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### Introduction: purpose and context
- Turnover taxes (taxes on sales) are used as a simple form of presumptive taxation of business income, especially in developing countries.
- Turnover taxes can reduce compliance costs for micro and small enterprises (MSEs) and encourage formalization when combined with taxpayer education and enforcement.
- Tax collection from turnover tax regimes is typically below 0.25 percent of GDP.
- More than 35 developing, emerging market, and transition economies use turnover as a presumptive income tax base for qualifying entrepreneurs and enterprises.
- Among developed countries, turnover-based presumptive income taxes are available for individual entrepreneurs or businesses in some cases (e.g., Austria, France, Italy, Portugal).
- The paper constructs a simple model to derive optimal design (tax rate t and threshold ̄Z).

### Model structure and key equations
- Firm environment and parameters:
  - Firms: maximum potential sales Z ∈ (0, Z_M), distribution H(Z) with density h(Z).
  - Firms can costlessly adjust sales downward.
  - Output price normalized to one; marginal cost C < 1.
  - Profit margin in regular regime = (1−C)Z.
- Tax regimes and parameters:
  - Regular regime: profit tax rate t_c, fixed compliance cost Γ; administrative cost A.
  - Presumptive regime: turnover tax rate t, fixed compliance cost Γ′; administrative cost A′.
  - Assumptions: Γ > Γ′ > 0; A > A′ > 0. Extra marginal cost in presumptive regime α. Informal sector extra marginal cost λ, with λ > α > 0 and 1 − C > λ > t_c (1 − C).
- After-tax net profit functions:
  - π_R(Z) = (1−t_c)(1−C)Z − Γ
  - π_A(̄Z) = (1−t − C − α) ̄Z − Γ′
  - π_P(Z) = (1−t − C − α) Z − Γ′
  - π_I(Z) = (1−C − λ) Z

### Partitioning firms: informality, separation, and bunching
- Informality cutoff:
  - Z_IP = Γ′ / (λ − t − α)
  - All firms with Z < Z_IP prefer informality over presumptive taxation.
  - Proportion of informal firms increases with t and with Γ′.
  - If λ − t − α ≤ 0, all firms prefer informality to the presumptive tax regime.
- Cutoffs and bunching:
  - Cut-off for adjusting relative to being regular:
    - Ẑ = [(1−t − C − α) ̄Z + Γ − Γ′] / [(1−t_c)(1−C)]
  - Separating equilibrium if Ẑ < ̄Z: firms produce at maximum output; threshold separates tax groups.
  - Bunching equilibrium if Ẑ > ̄Z: firms bunch just below the threshold; measure of bunchers H(Ẑ) − H(̄Z).
- For H(Z) uniform on (0,ZM) proportion of bunchers:
  - (Ẑ − ̄Z) / ZM = [t_c(1−C) − (t+α)] ̄Z (1−t_c)(1−C)ZM + Γ − Γ′ (1−t_c)(1−C)ZM  (equation (9))

### Comparative statics and analytic characterizations
- Social welfare: SW(̄Z,t) = Π(̄Z,t) + δ G(̄Z,t) with δ>1.
- Separating equilibrium (̄Z ≥ Ẑ):
  - Optimal threshold (Proposition 1):
    - ̄Z = (Γ+δA)−(Γ′+δA′) / (δ−1)[t_c(1−C)−t] + α  (equation (15))
  - First-order condition for t (Proposition 2):
    - (δ−1) ∫_{Z_IP}^{̄Z} Z h(Z)dZ = δ(tZ_IP−A′) h(Z_IP) dZ_IP/dt  (equation (16))
  - If H(Z) uniform, optimal t solves a cubic (Proposition 3); Lemma 3: as Γ′→0, t* → λ−α.
  - Comparative statics (Proposition 4):
    - Optimal ̄Z and t both increasing in λ.
    - Optimal ̄Z and t both decreasing in t_c.
    - Optimal ̄Z and t both decreasing (increasing) in fixed cost of complying with presumptive (regular) regime.
    - Optimal ̄Z and t both decreasing in Γ′.
- Bunching equilibrium (̄Z < Ẑ):
  - Welfare first-order conditions involve Ẑ and derivatives dẐ/d̄Z, dẐ/dt (Propositions 5 and 6).
  - For H(Z) uniform, optimal ̄Z* expressed in closed form (equation (22)); optimal t solves a quartic (Proposition 7).
  - Lemma 6: if interior solution exists under uniform H(Z), it is unique.
  - Comparative statics in bunching case are more complex; under sufficient conditions (h′(Z) non-negative at key points and t_c(1−C−α) > λ − α) both optimal ̄Z and t increase with λ (Proposition 8).

### Calibration, simulations, and quantitative findings
- Calibration targets and baseline parameters:
  - Additional marginal cost of informality λ = 0.075.
  - Additional marginal cost for presumptive regime α = 0.025.
  - C = 0.85 (profit margin in regular regime (1−C) = 15%).
  - Presumptive regime profit margin (1−C−α) = 12.50%.
  - Informal sector profit margin (1−C−λ) = 7.50%.
  - Average potential sales E(Z) = $815,000.
  - Fixed compliance cost regular Γ = $1,400.
  - Fixed compliance cost presumptive Γ′ = $300.
  - Fixed administrative cost per firm: regular A = $900; presumptive A′ = $25.
  - Lognormal distribution parameters: μ = 8, σ = 3.35.
- Base-case outcomes (profit tax rate = 25% base reference):
  - Optimal turnover threshold ≈ $78,970 (noted as 78.9 thousand $).
  - Optimal turnover tax rate = 2.41%.
  - Equilibrium type: Bunching (at profit tax rates of 25% or higher bunching equilibria are observed).
- Quantitative impacts (model calibrated for countries with large informal sectors):
  - Introducing an optimally designed turnover tax reduces the rate of informality of businesses by about 12 percentage points.
  - Optimal turnover tax rate found to be around 2.5% across most scenarios (range low = 2.1 percent, high = 3.6 percent).
  - Optimal threshold generally lies between $65,000 and $95,000 (range low = $53,000, high = $112,000).
- Detailed base-case table highlights (selected figures):
  - For profit tax rates 15%, 20%, 25%, 30%, 35% optimal turnover tax rates: 2.81%, 2.61%, 2.41%, 2.61%, 2.87%.
  - Corresponding optimal ̄Z (thousand $): 95.8; 85.4; 78.9; 70.1; 66.1.
  - Proportion of informal firms with presumptive tax regime (25% case): 66.55% and percentage point reduction in informality relative to no presumptive regime: 11.75% (25% case).
  - Aggregate outcomes listed as exact values in simulation tables (Total private income, Total tax revenue, Total welfare reported numerically per scenario).

