## Appendix I. Optimal VDP penalty if detected evaders are forced to comply in period 2

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### Context and motivation
- Governments seek improved revenue performance post-COVID, in particular by raising taxes from the most affluent using Voluntary Disclosure Programs (VDPs) and Tax Amnesty Programs (TAPs).
- VDPs/TAPs aim to generate additional public revenue:
  - Short run: immediate expansion of the tax base through increased voluntary compliance.
  - Medium run: potential sustainable compliance improvements if disclosure information is used for enforcement.
- VDP design features noted:
  - VDPs allow previous tax evaders to disclose assets and income at reduced punitive action (penalty ν < φ).
  - Outside VDPs detected evaders typically pay full tax on current and past obligations plus interest and penalties; reported penalty ranges: between 20 and 200 percent of the tax obligation.
  - In practice, interest is often reduced under VDPs; in 19 out of 47 cases penalties were waived, and in 26 cases criminal prosecution was waived.
- Global context:
  - Zucman (2013) estimate: $6 trillion of global wealth hidden offshore.
  - Alstadsæter et al. (2018) estimate: amount equivalent to 10 percent of world GDP.
  - Offshore evasion concentrated at the top: around 10 percent of all households evade taxes generally; rises to between 30 and 40 percent for the top percentile.
  - AEOI reduced foreign-owned deposits in offshore jurisdictions by 25 percent (Beer et al. (2019)).
- Empirical assessments (Baer and Le Borgne (2008); Benedek et al. (2022)):
  - Amnesties often yield modest short-term gross revenue gains (US states’ amnesties on average had raised 0.7 percent of the relevant tax).
  - Net revenue effects can be smaller or negative in the medium term due to increased evasion, reduced penalties, and administrative costs.
  - Repeated amnesties can be counterproductive (time inconsistency problem).

### Basic model and individual decision (two-period baseline)
- Population: continuum of wealthy individuals with offshore asset return y and tax rate τ.
- Individuals are risk-neutral with heterogeneous guilt g_i ~ Uniform[0,1].
- Utility if compliant: u_c = (1 − τ) y.
- Expected utility if evading with detection probability p and fine multiple φ:
  - u_e(p) = [1 − p τ φ − (1 − p) g_i] y.
- Threshold guilt ḡ(p) (indifference between compliance and evasion):
  - ḡ(p) = τ (1 − p φ) / (1 − p).
  - Example: φ = 3, τ = 0.3, p = 0.1 ⇒ ḡ = 0.23 (23 percent evade).
- Government chooses p to maximize revenue net of enforcement costs:
  - max_p R = [(1 − ḡ) + ḡ p φ − C(p,ḡ)] τ y.
- Administrative cost function:
  - C(p,ḡ) = λ/2 p^2 + α ḡ, with α ≥ 0 and λ > 0.
- Marginal deterrence: ∂ḡ/∂p = − τ (φ − 1) / (1 − p)^2 < 0.
- Comparative statics: optimal p* rises when λ falls; elasticity ε satisfies 0 < ε < 1 (p* rises less than proportionally as λ falls).

### Simulations — steady-state parameter grid (τ = 0.3; α ∈ {0.2, 0.4}; φ ∈ {2.5, 4}; λ ∈ {15, 30})
- Selected steady-state outcomes (p* solved from first-order condition):
  - λ = 15, φ = 2.5, α = 0.2 → p* = 0.08, ε = 0.91, ḡ = 0.26, R / (τ y) = 0.69, C / (τ y) / R = 0.14
  - λ = 30, φ = 2.5, α = 0.2 → p* = 0.04, ε = 0.96, ḡ = 0.28, R / (τ y) = 0.67, C / (τ y) / R = 0.12
  - λ = 15, φ = 4, α = 0.2 → p* = 0.11, ε = 0.68, ḡ = 0.19, R / (τ y) = 0.77, C / (τ y) / R = 0.17
  - λ = 30, φ = 4, α = 0.2 → p* = 0.06, ε = 0.83, ḡ = 0.24, R / (τ y) = 0.71, C / (τ y) / R = 0.15
  - λ = 15, φ = 2.5, α = 0.4 → p* = 0.09, ε = 0.92, ḡ = 0.26, R / (τ y) = 0.64, C / (τ y) / R = 0.25
  - λ = 30, φ = 2.5, α = 0.4 → p* = 0.04, ε = 0.96, ḡ = 0.28, R / (τ y) = 0.61, C / (τ y) / R = 0.23
  - λ = 15, φ = 4, α = 0.4 → p* = 0.12, ε = 0.69, ḡ = 0.18, R / (τ y) = 0.73, C / (τ y) / R = 0.24
  - λ = 30, φ = 4, α = 0.4 → p* = 0.07, ε = 0.84, ḡ = 0.23, R / (τ y) = 0.67, C / (τ y) / R = 0.25
- Aggregate implications:
  - Optimal p* increases when λ falls (improved detection efficiency) but with ε < 1.
  - Share of evaders ḡ lies between 17.8 percent and 28.1 percent across scenarios.
  - Revenue net of enforcement R / (τ y) ranges from 61.1 percent to 76.6 percent of potential gross revenue.
  - Enforcement costs range between 12.3 percent and 24.9 percent of tax revenue in simulated cases.

### Unanticipated VDP (period-2 VDP unanticipated in period 1)
- Setup:
  - Period 1 behavior equals steady-state; in period 2 a shock raises detection probability p2 and an unanticipated VDP with penalty ν < φ is offered to first-period evaders.
  - Analytical assumption: first-period audit outcomes do not affect VDP eligibility; first-period evaders may participate irrespective of being caught in period 1.
- Participation threshold for first-period evaders:
  - g_l = τ(ν − p2 φ) / (1 − p2)    (equation (9))
  - VDP unanticipated is incentive compatible only if g_l < ḡ(p1); mass entering VDP = ḡ(p1) − g_l > 0.
- Reservation penalty for unanticipated VDP:
  - ν < 1 + (p2 − p1)(φ − 1) / (1 − p1) ≡ ν_rerer_u    (equation (10))
  - If p2 = p1, VDP attracts participants only if ν < 1.
- Second-period net revenue (period-2):
  - R2 = [1 − ḡ(p1) + [ḡ(p1) − g_l] ν + φ p2 g_l − C(p2, g_l)] τ y    (equation (11))
- First-order condition for optimal ν yields (after substitution):
  - g_l = 1/2 [ ḡ(p1) − α τ / (1 − p2) ]    (equation (14))
  - Explicit optimal VDP fine:
    - ν* = 1/2 [ 1 + (p2 − p1)(φ − 1)/(1 − p1) + p2 φ − α ] = 1/2 [ ν_rerer_u + p2 φ − α ]    (equation (15))
  - Interpretation:
    - ν* is half the reservation penalty plus upward adjustment from expected penalty p2 φ and downward adjustment from external cost α.
    - If α = 0, period-2 share of evaders is half the period-1 share.
- Policy trade-off:
  - Higher ν induces the government to choose higher p2; lower ν allows scaling back enforcement.
  - If detection efficiency is unchanged (λ2 = λ1) and unconstrained, ν* can be negative; if constrained to ν ≥ 1, optimal is ν = 1 and detection probability unchanged.
- Condition for ν* ≥ 1 (from text example):
  - Required increase p2 − p1 can be substantial depending on φ and α; examples given:
    - If φ = 4, p1 = 0.2, α = 0.5 → required increase 9 percentage points (20% → 29%).
    - If φ = 2.5, p1 = 0.1, α = 1 → required increase 42 percentage points (10% → 52%).

