (9)):
    - Regression allows different slopes for countries with credit-to-GDP below 60.8% (dummy = 1) versus above.
    - Estimated results (Table 5):
      - Dependent variable (A) Size of informal sector:
        - constant: 25.97∗∗∗ (3.96)
        - credit: −0.06 (0.03)
        - credit * dummy: −0.19∗∗ (0.07)
        - dummy: 15.16∗∗∗ (4.41)
        - Adj. R^2: 0.41 ; Num. obs.: 155 ; RMSE: 9.26
      - Dependent variable (B) GDP per capita (logs):
        - constant: 9.72∗∗∗ (0.38)
        - credit: 0.00 (0.00)
        - credit * dummy: 0.04∗∗∗ (0.01)
        - dummy: −2.21∗∗∗ (0.43)
        - Adj. R^2: 0.50 ; Num. obs.: 148 ; RMSE: 0.89
      - Note: Standard Errors in parenthesis; significance levels: ∗∗∗ p<0.001, ∗∗ p<0.01, ∗ p<0.05.
    - Interpretation: interaction coefficients significant and signs consistent with model predictions — higher credit-to-GDP is associated with lower informality and higher GDP per capita only in less financially developed countries.

- Mechanism with three informal firm types:
  - (i) High-ability but poor entrepreneurs (switch to formality when credit relaxes).
  - (ii) Parasite firms: productive enough to be formal but choose informality to evade taxes.
  - (iii) Low-skilled survival firms that cannot be formal.
  - Relaxing collateral constraints has two countervailing effects on informality:
    - Reduction via (i) and via higher wages inducing low-skilled agents to remain workers.
    - Potential increase in parasite firms because higher production costs raise gains from tax evasion.
  - Net effect: at low credit-to-GDP, first effect dominates; at high credit-to-GDP, marginal gains from further easing are small because remaining informal firms are mostly parasite or low-skilled survival firms.

- Policy comparisons: financial deepening vs flat tax reform
  - Financial deepening dominates a flat personal income tax (τ = 0) in economies with very tight borrowing constraints in terms of:
    - Lower informality and tax evasion;
    - Higher total GDP, TFP, and fiscal revenues.
  - In financially developed economies, the flat tax can outperform financial deepening.
  - Efficiency gains from a flat tax are larger the lower financial development: reducing tax progressivity raises formal-scale operation of top entrepreneurs (intensive margin) and reduces relative gains from tax evasion among the richest (extensive margin), the latter amplified when financial frictions are tight.
  - Absence of progressive taxation would understate the role of financial deepening: with flat taxation and nearly perfect credit markets, lack of tax enforcement alone is insufficient to generate a large informal sector.

- Policy implication summary:
  - Deepening financial markets can be an effective tool to reduce firms informality and tax evasion in developing countries, especially where credit-to-GDP ratios are below about 60%.
  - For financially developed countries, fiscal interventions (e.g., changes in tax progressivity) may be relatively more effective in reducing informality.
  - Perfect credit markets are a lower bound on informality absent fiscal policy changes; targeted fiscal measures are necessary to eliminate informality driven by income taxation.

*Italic: Source: wpiea2020211-print-pdf - 4. Results*

### 5. Conclusion

### 5. Conclusion

### Model and calibrated experiment (Brazil)
- Model features:
  - Occupational choice with progressive income taxation and informal production.
  - Informal entrepreneurs have no access to credit and face an endogenous probability of detection by fiscal authorities.
- Calibration and benchmark policy experiment (removing financial frictions):
  - Size of the informal economy decreases from 37% to 23%.
  - Official GDP increases by +27%.
  - Measured aggregate TFP increases by +4%.
  - Tax evasion falls by -17%.
  - Fiscal revenues increase by +15%.

### Financial development, non-linearities, and empirical testing
- Theoretical finding:
  - A non-linear relation exists between the credit-to-GDP ratio and both the size of the informal economy and official GDP.
  - After a certain threshold value of the credit-to-GDP ratio, further financial deepening has a more limited effect on reducing informality and increasing official output.
- Empirical strategy:
  - Difference-in-difference analysis comparing cross-country variations in 1991 and 2003.
  - Financial deepening defined as a positive change in the credit controls index (IMF Financial Reforms database) between 1991 and 2003.
  - Regression specification: Inf_{i,t} = β_0 + γ_i + β_2 PostTreatment_t + β_1 did_{i,t} + β_3 credit_{i,t}, where Inf is informal economy share of GDP, PostTreatment is a dummy for 2003, and did is PostTreatment × treatment dummy (treatment = financial deepening).
- Empirical findings:
  - Strong association between financial deepening and a reduction in the size of the informal economy as a share of GDP (see Table 7, Table 8).
  - Splitting the sample at the 60 percent private credit-to-GDP threshold in 1991:
    - The association between financial deepening and reduction in informality is stronger for the least financially developed countries (private credit-to-GDP < 60).

