## wpiea2019098

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---

### Introduction — policy context and main empirical observations
- Turnover (sales) taxes are widely used as “presumptive” or “simplified” income tax regimes for small and medium-sized enterprises (SMEs) to reduce compliance and administration costs.
- Observed country practices (threshold, turnover tax rate, corporate income tax rate):
  - Kenya: threshold $49,000, turnover tax rate 3%, corporate income tax rate 30%.
  - Seychelles: threshold $74,000, turnover tax rate 1.5%.
  - Mauritania: threshold $84,000, turnover tax rate 3%.
  - Guinea: threshold $16,500, turnover tax rate 5%.
  - Belarus: threshold $625,000, turnover tax rate 5% (with 3% for VAT payers).
- Common policy heuristics:
  - Use the VAT threshold as the presumptive income tax threshold.
  - Make the effective tax burden at the turnover threshold more onerous than under the regular corporate income tax to encourage “graduation”.
  - Aim for “neutrality” by equalizing after-tax profit margins across regimes.
- Caveats:
  - Heuristics ignore compliance costs and firm heterogeneity.
  - Turnover taxes distort input choices because costs are non-deductible, potentially reducing productivity and pushing firms into informal or alternative fixed-fee regimes.
- Note: Table 1 uses US dollar exchange rate as of December 2018 for conversions.

### Main findings (analytical and numerical)
- First formal analysis of the optimal sales threshold separating presumptive (turnover) and regular (corporate income) tax regimes and the corresponding optimal tax rates.
- Key qualitative results:
  - Optimal threshold depends on the marginal cost of public funds, administrative costs, and productivity (value added per firm).
  - Optimal threshold and turnover tax rate respond to changes in the corporate income tax rate.
  - Firms’ heterogeneity in marginal costs (unit costs c) is crucial: non-deductibility of costs creates distortions that vary across firms.
  - Depending on parameters, both higher-cost and lower-cost firms may locate in the presumptive regime while middle-cost firms sit in the regular regime.
- Numerical benchmarks:
  - Optimal threshold generally between about $100,000 and $150,000 (depending on value added per firm).
  - Optimal turnover tax rate close to 3% in the benchmark when a single tax rate applies to all sectors.
  - For Sub-Saharan Africa, model predicts a higher optimal turnover tax rate and a lower optimal threshold compared with the benchmark.
  - Many observed country policies align with model prescriptions; others deviate substantially.

### Model setup — agents, technologies, and tax regimes
- Time/labor allocation:
  - Each individual supplies one unit of labor between formal production L and informal production 1−L.
  - Formal-sector output: f(L), with f(0) = 0, f′ > 0, f′′ < 0.
  - Informal-sector productivity: constant rate w; informal earnings w(1−L) are untaxable.
- Inputs and heterogeneity:
  - Formal production requires λ > 0 units of an imported intermediate per unit of output; individual-specific unit cost c = pI λ.
  - Unit costs c distributed with twice differentiable CDF H(c), density h(c), support c ∈ [0,1].
- Prices:
  - Country is small internationally: price of final good p and imported intermediate pI are fixed.
- Two linear tax regimes:
  - Regular regime (corporate income tax): tax rate t_c < 1; intermediate input cost is tax-deductible; firm faces fixed compliance cost Γ ≥ 0 and imposes fixed administrative cost A > 0 on the authority.
  - Presumptive regime (turnover tax): tax rate t < 1; costs are not deductible; no compliance or administrative costs for presumptive regime.
- Net price definitions:
  - p_P ≡ (1−t)p − c
  - p_R ≡ (1−t_c)(p − c)
- Profits and equilibrium:
  - Net profits: π(L) ≡ ρ f(L(ρ)) + w(1 − L(ρ)) − I_R Γ, where I_R = 1 if in regular regime, 0 otherwise.
  - Unconstrained FOC: ρ f′(L*) − w = 0.
  - Optimized profit π*(ρ) is increasing and strictly convex in ρ.

### Analytical comparison: single-regime benchmarks (square-root f illustration)
- Production f(L) = L^{1/2} yields closed-form expressions.
- Regular regime (all firms):
  - π_R(c) = (1/(4w))(p_R)^2 + w − Γ
  - Optimal corporate income tax rate (Lemma 1): t_c* = (δ−1)/(2δ−1) — increasing in marginal value of public funds and independent of unit-cost distribution.
- Presumptive regime (all firms):
  - π_P(c) = (1/(4w))(p_P)^2 + w
  - Optimal presumptive tax rate (Lemma 2): t* = (δ−1)/(2δ−1) (p − E(c))/p = t_c* (1 − E(c)/p); hence 0 < t* < t_c*.
- Welfare difference (Proposition 1; square-root f):
  - SW_R − SW_P = −(Γ + δA) + 1
    4w
    (δ−1)
    2
    (2δ−1)
    σ
    2
  - Interpretation: presumptive regime favored when dispersion in unit costs (σ^2) is small or compliance/administrative costs are large.