### Simulations — unanticipated VDP (period-2 detection efficiency improvement: λ2 = 3)
- Assumption: λ2 = 3 implying reduction by 90 percent if λ1 = 30, reduction by 80 percent if λ1 = 15.
- Selected simulation outcomes (τ = 0.3):
  - λ1 = 15, λ2 = 3, φ = 2.5, α = 0.2, p1* = 0.08 → p2* = 0.20, ν* = 0.75, ḡ(p1) = 0.26, g_l(p2) = 0.09, R1/(τ y) = 0.69, R2/(τ y) = 0.83, C1/(τ y) = 0.14, C2/(τ y) = 0.09
  - λ1 = 30, λ2 = 3, φ = 2.5, α = 0.2, p1* = 0.04 → p2* = 0.21, ν* = 0.80, ḡ(p1) = 0.28, g_l(p2) = 0.10, R1/(τ y) = 0.67, R2/(τ y) = 0.83, C1/(τ y) = 0.12, C2/(τ y) = 0.11
  - λ1 = 15, λ2 = 3, φ = 4, α = 0.2, p1* = 0.11 → p2* = 0.25, ν* = 1.12, ḡ(p1) = 0.19, g_l(p2) = 0.06, R1/(τ y) = 0.77, R2/(τ y) = 0.91, C1/(τ y) = 0.17, C2/(τ y) = 0.11
  - λ1 = 30, λ2 = 3, φ = 4, α = 0.2, p1* = 0.06 → p2* = 0.30, ν* = 1.39, ḡ(p1) = 0.24, g_l(p2) = 0.08, R1/(τ y) = 0.71, R2/(τ y) = 0.93, C1/(τ y) = 0.15, C2/(τ y) = 0.17
- Simulation insights:
  - Improved detection efficiency unambiguously increases detection probability, reduces evasion, raises revenue, and typically reduces administrative costs.
  - Optimal ν* ranges from 0.65 to 1.39 across simulations: previous evaders may pay either lower or higher effective tax than always-compliant taxpayers depending on parameters.
  - Magnitude of detection-probability increase depends on initial λ1: if detection initially costly (λ1 = 30), government boosts enforcement more and can set higher ν*; if detection initially less costly (λ1 = 15), government relies more on lower ν* to induce compliance.

### Anticipated VDP: T-period extension and anticipated effects
- T-period setup where period-2 VDP is anticipated alters behavior via option value and strategic evasion.
- Key thresholds in T = 2 analysis:
  - g_l = τ(ν − p2 φ) / (1 − p2)    (equation (22))
  - g_h = τ(ρ − p1 φ) / (1 − p1) with ρ = 2 − ν    (equation (23))
  - VDP is incentive compatible iff g_l < g_h, which leads to reservation penalty:
    - ν < 1 + 1/2 (p2 − p1) / (1 − p̄) (φ − 1) = ν_rerer_a    (equation (24))
  - If p2 = p1, ν_rerer_a = 1: with constant detection probability, no positive penalty VDP can induce coming clean.
  - Example in text: φ = 3, p2 = 0.2, p1 = 0.1 → ν_rerer_a = 1.
- Government two-period net revenue (net of collection costs):
  - R = [ 2 (1 − g_h) + (g_h − g_l) ν + (p1 g_h + p2 g_l) φ − C(p1, g_h) − C(p2, g_l) ] τ y    (equation (25))
- First-order condition and simplified optimality condition:
  - 2(g_h − g_l) = α τ (p2 − p1) / [ (1 − p1) (1 − p2) ]    (equation (27))
  - If α = 0, optimal g_h = g_l implying optimal VDP is not incentive compatible (no participation).
  - If α > 0 and p2 > p1, an incentive-compatible optimal VDP can exist.
- Closed-form optimal fine:
  - ν** = 1 + 1/2 (p2 − p1) / (1 − p̄) [ φ − 1 − α/2 ] = ν_rerer_a − 1/4 (p2 − p1) / (1 − p̄) α    (equation (28))
  - Interpretation:
    - α = 0 ⇒ ν** = ν_rerer_a (no participation).
    - If α > 0 and p2 > p1, ν** < ν_rerer_a to induce participation; sign of φ − 1 − α/2 determines whether ν** > 1.
    - Example: φ = 3, α = 2, p2 = 0.2, p1 = 0.1 ⇒ ν** = 1.06. For α > 4, ν** < 1.
- Detection probability first-order conditions:
  - Φ1(p1, ν, λ1) = 0 and Φ2(p2, ν, λ2) = 0 given by equations (29.a) and (29.b).
  - If λ1 = λ2, revenue-maximizing solution often sets ν** = 1 and p1* = p2* (no effective anticipated VDP).
  - Linear approximation (equation (30)) shows detection efficiency improvements and higher ν raise p2* relative to p1*; typical optimal solution satisfies p2* > p1* and ν** > 1 when enforcement efficiency rises.
- Average evaders:
  - Without VDP: share of evaders across two periods = 2 ḡ(p̄).
  - With VDP: period-1 evasion increases (g_h > without VDP), period-2 evasion decreases (g_l < without VDP); on balance average evaders across two periods is lower with VDP if ν < reservation penalty.
- Simulations solving (28) and (29) jointly (τ = 0.3; λ declines from 30 or 15 to 3):
  - Improvements in period-2 enforcement raise p2* markedly (examples: p1 6% → p2 32%; p1 4% → p2 37%).
  - ν** positive in all parameter configurations shown (examples: ν** = 1.2; highest ν** = 1.5).
  - Compared to no VDP, p1* is slightly higher with a VDP; p2* tends to be slightly lower with a VDP (VDP allows enforcement savings).
  - VDP increases period-1 evasion and reduces period-2 evasion; share of VDP entrants varies across scenarios.
  - In some scenarios period-2 government revenue exceeds revenue under full compliance due to penalty payments under the VDP.