### Flat tax policy experiment
- Evaluated as an alternative policy across different levels of financial development.
- Key results:
  - Decreasing tax progressivity:
    - Decreases the size of the informal sector.
    - Reduces tax evasion.
    - Boosts total GDP and measured aggregate TFP.
  - The policy effect is amplified in the presence of financial frictions.
- Comparative policy implication:
  - Financial market reforms are particularly effective in developing countries (low financial development) to fight informality.
  - A flat tax reform of personal income taxation appears more effective in countries with medium to high levels of financial market development.

### Aggregate implications and policy message
- Financial markets interact with informality producing important aggregate consequences up to a certain degree of financial development.
- Policy prioritization suggestion (based on model results):
  - In developing countries: prioritize financial market reforms (to relax credit constraints).
  - In medium- to high-financial-development countries: consider tax progressivity reforms (e.g., flat tax) to reduce informality and raise measured output and TFP.

### Empirical appendix — summary statistics and estimation (Tables 6–8)
- Table 6 : Summary Statistics (N, mean, sd, min, max)
  - Inf as share of GDP in 1991: N = 82; mean = 33.43; sd = 14.67; min = 7.56; max = 68.09
  - Inf as share of GDP in 2003: N = 82; mean = 31.14; sd = 14.48; min = 7.78; max = 69.01
  - Credit Controls Index in 1991: N = 82; mean = 1.613; sd = 0.978; min = 0; max = 3
  - Credit Controls Index in 2003: N = 82; mean = 2.451; sd = 0.750; min = 0; max = 3
  - Private Credit-to-GDP in 1991: N = 62; mean = 62.08; sd = 55.62; min = 1.452; max = 209.9
  - Private Credit-to-GDP in 2003: N = 78; mean = 70.55; sd = 60.61; min = 4.725; max = 231.6
- Table 7 : Summary Statistics for Changes Over 1991-2003 (Control Group: Δ Credit Controls = 0; Treatment Group: Δ Credit Controls > 0)
  - Change in Inf as share of GDP: Control = -6.20; Treatment = -8.51
  - Change in Credit-to-GDP ratio: Control = 40.84; Treatment = 63.32
  - Change in Credit Controls Index: Control = 0; Treatment = 1.563
- Table 8 : Estimation Results (Dependent Variable = Inf)
  - All Countries (Credit-to-GDP<60 and >60 columns reported separately)
  - Diff-and-diff Estimate:
    - All Countries: -2.479*** (0.712)
    - Least Financially Developed Countries: -2.817** (1.228)
    - Most Financially Developed Countries: -2.276*** (0.648)
  - PostTreatment Dummy:
    - All Countries: -1.034* (0.533)
    - Least Financially Developed Countries: -0.673 (0.419)
    - Most Financially Developed Countries: -1.337** (0.533)
  - Private Credit-to-GDP ratio coefficients:
    - All Countries: -0.0143 (0.0116)
    - Least Financially Developed Countries: -0.0168 (0.0334)
    - Most Financially Developed Countries: -0.0100 (0.0116)
  - Constants:
    - All Countries: 33.43*** (0.644)
    - Least Financially Developed Countries: 41.52*** (0.755)
    - Most Financially Developed Countries: 24.53*** (1.060)
  - Goodness of fit and sample sizes:
    - R-squared: All Countries = 0.505; Least = 0.586; Most = 0.664
    - Observations: All Countries = 140; Least = 72; Most = 68
    - Number of ifs: All Countries = 78; Least = 36; Most = 38
    - Size of Treatment Group: All Countries = 24; Least = 11
    - Country FE: YES (all specifications)
    - Robust standard errors in parentheses
    - Significance: *** p<0.01, ** p<0.05, * p<0.1

### Computational appendix — numerical solution approach (brief)
- Solution method:
  - Use method from Fella (2014) combining the endogenous grid method (EGM) with value function iteration (VFI) to handle non-concavities from discrete occupational choices.
- State space and discretization:
  - State space ω = {e, a, b} where a = savings, e = entrepreneurial ability, b = occupation choice (worker, entrepreneur in formal sector, entrepreneur in informal sector).
  - Assets, a, grid: 250 points.
  - Ability, e, discretized over 40 entrepreneurial abilities.
- Key steps in algorithm:
  1. Guess initial wage, w_0, and interest rate, r_0.
  2. Given w_0 and r_0, solve corporate sector capital and labor demand.
  3. Given w_0 and r_0, solve household policy functions (occupational choice and savings).
  4. Simulate economy over probability of detection in informal sector and compute aggregates: capital demand K_d; capital supply K_s; labor demand L_d; labor supply L_s.
  5. Check for convergence in capital and labor markets.
  6. Update w_1 and r_1 if necessary and iterate until convergence.
- Life-cycle solution:
  - Solve by backward induction from the oldest generation to the first, iterating on the Euler Equation and applying Fella (2014) for non-concave regions until policy and value functions are obtained for all states.

*Source: wpiea2020211-print-pdf - 5. Conclusion*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020211-print-pdf.pdf_