### Presumptive regime with a sales threshold — firm behavior and partitioning
- Reformulation in sales Z = p f(L):
  - Desired sales Z*(ρ) satisfy dπ/dZ = ρ/p − (w/p) f′(f^{-1}(Z*/p)) = 0.
  - Desired sales are decreasing in unit cost c: dZ(ρ)/dc < 0.
- Square-root example:
  - Z(ρ) = (p/(2w)) ρ.
  - Z_R(c) = (p/(2w)) [(1−t_c)(p − c)]
  - Z_P(c) = (p/(2w)) [(1−t)p − c]
- Threshold constraint and firm partitions (for given {t, t_c, Z̄}):
  - S1: Z_P(c) ≥ Z̄ and Z_R(c) ≥ Z̄
  - S2: Z_P(c) ≥ Z̄ and Z_R(c) < Z̄
  - S3: Z_P(c) < Z̄ and Z_R(c) ≥ Z̄
  - S4: Z_P(c) < Z̄ and Z_R(c) < Z̄
- Define crossover ˆc = (t_c − t) p / t_c and ˆZ = Z_R(ˆc) = Z_P(ˆc).
  - Case A: Z̄ ≥ ˆZ ⇒ S1 = [0, c′′), S2 = [c′′, c′), S3 = ∅, S4 = [c′, 1]
  - Case B: Z̄ < ˆZ ⇒ S1 = [0, c′), S2 = ∅, S3 = [c′, c′′], S4 = (c′′, 1]
- Behavioral implication: firms may “bunch” at or just below the sales threshold; constrained firms choose corner solutions relative to the threshold.

### Profit-difference functions and regime choice (Proposition 2)
- Profit-difference definitions:
  - S1: D1(c) = πR(c) − πB(c)
  - S2: D2(c) = πA(c) − πB(c)
  - S3: D3(c) = πR(c) − πP(c)
  - S4: D4(c) = πA(c) − πP(c)
- Properties:
  - D1(c): strictly convex, at most two real roots.
  - D2(c): linear, positively sloped for any t_c > 0; single root c∗; firms with c∗ ≤ c ≤ 1 bunch.
  - D3(c): strictly quasi-concave, at most two real roots under a curvature condition (3(f′′)2 − f′ f′′′ > 0).
  - D4(c): strictly quasi-concave, at most two real roots.
- Illustration for f(L) = L^{1/2}:
  - Explicit quadratic expression for πR − πP and closed-form roots when Γ ≤ Γ′ where
    - Γ′ = p2/(4w) ( t2(1 − tc)2 / [1 − (1 − tc)2] ).
  - If Γ = 0, roots simplify to cl = (t_c − t)p / t_c and ch = (2 − t_c − t)p / (2 − t_c).

### Characterizing equilibrium across S1–S4 and global patterns
- Stitch D1–D4 across S segments to obtain overall profit-difference curve over c ∈ [0,1].
- Numerical simulations commonly yield a global-optimum pattern R-B-P:
  - Low-cost firms c ∈ [0, c1): unconstrained in regular regime (R).
  - Mid-cost firms c ∈ [c1, c′]: bunch just below the threshold (B).
  - High-cost firms c ∈ (c′, 1]: unconstrained in presumptive regime (P).
- Aggregate expressions:
  - Aggregate net profit Π(̄Z, t, t_c): integral decomposition across regions (equation (33)).
  - Government spending G(̄Z, t, t_c): revenues net of administrative costs (equation (34)).
  - Social welfare SW = Π + δ G + V(p) (equation (35)).

### First-order conditions for optimal policies
- Optimal threshold ̄Z (Proposition 4) — equation (36):
  - Left-hand side: extensive margin (EM) net effect of raising threshold by $1 (mass of firms moving from regular to bunching × net revenue change).
  - Right-hand side: intensive margin (IM) net effect of raising threshold by $1 (production efficiency gains + additional presumptive-tax revenue from bunching firms).
  - At optimum: EM and IM balance. Note dc1/d ̄Z < 0.
- Optimal presumptive tax rate t (Proposition 5) — equation (37):
  - Left-hand side: EM effect of increasing t by 1% (mass at c1 moving to regular × net revenue change).
  - Right-hand side: IM — revenue gains from bunchers and unconstrained presumptive firms minus welfare loss from reduced output in presumptive regime due to higher t.
  - Practical considerations when raising t slightly:
    1. Net revenue change from firms switching to regular regime.
    2. Greater administrative costs from firms switching to regular regime.
    3. Tax revenue gain from firms remaining in presumptive regime.

### The reduction in presumptive-regime output and comparative statics
- Marginal firm c1 balances discontinuous gain from moving to regular regime with compliance cost Γ; Γ does not enter SW first-order conditions because of this balance.
- πR − πB has inverted hump shape; πR independent of t, πB decreasing in t ⇒ dc1/dt > 0.
- Term −t p^2 f′ dL/d pP captures output reduction effect of raising t in presumptive regime.
- First-order conditions (equations (51)–(56)) show welfare changes depend on integrals over firm types and weights δ and (δ − 1).

### Numerical calibration and baseline parameters
- Production: f(L) = β L^α.
- Calibration targets:
  - Average value added per firm ≈ $120,000.
  - Some bunching just below optimal threshold.
  - Compliance cost ≈ 0.4% of average turnover of firms in regular regime.
  - Formal sector output price = 1 USD; informal good normalized to 1.
  - Unit cost distribution: H(c) = 0.2 c^2 + 0.8 c^3, c ∈ [0,1].
  - Ratio of average value-added to sales = 26%.
  - Marginal value of public funds δ = 1.3 (baseline).
- Baseline parameter cell (Table 2): Γ = 1200, A = 240, δ = 1.3, α = 0.45, β = 800.
- Corporate income tax compliance cost empirical context: between 0.05% and 15% of taxable turnover in developing and transition economies.