### Appendix I — alternative assumption: detected evaders forced to comply in period 2
- A. Unanticipated VDP with forced compliance for detected evaders:
  - If fraction p1 ḡ are detected in period 1 and forced to comply in period 2, period-2 net revenue:
    - R2 = 1 − ḡ + (ḡ − g_l) ν + g_l φ p2 + p1{ ḡ − (ḡ − g_l) ν − g_l φ p2 } − C(p2, g_l)
  - First-order condition for optimal ν identical to Eq. (13) in main text; yields same ν* as Eq. (15). Conclusion: optimal VDP penalty unchanged by forcing detected evaders to comply, although revenue levels differ.
- B. Anticipated VDP with forced compliance for detected evaders:
  - Utilities and thresholds adjusted; ḡ, g_l, g_h expressions recalculated:
    - ḡ = τ[2 − p1 − φ (p1 + p2 − p1 p2)] / [ (1 − p1)(2 − p2) ]
    - g_l = τ(ν − p2 φ) / (1 − p2)
    - g_h = τ[2 − ν(1 − p1) − p1(φ + 1)] / (1 − p1)
  - Reservation penalty ensuring g_h > g_l:
    - ν_rerer = 1 + [ p2 − p1 ] / [ (1 − p1)(2 − p2) ] (φ − 1)
  - ν_rerer is unambiguously larger than reservation in main text if p2 > p1; example: p1 = 0.1, p2 = 0.3, φ = 2.5, α = 0.5 gives ν_rerer = 1.196 vs 1.188 in main text.
  - Revenue expressions R1 and R2 adapted to forced-compliance case (curly-brace terms account for detected evaders).
  - First-order condition for total revenue simplifies to:
    - 2(g_h − g_l) = α τ (p2 − p1) / [ (1 − p1)(1 − p2) ] — matches Eq. (27).
  - Optimal penalty:
    - ν** = ν_rerer − α (p2 − p1) / [ 2(1 − p1)(2 − p2) ] = 1 + [ p2 − p1 ] / [ (1 − p1)(2 − p2) ](φ − 1 − 1/2 α)
  - ν** is larger than optimal penalty in main text if p2 > p1; example values: ν** = 1.163 here vs 1.156 in main text for comparable parameters.
  - Intuition: when detected evaders are ineligible for VDP, a low VDP penalty is less effective to induce coming clean, so optimal ν rises.

### Conclusion — policy implications and trade-offs
- If individuals do not anticipate future policies, a one-off unanticipated VDP can be attractive to maximize revenue from offshore wealth; VDP must offer a low or even negative effective penalty to induce previous evaders to come clean.
- If individuals anticipate a VDP, anticipation effects can make the program ineffective or counterproductive: otherwise-compliant taxpayers may evade prior to the VDP to qualify (time inconsistency).
- A VDP can be optimal only if tax evasion imposes external costs beyond direct revenue loss (parameter α > 0), such as reduced tax morale or litigation costs; then the government may set a VDP that attracts previous evaders by setting ν below reservation levels.
- Key policy trade-off: carrot (low VDP penalties to attract disclosures) versus stick (higher enforcement effort). Improved detection efficiency (via AEOI and digitalization) increases the range of VDP designs that can be effective but also raises optimal penalties and enforcement choices.
- Design considerations: eligibility rules (e.g., excluding detected evaders), interaction with enforcement capacity (λ), and political/moral constraints on ν (e.g., ν ≥ 1) materially affect optimal policy and revenue outcomes.

*Source: IMF Working Paper — Appendix I and related appendices from wpiea2023006-print-pdf*

### Appendix I. Optimal VDP penalty if detected evaders are forced to comply in period 2 ........................... 30

### Appendix I. Optimal VDP penalty if detected evaders are forced to comply in period 2

### Context and motivation
- Many governments need to improve revenue performance during the post-COVID recovery, with particular interest in raising more taxes from the most affluent.
- Voluntary disclosure programs (VDPs) and tax amnesty programs (TAPs) have been used widely; examples cited include Kenya, Nigeria, Sri Lanka (2021) and Albania, Honduras, Indonesia, Trinidad and Tobago (2022).
- The main goal of VDPs and TAPs is to generate additional public revenue in the short and medium term:
  - Short run: immediate expansion of the tax base through increased voluntary compliance.
  - Medium run: potential sustainable improvements in tax compliance if acquired disclosure information is used for enforcement.

### Features of VDPs and TAPs (as described)
- VDP: allows previous tax evaders to disclose assets and income at reduced punitive action.
- Payments and punitive measures under VDPs:
  - Generally higher payment obligations than compliant taxpayers (through interest and penalties).
  - Payments and punitive measures are generally less than for evaders who are detected.
- Variations in design:
  - Vary in eligibility and payment liability.
  - Similarities with TAPs, which usually forgive full tax liability in exchange for some fixed payment; “extensive amnesties” can imply payment lower than the principal tax (Franzoni 1996).
- Typical punitive regime outside VDP:
  - Detected evaders obliged to pay full tax on current and past obligations, plus interest and penalties.
  - Penalty ranges reported: between 20 and 200 percent of the tax obligation.
  - Possibility of criminal prosecution.
- Under VDPs:
  - Interest often reduced.
  - In 19 out of 47 cases, penalties waived altogether.
  - In 26 cases, criminal prosecution of evaders is waived.
- Note: Some countries exclude use of acquired information in subsequent tax audits (closed/sealed fiscal years), reducing likelihood of sustained revenue increases.

### Evidence on prevalence and magnitude of offshore evasion
- Zucman (2013) estimate: $6 trillion of global wealth hidden in offshore locations.
- Alstadsæter et al. (2018) estimate: amount equivalent to 10 percent of world GDP.
- Offshore tax evasion concentrated among top of income distribution:
  - Around 10 percent of all households evade taxes generally.
  - This share rises to between 30 and 40 percent for the top percentile.
- Beer et al. (2019) find automatic exchange of information (AEOI) reduced foreign-owned deposits in offshore jurisdictions by 25 percent.

### Empirical assessments of VDPs/TAPs and risks
- Baer and Le Borgne (2008):
  - Amnesties tend to produce mostly modest short-term gross revenue gains.
  - Example: US states’ amnesties on average had raised 0.7 percent of the relevant tax.
  - Net revenue effects are smaller than gross numbers and can be negative in the medium term due to:
    - Increased evasion,
    - Reduced revenue from penalties,
    - Additional administrative costs.
  - Amnesties often coincide with improved enforcement efforts, complicating attribution of revenue gains.
  - Repeated amnesties can be counterproductive by inducing incentives for evasion by currently compliant taxpayers.
- Benedek et al. (2022) on VDP design, principles, and risks:
  - Listed advantages:
    - (i) early revenue from base broadening;
    - (ii) sustained revenue increases if disclosure information used for compliance;
    - (iii) reduced cost of litigation/prosecution;
    - (iv) possibly lower enforcement costs through improved voluntary compliance.
  - Listed risks:
    - (i) reduced taxpayer morale and perception of fairness and trust by rewarding evaders;
    - (ii) increased evasion if VDPs are repeated frequently, reflecting time inconsistency problem;
    - (iii) reduced revenue from penalties compared to detecting fraudulent evaders.
  - Effectiveness depends on legal and administrative design, including anti-money laundering and combating financing of terrorism provisions.

### Policy environment and enforcement capacity
- Global transparency initiatives:
  - Global Forum on Transparency and Exchange of Information for Tax Purposes with 163 members implementing information exchange standards.
  - Automatic exchange of information (AEOI) has become operational since 2018, affecting asset portfolios.
- Digitalization developments are revolutionizing data management in tax administrations and raise probability of detection of offshore evaders.
- Heterogeneity in capacity:
  - These effects are likely most relevant for advanced economies.
  - Developing countries generally have more limited capacity to receive and actively use information from abroad (see IMF 2022).

