Voluntary Disclosure Schemes for Offshore Tax Evasion

Tax authorities worldwide are implementing voluntary disclosure schemes to recover tax on o⁄shore investments. The US and UK, in particular, have implemented such schemes in response to bulk acquisitions of information on o⁄shore holdings, recent examples of which are the (cid:147)Paradise(cid:148) and (cid:147)Panama(cid:148) papers. Schemes o⁄er a⁄ected investors the opportunity to make a voluntary disclosure, with reduced (cid:133)ne rates for truthful disclosure. Might such incentives, once anticipated by investors, simply encourage evasion in the (cid:133)rst place? We characterize the investor/tax authority game with and without a scheme, allowing for the possibility that some o⁄shore investment has legitimate economic motives. We show that a scheme increases net expected tax revenue, decreases illegal o⁄shore investment, increases onshore investment, but could either increase or decrease legal o⁄shore investment. The optimal disclosure scheme o⁄ers maximal incentives for truthful disclosure by imposing the minimum allowable rate of (cid:133)ne.


Introduction
An estimated ten percent of world GDP is held in tax havens, much, though by no means all, of which goes unreported (Alstadsaeter et al. 2018;Zucman, 2013). The loss of tax receipts due to o¤shore tax evasion by individuals in the United States (US) alone has been estimated at $30-40 billion per annum (Gravelle, 2009). In recent years, data breaches have allowed tax authorities around the world to acquire information on thousands of o¤shore investments. To recover any tax owing on these investments, tax authorities have, in many instances, o¤ered a¤ected investors a one-o¤ and time-limited opportunity to make a voluntary disclosure through a bespoke facility giving overt incentives for honesty (usually in the form of a lower …ne rate). We term facilities of this form Incentivized O¤shore Voluntary Disclosure Schemes, or just "schemes". The net revenues arising from such schemes have been signi…cant: in 2009 a US scheme raised $3.4 billion (GAO, 2013) and a UK scheme netted nearly £500 million (Treasury Committee, 2012: 14). The UK scheme is estimated to have cost £6 million to administer (Committee of Public Accounts, 2008: 9), implying a return of 67:1. This compares favorably with reported yield/cost ratios in the UK of around 8:1 for traditional audit-based enforcement programs (HMRC, 2006). 1 The advent of o¤shore data leakages, and the associated implementation of voluntary disclosure schemes, may have come as a surprise to holders of legacy investments, but such developments are by now well understood by today's prospective o¤shore investors. Given that such schemes are by now largely anticipated, this raises the question of whether the continued use of such schemes is gainful to tax authorities. In particular, in o¤ering incentives for voluntary disclosure, might such schemes simply encourage illegal o¤shore investment in the …rst place -a concern pointed to by some recent empirical evidence. We shed light on this concern.
In this paper we appraise the use of anticipated o¤shore disclosure schemes using game theoretic tools. The model has two key features. First, we consider disclosure schemes that are implemented retrospectively in response to an information leak, as we argue characterizes practice in the UK and US. By the time of the information leak, however, the act of illegal o¤shore evasion has already taken place. As it cannot in ‡uence the illegal act retrospectively, the best a tax authority can do is seek to recover any tax owed. The importance of this observation lies in the fact that, in implementing incentivized schemes to recover e¢ciently 1 The ratio of 8:1 is the estimated yield/cost ratio for self-assessment non-business enquiry work in 2005-06. tax owed from past evasion, the tax authority may inadvertently change the incentives for future acts of o¤shore evasion. Second, we recognize that there can be legitimate economic reasons for holding money in o¤shore accounts. Accordingly, not all investors who appear in data on o¤shore holdings owe tax. Pritchard and Khan (2005), the only published work we are aware of by tax authority insiders with unfettered access to the UK o¤shore data, reports that even among those entities ‡agged as the highest risk category in o¤shore data only 70 percent were expected to owe tax.
Why invest o¤shore if not to evade tax? As well as potential pecuniary bene…ts in the form of higher pre-tax rates of interest than available onshore, o¤shore investments can also o¤er legitimate tax advantages. Pension funds routinely invest via funds domiciled o¤shore, for they enable investors from di¤erent countries to invest in the same fund, and can also legally prevent instances of double taxation. Most major onshore hedge funds have an accompanying o¤shore vehicle. For US based tax-exempt organizations, such vehicles provide some legitimate relief from taxation of unrelated business income tax. As well as legitimate tax advantages, o¤shore investments potentially o¤er a range of non-pecuniary bene…ts: o¤shore providers are known to o¤er greater convenience and sophistication, presumably as they face lighter regulatory controls as compared with their onshore counterparts (Helm, 1997: 414). 2 Recent leakages reveal that, in early 2000s, the Queen of the United Kingdom held around £10 million of her private money o¤shore: such investments had no tax motivation as the Queen is exempt from UK income and capital gains taxes. DEG, a development …nance institution wholly owned by the German state, is known to have used o¤shore accounts for a number of years, citing non-pecuniary factors it utilized for legitimate operational purposes. 3 Professional poker players, and other individuals who must transact regularly in many di¤erent currencies, are also known to make legitimate use of o¤shore bank accounts (see O'Reilly, 2007).
In order to appraise the impact of anticipated disclosure schemes we …rst model the strategic interaction between investors and the tax authority in the absence of a scheme. 4 We then 2 Relative to their onshore counterparts in the US, Helm argues that o¤shore funds have greater ‡exibility and less procedural delays in changing the nature, structure, or operation of their products, and they face fewer investment restrictions, short-term trading limitations, capital structure requirements, and governance provisions. For evidence on the impact of these di¤erences on the behavior of onshore and o¤shore …nancial institutions see Kim and Wei (2002).
3 See DEG (2015) wherein accounts held in Mauritius are disclosed on p. 57. For the operational justi…cation see https://www.welt-sichten.org/artikel/32312/deg-ohne-offshore-geht-es-nicht. 4 In this paper we focus solely on e¢ciency. There is, however, an equity concern when o¤ering incentives introduce a scheme into the model and compare the results. A investor can decide to invest an exogenous lump-sum either onshore or o¤shore. An onshore investment must be made legally, but an o¤shore investment may be made either legal or illegally. As such, not all investments tax authorities observe in o¤shore data owe tax. If an investor invests o¤shore, the investment is subsequently observed by the tax authority with a positive probability. In the absence of a scheme, if an investor's o¤shore investment is observed, the tax authority can, if it chooses, verify whether any tax is owed, but at a cost. Following veri…cation of a tax liability, the tax authority can recover outstanding taxes and levy …nes. An equilibrium of this game is ine¢cient to the extent that the tax authority struggles to achieve a credible threat to verify, owing to its inability to distinguish between legal and illegal o¤shore investments. In the presence of a scheme, the tax authority chooses an incentivized …ne rate that will apply to liabilities disclosed within the scheme, and investors decide whether or not to make a disclosure within the scheme. If an investor does make a disclosure they can either disclose their o¤shore investment to be illegal and pay the tax owed plus a …ne at the incentivized rate, or disclose their investment as legal. The tax authority can choose to verify the investments of those investors who disclose their o¤shore investment to be legal (for an illegal investment might be falsely disclosed as legal). Even if an investor decides not to make a disclosure within the scheme the tax authority can nevertheless choose to verify their investment and, where appropriate, levy …nes.
We …nd that the introduction of a disclosure scheme induces fewer investors to invest o¤shore illegally. Key to this …nding is the idea that disclosure schemes induce endogenous decisions by investors that act to lower the marginal cost of enforcement for tax authorities. Our …ndings imply that the number of investors investing onshore increases, but so too may the number investing o¤shore legally. Thus, our model suggests that empirical evidence pointing to increased o¤shore investment following the introduction of a scheme may not be evidence that such schemes generate additional o¤shore evasion, but instead evidence that such schemes generate additional legal o¤shore investment. Tax authorities also bene…t from schemes: expected net revenue increases due to the additional voluntary compliance that occurs when some investors switch from investing o¤shore illegally to investing legally.
Consistent with the design of schemes in the UK, the model predicts that the optimal scheme to tax evaders. Moreover, only a subset of evaders (i.e., those that evade through an o¤shore investment) bene…t. See, e.g., Bordignon (1993) and Rablen (2010) for studies of the role of equity in in ‡uencing tax evasion. There are also moral and legal concerns where information on o¤shore investments that was obtained by illegal means has been purchased by tax authorities (see, e.g., P…sterer, 2013).
o¤ers the lowest allowable …ne rate permitted in legislation for truthful disclosure within the scheme.
The paper proceeds as follows: Section 2 gives an overview of the use and design of disclosure schemes in the recovery of o¤shore tax evasion, and section 3 casts our contribution in the context of the existing literature. Section 4 presents the model, which is developed in the absence of a scheme in section 5, and in the presence of a scheme in section 6. Section 7 gives a comparative analysis of the consequences of the introduction of a scheme for investment behavior, welfare, and for tax revenue; and Section 8 concludes.