### Simulation results (Table 2 summary and baseline optimum)
- Table 2 reports optimal presumptive tax rates and thresholds for exogenous t_c ∈ {11.0%, 14.0%, 17.0%, 20.0%, 23.0%, 26.0%, 29.0%}.
  - t_c: 11.0% | 14.0% | 17.0% | 20.0% | 23.0% | 26.0% | 29.0%
  - Optimal presumptive t: 6.00% | 4.40% | 2.40% | 2.60% | 2.80% | 3.10% | 6.00%
  - Optimal sales threshold ($ thousand): 9243 | 3914 | 1120 | 978 | 692 | 4 | 924
  - Placement pattern (from lowest-cost to highest): PR - A - PR - B - PR - B - PR - B - PR - B - PR - B - PP
  - Proportion of firms (examples): 100% | 42% - 14% - 44% | 65% - 9% - 26% | 68% - 8% - 24% | 73% - 6% - 21% | 75% - 4% - 21% | 100%
  - Average value-added per firm: 1.26E+05 | 1.27E+05 | 1.23E+05 | 1.23E+05 | 1.16E+05 | 1.12E+05 | 1.26E+05
  - Average compliance cost / turnover: n.a. | 0.39% | 0.39% | 0.41% | 0.42% | 0.43% | n.a.
  - Compliance cost / turnover at the threshold: n.a. | 0.35% | 0.85% | 1.00% | 1.24% | 1.40% | n.a.
  - Social Welfare: 82185.22 | 82185.68 | 82476.49 | 82626.64 | 82611.02 | 82419.20 | 82185.22
- Baseline global optimum (last rows of Table 2):
  - Overall optimal policy: corporate income tax rate t_c∗ = 20.9%, optimal turnover tax rate t∗ = 2.7%, optimal threshold ̄Z∗ = $113 (thousand).
  - About 7% of firms bunch just below the threshold.
  - Baseline placement: R - B - P; Proportion: 70% - 7% - 23%.
  - Average value-added per firm: 1.18E+05.
  - Average compliance cost / turnover: 0.41%.
  - Compliance cost / turnover at the threshold: 1.06%.

### Comparative statics (selected cases)
- Case 1. Γ = 1300, A = 260:
  - t_c∗ = 21.1%, t∗ = 2.8%, ̄Z∗ = $112 (thousand); Proportion: 70% - 7% - 23%; Average compliance cost / turnover: 0.45%; Compliance cost at threshold: 1.16%
  - Effect: increase in compliance/administrative costs → decrease in optimal threshold, increase in corporate and presumptive tax rates.
- Case 2. δ = 1.35:
  - t_c∗ = 23.1%, t∗ = 2.8%, ̄Z∗ = $99 (thousand); Proportion: 72% - 7% - 21%; Average compliance cost / turnover: 0.42%; Compliance cost at threshold: 1.21%
  - Effect: higher marginal value of public funds → decrease in threshold, increase in corporate and turnover tax rates.
- Case 3. β = 810:
  - t_c∗ = 21.3%, t∗ = 2.7%, ̄Z∗ = $114 (thousand); Proportion: 70% - 8% - 22%; Average compliance cost / turnover: 0.40%; Compliance cost at threshold: 1.05%
  - Effect: higher productivity parameter β → increase in threshold and in tax rates.
- Additional observed patterns:
  - When t_c = 11.0%, all firms allocated to presumptive regime.
  - For t_c = 14.0% there is bunching at the threshold in the regular regime; for higher t_c there is bunching just below the threshold.
  - As t_c increases from 17% to 26%, optimal threshold falls and optimal turnover tax rate rises.
  - When t_c = 29%, optimal to have all firms in the presumptive regime by setting an elevated threshold.
  - Gross profit margins of firms bunching just below the threshold vary from 9% to 12% at equilibrium.

### Country/regional comparisons (Table 3 group averages)
- Approach: fix t_c = 20% and vary β to target average value added per firm using GDP per person employed from ILOSTAT.
- Selected group results:
  - Sub-Saharan Africa:
    - t∗ = 6.8%
    - ̄Z∗ ($ thousand) = 46
    - Placement: R - B - P
    - Proportion: 34% - 14% - 52%
    - Average compliance cost / turnover = 2.55%
    - Compliance cost / turnover at threshold = 2.61%
    - Note: lower average value added per firm → lower optimal threshold and higher turnover tax rate.
  - Latin America:
    - t∗ = 3.6%
    - ̄Z∗ ($ thousand) = 92
    - Proportion: 57% - 11% - 32%
    - Average compliance cost / turnover = 0.72%
    - Compliance cost / turnover at threshold = 1.30%
  - Euro Area:
    - t∗ = 2.0%
    - ̄Z∗ ($ thousand) = 150
    - Proportion: 76% - 6% - 18%
    - Average compliance cost / turnover = 0.25%
    - Compliance cost / turnover at threshold = 0.80%
- Empirical example: in Madagascar, firms with sales below $56,000 face a turnover tax rate of 5% while corporate income tax rate is 20%.

### Policy implications and conclusions
- Theoretical contribution: characterization of optimal turnover threshold and tax rate in presence of regular corporate taxation and presumptive turnover taxation.
- Calibrated model suggests:
  - Optimal turnover tax rate between 2 and 3 percent (central estimate 2.7%).
  - Optimal threshold around $115,000 (baseline ̄Z∗ = $113,000).
- Key design margins:
  - Interaction between corporate income tax rate and turnover threshold/tax rate: higher corporate rates increase bunching incentives and affect optimal threshold.
  - Importance of marginal value of public funds δ and compliance/administrative costs Γ and A.
- Open issues:
  - Joint determination of VAT and presumptive income tax thresholds omitted; potential economies of scope in compliance and administration.
  - Model assumes full compliance; concealing sales and additional avoidance behaviors not modeled.
- Practical takeaway: turnover-based presumptive regimes can be optimal with modest turnover tax rates (≈2–3%) and thresholds on the order of $100k–$150k, but optimal settings vary substantially with country-level productivity and cost/compliance structures.