### Key policy questions highlighted
- What conditions determine whether a VDP/TAP is desirable and effective?
- How do AEOI and digitalization affect the assessment of VDP/TAP desirability and design?
- What should be the terms of a VDP if detection probabilities increase?
- How should the... [content ends at this point in the provided excerpt]

*IMF Working Paper — Appendix I. Optimal VDP penalty if detected evaders are forced to comply in period 2*

### introduction of a VDP/TAP affect the enforcement effort to fight tax evasion?

### introduction of a VDP/TAP affect the enforcement effort to fight tax evasion?

### Literature background and key insights
- Early studies (Malik and Schwab (1991); Andreoni (1990); Stella (1991); Langenmeyr (2017)) find that VDPs/TAPs do not directly increase compliance unless accompanied by a shock that changes agents' incentives.
- Mechanisms identified in prior work:
  - Shocks to future consumption (Andreoni (1990)) can make a permanent VDP effective because it reduces uncertainty about future penalties.
  - Ex-ante underestimation of disutility from evasion (Malik and Schwab (1991)) can create scope for a TAP to induce disclosures as agents revise beliefs.
  - Temporary TAPs may work if paired with an announced increase in future enforcement, but credibility problems mean administrations often cannot commit and may exaggerate future effort, undermining TAP effectiveness (Stella (1991)).
  - A fully anticipated VDP can create option value for evasion and encourage previously compliant taxpayers to evade; VDPs may be attractive only if detection of evaders is much more administratively costly than voluntary disclosures (Langenmeyr (2017)).

### Contributions of this paper
- The paper extends the literature in four ways:
  - The shock that induces VDP use is an exogenous increase in enforcement efficiency that raises the probability of detection—relevant given AEOI and new digital technologies.
  - It analyzes both unanticipated and anticipated VDPs in a single two-period framework, allowing comparison of optimal design under both informational regimes.
  - It explores the relationship between an optimally designed VDP and optimal enforcement effort, showing VDPs can reduce marginal returns to enforcement and lead the government to save on administrative effort.
  - It presents simulations calibrated to existing tax-system parameters and plausible enforcement probabilities to provide policy guidance.

### Basic model: setup and individual decision
- Population: continuum of wealthy individuals, each with offshore asset return y, taxed at rate τ.
- Individuals are risk-neutral and choose between compliance and evasion based on heterogeneous guilt gi (uniform on [0,1]) and detection probability p.
- Utility if compliant:
  - uc = (1−τ) y
- Expected utility if evading (caught with probability p, fine φ>1 times tax due; if uncaught, utility reduced by guilt gi):
  - ue(p) = [1 − p τ φ − (1 − p) gi] y
- Threshold guilt ḡ(p) where individuals are indifferent:
  - ḡ(p) = τ (1 − p φ) / (1 − p)
- Examples and implications:
  - If φ = 1, only those with gi > τ truthfully report.
  - Numerical example: φ = 3, τ = 0.3, p = 0.1 ⇒ ḡ = 0.23, implying 23 percent of individuals choose evasion (uniform guilt).

### Government objective and enforcement costs
- Government maximizes revenue net of enforcement costs by choosing p:
  - max_p R = [(1 − ḡ) + ḡ p φ − C(p,ḡ)] τ y
- Administrative cost function:
  - C(p,ḡ) = λ/2 p^2 + α ḡ
  - Parameters: α ≥ 0 captures negative externality from evasion; λ > 0 captures enforcement (in)efficiency.
- First-order condition for optimal p:
  - ḡ(p) φ − (∂ḡ/∂p) [1 − p φ] = ∂C/∂p
- Marginal deterrence effect:
  - ∂ḡ/∂p = − τ (φ − 1) / (1 − p)^2 < 0

### Improvement in detection efficiency: comparative statics
- Optimal detection probability rises with improved efficiency (lower λ). Implicit differentiation yields:
  - − ∂p* / ∂λ = p* / [λ + 2 τ (φ − 1) / (1 − p)^2 (φ + α / (1 − p))] > 0
- Elasticity of optimal p with respect to λ, ε, satisfies 0 < ε < 1 (p* rises less than proportionally as λ falls).
- Total differential of administrative cost with respect to λ:
  - dC(p*,ḡ)/dλ = [1/2 − ε] p^2 + α τ (φ − 1) / (1 − p)^2 p ε / λ
  - Interpretation:
    - Efficiency improvement (dλ < 0) has offsetting effects: it lowers direct cost of existing enforcement (−1/2 p^2) but induces higher optimal p (cost-increasing, measured by ε p^2).
    - Administrative budget increases if ε > 1/2; otherwise it shrinks.
    - Second term modifies the effect via reduced share of evaders (α) when p rises.

### Simulations: steady-state outcomes for parameter grid
- Simulation assumptions: τ = 0.3; α ∈ {0.2, 0.4}; φ ∈ {2.5, 4}; λ ∈ {15, 30}.
- Table of outcomes (solution for p* using equation (6) with τ = 0.3):
  - λ = 15, φ = 2.5, α = 0.2 → p* = 0.08, ε = 0.91, ḡ = 0.26, R / (τ y) = 0.69, C / (τ y) / R = 0.14
  - λ = 30, φ = 2.5, α = 0.2 → p* = 0.04, ε = 0.96, ḡ = 0.28, R / (τ y) = 0.67, C / (τ y) / R = 0.12
  - λ = 15, φ = 4, α = 0.2 → p* = 0.11, ε = 0.68, ḡ = 0.19, R / (τ y) = 0.77, C / (τ y) / R = 0.17
  - λ = 30, φ = 4, α = 0.2 → p* = 0.06, ε = 0.83, ḡ = 0.24, R / (τ y) = 0.71, C / (τ y) / R = 0.15
  - λ = 15, φ = 2.5, α = 0.4 → p* = 0.09, ε = 0.92, ḡ = 0.26, R / (τ y) = 0.64, C / (τ y) / R = 0.25
  - λ = 30, φ = 2.5, α = 0.4 → p* = 0.04, ε = 0.96, ḡ = 0.28, R / (τ y) = 0.61, C / (τ y) / R = 0.23
  - λ = 15, φ = 4, α = 0.4 → p* = 0.12, ε = 0.69, ḡ = 0.18, R / (τ y) = 0.73, C / (τ y) / R = 0.24
  - λ = 30, φ = 4, α = 0.4 → p* = 0.07, ε = 0.84, ḡ = 0.23, R / (τ y) = 0.67, C / (τ y) / R = 0.25
- Aggregate implications from simulations:
  - Optimal p* increases when λ falls (detection efficiency improves) but less than proportionally (ε < 1).
  - Share of evaders ḡ ranges between 17.8 percent and 28.1 percent across parameter combinations.
  - Revenue net of enforcement cost R / (τ y) ranges from 61.1 percent to 76.6 percent of potential gross revenue across scenarios.
  - Enforcement costs range between 12.3 percent and 24.9 percent of tax revenue in the simulated cases.

### Unanticipated Voluntary Disclosure Program (VDP) — setup
- Two-period extension: taxpayers decide in period 1 (steady-state behavior), then in period 2 a shock increases detection probability and the government may introduce a VDP.
- Unanticipated VDP: taxpayers in period 1 do not foresee the availability of a VDP in period 2, so first-period behavior equals steady-state from the basic model.
- In period 2, the government introduces an unanticipated VDP allowing first-period evaders to come clean at reduced penalty ν < φ.
- Analytical convenience assumption: audit outcomes in period 1 do not affect VDP eligibility; first-period evaders may participate in the VDP irrespective of whether they were caught in period 1.