O¤shore Disclosure Schemes
Bulk leakages of o¤shore holdings data have in recent decades a¤ected investors in almost all major economies: Table 1 in Langenmayr (2017), which summarizes and updates information provided in OECD (2010) authorities are taking steps to improve international cooperation through the signing of tax information exchange agreements, with the G20 countries leading in this regard. 6 The creation in 2013 of an OECD Common Reporting Standard (OECD, 2013) and, in 2010, the adoption in the US of the Foreign Account Tax Compliance Act (FATCA), are leading to continuing information ‡ows regarding o¤shore investments. 7 5 A subset of the former list is the so-called "Lagarde List" -which contains 1,991 names of Greeks with accounts in Switzerland. It was passed to the Greek authorities in 2010 by the then French Finance Minister, Christine Lagarde (Boesler, 2012). 6 Within eight months of the G20 summit of April 2009 tax havens had signed more than 300 treaties (Johannesen and Zucman, 2014). See Konrad and Stolper (2016) for a more general model of the problem of coordinating against tax havens. 7 For more on the economic impact of FATCA see Dharmapala (2016).
Some tax authorities have opted to address data leakages through standing generic mechanisms for voluntary disclosure, rather than implement bespoke o¤shore disclosure schemes. According to Langenmayr (2017: Table 1) countries such as Australia, Canada, Germany and Japan have utilized standing mechanisms -but countries such as France, Israel, the UK, and the US, have opted for bespoke schemes. In these latter set of countries, the impetus for each scheme may be traced to speci…c data leakages. For instance, one of the very …rst schemes, the 2007 O¤shore Disclosure Facility (ODF), was implemented in the UK following legal action to force …ve major UK banks to disclose details of the o¤shore accounts held by their customers. The ODF o¤ered a¤ected investors time-limited access to a ten percent …ne rate (the minimum allowable penalty under UK civil legislation) if they made a full disclosure.
In 2009