*Source: wpiea2019098 — References (appendix, proofs, and body material on optimal presumptive tax regimes).*

### References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### wpiea2019098 - References

### Introduction — policy context and main empirical observations
- Turnover (sales) taxes are widely used as “presumptive” or “simplified” income tax regimes for small and medium-sized enterprises (SMEs) to reduce compliance and administration costs.
- Examples of observed country practices (threshold, turnover tax rate, corporate income tax rate) include:
  - Kenya: threshold $49,000, turnover tax rate 3%, corporate income tax rate 30%.
  - Seychelles: threshold $74,000, turnover tax rate 1.5%.
  - Mauritania: threshold $84,000, turnover tax rate 3%.
  - Guinea: threshold $16,500, turnover tax rate 5%.
  - Belarus: threshold $625,000, turnover tax rate 5% (with 3% for VAT payers).
- Table 1 (international practices) reports many country-specific thresholds and turnover tax rates; US dollar exchange rate as of December 2018 is used for conversions.
- Common policy heuristics:
  - Use the VAT threshold as the presumptive income tax threshold (rule of thumb).
  - Make the effective tax burden at the turnover threshold more onerous than under the regular corporate income tax to encourage “graduation” to the regular system.
  - Aim for “neutrality” by equalizing after-tax profit margins across regimes.
- Caveats noted:
  - These heuristics ignore compliance costs and firm heterogeneity; without a detailed model (and behavioral responses), such recommendations can be vacuous.
  - Turnover taxes distort input choices because costs are non-deductible, potentially reducing productivity and pushing firms into informal or alternative fixed-fee regimes.

### Main findings (analytical and numerical)
- The paper provides the first formal analysis of the optimal sales threshold separating presumptive (turnover) and regular (corporate income) tax regimes and the corresponding optimal tax rates.
- Key qualitative results:
  - The optimal threshold depends on the marginal cost of public funds, administrative costs, and productivity (value added per firm).
  - The optimal threshold and turnover tax rate respond to changes in the corporate income tax rate.
  - Firms’ heterogeneity in marginal costs (unit costs c) is crucial: non-deductibility of costs creates distortions that vary across firms.
  - Depending on parameters, both higher-cost and lower-cost firms may locate in the presumptive regime while middle-cost firms sit in the regular regime.
- Numerical benchmarks and summary comparisons with observed practices:
  - The optimal threshold is generally between about $100,000 and $150,000 (depending on value added per firm).
  - The optimal turnover tax rate is close to 3% in the benchmark calculation when a single tax rate applies to all sectors.
  - For Sub-Saharan Africa, the model predicts a higher optimal turnover tax rate and a lower optimal threshold compared with the benchmark.
  - Many countries’ observed policies are consistent with model prescriptions, but others deviate substantially.

### Model setup — agents, technologies, and tax regimes
- Time/labor allocation:
  - Each individual supplies one unit of labor between formal production L and informal production 1−L.
  - Formal-sector output: f(L), with f(0) = 0, f′ > 0, f′′ < 0.
  - Informal-sector productivity: constant rate w; informal earnings w(1−L) are untaxable.
- Inputs and heterogeneity:
  - Formal production requires λ > 0 units of an imported intermediate per unit of output; individual-specific unit cost c = pI λ.
  - Unit costs c are distributed with twice differentiable CDF H(c), density h(c), support c ∈ [0,1].
- Prices:
  - Country is small internationally: price of final good p and imported intermediate pI are fixed.
- Two linear tax regimes (formal-sector income):
  - Regular regime (corporate income tax): tax rate t_c < 1; intermediate input cost is tax-deductible; firm faces fixed compliance cost Γ ≥ 0 and imposes fixed administrative cost A > 0 on the authority.
  - Presumptive regime (turnover tax): tax rate t < 1; costs are not deductible; no compliance or administrative costs assumed for presumptive regime.
- Net price ρ depends on regime:
  - p_P ≡ (1−t)p − c (presumptive regime)
  - p_R ≡ (1−t_c)(p − c) (regular regime)
- Profits and equilibrium conditions:
  - Net profits: π(L) ≡ ρ f(L(ρ)) + w(1 − L(ρ)) − I_R Γ, where I_R = 1 if in regular regime, 0 otherwise.
  - Unconstrained first-order condition: ρ f′(L*) − w = 0.
  - Optimized profit function: π*(ρ) and, as functions of c: π_R(c) ≡ π*(p_R(c)), π_P(c) ≡ π*(p_P(c)).
- Envelope and convexity results:
  - dπ*/dρ = f(L*) > 0.
  - d^2π*/dρ^2 = f′(L*) dL*/dρ > 0, so π* is increasing and strictly convex in ρ.
  - Because dρ/dc differs across regimes (d p_P/dc = −1; d p_R/dc = −(1−t_c)), the sensitivity of profits to c differs.