*International Monetary Fund. wpiea2023006-print-pdf — introduction of a VDP/TAP affect the enforcement effort to fight tax evasion?*

### Appendix I relaxes this assumption by restricting

### wpiea2023006-print-pdf - Appendix I relaxes this assumption by restricting

### A. Tax Evasion and VDP Participation
- Figure 2 setup: utility in period 2 as a function of guilt parameter 푔푔푖푖, with constant detection probability over time (푝푝2 = 푝푝1 = 푝푝). Horizontal intercept (1−휏휏) = utility from compliance; downward sloping line = utility from evasion. Individuals with guilt below 푔푔̅ evade in period 2; those with 푔푔푖푖 between 푔푔̅ and 1 comply.
- Introduction of a VDP in period 2: utility from entering VDP = horizontal line with intercept (1−휏휏휈휈), reflecting VDP penalty 휈휈.
- Key observation on incentive to enter VDP:
  - For previous evaders (guilt < 푔푔̅) to enter VDP, utility in VDP must exceed compliance: (1−휈휈휏휏) > (1−휏휏), which requires 휈휈 < 1.
  - Thus, VDP induces coming clean only if it provides a discount relative to standard tax regime (extensive amnesty).
- Definition: VDP is incentive compatible if some individuals find it attractive to enter the program.
- Lower-bound guilt threshold 푔푔𝑙𝑙 defined by equality of VDP utility and expected utility from continued evasion:
  - 푢푢𝑣𝑣𝑣𝑣𝑝푝 = (1−휈휈휏휏)푦푦
  - 푢푢𝑖𝑖𝑒𝑒(푝푝2) = [1−푝푝2휏휏휑휑−(1−푝푝2)푔푔𝑖𝑖]푦푦
  - Solving gives:
    - 푔푔𝑙𝑙 = 휏휏(휈휈−푝푝2휑휑) / (1−푝푝2)    (9)
- Incentive compatibility condition:
  - VDP unanticipated is incentive compatible only if 푔푔𝑙𝑙 < 푔푔̅(푝푝1) (where 푔푔̅(푝푝1) given by equation (3)).
  - Mass of first-period evaders entering VDP = 푔푔̅(푝푝1) − 푔푔𝑙𝑙(푝푝2) > 0.
- Reservation penalty for unanticipated VDP (expressed via equations (3) and (9)):
  - 휐휐 < 1 + (푝푝2 − 푝푝1)(휑휑 − 1) / (1 − 푝푝1) ≡ 휐휐𝑟𝑟𝑒𝑒𝑟𝑟_𝑢𝑢    (10)
  - If 푝푝2 = 푝푝1, VDP attracts participants only if penalty < 1.

### B. Optimal Policy Under Constant Detection Efficiency
- Second-period revenue net of administrative costs:
  - 푅푅2 = [1 − 푔푔̅(푝푝1) + [푔푔̅(푝푝1) − 푔푔𝑙𝑙]휈휈 + 휑휑푝푝2푔푔𝑙𝑙 − 퐶퐶(푝푝2, 푔푔𝑙𝑙)] 휏휏푦푦    (11)
- First-order condition w.r.t. 푝푝2 (optimal detection probability):
  - 푔푔𝑙𝑙 휑휑 − (휕휕푔푔𝑙𝑙/휕휕푝푝2) [휈휈 − 푝푝2휑휑] = 푑푑퐶퐶(푝푝2, 푔푔𝑙𝑙)/휕휕푝푝2    (12)
  - Differences from basic model: net revenue effect from reducing evasion is 휈휈 − 푝푝2휑휑 (not 1 − 푝푝1휑휑); share of evaders is 푔푔𝑙𝑙 (not 푔푔̅).
  - If 휈휈 = 1, optimal detection probability unchanged between periods.
- First-order condition for optimal VDP penalty:
  - [푔푔̅(푝푝1) − 푔푔𝑙𝑙] − (휕휕푔푔𝑙𝑙/휕휕휈휈)(휈휈 − 휑휑푝푝2) = 훼훼 (휕휕푔푔𝑙𝑙/휕휕휈휈)    (13)
  - Using 휕휕푔푔𝑙𝑙/휕휕휈휈 = 휏휏/(1 − 푝푝2) and definition of 푔푔𝑙𝑙 yields:
    - 푔푔𝑙𝑙 = 1/2 [푔푔̅(푝푝1) − 훼훼휏휏/(1 − 푝푝2)]    (14)
  - Implications:
    - If 훼훼 = 0: optimal VDP sets period-2 share of evaders equal to half the period-1 share.
    - If 훼훼 > 0: optimal share of evaders is further reduced.
- Explicit optimal VDP fine:
  - 휐휐∗ = 1/2 [1 + (푝푝2 − 푝푝1)(휑휑 − 1)/(1 − 푝푝1) + 푝푝2휑휑 − 훼훼] = 1/2 [휐휐𝑟𝑟𝑒𝑒𝑟𝑟_𝑢𝑢 + 푝푝2휑휑 − 훼훼]    (15)
  - Interpretation: 휐휐∗ is half the reservation penalty plus adjustments: upward from expected penalty 푝푝2휑휑, downward from external cost 훼훼.
- Relationship between optimal detection probability and VDP penalty:
  - Define Φ (equation (16)):
    - Φ ≡ 푔푔𝑙𝑙[ (휑휑 − 휐휐)/(1 − 푝푝2) + 휑휑 ] + 훼훼휏휏 (휑휑 − 휐휐)/(1 − 푝푝2)^2 − 휆휆2 푝푝2 = 0    (16)
  - Implicit derivative yields:
    - 휕휕푝푝2/휕휕휈휈 = − Φ_휕휕푝푝푒푒푟푟푖푖_푝푝푣푣2 / Φ_푝푝2_푝푝푒푒푟푟푖푖_푝푝푣푣2 > 0    (17)
  - Policy trade-off (carrot vs stick): higher VDP penalty leads government to choose higher detection probability; lower VDP penalty allows scaling back enforcement.
- If detection efficiency unchanged (휆휆2 = 휆휆1) and unconstrained, optimal VDP penalty can be negative (previous evaders face lower effective tax than compliant taxpayers). If politically/morally constrained to 휐휐 ≥ 1, optimal is 휐휐 = 1 and detection probability unchanged.