### Policy implications and recommendations
- Turnover taxes can promote inclusiveness and reduce informality when combined with taxpayer education and enforcement.
- Suggested optimal policy for countries with large informal sectors:
  - Optimal turnover threshold in the range $65,000 to $95,000.
  - Turnover tax rate of around 2.5 percent.
  - Expected migration from informal sector to registration: about 12 percentage points of enterprises.
- Design guidance:
  - Prefer few tax rates (ideally a single rate) and anti-splitting rules to avoid arbitrage and reclassification.
  - Recognize that a proportional turnover tax produces horizontal inequity unless firms have identical profit margins.
  - Consider alternatives: profit tax with simplified accounting rules (e.g., cash accounting) or presumptive cost deduction as proportion of turnover.
  - For near-subsistence sellers, fixed fee systems may be more appropriate than turnover taxes.
- Administrative measures to encourage formalization:
  - Reduce the cost of tax filing and tax payments.
  - Strengthen campaigns and tax inspections to increase detection of informal enterprises.
  - Improve taxpayer education.
- Implementation feasibility:
  - Closed-form formulas (uniform H(Z)) and numerical simulations are implementable in spreadsheets for policy analysts.

### Caveats and limitations
- Model limits of evasion: tax evasion modeled only as firms producing informally; formal under-reporting is abstracted from.
- VAT interactions are abstracted from; coincidence of VAT and turnover thresholds complicates incentives.
- λ is treated as fixed; if marginal cost of informality rises with output results would differ.
- Presumptive tax treated as obligatory for below-threshold firms; in practice regimes may be voluntary.
- General equilibrium and sectoral effects are abstracted from; firms’ maximum supply is exogenous and unaffected by turnover tax rate.
- Future work suggested to incorporate under-reporting, VAT interplay, voluntary regimes, and general equilibrium effects.

*Source: wpiea2023267-print-pdf - REFERENCES*

### REFERENCES .............................................................................................................

### wpiea2023267-print-pdf - REFERENCES

### Introduction: purpose and context
- Turnover taxes (taxes on sales) are used as a simple form of presumptive taxation of business income, especially in developing countries.
- Turnover taxes can reduce compliance costs for micro and small enterprises (MSEs) and encourage formalization when combined with taxpayer education and enforcement.
- Tax collection from turnover tax regimes is typically below 0.25 percent of GDP.
- More than 35 developing, emerging market, and transition economies use turnover as a presumptive income tax base for qualifying entrepreneurs and enterprises (see Table 1).
- Among developed countries, turnover-based presumptive income taxes are available for individual entrepreneurs or businesses in some cases (e.g., Austria, France, Italy, Portugal).
- The tax literature offers limited guidance on choosing the turnover tax rate and threshold; this paper constructs a simple model to derive optimal design.

### Model structure and key equations
- Population of firms endowed with maximum potential sales Z ∈ (0, Z_M), distributed with CDF H(Z) and density h(Z) (>0), with h′(Z) denoting derivative.
- Firms can costlessly adjust sales downward from their maximum potential to any level below that maximum.
- Output price normalized to one; marginal cost of production is constant C < 1. Pre-tax profit at sales Z is (1−C)Z.
- Tax regimes and parameters:
  - Regular regime: firms with actual sales ≥ Z̄ face a profit tax at rate t_c and fixed compliance cost Γ; government administrative cost A.
  - Presumptive regime: firms with sales < Z̄ taxed on sales at rate t, fixed compliance cost Γ′; government administrative cost A′.
  - Assumptions: Γ > Γ′ > 0; A > A′ > 0. Profit tax rate t_c is exogenous; policy choices are t and Z̄.
  - Extra marginal cost in presumptive regime: α (on top of C).
  - Informal sector marginal cost increase: λ (on top of C), with λ > α > 0 and 1 − C > λ > t_c (1 − C).
- After-tax net profit functions (as presented):
  - π_R(Z) = (1−t_c)(1−C)Z − Γ
  - π_A(Z̄) = (1−t − C − α) Z̄ − Γ′
  - π_P(Z) = (1−t − C − α) Z − Γ′
  - π_I(Z) = (1−C − λ) Z

### Partitioning firms: informality, separation, and bunching
- Informality cutoff for firms below threshold (Z_IP):
  - Z_IP = Γ′ / (λ − t − α)
  - All firms with Z < Z_IP prefer informality over presumptive taxation.
  - The proportion of informal firms increases with the turnover tax rate t and with the cost of tax compliance Γ′.
  - Denominator λ − t − α must be positive or else all firms prefer informality to the presumptive tax regime.
- Cutoffs for informality versus regular regime:
  - Define Z_IR such that all Z < Z_IR choose the informal sector over the regular regime.
  - Define Z_AI such that firms with Z̄ < Z < Z_AI prefer adjusting sales down to just below Z̄ rather than producing in the informal sector.
- Separating vs. bunching equilibrium:
  - Adjusting (bunching below threshold) dominates the regular regime when:
    - (1−t − C − α) Z̄ − Γ′ > (1−t_c)(1−C) Z
  - The cut-off sales level for adjusting relative to being regular is:
    - Ẑ = [(1−t − C − α) Z̄ + Γ − Γ′] / [(1−t_c)(1−C)]
  - If Ẑ < Z̄, bunching cannot occur and the equilibrium is separating: firms produce at their maximum sales and the threshold separates tax groups.
  - If Ẑ > Z̄, a bunching equilibrium occurs (firms bunch just below the threshold).

### Comparative statics and conceptual results
- Optimal turnover tax rate and threshold vary with:
  - The profit tax rate t_c.
  - The cost of informal sector production λ.
  - The costs of tax compliance Γ and Γ′.
- Two behavioral margins shape the optimal t:
  - A higher t increases incentives for firms to escape into informality (raising informality).
  - A lower t increases incentives for firms to bunch immediately below the threshold and leave the regular regime.
- The optimal policy may lead to either bunching behavior or absence of bunching; bunching arises only at relatively high profit tax rates.
- The effective tax rate on profit implied by the optimal turnover tax rate can be lower than the effective or statutory CIT rate at the threshold. Thus, the optimal presumptive regime need not replicate the effective tax burden of the regular regime.

### Calibration, simulations, and quantitative findings
- Model calibrated largely on evidence from countries in Sub-Saharan Africa generates realistic shares of informal enterprises and numerical values for the optimal turnover tax design.
- Quantitative outcomes reported:
  - Introducing an optimally designed turnover tax reduces the rate of informality of businesses by about 12 percentage points in the model calibrated for countries with large informal sectors.
  - The optimal turnover tax rate is found to be around 2.5% across most scenarios.
  - The threshold separating the turnover tax regime from profit tax lies for the most part between $65,000 and $95,000.
- The model abstracts from VAT interactions but notes that if the turnover tax threshold is pegged to the VAT registration threshold, the model can be used to find the optimal turnover tax rate for the given threshold.

### Policy implications and guidance
- Turnover taxes can be a useful tool to promote inclusiveness and reduce informality when combined with taxpayer education and enforcement measures.
- Optimal design requires balancing:
  - The trade-off between encouraging firms to register (by limiting incentives to remain informal) and avoiding excessive bunching just below the threshold.
  - The multi-margin effects from interactions between t, Z̄, compliance costs, and informality costs.
- Conventional policy advice that the effective tax rate in the presumptive regime should equal or exceed that in the regular regime is not generally optimal; the turnover tax must address both the threshold margin and the informality-registration margin.
- If policy-makers peg turnover tax thresholds to VAT registration thresholds, separate optimization of the turnover tax rate remains important because a common threshold can exacerbate bunching.