### Analytical comparison: all firms in a single regime
- Square-root production function illustration: f(L) = L^{1/2} yields closed-form expressions.
- All firms in regular regime:
  - π_R(c) = (1/(4w))(p_R)^2 + w − Γ
  - g_R(c) = t_c (p − c) (1/(2w))(p_R)^2 − A
  - SW_R = v(p) + (1/(4w)) E([(1−t_c)(p−c)]^2) + δ (1/(2w)) t_c E[(p−c)(1−t_c)(p−c)] + w − (Γ + δA)
- Lemma 1:
  - The optimal corporate income tax rate in the regular regime is an increasing function of the marginal value of public funds and is independent of the distribution of unit costs.
  - Optimal t_c* = (δ−1)/(2δ−1)
- All firms in presumptive regime:
  - π_P(c) = (1/(4w))(p_P)^2 + w
  - g_P(c) = (t p /(2w)) ((1−t)p − c)
  - SW_P = v(p) + (1/(4w)) E([(1−t)p − c]^2) + δ (t p /(2w)) E[(1−t)p − c] + w
- Lemma 2:
  - The optimal presumptive tax rate is an increasing function of the average profit margin and is smaller than the optimal rate in the regular regime.
  - t* = (δ−1)/(2δ−1) (p − E(c))/p = t_c* (1 − E(c)/p)
  - Hence 0 < t* < t_c* because p > E(c).
- Welfare comparison (square-root f):
  - Proposition 1:
    - The welfare difference SW_R − SW_P is increasing in the variance of unit costs σ^2, but decreasing in fixed compliance and administrative costs.
    - Depending on these forces, either regime could be optimal if only one regime is possible and tax rates are linear.
    - Exact expression reported:
      SW_R − SW_P = −(Γ + δA) + 1
      4w
      (δ−1)
      2
      (2δ−1)
      σ
      2
    - Interpretation: the presumptive regime works best when there is little dispersion in unit costs (or profit margins). Categorizing presumptive-regime firms by activity types and applying different tax rates reduces variance but increases system complexity and potential administrative costs.

### Presumptive regime with a sales threshold — characterization of firm behavior
- Reformulation in terms of sales Z = p f(L):
  - Profit maximization over Z: π(Z, ρ) ≡ ρ Z / p + w [1 − f^{-1}(Z/p)] − I_R Γ.
  - First-order condition for desired sales Z*(ρ):
    dπ/dZ = ρ/p − (w/p) f′(f^{-1}(Z*/p)) = 0.
  - Desired sales are decreasing in unit cost c: dZ(ρ)/dc < 0.
- Square-root example (f(L) = L^{1/2}):
  - Z(ρ) = (p/(2w)) ρ.
  - π(ρ) = ρ^2/(4w) + w − I_R Γ.
  - Desired sales as functions of c:
    - Z_R(c) = (p/(2w)) [(1−t_c)(p − c)]
    - Z_P(c) = (p/(2w)) [(1−t)p − c]
- Definition of threshold constraint:
  - A firm is constrained if its desired sales under the regime it faces exceed the threshold (when in presumptive regime) or fall below the threshold (when in regular regime).
- Partition of firms into four mutually exclusive sets (for given {t, t_c, Z̄}):
  - S1(c): Z_P(c) ≥ Z̄ and Z_R(c) ≥ Z̄ (Only firms in the presumptive regime are constrained)
  - S2(c): Z_P(c) ≥ Z̄ and Z_R(c) < Z̄ (Firms are constrained in both regimes)
  - S3(c): Z_P(c) < Z̄ and Z_R(c) ≥ Z̄ (Firms are unconstrained in both regimes)
  - S4(c): Z_P(c) < Z̄ and Z_R(c) < Z̄ (Only firms in the regular regime are constrained)
- Graphical characterization (linear case) and two cases depending on relative position of Z̄ and crossover ˆc:
  - Define ˆc = (t_c − t) p / t_c where p_R(c) = p_P(c) and ˆZ = Z_R(ˆc) = Z_P(ˆc).
  - Case A: Z̄ ≥ ˆZ ⇒ 0 < c′′ ≤ c′ ≤ ˆc and sets:
    - S1 = [0, c′′)
    - S2 = [c′′, c′)
    - S3 = ∅
    - S4 = [c′, 1]
  - Case B: Z̄ < ˆZ ⇒ 0 < ˆc < c′ < c′′ and sets:
    - S1 = [0, c′)
    - S2 = ∅
    - S3 = [c′, c′′]
    - S4 = (c′′, 1]
- Important behavioral implication: firms may “bunch” at or just below the sales threshold; constrained firms choose corner solutions relative to the threshold.

### Key analytical expressions (selected)
- Net price definitions:
  - p_P ≡ (1−t)p − c
  - p_R ≡ (1−t_c)(p − c)
- Regular-regime optimal tax (Lemma 1):
  - t_c* = (δ−1)/(2δ−1)
- Presumptive-regime optimal tax (Lemma 2):
  - t* = (δ−1)/(2δ−1) (p − E(c))/p = t_c* (1 − E(c)/p)
- Welfare difference (Proposition 1; square-root f):
  - SW_R − SW_P = −(Γ + δA) + 1
    4w
    (δ−1)
    2
    (2δ−1)
    σ
    2
- Desired sales (square-root f):
  - Z(ρ) = p/(2w) ρ
  - Z_R(c) = p/(2w) [(1−t_c)(p − c)]
  - Z_P(c) = p/(2w) [(1−t)p − c]

*Source: wpiea2019098 — References (appendix, proofs, and body material on optimal presumptive tax regimes).*

### 1.  Unconstrained in presumptive regime:π

### 1.  Unconstrained in presumptive regime:π

### Key model definitions and profit functions
- Unconstrained in presumptive regime:
  - πP(pP) = pP Z(pP) p + w[1 − f−1(Z(pP) p)]
- Constrained in presumptive regime:
  - πB(pP) = pP ̄Z p + w(1 − μ(̄Z))
- Unconstrained in regular regime:
  - πR(pR) = pR Z(pR) p + w[1 − f−1(Z(pR) p)] − Γ
- Constrained in regular regime:
  - πA(pR) = pR ̄Z p + w(1 − μ(̄Z)) − Γ
- μ = f−1(̄Z/p) is the labor supply that makes sales just equal to the threshold.
- πB(pP) and πA(pR) are profits of “bunching” firms just below the threshold; if sales rose they would switch regimes.