### C. Optimal Policy Under Increasing Detection Efficiency
- Marginal responses to detection efficiency 휆휆:
  - 휕휕휐휐∗/휕휕휆휆2 = 1/2 [ (휑휑 − 1)/(1 − 푝푝1) + 휑휑 ] (휕휕푝푝2/휕휕휆휆2)    (18.a)
  - Φ_푝푝2_푝푝푒푒푟푟𝑖𝑖 ∙ (휕휕푝푝2/휕휕휆휆2) + Φ_휕휕푝푝𝑒𝑒𝑟푟𝑖𝑖_푝푝푣푣2 ∙ (휕휕휐휐∗/휕휕휆휆2) − 푝푝2 = 0    (18.b)
  - Conclusion: change in optimal VDP penalty has same sign as change in enforcement effort; higher detection efficiency → higher optimal penalty.
- If VDP penalty fixed, response in optimal detection probability to higher detection efficiency:
  - −푝푝2 / Φ_푝푝2_푝푝푒푒푟푟𝑖𝑖_푝푝푣푣2 > 0, resembling steady-state response.
  - If VDP penalty adjustable, sensitivity of enforcement to efficiency shocks is larger (see equation (19)).
- Condition for optimal unanticipated VDP penalty to be ≥ 1:
  - 푝푝2 − 푝푝1 ≥ (1 − 푝푝1)(휑휑 + 훼훼)/(휑휑 − 1) / [ (1 − 푝푝1)/(휑휑 + 휑휑) ? ] — from text:
    - Explicit condition stated:
      - 푝푝2 − 푝푝1 ≥ (1 − 푝푝1)(휑휑 + 훼훼) / (휑휑 − 1)(1 − 푝푝1 + 휑휑)  (presented in text as)
      - Text gives operational examples: required increase 9 percentage points if 휑휑 = 4, 푝푝1 = 0.2, 훼훼 = 0.5 (20% → 29%); required increase 42 percentage points if 휑휑 = 2.5, 푝푝1 = 0.1, 훼훼 = 1 (10% → 52%).  (20)
  - If condition (20) not met and constraint 휈휈∗ ≥ 1 binds, government may still introduce VDP with 휐휐 = 1; Appendix III shows this unambiguously raises revenue.

### D. Simulations
- Table 2 summary: numerical simulations for optimal policy in period 2 using same parameter configurations as Table 1; assume λ2 = 3.
  - Implies reduction by 90 percent if λ1 = 30, reduction by 80 percent if λ1 = 15.
- Simulation qualitative findings:
  - Improved detection efficiency in period 2:
    - Unambiguously increases detection probability.
    - Reduces number of tax evaders.
    - Raises revenue.
    - Reduces administrative costs in most cases.
  - Optimal effective penalty under VDP in period 2 ranges from 0.65 to 1.39 — previous evaders may pay either higher or lower effective tax than all-time compliers.
  - Magnitude of increase in detection probability depends on whether detection efficiency increases moderately (λ1 = 15) or extensively (λ1 = 30):
    - If detection more costly initially (λ1 = 30 → initially lower 푝푝1∗ and higher share of evaders), government boosts enforcement substantially in period 2 and can set relatively higher VDP penalty.
    - If detection efficiency increases moderately (λ1 = 15), detection probability rises less; government relies on lower VDP penalty to induce compliance.
- Selected entries from Table 2 (note: solution for 푝푝2∗ using equation (16) and for 휐휐∗ expression (15), with 휏휏 = 0.3):
  - Parameter set 1: λ1 = 15, λ2 = 3, 휑휑 = 2.5, 훼훼 = 0.2, 푝푝1∗ = 0.08 → 푝푝2∗ = 0.20, 휐휐∗ = 0.75, 푔푔̅(푝푝1) = 0.26, 푔푔𝑙𝑙(푝푝2) = 0.09, 푅푅1/휏휏푦푦 = 0.69, 푅푅2/휏휏푦푦 = 0.83, 퐶퐶1/휏휏푦푦 = 0.14, 퐶퐶2/휏휏푦푦 = 0.09
  - Parameter set 2: λ1 = 30, λ2 = 3, 휑휑 = 2.5, 훼훼 = 0.2, 푝푝1∗ = 0.04 → 푝푝2∗ = 0.21, 휐휐∗ = 0.80, 푔푔̅(푝푝1) = 0.28, 푔푔𝑙𝑙(푝푝2) = 0.10, 푅푅1/휏휏푦푦 = 0.67, 푅푅2/휏휏푦푦 = 0.83, 퐶퐶1/휏휏푦푦 = 0.12, 퐶퐶2/휏휏푦푦 = 0.11
  - Parameter set 3: λ1 = 15, λ2 = 3, 휑휑 = 4, 훼훼 = 0.2, 푝푝1∗ = 0.11 → 푝푝2∗ = 0.25, 휐휐∗ = 1.12, 푔푔̅(푝푝1) = 0.19, 푔푔𝑙𝑙(푝푝2) = 0.06, 푅푅1/휏휏푦푦 = 0.77, 푅푅2/휏휏푦푦 = 0.91, 퐶퐶1/휏휏푦푦 = 0.17, 퐶퐶2/휏휏푦푦 = 0.11
  - Parameter set 4: λ1 = 30, λ2 = 3, 휑휑 = 4, 훼훼 = 0.2, 푝푝1∗ = 0.06 → 푝푝2∗ = 0.30, 휐휐∗ = 1.39, 푔푔̅(푝푝1) = 0.24, 푔푔𝑙𝑙(푝푝2) = 0.08, 푅푅1/휏휏푦푦 = 0.71, 푅푅2/휏휏푦푦 = 0.93, 퐶퐶1/휏휏푦푦 = 0.15, 퐶퐶2/휏휏푦푦 = 0.17
  - Additional parameterizations in Table 2 produce 휐휐∗ values of 0.65, 0.71, 0.97, 1.28 with corresponding outcomes reported in the table.

*Appendix I relaxes this assumption by restricting VDPs only to evaders who were not caught in the first period.*

### Appendix V extends our two-period model to a T-period setup, whereby the VDP in period 2 is followed by T-2 periods in w

### Appendix V extends our two-period model to a T-period setup, whereby the VDP in period 2 is followed by T-2 periods in which those who opt in for the VDP are treated as fully compliant taxpayers in subsequent years

### Model setup and evader utility
- Utility from evading in both periods is given as:
  - 푈푈푒푒 = (1−푝푝1 휑휑휏휏 − (1−푝푝1) 푔푔푖푖) + (1−푝푝2 휑휑휏휏 − (1−푝푝2) 푔푔푖푖)
  - Rewritten as: 2 − 휑휑휏휏 (푝푝1 + 푝푝2) − 푔푔푖푖 (2 − 푝푝1 − 푝푝2) = 2푢푢푖푖푒푒(푝푝̅)
- Assumptions and variants:
  - Evaders caught in period 1 may be allowed to evade again in period 2; if that option is ruled out expressions are modified (see Appendix I).
  - Anticipation effects: behavior in both periods depends on the average probability when agents cannot switch or come clean in period 2.