Literature Review
To our knowledge, the only theoretical analysis dedicated to o¤shore disclosure schemes is found in Langenmayr (2017). In her model, the tax authority is a …rst mover, deciding on the incentivized …ne rate before investors decide whether or not to evade tax. Treating the tax authority as a …rst mover is appropriate to modelling the implementation of schemes in those countries which have chosen to handle o¤shore data acquisitions through standing generic mechanisms for voluntary disclosure. To our knowledge, however, no existing analysis addresses practice in, e.g., the UK and US, which -as discussed previously-have implemented bespoke schemes in reaction to speci…c data leakages. 9 We address this la-cuna: in our analysis the tax authority is assumed to move after investors have made their investment choice. This case is of interest as when the tax authority is endowed with the advantage associated with moving …rst an optimal scheme cannot lower net revenue, but when the …rst-mover advantage is handed to investors the desirability of such schemes is not a priori obvious.
Two other di¤erences relative to Langenmayr's study are worthy of mention. First, Langenmayr …nds the introduction of a scheme increases o¤shore tax evasion. This e¤ect arises at the discretion of the tax authority as a consequence of its revenue maximizing strategy. That is, in equilibrium, the tax authority "permits" an increase in evasion as the loss of revenue through voluntary compliance is more than recouped through additional …ne payments. 10 In our model the tax authority takes evasion behavior as …xed, for it has already taken place when the scheme is conceived. In this context, these apparently perverse incentives on the part of the tax authority do not arise. Rather, we …nd that the introduction of a scheme unambiguously reduces illegal o¤shore evasion, albeit legal o¤shore investment could indeed be increased by a scheme). Second, while Langenmayr makes the important point that disclosure schemes may reduce the per-investor veri…cation cost (as the investor freely supplies the necessary information) we show that a case for such schemes exists even neglecting this consideration. Instead, we highlight how the design of a scheme reduces the number of investments that must be veri…ed. As a consequence, the marginal cost of increasing the probability of veri…cation falls, for this probability applies to a smaller base of investments.
Our analysis relates to a number of other literatures. We connect to a literature on the use by tax authorities of pre-audit settlements in which investors can acquire full (e.g., Chu, 1990;Glen Ueng and Yang, 2001) or partial (Goerke, 2015) insurance from audit risk. These settlements are shown to yield a Pareto improvement relative to random auditing as (i) the tax authority captures the positive risk premium of a risk averse investor and (ii) the tax authority conducts fewer random audits. Such audit settlement schemes, however, rely on the tax authority moving …rst, before the investor makes the evasion choice. They are, therefore, not directly applicable in our framework. It is also notable that, even were we to allow the tax authority to move …rst, such settlement procedures would not induce a Pareto improvement in our framework. We consider risk neutral investors, so the tax authority is not veri…cation, it must be strictly gainful in expectation. This leads to tax authority to adopt a pure strategy, whereas Graetz et al. consider a mixed strategy for the tax authority. 10 For another context in which a revenue-maximizing tax authority does not maximize voluntary compliance see Rablen (2014). able to extract a positive risk premium; and we assume the tax authority audits optimally with and without a scheme, which rules out random auditing. In particular, in our model the tax authority does not gain from a reduction in the number of audits it performs per se, as it only ever audits when it is strictly gainful in expectation to do so.
As our model examines both the initial decision by the investor to evade, as well as the investor's subsequent disclosure decision, it is closely associated with the literature investigating anticipated tax amnesties, by which we mean voluntary disclosure schemes run in the absence of new information, which nevertheless o¤er investors reduced penalties if they wish to disclose an illegal o¤shore investment (see, e.g., Bayer et al. (2015) and the references therein). Empirical evidence demonstrates clearly that there exists a signi…cant body of investors who will not disclose under an amnesty who will disclose under a scheme, presumably because the latter entails the credible threat of sanctions in the event of non-disclosure. Londoño-Vélez and Ávila-Mahecha (2018) document how participation in a pre-existing Colombian mechanism for voluntary disclosure increased more than eightfold following the publication of the Panama papers, while Johannesen et al. (2018) and Bethmann and Kvasnicka (2016) document similarly large e¤ects on the use of standing voluntary disclosure mechanisms in the US and Germany respectively following o¤shore data leakages. Consistent with this evidence, the investors in our model would never make a voluntary disclosure in the absence of new information, but do make a disclosure when, following the receipt of information, a scheme is o¤ered. Whereas the literature has cast doubt on the desirability to tax authorities of anticipated amnesties, our analysis of voluntary disclosure schemes arrives at more positive conclusions. An optimally designed scheme, even when anticipated, increases net revenue and reduces illegal o¤shore evasion.
Our work also connects to the literature on law enforcement with self-reporting (e.g., Kaplow and Shavell, 1994). In this literature truthful disclosure is induced by allowing those who report to pay a sanction equal to the certainty equivalent of the expected sanctions they would otherwise face by not self-reporting. The insights of Kaplow and Shavell are su¢cient to establish that, if a tax authority moves …rst, then a scheme can always be made unambiguously bene…cial: it can be chosen, for instance, to lower enforcement costs while holding incentives to commit evasion …xed. While our model also utilizes this insight, the key di¤erence between our model and this literature is that the tax authority moves second, after the crime is committed. In this setting it is unclear that the desirable properties of self-reporting when the law enforcer moves …rst are retained.
A further related literature is that on optimal auditing in the presence of signals (e.g., Scotchmer, 1987;Macho-Stadler and Pérez-Castrillo, 2002;Bigio and Zilberman, 2011). Under a scheme both the very act of making a disclosure, as well as its content, are signals the tax authority observes before deciding whether to audit (verify). Last, as the ability of tax authorities to extract revenue from whistleblower data in ‡uences the degree to which they should incentivize such behavior, our …ndings inform the literature on the optimal incentivization of whistleblowing (Yaniv, 2001) and complement studies that analyze the e¤ects on compliance of the presence of potential whistleblowers (Mealem et al., 2010;Bazart et al., 2014;Johannesen and Stolper, 2017).