### Profit-difference functions and firms' regime choices
- For firms in S1:
  - D1(c) = πR(c) − πB(c)
  - Firms choose the regular regime iff D1(c) > 0.
- For firms in S2:
  - D2(c) = πA(c) − πB(c)
  - Firms bunch at the threshold iff D2(c) > 0.
- For firms in S3:
  - D3(c) = πR(c) − πP(c)
  - Firms choose the regular regime iff D3(c) > 0.
- For firms in S4:
  - D4(c) = πA(c) − πP(c)
  - Firms bunch at the threshold iff D4(c) > 0.
- Roots of D_i(c) partition the cost space into firms preferring each regime, given tax policies {t, tc, ̄Z} and compliance cost Γ.

### Proposition 2 — properties of profit-difference curves
- D1(c):
  - Graph is strictly convex and has at most two real roots.
  - If roots are imaginary, all firms in S1 choose the regular regime.
- D2(c):
  - Graph is linear and positively sloped for any tc > 0.
  - Let c∗ denote its single root.
  - Firms in S2 with c∗ ≤ c ≤ 1 choose to bunch at the threshold. If c∗ > 1, all firms in S2 bunch just below the threshold.
- D3(c):
  - Graph is strictly quasi-concave and has at most two real roots if 3(f′′)2 − f′ f′′′ is positive for all L > 0.
  - If roots are imaginary, all firms in S3 choose the presumptive regime.
- D4(c):
  - Graph is strictly quasi-concave and has at most two real roots.
  - If roots are imaginary, all firms in S4 choose the presumptive regime.
- Hump-shaped or inverted hump-shape in D3 or D1 can yield two real roots; monotone shapes yield single roots.

### Illustration: D3(c) with f(L) = L1/2
- For the interval where pP(c) ≥ 0:
  - πR − πP =
    1/(4w) [(1 − tc)2 − 1] c2 + p2/(w) [(1 − t) − (1 − tc)2] c + { p2/(4w) [(1 − tc)2 − (1 − t)2] − Γ }  (equation (28))
- For the interval where pP(c) < 0 and hence πP = w:
  - πR − πP = 1/(4w) [(1 − tc)(p − c)]2 − Γ  (equation (29))
- Roots of (28):
  - c = p[(1 − t) − (1 − tc)2] ± p √{ t2(1 − tc)2 − 4w/p2 [1 − (1 − tc)2] Γ } / [1 − (1 − tc)2]  (equation (30))
- Real roots exist iff Γ ≤ Γ′, where
  - Γ′ = p2/(4w) ( t2(1 − tc)2 / [1 − (1 − tc)2] )  (equation (31))
- If Γ = 0, roots become:
  - cl = (tc − t)p / tc ; ch = (2 − tc − t)p / (2 − tc)  (equation (32))
  - Note: ch may be inadmissible if pP(ch) < 0.
- Cases by Γ:
  - (a) Γ ∈ [0, Γ′′): single root cl; some firms switch to corner solution L∗ = 0 when unit costs exceed (1 − t)p.
  - (b) Γ ∈ [Γ′′, Γ′]: both roots admissible.
  - (c) Γ ∈ (Γ′, ∞): no real roots; all firms prefer the presumptive regime.
- When t < tc and cl > 0: firms with c ∈ [0, cl) prefer presumptive; c ∈ [cl, 1] prefer regular. If t ≥ tc then cl ≤ 0 and all firms weakly prefer regular.

### Characterizing equilibrium across cost partitions S1–S4
- The S1–S4 sets partition cost space [0,1]. Profit-difference roots within each set determine firm behavior.
- Case A: S3 empty; join D1 to D2 and D2 to D4 via equalities πR = πA and πP = πB at boundary costs.
- Case B: S2 empty; join D1 to D3 where πP = πB and D3 to D4 where πR = πA.
- Overall profit-difference curve obtained by stitching D1–D4 across their relevant S segments.

### Numerical simulation equilibrium (global optimum pattern)
- Simulation results indicate a globally optimal tax policy often generates the configuration analogous to Figure 5 with a single overall root c1.
- Equilibrium type R-B-P (from Figure 6):
  - Relatively low-cost firms c ∈ [0, c1) are unconstrained in the regular regime (R).
  - Mid-cost firms c ∈ [c1, c′] bunch just below the threshold (B).
  - Higher-cost firms c ∈ (c′, 1] are unconstrained in the presumptive regime (P).

### Aggregate profit, government spending, and social welfare
- Aggregate net profit:
  - Π(̄Z, t, tc) =
    ∫0 c1 { pR f(L(pR)) + w[1 − L(pR)] − Γ } h(c) dc
    + ∫c1 c′ { pP ̄Z/p + w[1 − μ(̄Z)] } h(c) dc
    + ∫c′1 { pP f(L(pP)) + w[1 − L(pP)] } h(c) dc  (equation (33))
  - μ(̄Z) is labor supply at which firm’s sales equals the threshold.
- Government spending (revenues net of administrative costs):
  - G(̄Z, t, tc) =
    ∫0 c1 [ tc (p − c) f(L(pR)) − A ] h(c) dc
    + ∫c1 c′ t ̄Z h(c) dc
    + ∫c′1 t p f(L(pP)) h(c) dc  (equation (34))
- Social welfare:
  - SW = Π(̄Z, t, tc) + δ G(̄Z, t, tc) + V(p)  (equation (35))
  - V(p) is aggregate consumer surplus from formal-sector sales (constant for optimization).