*Source: wpiea2023267-print-pdf - REFERENCES*

### 2. Firms in the segment[

### 2. Firms in the segment[

### Bunching versus separating equilibria
- When πA( ̄Z)>πR(Z) firms adjust production to (just below) the threshold, earning (arbitrarily close to) πA( ̄Z); sales levels between  ̄Z and ˆZ will not be observed.
- When ˆZ> ̄Z the measure of firms bunching is H(ˆZ)−H( ̄Z).
- For H(Z) uniform on (0,ZM) the proportion of bunchers is
  - ˆZ− ̄Z
  - ZM
  - = [t c(1−C)−(t+α)] ̄Z (1−t c)(1−C)ZM + Γ−Γ′ (1−t c)(1−C)ZM  (equation (9))
- Bunching is driven by differences in tax rates and compliance costs between the regular and presumptive regimes; other things equal, bunching is expected at relatively high rates of profit tax.
- In empirical work, a separating equilibrium is identified by an absence of bunching in the turnover-size distribution.
- The analysis assumes the slope of πR exceeds the slope of πP; if πP is steeper than πR then ˆZ must exceed ̄Z and only bunching equilibria exist.

### Partitions of the sales space and firm choices
- Focus: smallest firms in informal sector, intermediate firms in presumptive regime, largest firms in regular regime.
- Lemma 1 (ˆZ< ̄Z with 0<ZIP< ̄Z and ZIR< ̄Z): separating case — all firms produce maximum output and
  - ∀Z∈(0,ZIP), firms locate in the informal sector
  - ∀Z∈[ZIP, ̄Z), firms locate in the presumptive regime
  - ∀Z∈[ ̄Z,ZM), firms locate in the regular regime
- Lemma 2 (0<ZIP< ̄Z<ˆZ≤ZAI): bunching case — some firms restrict output and
  - ∀Z∈(0,ZIP), firms locate in the informal sector
  - ∀Z∈[ZIP, ̄Z), firms locate in the presumptive regime
  - ∀Z∈[ ̄Z,ˆZ), firms bunch just below the threshold
  - ∀Z∈[ˆZ,ZM), firms locate in the regular regime
- ZIR ∈ [0,ˆZ] in bunching partition; its precise location is irrelevant for equilibrium because ˆZ≤ZAI prevents contradictions in preference orderings.

### Social welfare setup
- Social welfare SW( ̄Z,t) = Π( ̄Z,t) + δ G( ̄Z,t) with δ>1 representing marginal social value of tax revenues (equation (10)).
- Welfare integrals are constructed according to the relevant partition (Lemma 1 or Lemma 2).

### Welfare in a separating equilibrium ( ̄Z ≥ ˆZ)
- Aggregate profit:
  - Π( ̄Z,t) = ∫0 ZIP πI h(Z)dZ + ∫ZIP ̄Z πP h(Z)dZ + ∫ ̄Z ZM πR h(Z)dZ  (equation (11))
- Total net tax revenue:
  - G( ̄Z,t) = ∫ZIP ̄Z (tZ−A′)h(Z)dZ + ∫ ̄Z ZM [t c(1−C)Z−A]h(Z)dZ  (equation (12))
- Social welfare:
  - SW( ̄Z,t) = Π terms + δ{G terms}  (equation (13))
- First-order condition for threshold ̄Z:
  - {[(1−t−C−α)+δt] ̄Z−Γ′−δA′}h( ̄Z) = {[(1−t c)(1−C)+δt c(1−C)] ̄Z−Γ−δA}h( ̄Z)  (equation (14))
- Proposition 1: For given t and t c, optimal threshold in a separating equilibrium ( ̄Z>ˆZ) is
  - ̄Z = (Γ+δA)−(Γ′+δA′) / (δ−1)[t c(1−C)−t] + α  (equation (15))
- Proposition 2: For given ̄Z and t c, the optimal turnover tax t satisfies
  - (δ−1) ∫ZIP ̄Z Z h(Z)dZ = δ(tZIP−A′)h(ZIP) dZIP/dt  (equation (16))
  - Left side: marginal social benefit of increased tax revenues from presumptive firms; right side: marginal social cost from firms moving to informality, weighted by density at margin ZIP.
- If H(Z) is uniform on (0,ZM), first-order condition for t is a cubic; unique real root exists when cubic discriminant is positive.
- Proposition 3: explicit cubic solution for t when H(Z) uniform; t = λ−α + S + T, with S and T built from Q, R and coefficients a,b,c,d where
  - a = −M0
  - b = 3M0(λ−α)
  - c = −3(λ−α)2 M0 − M1 − M2
  - d = M0(λ−α)3 + M1(λ−α) + M3
  - M0 = (δ−1) ̄Z2
  - M1 = Γ′2 + δA′ Γ′ − (δ−1)/2 Γ′2
  - M2 = (δ−1) Γ′2
  - M3 = (α−C) Γ′2
  - Discriminant ∆ = Q3 + R2  (Proposition 3; equation (17) context)
- Lemma 3: If H(Z) uniform, lim Γ′→0 t* → λ−α.
- Comparative statics (separating equilibrium):
  - Lemma 4 (effects on optimal threshold ̄Z, holding t fixed; h′(Z) non-negative at optimal threshold):
    1. ̄Z unaffected by marginal cost of informal production (λ).
    2. ̄Z decreasing in profit tax rate (t c).
    3. ̄Z decreasing (increasing) in fixed cost of complying with presumptive (regular) regime (A′ and A respectively).
    4. ̄Z decreasing in marginal cost of producing in presumptive regime (Γ′).
  - Lemma 5 (effects on optimal t, holding ̄Z fixed; h′(Z) non-negative at ZIP):
    1. t increasing in marginal cost of informal production (λ).
    2. t unaffected by profit tax rate (t c).
    3. t decreasing in fixed cost of complying with presumptive regime (A′); independent of fixed cost of regular regime (A).
    4. t decreasing in marginal cost of producing in presumptive regime (Γ′).
  - Proposition 4 (full comparative statics, h′(Z) non-negative at ̄Z and ZIP):
    1. Optimal ̄Z and t both increasing in λ.
    2. Optimal ̄Z and t both decreasing in t c.
    3. Optimal ̄Z and t both decreasing (increasing) in fixed cost of complying with presumptive (regular) regime.
    4. Optimal ̄Z and t both decreasing in Γ′.
  - Interpretations:
    - Higher λ (informal cost) allows higher t and higher ̄Z.
    - Higher t c lowers ̄Z and t; higher t c makes regular regime more lucrative to government.
    - Higher Γ′ lowers t and ̄Z to attract informal firms.
    - Ambiguities arise for effects of A′ and δ in some cases due to offsetting forces.