Model
In this section we model o¤shore disclosure schemes as a strategic interaction between investors, who can invest either onshore or o¤shore, and the domestic tax authority.
Each investor i belonging to the set T receives a lump-sum w i > 0, unobserved by the tax authority. The lump-sum is distributed across investors according to the function W : [w, w] ∈ R >0 7 → (0, 1). Each investor should, by law, declare the lump-sum for taxation at the marginal rate θ ∈ (0, 1). We assume, however, that investors have three possible actions (i) invest the lump-sum o¤shore without declaring it for domestic taxation (illegal o¤shore investment); (ii) declare the lump-sum for domestic taxation and invest the remaining amount [1 − θ] w o¤shore (legal o¤shore investment); or (iii) declare the lump-sum for domestic taxation and invest the remainder onshore. In considering these actions we stress that investing money o¤shore is not an illegal act: what makes an o¤shore investment illegal in our model is the failure to previously declare the source capital for domestic taxation.
Amounts invested o¤shore earn a rate of return r OF F > 0, and amounts invested onshore earn a rate of return r ON > 0. 11 Investors consume the investment (plus earned interest), upon its maturity.
We shall assume, for simplicity, that interest income accruing from investment is untaxed.
That is, we focus on the evasion of tax on the source capital rather than the evasion ("sheltering") of interest income. As well as giving tractability, we note that the former is of greater economic signi…cance: the amount of source capital is typically many times the annual interest ‡ow such that only when undeclared interest has accrued over many years does the tax liability from this source become of a comparable magnitude to that on the undeclared capital. 12 As discussed in the introduction, o¤shore investments may di¤er from onshore investments both in the pecuniary and non-pecuniary dimensions. We capture the former dimension through the separate rates of return, r ON and r OF F ; and the latter dimension, for each investor i, by a parameter b i > 0, where b i < 1 signi…es that the non-pecuniary bene…ts to i from investing o¤shore exceed those from investing onshore, while b i > 1 signi…es the reverse. b i is independent of w i , and is distributed across investors according to the function An o¤shore investment (legal or illegal) is subsequently observed by the tax authority with probability p ∈ (0, 1). In the long run, p is endogenous to the e¤orts of tax authorities in, e.g., improving international cooperation and incentivizing whistleblowing. In the short-run, however, tax authorities must take p as …xed, as we shall suppose.
The underlying inference problem for the tax authority is as follows: if it observes an o¤shore investment of amount y, this could be the illegal investment of an investor with lumpsum w = y or the legal investment of an investor with lump-sum w = y/ [1 − θ]. While the simplicity of our model confers many advantages, one disadvantage is that it might lead the reader to underestimate the practical complexities to a tax authority of making this inference: investors a¤ected by o¤shore schemes are, in most cases, high net-worth individuals with often extremely complex …nancial arrangements, frequently involving the use of intermediary trust structures that make even mapping investments to their "true" owners a prolonged and labor-intensive process. For this demographic, the idea that the lump-sum -even when declared -will appear in a transparent and separately itemized form within the tax return for a known individual in a known tax year is in most cases unduly optimistic. Rather -as evidenced by the fact that tax authorities are routinely observed to seek external information from both the a¤ected taxpayer and other …nancial institutionstax authorities are typically unable to verify the legality of an investment solely on the basis of their internal information. Moreover, even once the lump-sum has been pinpointed, its nature (e.g., bequest, income, capital gain) must be established to verify that the correct tax (inheritance, income, or capital gains tax) was applied. Bearing these points in mind, we therefore suppose the tax authority must sink a veri…cation cost c > 0 to reveal the nature of an o¤shore investment. 13 If a tax liability is veri…ed, the tax authority can levy a …ne on the undeclared tax at a rate f ∈ [f , f ], where these upper and lower bounds are interpreted as being speci…ed in legislation. Standard arguments (e.g., Kaplow and Shavell, 1994) ensure that a revenuemaximizing tax authority will choose f = f . At the …ne rate f , the amount an investor must pay in tax and …nes on a veri…ed illegal investment y is denoted by (1) To simplify aspects of the analysis we make the following assumptions: Assumption 1 Q f , w > c.
Assumption 1 may be interpreted as requiring the lump-sum w to be su¢ciently large that it is gainful for the tax authority to verify an illegal o¤shore investment. Empirically, this assumption is very likely satis…ed, for observed o¤shore investments are typically large. 14 Moreover, to the extent that some observed o¤shore holdings are too small to be worthwhile investigating, such holdings can be screened almost costlessly by the tax authority. Assumption 2 implies that, at the maximum …ne rate, f , it is not gainful (in expectation) to invest o¤shore illegally if the tax authority, conditional on observing the investment, will verify with certainty. Conversely, at the minimum …ne rate speci…ed under legislation, f , it is gainful to invest o¤shore illegally even if, conditional on observing the investment, the tax authority will verify with certainty. If the former inequality is not satis…ed, illegal o¤shore investment is a one-way bet, for it pays even when the tax authority's enforcement is maximal. If the 13 In our analysis, the cost c applies always, irrespective of whether an investor makes a voluntary disclosure. In Langenmayr (2017), by contrast, veri…cation is assumed to cost the tax authority less if the investor makes a voluntary disclosure. As this alternative assumption -which can be readily be introduced into our model -adds to the case for disclosure schemes, it only strengthens our results when adopted.
14 According to Watt et al. (2012), the list of HSBC Jersey account holders obtained by HMRC in 2012 identi…es 4,388 people holding £699 million in o¤shore current accounts, which implies an average holding of £159,000. The median account balance of more than 10,000 closed cases from the 2009 OVDP in the US is reported as $570,000 in GAO (2013). latter inequality is not met, the tax authority's enforcement is so strong that it can eliminate all o¤shore investment in the presence of a scheme. Note that Assumption 2 rules out the pure amnesty case p = 0 in which there is no threat that an illegal o¤shore investment will subsequently be observed.
Investors behave so as to maximize expected consumption, while the tax authority behaves so as to maximize revenue (comprising voluntary compliance, recovered tax, and …nes) net of enforcement costs. While the implied risk neutrality of the tax authority is standard, the risk neutrality of investors might seem restrictive. Allowing for risk averse investors can only strengthen the case for voluntary disclosure schemes, however. In the absence of a scheme, risk averse investors would pay a premium to insure against the risk associated with possible tax authority veri…cation. When a scheme is o¤ered, however, investors can avoid uncertain veri…cation by disclosing truthfully. In this way the tax authority is able to capture the investor's risk premium within the scheme. To establish an economic case for the use of such schemes it is therefore su¢cient to examine the risk neutral case.
For simplicity, we de-emphasize intertemporal considerations by assuming a time preference rate of unity (for both investors and the tax authority). 15 Denote the expected consumption from choosing an investment of type k as C k , where k = ON is for onshore investment, k = L is for legal o¤shore investment, and k = I is for illegal o¤shore investment. We may then partition the set T into those investors that invest onshore, o¤shore legally, and o¤shore illegally, Conditional on having chosen to invest o¤shore, the probability that an investor who has invested an amount y chooses to do so illegally is denoted φ = φ (y) ∈ [0, 1]. When the tax authority chooses its enforcement parameters φ (y) is already determined, though its value is not observed by the tax authority. We suppose, however, that the tax authority forms a (rational) expectation of this quantity,φ (y), satisfying E(φ (y)) = φ (y).