### First-order condition for optimal threshold ̄Z (Proposition 4)
- Optimal threshold characterized by:
  - δ [ tc (p − c1) f(L(pR(c1))) − A − t ̄Z ] h(c1) (dc1/d ̄Z) × (−1) =
    ∫c1 c′ [ pP/p − w μ ̄Z ] h(c) dc + δ t [ H(c′) − H(c1) ]  (equation (36))
- Interpretation:
  - Left-hand side (absolute) = extensive margin (EM) net effect of raising threshold by $1: mass of firms moving from regular to bunching times net revenue change per affected firm.
  - Right-hand side = intensive margin (IM) net effect of raising threshold by $1: increased production efficiency for previously constrained firms plus additional presumptive-tax revenue from firms already bunching, weighted by δ.
  - At optimum: extensive and intensive margins balance.
  - Additional identities:
    - μ ̄Z ≡ dμ/d ̄Z.
    - dc1/d ̄Z < 0 (c1 shifts left as ̄Z increases).

### First-order condition for optimal presumptive tax rate t (Proposition 5)
- Optimal t characterized by:
  - δ [ tc (p − c1) f(L(pR(c1))) − A − t ̄Z ] h(c1) (dc1/dt) × (−1) =
    (δ − 1) ̄Z [ H(c′) − H(c1) ] + (δ − 1) ∫1 c′ p f(L(pP)) h(c) dc − δ ∫1 c′ t p2 f′ (dL/dpP) h(c) dc  (equation (37))
- Interpretation:
  - Left-hand side (absolute) = extensive margin effect of increasing t by 1%: mass of firms at c1 moving to regular regime times net revenue change.
  - Right-hand side = intensive margin:
    - First pair of terms: welfare change from increased revenues from bunchers and unconstrained presumptive firms.
    - Last term: welfare loss from reduced output and hence reduced tax revenue in presumptive regime due to higher t.
- Practical considerations when raising t slightly:
  1. Net revenue change from firms switching to regular regime.
  2. Greater administrative costs from firms switching to regular regime.
  3. Tax revenue gain from firms remaining in presumptive regime.

*Source: wpiea2019098 - 1.  Unconstrained in presumptive regime:π*

### 4.  The reduction in the output of firms in the presumptive regime.

### 4.  The reduction in the output of firms in the presumptive regime.

### Model mechanisms and analytical results
- The marginal firm, c1, balances the discontinuous gain in net profit from moving to the regular regime with the compliance cost Γ; Γ does not enter social welfare first-order conditions explicitly because of this balance.
- πR(c) − πB(c) has an inverted hump shape; πR is independent of t, while πB is decreasing in t. Consequently, dc1/dt > 0.
- The term −t p^2 f′ dL/d pP arises from t p(f′ dL/d pP d pP/dt) with d pP/dt = −p; it can be written as ZP (d f/dt)(t f) where ZP = p f(L(pP)).
- Several profit-difference functions (D1(c) through D4(c)) are analyzed for curvature and roots:
  - D1(c) = πR − πB is strictly convex (d^2(πR − πB)/dc^2 > 0) and can have at most two real roots.
  - D2(c) = πA − πB is linear with unique real root c* = (pΓ/ ̄Z + (t c − t)p)/t c; πA > πB for all c ∈ [c*, 1].
  - D3(c) = πR − πP is strictly quasi-concave (at most one interior local maximum) under the stated assumptions.
  - D4(c) = πA − πP is strictly quasi-concave or decreasing throughout, depending on parameter values.
- First-order conditions for social welfare with respect to the threshold ̄Z and the turnover tax t are derived (equations (51)–(56)), showing how welfare changes depend on integrals over firm types and on the weights δ and (δ − 1) applied to revenue changes and profit taxation.

### Numerical calibration and baseline parameters
- Production function: f(L) = β L^α.
- Calibration targets and assumptions:
  - Average value added of a firm ≈ $120,000.
  - Some bunching of firms just below the optimal threshold.
  - Compliance cost ≈ 0.4% of the average turnover of firms in the regular regime.
  - Formal sector output price = 1 USD; informal sector good price normalized to 1.
  - Distribution of firms’ unit costs: H(c) = 0.2 c^2 + 0.8 c^3, c ∈ [0,1].
  - Ratio of average value-added to sales = 26%.
  - Marginal value of public funds δ = 1.3 (baseline).
- Baseline parameter cell in Table 2: Γ = 1200, A = 240, δ = 1.3, α = 0.45, β = 800.
- Corporate income tax compliance cost empirical context: between 0.05% and 15% of taxable turnover in developing and transition economies; surveys suggest compliance costs comparable to calibration for turnovers in the range of $150,000 to $1 million.