### Welfare in a bunching equilibrium ( ̄Z < ˆZ)
- Aggregate profit:
  - Π( ̄Z,t) = ∫0 ZIP πI h(Z)dZ + ∫ZIP ̄Z πP h(Z)dZ + ∫ ̄Z ˆZ πA h(Z)dZ + ∫ˆZ ZM πR h(Z)dZ  (equation (18))
- Total net tax revenue:
  - G( ̄Z,t) = ∫ZIP ̄Z [tZ−A′]h(Z)dZ + ∫ ̄Z ˆZ [t ̄Z−A′]h(Z)dZ + ∫ˆZ ZM [t c(1−C)Z−A]h(Z)dZ  (equation (19))
- Social welfare:
  - SW( ̄Z,t) = Π terms + δ{G terms}  (equation (20))
- First-order condition for ̄Z (Proposition 5):
  - [(1−t−C−α)+δt][H(ˆZ)−H( ̄Z)] + δ(t ̄Z−A′)h(ˆZ) dˆZ/d ̄Z = δ[t c(1−C) ˆZ−A] h(ˆZ) dˆZ/d ̄Z  (equation (21))
  - Left side: marginal benefits (increase in presumptive profits/tax revenues from existing adjusters + increases from firms becoming bunchers); right side: marginal cost from firms leaving regular regime.
- For H(Z) uniform, explicit optimal ̄Z* under bunching:
  - ̄Z* = (1−t−C−α)[(Γ+δA)−(Γ′+δA′)] + [δt − δ(t c/(1−t c))(1−t−C−α)](Γ−Γ′)
    / {[(1−t−C−α)+δt][(t+α)−t c(1−C)] + δ(t c/(1−t c))(1−t−C−α)2 − δt(1−t−C−α)}  (equation (22))
- First-order condition for t in bunching equilibrium (Proposition 6):
  - (δ−1) ∫ZIP ̄Z Z h(Z)dZ + (δ−1) ∫ˆZ ̄Z ̄Z h(Z)dZ − δ[t c(1−C) ˆZ−A]h(ˆZ) dˆZ/dt = −δ(t ̄Z−A′)h(ˆZ) dˆZ/dt + δ(tZIP−A′)h(ZIP) dZIP/dt  (equation (23))
  - New terms relative to separating case: gains from increased tax on existing bunchers and new revenues from former bunchers switching to regulars (dˆZ/dt<0); costs from fewer bunchers in presumptive regime.
- For H(Z) uniform, (23) is a quartic in t. If quartic discriminant is negative (as in simulations), there are two real roots; one maximizes welfare, one minimizes.
- Proposition 7: explicit (complex) quartic-root expression for t in uniform case involving coefficients a,b,c,d,e and derived quantities; discriminant ∆′ = −(∆12 − 4 ∆03)/27. (See proposition statement for full coefficient definitions.)
- Lemma 6: If H(Z) uniform and an interior solution exists (non-negative policies satisfying bunching partition of Lemma 2), it is unique.

### Key analytical insights and policy-relevant conclusions
- Optimal turnover threshold and turnover tax interact: closed-form for ̄Z given t in separating case (equation (15)), and explicit cubic/quartic characterizations of optimal t when H(Z) uniform (Propositions 3 and 7).
- Existence of informal sector constrains optimal t: as Γ′→0, t* → λ−α (Lemma 3).
- Separating equilibria: policy margins are determined primarily by the margin between presumptive regime and informality; bunching equilibria require trading off margins between presumptive–informal and presumptive–regular.
- Comparative statics indicate:
  - Higher informality costs (λ) support higher t and higher ̄Z.
  - Higher profit tax (t c) leads to lower t and lower ̄Z.
  - Higher compliance cost in presumptive regime (A′, Γ′) tends to reduce t and ̄Z; higher compliance/administrative cost in regular regime (A) tends to raise t and ̄Z.
- Empirically: absence/presence of bunching around thresholds can identify separating versus bunching equilibria and guide whether adjustments to turnover threshold or tax rates should be prioritized.

*Source: wpiea2023267-print-pdf - 2. Firms in the segment[*

### 1.   Comparative statics for a bunching equilibrium

### 1.   Comparative statics for a bunching equilibrium

### Overview
- Comparative statics for the bunching case are more complicated than for the separating case; some effects of parameter changes on the optimal turnover threshold or tax rate cannot be signed a priori.
- Assumption: the first- and second-order conditions for a welfare maximum are satisfied.
- Approach: first consider direct or partial effects of small parameter changes before full comparative statics.

### Partial comparative statics — optimal threshold (Lemma 7)
- Condition: results hold in a bunching equilibrium if h′(Z) is non-negative at Z̄, ZIP and Ẑ.
- Interior solution for t requires ZIP > 0, i.e., λ − t − α > 0. Consequently, the ‘second’ real root for the quartic equation in t minimizes welfare because it violates λ − t − α > 0.
- Partial effects (holding the turnover tax rate fixed):
  - The optimal threshold is independent of the marginal cost of informal production.
  - The optimal threshold is decreasing in the profit tax rate, if t_c(1−C−α) > λ − α.
  - The partial comparative statics effects of the fixed compliance costs and the additional marginal cost of production in the presumptive regime are ambiguous.

### Partial comparative statics — optimal turnover tax rate (Lemma 8)
- Condition: results hold in a bunching equilibrium if h′(Z) is non-negative at Z̄, ZIP and Ẑ.
- Partial effects (holding the threshold fixed):
  - The optimal turnover tax rate is increasing in the marginal cost of informal production.
  - The optimal turnover tax rate is increasing in the profit tax rate, if t_c(1−C−α) > λ − α.
  - The optimal turnover tax rate is decreasing in the marginal cost of producing in the presumptive regime.
  - The effects of the fixed compliance costs are ambiguous.

### Full comparative statics (Proposition 8)
- Sufficient condition: in a bunching equilibrium, if h′(Z) is non-negative at Z̄, ZIP and Ẑ, and if t_c(1−C−α) > λ − α, then:
  - The optimal turnover tax rate and threshold are both increasing in the marginal cost of informal production.
- Mechanism:
  - An increase in the marginal cost of informal sector production immediately induces a higher turnover tax rate, which in turn encourages a higher threshold.
  - Although the higher threshold affects the proportion of firms bunching and generates further adjustments, under the stated conditions the overall effect of a higher λ can be determined; the last condition ensures the marginal shift in the cutoff for bunching generates a net gain in tax revenues.
- Effect of an increase in the profit tax rate:
  - Mechanically (ignoring behavioral changes), improving the revenue-generating power of the regular regime reduces the optimal threshold, which mechanically lowers the optimal turnover tax rate because of reduced revenue-generating power of the presumptive regime.
  - Simultaneously, a higher profit tax rate encourages more firms to bunch, creating an incentive for the government to curb bunching by raising the turnover tax rate.
  - Net effects on bunching behavior depend on the levels of the threshold and turnover tax rate and on the mass of firms at each margin of behavioral response; in general, final effects of changes in t_c or fixed compliance costs are not signable, even with a uniform firm size distribution.