No Scheme
In order to appraise the use of disclosure schemes, we now model the "do nothing" benchmark case in which the tax authority does not o¤er a scheme (NS). The game in the absence of a scheme is set out in Figure 1. At the outset, nature determines each investor's lumpsum, w i , and his/her level of non-pecuniary bene…t, b i , but this action is unobserved by the tax authority. Next, investors make an investment choice as described previously. O¤shore investors have their investment subsequently observed by the tax authority with probability p ∈ (0, 1). 16 The distribution function of observed o¤shore investments is denoted by Y ( ).
If o¤shore holdings are not observed by the tax authority, any illegal o¤shore investment goes undetected with probability one, and the game ends. If o¤shore holdings are observed by the tax authority, it will verify each o¤shore investment with a probability α ∈ [0, 1]. Veri…ed undeclared liabilities are …ned at the rate f . It follows that expected investor consumption is given by (2) (3) where implicit in this formulation is that an investor holding an illegal o¤shore investment must repatriate some of their investment to meet the tax and …nes payable as a result of veri…cation, and therefore do not earn interest on this amount. Note from (2) and (3) that if r ON = r OF F and b i = 1 then an investor is exactly indi¤erent between a legal o¤shore investment and an onshore investment. More generally, the balance of pecuniary and nonpecuniary bene…ts favors a legal o¤shore investment when r OF F − r ON > b i − 1.
Figure 1 -see p. 29 16 We assume here, for simplicity, that the tax authority acquires o¤shore data at zero cost, as was indeed the case in many of the schemes discussed in the Introduction. Even when payments were made, the amounts involved -where known -appear relatively modest in relation to the revenue generated. Bradley Birkenfeld, a UBS employee who acted as an IRS informer, received a payment of $104 million, but in the context of some $3.4 billion that was eventually raised by the resulting scheme (GAO, 2013). The UK tax authority is reported to have paid a former Liechtenstein bank employee a fee of just £100,000 for information regarding more than £100 million of o¤shore funds (Oates, 2008). Clearly, however, any amount paid to acquire information must be set against any revenue accruing from the scheme.
The expected net revenue the tax authority will generate from the members of T is given by: where the …rst term is the revenue generated through voluntary compliance, and the second term, is the expected net revenue from verifying investors in OF F . Importantly, however, the tax authority only observes ex-post the realized investment amount y of each member of the set T OF F . It therefore takes as …xed the level of voluntary compliance, the total size of the set T OF F , and the decomposition of T OF F between investors who have invested o¤shore legally and illegally. Accordingly, choosing α to maximize R T (α; φ) becomes simply equivalent to choosing α to maximize R OF F (α; φ), i.e., the net revenue from verifying the investments of investors in T OF F . Di¤erentiating R OF F (α; φ) with respect to α we obtain Hence, when observing an o¤shore investment of amount y, the tax authority chooses where here we adopt the convention that, if the tax authority is indi¤erent between verifying and not-verifying, it does not verify. Equation (6) captures an important intuition of the model: if the propensity to invest o¤shore illegally, φ, is su¢ciently high then the tax authority will always choose to verify (α = 1). If, however, φ, is su¢ciently low that the expected gain from veri…cation, φQ(f , y) − c, falls to (or below) zero, the tax authority does not …nd it gainful to verify an observed o¤shore investment, hence α = 0. As shall become clear, the discreteness of the tax authority's veri…cation strategy induces some risk neutral investors to commit o¤shore evasion probabilistically as part of a mixed strategy.
It follows from (6) that expected consumption, conditional on choosing to invest o¤shore, can be written as We depict C N S OF F (φ, w) in Figure 2. We see that, when φ ≤ c/Q(f , w) the investor's payo¤ in (7) is strictly increasing in φ for, from (6), the tax authority will choose not to verify.
Immediately above φ = c/Q(f , w) the payo¤ C N S OF F jumps downward discretely, however, for at this higher level of φ the tax authority will verify. As a consequence of Assumption 2, once the tax authority can commit to verify, it is no longer gainful in expectation to invest o¤shore illegally. Accordingly, increases in φ above c/Q(f , w) are seen in Figure 2 to only reduce the payo¤ C N S OF F further. Thus, C N S OF F is maximized with respect to φ where at which point the tax authority is exactly indi¤erent between verifying and not-verifying. Substituting (8) into (7) we obtain The payo¤ in (9) to investing o¤shore is strictly preferred to the payo¤ from investing onshore in (2) Proposition 1 In the absence of a scheme, if b i <b N S (w i ) an investor i ∈ T invests o¤shore illegally with probability c Q(f ,w i ) and o¤shore legally with probability Q(f ,w i )−c Q(f ,w i ) ; and invests onshore with probability one otherwise.
A hallmark of the equilibrium outcome is that, owing to its inability to distinguish between legal and illegal o¤shore investments, the tax authority is only able to cap the propensity for illegal o¤shore investment at φ (w i ) = c/Q(f , w i ). Below this propensity it is unable to sustain a credible veri…cation threat.