### Guilt thresholds and incentive compatibility
- Define 푔푔ℓ as the guilt level where utility from entering the VDP equals utility from evasion in both periods:
  - 푔푔ℓ = 휏휏 (휈휈 − 푝푝2 휑휑) / (1 − 푝푝2)   (equation (22))
  - Individuals with guilt < 푔푔ℓ will evade in both periods (subject to parameter configurations and special cases noted in text).
- Define 푔푔ℎ as the guilt level where utility from entering the VDP equals utility from full compliance in both periods:
  - 푔푔ℎ = 휏휏 (휌휌 − 푝푝1 휑휑) / (1 − 푝푝1)   (equation (23)), where 휌휌 = 2 − 휈휈
  - Taxpayers with guilt > 푔푔ℎ prefer full compliance in both periods.
- Classification of VDP opt-ins:
  - Group 1: guilt in [푔푔ℓ, 푔푔̅(푝푝̅)] — would have evaded absent VDP, but VDP induces them to come clean in period 2 (target group).
  - Group 2: guilt in [푔푔̅(푝푝̅), 푔푔ℎ] — would have complied in period 1 absent VDP but are induced to evade in period 1 in anticipation of VDP (unintended effect).
  - Dividing line 푔푔̅(푝푝̅) is where agents are indifferent between evasion and compliance absent a VDP.
- Incentive compatibility definition:
  - An anticipated VDP is incentive compatible iff 푔푔ℓ < 푔푔ℎ (thresholds from (22) and (23)).
  - This inequality holds when:
    - 휐휐 < 1 + 1/2 (푝푝2 − 푝푝1) / (1 − 푝푝̅) (휑휑 − 1) = 휐휐푟푟푒푒푟푟_푎푎   (equation (24))
  - Implications:
    - If detection probability is constant over time (푝푝2 = 푝푝1), then 휐휐푟푟푒푒푟푟_푎푎 = 1, as with the unanticipated VDP: no VDP with a positive penalty can induce coming clean.
    - If detection probability rises between periods, a VDP with a positive penalty can be incentive compatible (휐휐푟푟푒푒푟푟_푎푎 > 1).
    - Example from text: if 휑휑 = 3, 푝푝2 = 0.2 and 푝푝1 = 0.1, we obtain 휈휈푟푟푒푒푟푟_푎푎 = 1.   11, i.e., with a doubling of the probability of detection, the maximum penalty under which a VDP can be incentive compatible is only 11 percent.

### Optimal VDP — government revenue maximization
- Government net revenue across two periods (net of collection costs) is:
  - 푅푅 = [ 2 (1 − 푔푔ℎ) + (푔푔ℎ − 푔푔ℓ) 휈휈 + (푝푝1 푔푔ℎ + 푝푝2 푔푔ℓ) 휑휑 − 퐶퐶(푝푝1, 푔푔ℎ) − 퐶퐶(푝푝2, 푔푔ℓ) ] 휏휏푦푦   (equation (25))
  - Interpretation of terms provided in text: voluntary compliers, VDP entrants paying effective tax 휈휈휏휏푦푦, penalties from detected evaders, and collection costs.
- First-order condition for optimal VDP penalty (휈휈) (equation (26) in text) captures trade-offs:
  - Marginal increase in 휈휈 directly raises revenue from VDP participants (mass 푔푔ℎ − 푔푔ℓ).
  - But higher 휈휈 reduces incentives to enter the VDP, affecting both groups of potential entrants differently; derivatives and marginal revenue changes are provided in text.
- Simplified first-order condition yields:
  - 2(푔푔ℎ − 푔푔ℓ) = 훼훼휏휏 [ 푝푝2 − 푝푝1 ] / ( (1 − 푝푝1) (1 − 푝푝2) )   (equation (27))
  - Implications:
    - If external cost parameter 훼훼 = 0, optimal choice is 푔푔ℎ = 푔푔ℓ, so the optimal VDP is not incentive compatible (no one joins the VDP), i.e., no reason to introduce a VDP.
    - If 훼훼 > 0 and detection probability rises (푝푝2 > 푝푝1), an incentive-compatible optimal VDP can exist.
- Optimal fine solution:
  - 휈휈∗∗ = 1 + 1/2 (푝푝2 − 푝푝1) / (1 − 푝푝̅) [ 휑휑 − 1 − 훼훼/2 ] = 휈휈푟푟푒푒푟푟_푎푎 − 1/4 (푝푝2 − 푝푝1) / (1 − 푝푝̅) 훼훼   (equation (28))
  - Interpretation:
    - 훼훼 = 0 implies 휈휈∗∗ = 휈휈푟푟푒푒푟푟_푎푎 (reservation penalty), so no participation.
    - If both 훼훼 > 0 and 푝푝2 > 푝푝1, then 휈휈∗∗ < 휈휈푟푟푒푒푟푟_푎푎: government sets fine below reservation level to induce evaders to come clean and reduce external cost.
    - Sign of 휑휑 − 1 − 훼훼/2 determines whether optimal penalty is > 1:
      - Example: if 휑휑 = 3, 훼훼 = 2, 푝푝2 = 0.2, 푝푝1 = 0.1, then 휈휈∗∗ = 1.06.
      - For 훼훼 > 4, 휈휈∗∗ < 1.

### Optimal administration — detection probabilities
- First-order conditions for detection probabilities 푝푝1 and 푝푝2 (with multipliers 휆휆1 and 휆휆2) are:
  - Φ1(푝푝1, 휈휈, 휆휆1) ≡ 푔푔ℎ [ (휑휑 − 휌휌) / (1 − 푝푝1) + 휑휑 ] + 훼훼휏휏 (휑휑 − 휌휌) / (1 − 푝푝1)^2 − 휆휆1 푝푝1 = 0   (equation (29.a))
  - Φ2(푝푝2, 휈휈, 휆휆2) ≡ 푔푔ℓ [ (휑휑 − 휈휈) / (1 − 푝푝2) + 휑휑 ] + 훼훼휏휏 (휑휑 − 휈휈) / (1 − 푝푝2)^2 − 휆휆2 푝푝2 = 0   (equation (29.b))
- Key analytical solutions and approximations:
  - If detection efficiency does not change between periods, an optimal solution has 휈휈∗∗ = 1 and 푝푝1∗ = 푝푝2∗: revenue-maximizing strategy is to set the penalty at the reservation rate and keep detection probability unchanged (implying no effective anticipated VDP).
  - Linear approximation around 푝푝̅ = 푝푝1 = 푝푝2 and 휈휈 = 1 yields:
    - 푝푝2∗ − 푝푝1∗ ≈ 휀휀 / 휆휆 [ 푝푝̅ (휆휆1 − 휆휆2) − 2 Φν1 (휈휈 − 1) ]   (equation (30))
    - Where 휀휀 ≡ − (휕휕푝푝̅ / 휕휕휈휈) (휕휕푝푝) > 0 and Φν1 = − 2 휏휏 (1 − 푝푝̅)^2 [ 휑휑 − 1 − 훼훼/2 ] < 0.
  - Insights from (30):
    - Increased detection efficiency (휆휆1 − 휆휆2 > 0) raises the optimal detection probability in period 2; magnitude scaled by 휀휀 and 푝푝̅.
    - The VDP penalty (휈휈) has an unambiguous positive impact on the optimal detection probability.
    - Optimal solution typically satisfies 푝푝2∗ > 푝푝1∗ and 휈휈∗∗ > 1 when enforcement efficiency rises.

### Average evaders and net effect of a VDP
- Without a VDP, the share of evaders across the two periods equals 2 푔푔̅(푝푝̅).
- With a VDP:
  - Period 1 evaders: 푔푔ℎ(푝푝1) > without VDP.
  - Period 2 evaders: 푔푔ℓ(푝푝2) < without VDP.
  - On balance, the average number of evaders across the two periods is lower with the VDP, as long as the VDP penalty is strictly below the reservation penalty.