### Numerical solutions for the optimal policies — methodology (Section IV)
- Numerical simulations identify the welfare-maximizing design of the turnover tax regime.
- Optimal policy determined by direct search optimization based on random numbers.
- Procedure:
  - Take 4000 draws from a given frequency distribution.
  - Determine the optimal behavior of each firm (informality, unconstrained in the presumptive regime, bunching, or unconstrained in the regular regime), accounting for the sales constraint imposed by the threshold.
  - Consider all possible types of partitions of the cutoffs.
  - Calculate total welfare by aggregating individual contributions of each firm.
  - Both ‘separating’ and ‘bunching’ cases can emerge as equilibria, depending on the value of the (exogenous) profit tax rate.

### Calibration (Section IV.A)
- Purpose: parameter values based on a synthesis of survey results for countries with turnover tax regimes; evidence mainly from countries in Sub-Saharan Africa. Results do not represent any particular country.

- 1. Sales
  - Relevant firms: micro, small, and medium-sized enterprises.
  - Survey evidence (LaPorta and Shleifer (2008), 2006 data):
    - Micro survey average sales: about $51,000 in 2006.
    - Enterprise survey average sales: about $1.1 million in 2006.
    - Overall weighted-average sales across the two surveys: about $815,000.
  - Calibration: fit a lognormal distribution to approximately this mean.

- 2. Compliance costs
  - Evidence: wide variation across countries; three surveys used to permit separate calculations for profit tax and turnover tax.
  - Smulders et al. (2012), South Africa (2011 survey):
    - Turnovers below ZAR 1 million (about $102,445 in 2022) could opt for a turnover tax.
    - Average in-house cost of complying with the profit tax: ZAR 15,822 ($1,621 in 2022).
    - Average in-house cost of complying with the turnover tax: ZAR 14,030 ($1,473 in 2022).
    - Average external cost of compliance for all taxes combined: ZAR 9,882 ($1,012 in 2022); smaller businesses tended not to incur these expenses.
    - Note: most outsourced tax compliance-related activities associated with income tax returns; total compliance costs for profit tax obtained by adding full external cost to the in-house cost.
    - Reported averages are after trimming the 5% extremes of survey responses.
    - Because South Africa’s turnover tax has a progressive rate structure, the cost of complying with it may be relatively high compared to a typical proportional turnover tax.
  - World Bank Group (2017), Kyrgyzstan (surveys 2012, 2014, 2016):
    - About 60 percent of sampled firms had turnovers below KGS 1 million ($18,223 in 2022), and 80 percent of these paid the turnover tax.
    - Average company cost of tax compliance (in-house plus external):
      - General regime without VAT: KGS 18,716 ($307 in 2022).
      - Turnover tax regime: KGS 13,343 ($220 in 2022).
    - Assessments represent lower bounds because in-house time-costs were evaluated using formal salaries.
  - CIAT et al. (2015), Brazil (2012 survey):
    - Distinguishes between corporate income taxpayers and payers of Simples Nacional.
    - Average total compliance costs:
      - Payers of the tax on actual income: BRL 19,770 ($5,908 in 2022).
      - Payers of Simples: BRL 7,563 ($2,260 in 2022).
    - Federation of Brazilian accounting firms attributes only one-quarter of total accounting costs to strictly tax compliance activities, suggesting average total compliance cost of $1,477 for the profit tax and $565 for the turnover tax in Brazil.
  - Scaling and calibration choices:
    - Keen and Mintz (2004) assume variable compliance costs for VAT are 0.1 percent of turnover.
    - Based on survey information and firm size distribution, variable costs equal to 0.1 percent of the average turnover of firms in each tax regime are deducted to derive the fixed cost of compliance.
    - Fixed compliance costs set to $1,400 for the profit tax and $300 for the turnover tax.
    - Simulations assume variable compliance costs are 0.1 percent of each firm’s turnover.

- 3. Administration costs
  - Basis: Nemec et al. (2015) study on Slovakia.
  - Slovakia estimates:
    - Cost for the presumptive form of PIT: 2.25 percent of the presumptive PIT revenues.
    - Cost of administering CIT: 1.65 percent of CIT revenues.
    - Note: Slovakia’s cost is considerably higher than the Czech Republic’s 0.67 percent of CIT revenues.
  - Calibration: fixed cost of tax administration set to achieve, in the base-case simulations, approximately the Czech Republic figure for profit tax revenues and the presumptive PIT figure for turnover tax revenues.

*Source: wpiea2023267-print-pdf - 1.   Comparative statics for a bunching equilibrium*

### 4.   Cost of informal production

### 4.   Cost of informal production

### Calibration and baseline parameters
- The additional marginal cost associated with producing informally, λ = 0.075, chosen to target a realistic relative size of the informal sector.
- The additional marginal cost associated with being in the presumptive regime, α = 0.025.
- C = 0.85, implying a profit margin in the regular regime of (1−C) = 15%.
- Profit margins implied:
  - Regular regime: (1−C) = 15%
  - Presumptive regime: (1−C−α) = 12.50%
  - Informal sector: (1−C−λ) = 7.50%
- Other baseline parameter values (Table 2):
  - Average potential sales E(Z) = $815,000
  - Fixed compliance cost in regular regime Γ = $1,400
  - Fixed compliance cost in presumptive regime Γ′ = $300
  - Fixed administrative cost (per firm) in regular regime A = $900
  - Fixed administrative cost (per firm) in presumptive regime A′ = $25
  - Mean of lognormal distribution μ = 8
  - Standard deviation of lognormal distribution σ = 3.35
- Base case outcome: about 67 percent of firms are in the informal, untaxed, sector (stated elsewhere as “about two-thirds”).

### Base case optimal policy and equilibrium types (summary of Table 3)
- For profit tax rate = 25% (base reference):
  - Optimal turnover threshold ≈ $78,970 (noting table entry: 78.9 thousand $)
  - Optimal turnover tax rate = 2.41%
  - Type of equilibrium: Bunching (at profit tax rates of 25% or higher bunching equilibria are observed)
- Base case across profit tax rates 15%–35% (Table 3):
  - Profit tax rate 15%:
    - Optimal turnover threshold (thousand $) = 95.8
    - Optimal turnover tax rate = 2.81%
    - Effective tax rate in regular regime = 16.62%
    - Effective tax rate in presumptive regime = 23.06%
    - Type of equilibrium = Separating
  - Profit tax rate 20%:
    - Optimal turnover threshold (thousand $) = 85.4
    - Optimal turnover tax rate = 2.61%
    - Effective tax rate in regular regime = 21.48%
    - Effective tax rate in presumptive regime = 22.45%
    - Type of equilibrium = Separating
  - Profit tax rate 25%:
    - Optimal turnover threshold (thousand $) = 78.9
    - Optimal turnover tax rate = 2.41%
    - Effective tax rate in regular regime = 28.35%
    - Effective tax rate in presumptive regime = 19.88%
    - Type of equilibrium = Bunching
  - Profit tax rate 30%:
    - Optimal turnover threshold (thousand $) = 70.1
    - Optimal turnover tax rate = 2.61%
    - Effective tax rate in regular regime = 34.61%
    - Effective tax rate in presumptive regime = 21.62%
    - Type of equilibrium = Bunching
  - Profit tax rate 35%:
    - Optimal turnover threshold (thousand $) = 66.1
    - Optimal turnover tax rate = 2.87%
    - Effective tax rate in regular regime = 40.75%
    - Effective tax rate in presumptive regime = 23.83%
    - Type of equilibrium = Bunching