The Scheme
We now suppose the tax authority o¤ers a scheme in the event that o¤shore investments are observed. The game is set out in Figure 3. The initial hidden action by nature and the subsequent investment decision are modelled in the same way as in the absence of a scheme. If o¤shore investments are observed, however, the tax authority chooses the terms of a scheme it then announces to investors. 17,18 Investors then choose either to enter or not-enter the scheme. If the investor enters s/he discloses a type d ∈ {L, I}. An investor disclosing d = I (illegal) accompanies their disclosure with a payment to the tax authority  Owing to the revelation principle, attention may be con…ned to schemes (mechanisms) in which investors disclose truthfully. Consider the subgame that arises when an investor enters the scheme. If an investment is illegal, falsely disclosing d = L results in an expected payment of α S Q(f , y), whereas disclosing d = I results in a sure payment of Q(f S , y). Hence truthful 17 Thus the investor faces uncertainty as to whether their investment will be observed, but know a scheme will be o¤ered if the investment will be observed. The model can be generalized to allow the tax authority to implement a scheme with a given probability in the event that information is observed. As, however, this probability turns out to be exactly one in equilibrium we omit this step without loss of generality. 18 In practice a tax authority may also face a second choice as to the set of investors with whom it communicates the scheme. For instance, prior to the OVDP in the US, the Swiss authorities agreed to hand the IRS the names of approximately 4,450 US clients with accounts at UBS. The IRS then had the choice of (i) requiring UBS to write to a¤ected clients informing them that the details of their o¤shore holding had been handed to the IRS; or (ii) requiring UBS to write to a wider set of its clients (up to the set of all UBS clients with o¤shore holdings) informing them that the details of their o¤shore holding might have been handed to the IRS. In actuality, the IRS chose the second option, and -to prevent investors from inferring whether their information had been handed over -negotiated a con…dentiality clause with the Swiss that concealed the criteria by which the accounts were selected until after the OVDP deadline had passed (GAO, 2013). We abstract from this issue here, but note it as a potentially interesting avenue for future research. disclosure requires f S to satisfy Q(f S , y) ≤ α S Q(f , y). 19 As, in equilibrium, the tax authority will never …nd it optimal to set f S below that required to achieve truthful disclosure, it follows that If it observes the set of o¤shore investments the tax authority chooses the parameters of the scheme, {α S , f S }, as well as the analogous parameters for investors who choose to not-enter the scheme {α O , f O }, to maximize the expected net revenue raised from investors belonging to T OF F .
An investor with an illegal o¤shore investment faces a sure payment Q(f S , w) = α S Q(f , w) if they enter the scheme, and an expected payment α O Q(f , w) if they choose to not-enter.
We assume that, in the case of perfect indi¤erence, investors enter the scheme. Accordingly, an investor with an illegal o¤shore investment will enter the scheme if α O ≥ α S . An investor with a legal o¤shore investment is indi¤erent between entering and not-entering the scheme, so will enter also.
To emphasize a key intuition of the model we focus on the case in which investors choose to enter the scheme (α O ≥ α S ), in which case -and using the equality in (10) -expected net revenue generated within the scheme is Focusing on the second term in the integral in (11), which is the cost of veri…cation, note that the veri…cation probability α S applies only to the proportion 1 − φ of o¤shore investors who have chosen to invest o¤shore legally, and therefore disclose d = L (the remaining proportion φ of o¤shore investors who invest illegally truthfully disclose d = I). Conversely, in the absence of a scheme, the veri…cation probability α applies to all o¤shore investors. Veri…cation costs therefore fall by a factor [1 − φ] as fewer investments need to be veri…ed.
A consequence of this observation is that the marginal cost of increasing the veri…cation probability also falls by a factor [1 − φ] under a scheme. As we shall see, this generates a set of values of φ for which the tax authority is able to maintain a credible veri…cation threat in the presence of a scheme, but is unable in the absence of a scheme.
Establishing the equilibrium of the game in the presence of a scheme proceeds through the same set of steps as performed in Section 5. Matters are made more complicated, however, by the existence of two distinct veri…cation probabilities {α O , α S } that are chosen by the tax authority simultaneously. Taxpayers again invest so as to make the tax authority indi¤erent between verifying and not-verifying, but -in light of the discussion above -this now occurs at a new threshold given by Relegating the proof to the Appendix, we arrive at the following: Proposition 2 In the presence of a scheme, if b i <b S (w i ) an investor i ∈ T invests o¤shore illegally with probability c c+Q(f ,w i ) , and o¤shore legally with probability Q(f ,w i ) c+Q(f ,w i ) ; and invests onshore with probability one otherwise.

Veri…cation
A result that underlies all of the remaining …ndings we shall present is that the introduction of a scheme enhances the ability of the tax authority to sustain a credible threat to verify, leading to a lower threshold value of φ at which the tax authority becomes indi¤erent between verifying and not-verifying: The underlying intuition for Proposition 3 is that, as noted previously, a scheme lowers the marginal cost of raising the veri…cation probability above zero. In particular, an increase in φ applies only to o¤shore investors who disclose d = L, whereas it applies to all o¤shore investors in the absence of a scheme. To ensure that veri…cation is not gainful in expectation for the tax authority, investors therefore become obliged to invest o¤shore illegally with a lower probability.

Investment and Evasion -Onshore and O¤shore
By comparing the respective equilibria in the absence (Proposition 1) and presence (Proposition 2) of a scheme, we now analyze the consequences of introducing a scheme for both onshore and o¤shore investment volumes, and for the decomposition of o¤shore investments between those that are legal, and those that are illegal.