### Simulations — Table 3 highlights
- Simulations solve (28) and (29) jointly (휏휏 = 0.3) and vary parameters consistent with earlier tables; detection efficiency parameter declines from 30 or 15 to 3.
- Main simulation findings:
  - Improvements in period-2 enforcement efficiency increase optimal detection probability in period 2 (examples: first-row first-period detection 6 percent → 32 percent in period 2; bottom-row 4 → 37 percent).
  - With a VDP, the optimal fine 휈휈∗∗ is positive in all parameter configurations shown (examples: first row 휈휈∗∗ = 1.2; bottom row highest optimal penalty = 1.5, i.e., 50 percent).
  - Compared to no VDP, the optimal first-period detection probability is slightly higher with a VDP; second-period detection probability tends to be slightly lower with a VDP (VDP allows savings on enforcement costs).
  - VDP increases period-1 evasion and reduces period-2 evasion; share of VDP entrants varies across scenarios (first row only 2 percent; last row evaders decline from 10 to 4 percent).
  - In some scenarios government revenue in period 2 exceeds revenue under full compliance due to penalty payments under the VDP.

### Conclusion (summary of implications)
- If individuals do not anticipate future policies, a one-off VDP can be attractive to maximize revenue from offshore wealth; VDP must offer a low or even negative penalty to induce previous evaders to come clean.
- If individuals anticipate a VDP, anticipation effects make a VDP neither optimal nor effective in reducing evasion: otherwise compliant taxpayers may evade prior to the VDP to qualify for it (time inconsistency).
- A VDP can be effective only if tax evasion imposes external costs beyond direct revenue loss (e.g., reduced tax morale, litigation costs); then the government may set a VDP that attracts previous evaders.
- The conditions for a socially beneficial VDP are stringent and depend on:
  - Rise in detection probabilities,
  - Anticipation effects,
  - External costs from evasion.
- Policy trade-off highlighted: carrot of low penalties versus stick of higher enforcement efforts; special design features and empirical analysis are needed to guide policy in practice.

*Source: IMF Working Paper — Appendix V (from the supplied content).*

### Appendix I. Optimal VDP penalty if detected

### Appendix I. Optimal VDP penalty if detected

### A. Unanticipated VDP
- Setup: Consider share g̅ of individuals who evade in period 1. Evaders not detected face utilities:
  - U_vdp = (1−τν)
  - U_e = p2(1−τφ) + (1−p2)(1−g_i)
- Indifference implies taxpayers with guilt levels above
  - g_l = τ(ν − p2 φ) / (1 − p2)
  will participate in the VDP. This reservation penalty is unchanged from the main text.
- If a fraction p1 g̅ are detected in period 1 and forced to comply in period 2, period-2 net revenue is:
  - R2 = 1 − g̅ + (g̅ − g_l) ν + g_l φ p2 + p1{ g̅ − (g̅ − g_l) ν − g_l φ p2 } − C(p2, g_l)
  - (the curly-brace term is revenue from evaders detected in period 1 who will comply in period 2)
- First-order effect on enforcement costs in period 2:
  - ∂C/∂ν = α τ / (1 − p2)
- First-order condition for optimal VDP penalty:
  - ∂R2/∂ν = (g̅ − g_l)(1 − p1) − (∂g_l/∂ν)(ν − p2 φ)(1 − p1) = α (∂g_l/∂ν)(1 − p1)
- Implication: This condition is identical to Eq. (13) in the main text and yields the same optimal penalty as Eq. (15). Thus, although revenue expands under the alternative assumption (detected evaders forced to comply), optimal policy rules for the VDP penalty are unchanged.

### B. Anticipated VDP
- Utilities when detected evaders are forced to comply in period 2:
  - U_c = 2(1 − τ)
  - U_e =
    - (1 − p1){ (1 − p2)(1 − g_i)/2 + p2[(1 − g_i) + (1 − φ τ)] } + p1[(1 − τ φ) + (1 − τ)]
    - Simplified: U_e = 2 − p1 τ − φ τ (p1 + p2 − p1 p2) − g_i (2(1 − p1) − p2 + p1 p2)
  - U_vdp = (1 − p1)[ (1 − g_i) + (1 − τ ν) ] + p1[(1 − τ φ) + (1 − τ)]
- Thresholds:
  - Indifference between U_c and U_e:
    - g̅ = τ[ 2 − p1 − φ (p1 + p2 − p1 p2) ] / [ (1 − p1)(2 − p2) ]
  - Indifference between U_e and U_vdp:
    - g_l = τ(ν − p2 φ) / (1 − p2)
  - Indifference between U_c and U_vdp:
    - g_h = τ[ 2 − ν(1 − p1) − p1(φ + 1) ] / (1 − p1)
- Reservation penalty (minimum ν ensuring g_h > g_l):
  - ν_rerer = 1 + [ p2 − p1 ] / [ (1 − p1)(2 − p2) ] (φ − 1)
- Comparative statement: ν_rerer is unambiguously larger than Eq. (24) in the main text if p2 > p1. Example parameters: p1 = 0.1, p2 = 0.3, φ = 2.5, α = 0.5 give:
  - ν_rerer = 1.196 compared to 1.188 in Eq. (24). Hence, VDP is more likely to be incentive compatible under forced compliance after detection.
- Revenue expressions:
  - R1 = (1 − g_h) + p1 g_h φ − C(p1, g_h)
  - R2 = (1 − g_h) + (g_h − g_l) ν + g_l φ p2 + p1{ g_h − (g_h − g_l) ν − g_l φ p2 } − C(p2, g_l)
  - (curly-brace term summarizes adjustments since some evaders were caught and pay full taxes)
- First-order condition for total revenue (R1 + R2) w.r.t ν:
  - (1 − p1)(g_h − g_l) − (∂g_h/∂ν)(2 − p1 φ − ν(1 − p1) − p1) − (∂g_l/∂ν)(ν − φ p2 − ν p1 + φ p1 p2) = α[ (∂g_h/∂ν) + (∂g_l/∂ν)(1 − p1) ]
- Using v_gh_vφ = −τ and v_gl_vφ = τ/(1 − p2), this simplifies to:
  - 2(g_h − g_l) = α τ (p2 − p1) / [ (1 − p1)(1 − p2) ]
  - This matches Eq. (27) in the main text.
- Optimal penalty:
  - ν** = ν_rerer − α (p2 − p1) / [ 2(1 − p1)(2 − p2) ]
  - Equivalently displayed in text as:
    - ν** = 1 + [ p2 − p1 ] / [ (1 − p1)(2 − p2) ](φ − 1 − 1/2 α)
- Comparatives and example:
  - ν** is larger than the optimal penalty in Eq. (28) of the main text if p2 > p1.
  - Numerical example earlier: optimal penalty is 1.163 here versus 1.156 in Eq. (28).
  - Intuition: If some detected evaders are ineligible for VDP, a low VDP penalty is less effective to induce evaders to come clean; optimal penalty rises.

*Source: IMF Working Paper — Appendix I, "Optimal VDP penalty if detected" from wpiea2023006-print-pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023006-print-pdf.pdf_