### Sensitivity analysis (Cases 1–8) — key comparative findings (from Table 3 and text)
- Case 1 — Deductible compliance cost (Γ deductible):
  - Optimal turnover tax rates range: 2.31%–2.49% across profit tax rates 15%–35%.
  - Equilibria: Separating for lower profit tax rates, bunching for higher ones.
  - Effective tax rate in regular regime equals statutory rate when Γ is deductible: 15.00%, 20.00%, 25.00%, 30.00%, 35.00% for profit tax rates 15%–35% respectively.
- Case 2 — Lower cost of compliance in regular regime (Γ = 1200):
  - Optimal turnover thresholds and tax rates fall relative to base case.
  - Examples: at 25% profit tax, threshold = 70.3 (thousand $), turnover tax rate = 2.31%.
- Case 3 — Higher cost of informality (λ = 0.07):
  - Optimal turnover threshold and tax rate both increase.
  - Examples: at 25% profit tax, threshold = 79.7 (thousand $), turnover tax rate = 2.81%.
- Case 4 — Higher extra marginal cost in presumptive regime (α = 0.03):
  - Induces a lower turnover tax rate and lower threshold.
  - Examples: at 25% profit tax, threshold = 57.7 (thousand $), turnover tax rate = 2.10%.
- Case 5 — Uniform distribution (H(Z) uniformly distributed on (0, 1,600,000)):
  - Optimal turnover tax rate substantially higher than base case.
  - Examples: at 25% profit tax, threshold = 85.7 (thousand $), turnover tax rate = 3.41%.
- Case 6 — Lognormal distribution with larger mean (μ = 8.68):
  - Minimal effect on turnover tax rate; threshold slightly changes depending on profit tax rate.
  - Examples: at 25% profit tax, threshold = 78.2 (thousand $), turnover tax rate = 2.59%.
- Case 7 — Higher administrative cost in presumptive regime (A′ = 100):
  - Results in lower threshold but higher turnover tax rate.
  - Examples: at 25% profit tax, threshold = 75.9 (thousand $), turnover tax rate = 2.73%.
- Case 8 — Higher marginal value of public funds (δ = 1.5):
  - In separating equilibria, increases optimal threshold and tax rate; in bunching equilibria, optimal threshold falls and turnover tax rate effects are unsystematic.
  - Examples: at 25% profit tax, threshold = 80.8 (thousand $), turnover tax rate = 2.81%.

- Aggregate sensitivity summary (text):
  - Optimal turnover tax rate found to be around 2.5% across most scenarios (range low = 2.1 percent, high = 3.6 percent).
  - Optimal threshold separating turnover tax regime from profit tax generally between $65,000 and $95,000 (range low = $53,000, high = $112,000).

### Detailed base case outcomes (Table 4)
- With presumptive tax regime, by profit tax rate (15%, 20%, 25%, 30%, 35%):
  - Optimal turnover tax rate: 2.81%, 2.61%, 2.41%, 2.61%, 2.87%
  - ̄Z (optimal turnover threshold, listed in table as integer): 95,800; 85,400; 78,900; 70,100; 66,100
  - ˆZ (ˆZ values): 81,435.45; 79,550.50; 80,542.31; 76,503.71; 76,568.51
  - ZIP (sales cutoff for informality with presumptive tax regime): 13,698.63; 12,552.30; 11,583.01; 12,552.30; 14,084.51
  - ZAI: 119,773.60; 108,614.13; 102,146.80; 88,438.53; 80,872.40
  - ZIR: 26,666.67; 31,111.11; 37,333.33; 46,666.67; 62,222.22
  - Total private income: 4.1309E+08; 3.8901E+08; 3.6484E+08; 3.4056E+08; 3.1626E+08
  - Total tax revenue: 7.2742E+07; 9.6855E+07; 1.2103E+08; 1.4530E+08; 1.6956E+08
  - Total welfare: 5.0766E+08; 5.1492E+08; 5.2218E+08; 5.2944E+08; 5.3670E+08
  - Proportion of informal firms with presumptive tax regime: 68.28%; 59.28%; 66.55%; 67.45%; 68.50%
  - Percentage point reduction in informality (relative to no presumptive regime): 7.13%; 17.66%; 11.75%; 12.31%; 13.23%
- Without presumptive tax regime (sales cutoff for informality and resulting informality rates):
  - Sales cutoff for informality: 34,583.26; 31,186.73; 37,436.34; 46,676.87; 62,357.35 (for profit tax rates 15%–35%)
  - Proportion of informal firms without presumptive regime: 75.40%; 76.93%; 78.30%; 79.76%; 81.73%
  - Net tax revenue from regular regime: 7.2131E+07; 9.6472E+07; 1.2078E+08; 1.4503E+08; 1.6917E+08

### Key mechanisms and interpretation
- Interaction between profit tax rate, turnover tax rate, and threshold:
  - When base case equilibria exhibit bunching, the optimal turnover tax rate increases with the profit tax rate (higher profit tax rate creates stronger bunching incentives).
  - In separating equilibria, the optimal turnover tax rate decreases with the profit tax rate because of a reduced optimal threshold.
  - Optimal thresholds are decreasing with the profit tax rate throughout the base case range.
- Effective tax rates:
  - At higher profit tax rates, the effective tax rate (taxes / before-tax profit) in the regular regime tends to exceed that in the presumptive regime when evaluated at the turnover threshold (especially if fixed compliance cost Γ is non-deductible).
  - Policy recommendation often aims to make the effective tax rate equal or higher in the presumptive regime to discourage bunching; the analysis emphasizes that tax compliance costs and incentives to be informal must also be accounted for.

### Policy implications and recommendations (from Conclusions)
- Suggested optimal policy for countries with large informal sectors:
  - Optimal turnover threshold in the range $65,000 to $95,000.
  - Turnover tax rate of around 2.5 percent.
  - The optimal turnover tax induces about 12 percentage points of enterprises to migrate from operating in the informal sector to registering for the turnover tax regime, compared to absence of turnover tax.
- Design considerations:
  - Set few tax rates (ideally a single rate) to avoid arbitrage and reclassification; accompany with anti-splitting rules.
  - A proportional turnover tax produces horizontal inequity unless firms have identical profit margins.
  - Consider alternatives: profit tax with simplified accounting rules (e.g., cash accounting) or presumptive cost deduction as proportion of turnover (functionally similar to turnover tax but applies statutory profit tax rate).
  - For near-subsistence sellers, fixed fee systems may be more appropriate than turnover taxes.
- Administrative actions to encourage formalization:
  - Reduce the cost of tax filing and tax payments to encourage fiscal formalization.
  - Strengthen campaigns and tax inspections to increase detection of informal enterprises.
  - Improve taxpayer education.
- Feasibility and implementation:
  - The analysis produces formulas for optimal threshold and tax rate (when firm size distribution is uniform) and numerical simulations; formulas are implementable with spreadsheets for policy analysts.