Parts (i) and (ii) of Proposition 4 focus on the proportion of investors who invest o¤shore
with and without a scheme. The proof of part (i) demonstrates that the enhanced veri…cation threat present under a scheme causes a fall in the critical level of relative non-pecuniary ben-e…ts required to induce investors to invest onshore, i.e.,b S (w) <b N S (w). This implies that the introduction of a scheme induces a set of investors -those with characteristics belonging to the shaded set in Figure 4 -to switch from investing o¤shore to investing onshore. 20 According to part (ii), the introduction of a scheme also unambiguously reduces the proportion of investors who invest o¤shore illegally. As, however, both T OF F and T I shrink, the proportion of investors who invest o¤shore legally could either increase or decrease. In particular, if T I shrinks proportionately more than does T OF F , then T L expands. Parts (iii) and (iv) of Proposition 4 show that analogous results to those in parts (i) and (ii) hold also for aggregate investment. In part (iii) the introduction of a scheme causes aggregate investment to fallsimply because some investors switch from investing w o¤shore illegally, to instead investing the reduced amount [1 − θ] w onshore legally. The fall of total investment, coupled with an increase of onshore investment, implies that o¤shore component of investment must fall.  The possibility that the legal component of o¤shore investment could be observed to increase following the introduction of a scheme is consistent with the evidence of Langenmayr (2017), who observes an increase in o¢cially recorded o¤shore investments by US citizens following the introduction of the 2009 OVDP. Within Langenmayr's framework -which does not allow for legal o¤shore investment -an increase in o¤shore investment can only be interpreted as an increase in illegal o¤shore evasion. Our model, which allows for legal o¤shore investment for legitimate economic purposes, o¤ers an alternative interpretation of this …nding.

Tax Revenue
Does the introduction of a scheme increase the expected net revenue of the tax authority?
Proposition 5 The expected net revenue collected by the tax authority from the set of investors T is increased by the introduction of a scheme: R S T > R N S T .
The intuition for Proposition 5 is that the increased propensity to invest legally raises the level of voluntary compliance. This increase in expected revenue from voluntary compliance is not o¤set by lower net revenues arising on amounts disclosed within the scheme (on account of the lower incentivized …ne rate being applied), for -both with and without a schemethe …rst-mover advantage enjoyed by investors permits them to make choices that leave the tax authority just indi¤erent between verifying and not-verifying. When this occurs the expected yield in tax and …nes from veri…cation is exactly o¤set by its cost.
Were we to have assumed that the tax authority could choose the scheme parameters before investors make their investment choice, the …nding that net revenue increases under a scheme would be unsurprising. As, however, we take the tax authority to move second, the implications for net revenue were initially uncertain. It is notable, therefore, that even when moving second, voluntary disclosure schemes still increase net revenue. 21 21 Whereas we consider a tax authority unfettered in its choice of …ne rate from the interval [f , f ], in many cases it is only in prescribed circumstances that the tax authority can levy the highest allowable …ne rate. In the UK, for instance, the …ne rate that is applied is conditional upon the "behavioral" nature of the observed non-compliance: the lower bound applies if the non-compliance is judged to be through "careless error", whereas the upper bound applies to "deliberate and concealed" inaccuracies (HMRC, 2012). A further potential bene…t of schemes, therefore, is that they may provide the legal grounds to apply a higher rate in cases where an investor either fails to respond to a disclosure opportunity, or makes a false disclosure within the scheme.

Investor Welfare
We now examine the impact of a scheme for expected investor consumption (utility): Proposition 6 For investors belonging to Part (i) of Proposition 6 is for investors who invest onshore irrespective of the provision of a scheme: such investors are wholly una¤ected. Part (ii) states that investors who invest o¤shore irrespective of the provision of a scheme lose consumption in the presence of a scheme. This loss arises as the probability φ S that an o¤shore investor chooses to invest illegally is lower in the presence of a scheme. Thus, the investor loses expected consumption on account of paying tax on the lump-sum with a greater probability. Part (iii) is for investors for whom the introduction of a scheme induces a switch from investing o¤shore to investing onshore. Such investors move from the higher payo¤ C N S OF F in the absence of a scheme to the lower payo¤ C ON in the presence of a scheme (continuing to invest o¤shore would yield the still lower payo¤ C S OF F < C ON ). That those investing o¤shore illegally lose utility appears desirable -after all, it is a consequence of a reduction in incentives for breaking tax law. More generally, were we to model explicitly the bene…ts from taxation in the form of the public services it pays for, the increased tax revenue generated by schemes would generate utility for all investors through increased provision.

Optimal Incentivized Fine Rate
For tax authorities seeking to understand the optimal design of disclosure schemes it is of interest to highlight a feature of the optimal scheme relating to the question of how to set the incentivized …ne rate for those that enter the scheme. We have the following result: Proposition 7 In the optimal scheme it holds that f S = f .
According to Proposition 7, the incentivized …ne rate is the lowest …ne rate allowed under legislation. This is consistent with the design of disclosure schemes in the UK, which have o¤ered those who disclose the minimum ten percent penalty permitted in law. The Netherlands -which implemented bespoke schemes in 2009 and 2013 -has a minimum …ne of zero (f = 0), and, consistent with our …nding, implemented its schemes on a no-…ne basis. The result in Proposition 7 may initially seem surprising as choosing a lower …ne rate would seem to reduce net revenue through a reduction in …ne revenue. This is not so, however, for although …ne revenue indeed falls, net revenue is left unchanged. Lowering the incentivized …ne rate makes truthful disclosure more attractive to investors, meaning that the tax authority can achieve truthful disclosure with less veri…cation. The reduction in veri…cation costs achieved in this way exactly o¤sets the loss in …ne revenue (as a consequence of the tax authority being indi¤erent between verifying and not-verifying), leaving net revenue constant. In this way, the same level of net revenue is achieved with least veri…cation activity by setting the incentivized …ne rate minimally. 22