### Simulation results (Table 2 summary)
- Table 2 reports optimal turnover tax rates and optimal thresholds for exogenous corporate tax rates t c ∈ {11.0%, 14.0%, 17.0%, 20.0%, 23.0%, 26.0%, 29.0%}.
- For each corporate tax rate, the table reports:
  - Tax rate in the regular regime (t c): 11.0% | 14.0% | 17.0% | 20.0% | 23.0% | 26.0% | 29.0%
  - Optimal tax rate in the presumptive regime: 6.00% | 4.40% | 2.40% | 2.60% | 2.80% | 3.10% | 6.00%
  - Optimal sales threshold ($ thousand): 9243 | 3914 | 1120 | 978 | 692 | 4 | 924
  - Placement of firms (from lowest-cost segment to highest): PR - A - PR - B - PR - B - PR - B - PR - B - PR - B - PP
  - Proportion of firms: 100% | 42% - 14% - 44% | 65% - 9% - 26% | 68% - 8% - 24% | 73% - 6% - 21% | 75% - 4% - 21% | 100%
  - Average value-added per firm: 1.26E+05 | 1.27E+05 | 1.23E+05 | 1.23E+05 | 1.16E+05 | 1.12E+05 | 1.26E+05
  - Average compliance cost / turnover: n.a. | 0.39% | 0.39% | 0.41% | 0.42% | 0.43% | n.a.
  - Compliance cost / turnover at the threshold: n.a. | 0.35% | 0.85% | 1.00% | 1.24% | 1.40% | n.a.
  - Social Welfare: 82185.22 | 82185.68 | 82476.49 | 82626.64 | 82611.02 | 82419.20 | 82185.22
- Baseline global optimum (last rows of Table 2):
  - Overall optimal policy: corporate income tax rate t c∗ = 20.9%, optimal turnover tax rate t∗ = 2.7%, optimal threshold ̄Z∗ = $113 (thousand), about 7% of firms bunch just below the threshold.
  - Baseline comparative statistics: Place of firms: R - B - P; Proportion of firms: 70% - 7% - 23%; Average value-added per firm: 1.18E+05; Average compliance cost / turnover: 0.41%; Compliance cost / turnover at the threshold: 1.06%.
- Comparative statics (three cases):
  - Case 1. Γ = 1300, A = 260:
    - t c∗ = 21.1%, t∗ = 2.8%, ̄Z∗ = $112 (thousand)
    - Placement: R - B - P; Proportion: 70% - 7% - 23%; Average value-added per firm: 1.18E+05; Average compliance cost / turnover: 0.45%; Compliance cost / turnover at the threshold: 1.16%
    - Effect: increase in compliance/administrative costs → decrease in optimal threshold, increase in corporate and presumptive tax rates.
  - Case 2. δ = 1.35:
    - t c∗ = 23.1%, t∗ = 2.8%, ̄Z∗ = $99 (thousand)
    - Placement: R - B - P; Proportion: 72% - 7% - 21%; Average value-added per firm: 1.16E+05; Average compliance cost / turnover: 0.42%; Compliance cost / turnover at the threshold: 1.21%
    - Effect: higher marginal value of public funds → decrease in threshold, increase in corporate and turnover tax rates.
  - Case 3. β = 810:
    - t c∗ = 21.3%, t∗ = 2.7%, ̄Z∗ = $114 (thousand)
    - Placement: R - B - P; Proportion: 70% - 8% - 22%; Average value-added per firm: 1.20E+05; Average compliance cost / turnover: 0.40%; Compliance cost / turnover at the threshold: 1.05%
    - Effect: higher productivity parameter β → increase in threshold and in tax rates.
- Additional equilibrium observations:
  - When t c = 11.0%, all firms are allocated to the presumptive regime (regular regime not worth compliance/administrative costs).
  - For t c = 14.0% there is bunching at the threshold in the regular regime; for higher t c there is bunching just below the threshold.
  - As t c increases from 17% to 26%, the optimal threshold falls and the optimal turnover tax rate rises.
  - When t c is sufficiently high (t c = 29%), it becomes optimal to have all firms in the presumptive regime by setting an elevated threshold.
  - Gross profit margins of firms bunching just below the threshold vary from 9% to 12% at equilibrium.

### Country/regional comparisons (Table 3)
- Approach: fix corporate income tax rate at 20% and vary β to target average value added per firm, using GDP per person employed from ILOSTAT to scale.
- Selected results reported in Table 3 (group averages):
  - Sub-Saharan Africa:
    - t∗ = 6.8%
    - ̄Z∗ ($ thousand) = 46
    - Placement of firms: R - B - P
    - Proportion of firms: 34% - 14% - 52%
    - Average compliance cost / turnover = 2.55%
    - Compliance cost / turnover at the threshold = 2.61%
    - Note: lower average value added per firm → lower optimal threshold and higher turnover tax rate; Sub-Saharan Africa example yields high turnover tax rate and large proportion of firms in presumptive regime.
  - Latin America:
    - t∗ = 3.6%
    - ̄Z∗ ($ thousand) = 92
    - Placement: R - B - P
    - Proportion: 57% - 11% - 32%
    - Average compliance cost / turnover = 0.72%
    - Compliance cost / turnover at the threshold = 1.30%
  - Euro Area:
    - t∗ = 2.0%
    - ̄Z∗ ($ thousand) = 150
    - Placement: R - B - P
    - Proportion: 76% - 6% - 18%
    - Average compliance cost / turnover = 0.25%
    - Compliance cost / turnover at the threshold = 0.80%
- Empirical note: in Madagascar, firms with sales below $56,000 face a turnover tax rate of 5% while corporate income tax rate is 20% (illustrative realism of results).

### Policy implications and conclusions
- The paper provides a theoretical analysis of optimal turnover (presumptive) threshold and tax rate in economies with both regular corporate taxation and presumptive turnover taxation.
- Calibrated model suggestion:
  - Optimal turnover tax rate between 2 and 3 percent (central estimate 2.7%).
  - Optimal threshold around $115,000 (baseline ̄Z∗ = $113,000).
- Key margins for policy design:
  - Relationship between the corporate income tax rate and the turnover threshold/tax rate: higher corporate tax rates tend to increase bunching incentives and affect optimal threshold decisions.
  - Importance of revenues to the government (marginal value of public funds δ) and the magnitude of compliance and administrative costs (Γ, A).
- Additional considerations and open issues:
  - Joint determination of VAT and presumptive income tax thresholds may create economies of scope in taxpayer compliance and tax administration; this is omitted and identified as an area for future research.
  - The model assumes full compliance but allows firms to restrict output (bunching) to remain below the threshold; concealing sales is an additional form of avoidance not modeled here and suggested as future work.
- Practical takeaway: turnover-based presumptive regimes can be optimal with modest turnover tax rates (≈2–3%) and thresholds on the order of $100k–$150k, but optimal settings vary substantially with country-level productivity and cost/compliance structures.

*Source: IMF Working Paper — Section 4 (numerical simulations and results) from the provided PDF.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019098.pdf_