### Caveats and limitations highlighted
- Tax evasion in the model only occurs via firms producing informally (completely outside tax authorities); in reality, formal firms may under-report sales.
- Model abstracts from a co-existing VAT threshold; coincidence of VAT and turnover thresholds has complex strategic incentives.
- The model assumes λ is fixed; a rising marginal cost of informality with output would likely reduce size and number of informal firms and increase optimal turnover tax rate.
- The model assumes the presumptive tax is obligatory for firms with sales below the threshold; in practice, regimes may be voluntary.
- General equilibrium effects across sectors are abstracted from (registered and unregistered firms produce same good).
- Firms’ maximum supply is assumed exogenous and unaffected by turnover tax rate.
- Future work could integrate these considerations.

*Source: IMF Working Paper — Section 4. Cost of informal production (wpiea2023267-print-pdf).*

### REFERENCES

### wpiea2023267-print-pdf - REFERENCES (APPENDIX: PROOFS AND COMPARATIVE STATICS)

### References
- Bibliographic list of cited works (Aga et al. 2020; Benjamin and Mbaye 2012; Best et al. 2015; Bird and Zolt 2008; Bruhn and Loeprick 2016; Bruhn and McKenzie 2014; CIAT, Receita Federal, and SEBRAE 2015; Coolidge and Ilic 2009; de Mel, McKenzie, and Woodruff 2010; Dharmapala, Slemrod, and Wilson 2011; Engelschalk and Loeprick 2016; Irving 2020; Kanbur and Keen 2014; Keen and Mintz 2004; Kleven and Waseem 2013; LaPorta and Shleifer 2008; Liu et al. 2021; Nelson 2008; Nemec, Pompura, and Sagat 2015; Sapiei, Abdullah, and Sulaiman 2014; Smulders et al. 2012; Thuronyi 1996; Waseem 2018; World Bank Group 2007; World Bank Group 2017.)

### Appendix — Structure
- The appendix contains proofs of Lemma 1, Lemma 2, Propositions 1–8, and Lemmas 3–8.
- Key displayed equations referenced by equation numbers: (25), (26), (27), (28), (29), (30), (31), (32), (33), (34), (35), (36), (37), (38), (39), (40), (41), (42), (43), (44), (45), (46), (47), (48), (49).

### Proposition and Lemma results — main analytical conclusions and conditions
- Lemma 1 and Lemma 2
  - Conclusions follow immediately from the definitions of Z_IP, ˆZ, ̄Z, Z_IR, and Z_AI (no additional algebra shown).

- Proposition 1
  - First-order condition for threshold ̄Z derived from derivative of welfare for separating equilibrium (equation (25)).
  - The expression dSW/d̄Z is set to zero and rearranged to solve for ̄Z (equation (25) displayed).

- Proposition 2
  - First-order condition for tax rate t in separating equilibrium derived from dSW/dt (equation (26)).
  - Uses π_I(Z_IP)=π_P(Z_IP) and sets derivative to zero to yield the result.

- Proposition 3
  - If H(Z) is uniform, substitution of Z_IP from (6) into (16) leads to equation (27):
    - (δ−1) ̄Z^2 = [(δ+1)Γ′2 + 2δΓ′A′]t − [2δΓ′A′ − (δ−1)Γ′2](λ−α) / (λ−t−α)^3
  - Equation (27) is a cubic at^3 + bt^2 + ct + d = 0 with coefficients given in the proposition.
  - Discriminant ∆ = Q^3 + R^2; if ∆>0 there is a unique real root given by Cardano’s formula (Nelson, 2008).

- Lemma 3
  - As Γ′ → 0, cubic coefficients simplify to:
    - a = −(δ−1) ̄Z^2
    - b = 3(δ−1) ̄Z^2 (λ−α)
    - c = −3(δ−1)(λ−α)^3 ̄Z^2
    - d = (δ−1)(λ−α)^3 ̄Z^3
  - Under these limits, S and T → 0 and t → λ−α.

- Lemma 4 (comparative statics for ̄Z in separating equilibrium)
  - Total derivative signs (evaluated at fixed t):
    1. d̄Z/dλ|_t = −(∂^2SW/∂̄Z∂λ) / (∂^2SW/∂̄Z^2) = 0 (since ∂^2SW/∂̄Z∂λ = 0).
    2. d̄Z/dt_c|_t = −(∂^2SW/∂̄Z∂t_c) / (∂^2SW/∂̄Z^2) < 0.
    3. d̄Z/dΓ′|_t = −(∂^2SW/∂̄Z∂Γ′) / (∂^2SW/∂̄Z^2) < 0; d̄Z/dΓ|_t = −(∂^2SW/∂̄Z∂Γ) / (∂^2SW/∂̄Z^2) > 0.
    4. d̄Z/dα|_t = −(∂^2SW/∂̄Z∂α) / (∂^2SW/∂̄Z^2) < 0.
  - Second-order partial derivatives listed; ∂^2SW/∂̄Z^2 assumed negative for optimality.
    - ∂^2SW/∂̄Z∂t = (δ−1) ̄Z h(̄Z) > 0
    - ∂^2SW/∂̄Z∂λ = 0
    - ∂^2SW/∂̄Z∂Γ′ = −h(̄Z) < 0
    - ∂^2SW/∂̄Z∂Γ = h(̄Z) > 0
    - ∂^2SW/∂̄Z∂t_c = −(δ−1)(1−C) ̄Z h(̄Z) < 0
    - ∂^2SW/∂̄Z∂α = − ̄Z h(̄Z) < 0

- Lemma 5 (comparative statics for t in separating equilibrium)
  - Total derivative signs (evaluated at fixed ̄Z):
    1. dt/dλ|_̄Z = −(∂^2SW/∂t∂λ) / (∂^2SW/∂t^2) > 0.
    2. dt/dt_c|_̄Z = −(∂^2SW/∂t∂t_c) / (∂^2SW/∂t^2) = 0.
    3. dt/dΓ′|_̄Z = −(∂^2SW/∂t∂Γ′) / (∂^2SW/∂t^2) < 0; dt/dΓ|_̄Z = −(∂^2SW/∂t∂Γ) / (∂^2SW/∂t^2) = 0.
    4. dt/dα|_̄Z = −(∂^2SW/∂t∂α) / (∂^2SW/∂t^2) < 0.
  - Key second derivatives:
    - ∂^2SW/∂t∂̄Z = (δ−1) ̄Z h(̄Z) > 0
    - d^2SW/dt^2 given by equation (28) and assumed negative under h′(Z_IP) ≥ 0 and dZ_IP/dt > 0.
    - ∂^2SW/∂t∂λ > 0 (equation (29)) under stated sign conditions.
    - ∂^2SW/∂t∂Γ′ < 0 (equation (30)) under h′(Z_IP) > 0.
    - ∂^2SW/∂t∂Γ = 0
    - ∂^2SW/∂t∂t_c = 0
    - ∂^2SW/∂t∂α < 0 (equation (31)) under h′(Z_IP) > 0.