Conclusion
Tax authorities around the world are using incentivized voluntary disclosure schemes to recover tax on o¤shore investments. Such schemes o¤er discounted …ne rates for those who voluntarily disclose (albeit in the shadow of subsequent enforcement against those who do not). International initiatives such as the OECD Common Reporting Standard are expected to result in their continued use. As, however, the use of such schemes by tax authorities in response to data leakages is by now anticipated, the stellar returns observed for the earliest such schemes should not be expected to continue. As our model highlights, rational investors who anticipate being o¤ered a scheme behave in a such a manner as to make the tax authority indi¤erent between verifying o¤shore investments and not. In this paper we examined whether indeed such anticipated schemes continue to be of value to tax authorities, or could actually be incentivizing o¤shore evasion in the …rst place.
We consider an environment in which investors can invest a lump-sum onshore or o¤shore.
Should they choose to invest o¤shore, they may do so legally or illegally -neither is o¤shore investment in itself illegal, nor is all o¤shore investment driven by illegal tax motives.
After investments have been made, the tax authority may potentially observe the o¤shore investments, but does not observe which were made legally, and which illegally. Investors make their investment decision knowing that, if they invest o¤shore and the investment is subsequently observed a scheme will be o¤ered. The terms of the scheme, however, are determined only after the investment is observed, as we have argued characterizes the schemes operated in the UK, US and elsewhere.
In this context, we …nd that the tax authority can increase its expected net revenue by implementing a disclosure scheme, rather than by simply using its regular veri…cation regime. A hallmark of the optimal disclosure scheme is that it o¤ers the minimum allowable …ne rate in law to those that disclose truthfully. The particular bene…t the implementation of a scheme a¤ords tax authorities in our model is a reduction in the base of investments that require costly veri…cation. This lowers the marginal cost of veri…cation, permitting the tax authority to present investors with a stronger threat to enforce the tax law. Although the implementation of disclosure schemes is consistent with a rise in legal o¤shore investment, importantly our model predicts that the illegal component of o¤shore investment always falls.
Thus, in a sense our model helps makes precise, it is possible to o¤er ex-post inducements for truthful disclosure without simply incentivizing the underlying criminal activity.
We o¤er the following suggestions for future research. One extension would be to would be to extend the model to allow for the possible sheltering of interest in o¤shore accounts, alongside the possibility of tax evasion on the source capital. Second, imperfect veri…cation technology might be allowed for, as in Rablen (2014). Third, communication between a¤ected investors through a network, as in Hashimzade et al. (2014), might be introduced. Last,  …nd that many US investors did not make use of the 2009 OVDP scheme but chose to make "quiet" disclosures through standing voluntary disclosure mechanisms following the leak of o¤shore data. The model could be extended to allow for this possibility as one of the investors' choices. While each of these avenues must await a dedicated treatment, we hope to have shed some further light on the economic e¤ects and optimal design of disclosure schemes.
R OF F (α S , y; φ) achieves a maximum at α S = 1. Hence With the nature of enforcement now determined, we analyze the investor's investment decision. Expected consumption, conditional upon investing o¤shore illegally with probability φ ∈ [0, 1], can be written using (A.3) as The shape of C S OF F (φ, w) as a function of φ has the same qualitative features as the equivalent function in the absence of a scheme shown in Figure 2. In particular, for φ ≤ c/[c + Q(f , w)] the investor's payo¤ is strictly increasing in φ, as the tax authority cannot credibly commit to veri…cation. For φ > c/[c + Q(f , w)] the investor's payo¤ initially falls discreetly, and becomes strictly decreasing in φ thereafter, as the tax authority will now verify. It follows that C S OF F (φ) obtains a maximum in φ at The payo¤ C S OF F (w) in (A.6) to investing o¤shore is strictly preferred to the payo¤ from investing onshore in (2) from which the Proposition follows. Proof of Proposition 3. We have φ S (w) = c c + Q(f , w) < c Q(f , w) = φ N S (w) .
Proof of Proposition 4. The expected proportion of investors with lump-sum w who invest o¤shore legally, τ L (w), and illegally, τ I (w), are given, respectively, by τ k L (w) = 1 − φ k B(b k (w)); τ k I (w) = φ k B(b k (w)); where k ∈ {N S, S}, and φ k is the value of φ in state k. Hence, in aggregate, the expected proportions of investors choosing each investment type are given by τ k I (w) dW. Expected aggregate net onshore and o¤shore investment are given by where the latter may be further decomposed into its legal and illegal components: Next, we establish thatb S (w) <b N S (w):

Proof of Proposition 5.
As the choices of investors in T OF F make the tax authority indi¤erent between verifying and not-verifying (both with and without a scheme), it is straightforward to show that, in equilibrium, R S OF F (y) = R N S OF F (y) = 0. Hence, using (5) and (A.1), we have where k ∈ {N S, S}. The result then follows from the inequalities in Proposition 3.
Proof of Proposition 6.
(i) Immediate from (2); (ii) In equilibrium C L = C I − [1 + r OF F ] θw. Hence C OF F φ k = φ k C I + [1 − φ k ]{C I − [1 + r OF F ] θw} = C I − [1 − φ k ] [1 + r OF F ] θw. It follows that C OF F φ S < C OF F φ N S ⇔ φ S < φ N S , where the rightside holds by Proposition 3; (iii) As C ON is una¤ected by a scheme, investors who invest o¤shore in the absence of a scheme but switch to investing onshore in the presence of a scheme must switch to a lower payo¤. Proof of Proposition 7. Using the relationship Q(f S , y) = α S Q(f , y) established in (10) Figure 4: The critical valuesb N S (w) andb S (w) at which an investor is indi¤erent between investing onshore or o¤shore.