- Proposition 4 (joint comparative statics separating equilibrium)
  - System in matrix form (equation (32)); Hessian H and determinant |H| (equation (33)).
  - H negative definite if |H|>0 and ∂^2SW/∂̄Z^2 < 0 and ∂^2SW/∂t^2 < 0.
  - Cramer’s-rule results for signs:
    1. d̄Z/dλ > 0
    2. d̄Z/dt_c < 0
    3. d̄Z/dΓ′ < 0
    4. d̄Z/dΓ > 0
    5. d̄Z/dα < 0
    - Similarly for dt derivatives:
      1. dt/dλ > 0
      2. dt/dΓ′ < 0
      3. dt/dΓ > 0
      4. dt/dt_c < 0
      5. dt/dα < 0

- Proposition 5 (FOC for threshold ̄Z in bunching equilibrium)
  - dSW/d̄Z expression for bunching equilibrium given in (34).
  - Using π_P(̄Z) = π_A(̄Z) and π_A(̄Z) = π_R(ˆZ), simplifies to equation (35):
    - Z_ˆZ^̄Z (1−t−C−α+δt) h(Z) dZ + δ{(t̄Z−A′) − [t_c(1−C)ˆZ − A]} h(ˆZ) dˆZ/d̄Z = 0

- Proposition 6 (FOC for t in bunching equilibrium)
  - dSW/dt given by equation (36); simplification leads to equation (37):
    - (δ−1) ∫_{Z_IP}^{̄Z} Z h(Z) dZ − δ(tZ_IP−A′) h(Z_IP) dZ_IP/dt + (δ−1) ∫_{ˆZ}^{̄Z} ̄Z h(Z) dZ + δ{(t̄Z−A′) − [t_c(1−C)ˆZ − A]} h(ˆZ) dˆZ/dt = 0

- Proposition 7 (uniform H(Z) and quartic for t)
  - For uniform H(Z), substitution yields quartic equation (38), rearranged to standard quartic 4 + b t^3 + c t^2 + d t + e = 0.
  - Discriminant ∆′ = −(∆_1^2 − 4 ∆_0^3)/27; if ∆′ < 0 then two distinct real roots and two imaginary roots (Irving, 2020, Theorem 6.5). One root welfare-maximizing, the other welfare-minimizing.

- Lemma 6 (existence and uniqueness in bunching equilibrium)
  - Equation (21) rewritten as (39); left-hand side positive because 1−t−C−α>1−C−λ>0 and ˆZ>̄Z; thus right-hand side must be positive.
  - Equation (23) rewritten as (40); right-hand side negative because h(ˆZ) dˆZ/dt < 0; left-hand side must be negative implying tZ_IP − A′ > 0.
  - Under h′(Z_IP) non-negative, left-hand side strictly decreasing in t and right-hand side strictly increasing in t, so unique interior t if it exists; (22) ensures unique ̄Z* for optimal t.

- Lemma 7 (comparative statics for ̄Z in bunching equilibrium)
  - Total derivative signs (evaluated at fixed t):
    1. d̄Z/dλ|_t = −(∂^2SW/∂̄Z∂λ) / (∂^2SW/∂̄Z^2) = 0.
    2. d̄Z/dt_c|_t = −(∂^2SW/∂̄Z∂t_c) / (∂^2SW/∂̄Z^2) < 0.
  - Second derivatives for bunching case provided; key signs:
    - ∂^2SW/∂̄Z∂t > 0 under stated parameter assumptions (equations (41), (42)).
    - ∂^2SW/∂̄Z∂λ = 0
    - ∂^2SW/∂̄Z∂Γ′ and ∂^2SW/∂̄Z∂Γ have ambiguous signs (depend on derivatives of ˆZ).
    - ∂^2SW/∂̄Z∂t_c < 0 (equations (43), (44)) under t_c(1−C−α) > λ−α.
    - ∂^2SW/∂̄Z∂α ambiguous.

- Lemma 8 (comparative statics for t in bunching equilibrium)
  - Total derivative signs (evaluated at fixed ̄Z):
    1. dt/dλ|_̄Z = −(∂^2SW/∂t∂λ) / (∂^2SW/∂t^2) > 0.
    2. dt/dt_c|_̄Z = −(∂^2SW/∂t∂t_c) / (∂^2SW/∂t^2) > 0.
    3. dt/dα|_̄Z = −(∂^2SW/∂t∂α) / (∂^2SW/∂t^2) < 0.
  - Key second derivatives:
    - ∂^2SW/∂t∂̄Z > 0 (symmetry).
    - ∂^2SW/∂t^2 expression given and assumed negative for optimality.
    - ∂^2SW/∂t∂λ > 0 (equation (45)) under h′(Z_IP) ≥ 0.
    - ∂^2SW/∂t∂Γ′ ambiguous (equation (46)).
    - ∂^2SW/∂t∂Γ ambiguous.
    - ∂^2SW/∂t∂t_c > 0 (equation (46)) under first-order conditions.
    - ∂^2SW/∂t∂α < 0 (equation (47)) under h′ conditions.

- Proposition 8 (joint comparative statics for bunching equilibrium)
  - Matrix system in equation (48); Hessian determinant |H| as in (49).
  - H negative definite when |H|>0 and ∂^2SW/∂̄Z^2 < 0 and ∂^2SW/∂t^2 < 0 (condition on h′(Z) non-negative at optimum).
  - Cramer’s-rule comparative statics:
    1. d̄Z/dλ > 0.
    2. d̄Z/dt_c sign ambiguous (∂^2SW/∂̄Z∂t_c < 0, ∂^2SW/∂t∂t_c > 0).
    - dt/dλ > 0.
    - dt/dt_c sign ambiguous.

### Key mathematical/analytical notes
- Many results rely on sign conditions for h(Z), h′(Z), and derivatives of Z_IP and ˆZ with respect to parameters.
- Existence/uniqueness arguments use monotonicity of left- and right-hand sides of first-order conditions and sign properties (e.g., dZ_IP/dt > 0, dˆZ/dt < 0).
- Closed-form solutions derived where H(Z) uniform: cubic (Proposition 3) and quartic (Proposition 7) equations with discriminants ∆ and ∆′ controlling multiplicity of real roots.
- Second-order conditions for welfare maxima require ∂^2SW/∂̄Z^2 < 0 and ∂^2SW/∂t^2 < 0 and Hessian determinant |H|>0.

*Source: wpiea2023267-print-pdf - REFERENCES (Appendix: Proofs and Comparative Statics).*

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