## Abstract

A tax buyout is a contract between tax authorities and a tax payer which reduces the marginal income tax rate in exchange for a lump-sum payment. While previous contributions have focussed on labour supply, we consider the interaction with tax evasion and show that a buyout can increase expected tax revenues. This will be the case if (1) the audit probability is constant and the penalty for evasion is a function of undeclared income or (2) the penalty depends on the amount of taxes evaded, and authorities use information about income generated by the decision about a tax buyout offer when setting audit probabilities. Since individuals will only utilise a tax buyout if they are better off, higher tax revenues imply that such contracts can be Pareto improving.

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## Notes

See, for example, Bigio and Zilberman (2011) Carillo et al. (2012) and Tonin (2013). Models with a continuum of tax payers and cut-off rules for auditing are usually characterised by bunching (at the cut-off), but not necessarily by the absence of a whole interval of income declarations (see Franzoni 2009 for a survey). Similar features, that is, bunching and the absence of certain income declarations, will result if there is no evasion while taxation entails fixed administrative cost, such that it may be optimal to effectively exempt those with low (potential) tax bases from taxation (cf. Keen and Mintz 2004; Dharmapala et al. 2011).

Alternatively, the initial tax system could be progressive and the official tax burden given by \((Y-s)t\), where \(s, Y > s > 0\), represents the level of tax exemption. The findings derived below for a setting with \(s = 0\) carry over to a model in which \(s>0\) holds. This is the case because the level of taxable income, \(Y-s\), does not qualitatively affect the merits of tax buyouts. A derivation of the respective results is available upon request.

Setting \(t = r\) and \(B>0\) is equivalent to the offer of lump-sum income taxation. A tax payer who accepts such a buyout can no longer evade taxes because s/he has committed to paying the entire tax obligation

*B*. Consequently, we focus on the case of \(r < t\).This assertion relies on the assumption that higher expected tax revenues make the government better off and that higher expected utility of individuals does not reduce the government’s payoff. Consequently, we need no further restrictions on the specification of the government objective in order to establish the possibility that a tax buyout can be a Pareto improvement. I am grateful to an anonymous referee for the suggestion to make explicit this assumption which underlies the above argument with respect to a Pareto improvement.

For similar results, see Chu (1990) and Sleet (2010), who discuss the contribution by Del Negro et al. (2010). In Alesina and Weil (1992), there is an intermediate range of productivities for which the response to a tax buyout offer is ambiguous because marginal utilities from consumption and leisure change.

The findings of this paper will basically be valid also if individuals do not differ in income but in another characteristic, such as the attitude towards risk, as long as their response to a tax buyout offer reveals information about the true tax base. To illustrate, suppose that people differ in the degree of risk aversion, so that the extent of evasion, given identical gross income, is monotonically related to risk attitudes. Since, moreover, the willingness to accept a tax buyout offer will be related to the degree of risk aversion, accepting or declining a buyout offer will be informative about evasion behaviour.

We know that a reduction in tax rate from

*t*to \(t-r\) unambiguously reduces optimal tax payments for \(\alpha \) = 0, since \(\partial {V}^{*}/\partial {r} <\) 0 (cf. Eqs. (2.4) and (2.6)). The change in \(V^{*}\) due to a rise in*B*is zero for a constant level of absolute risk aversion (cf. (2.5)). Therefore, the proposed tax reform can actually reduce optimal payments.This ‘missing middle’ of tax declarations is a further feature of a tax buyout, in addition to those mentioned in the Sect. 1, which distinguishes a buyout from the mechanism proposed by Chu (1990). If high-income individuals do not evade taxes, as in Chu (1990), there is no need to avoid certain income declarations in order to escape being detected evading taxes.

A complete and meaningful description of the tax buyout which maximises

*T*, i.e. an explicit derivation of*B**and*r**, would require a more elaborate specification, for example, of payoffs and the distribution function*G*. Note, however, that the first term in the second line of (4.5) is deducted while the remaining terms in the second line are positive for \(Y_{2}<Y_{3}\). Accordingly, if tax authorities can set*B*and*r*and thereby \(Y_{1}\) and*r*, they are able to maximise expected tax revenues*T*.Reinganum and Wilde (1985), for example, postulate an audit rule according to which an audit is certain if the income declaration falls below a predetermined level and is zero otherwise. I am extremely grateful to an anonymous referee for bringing the issue analysed in this section to my attention.

Note that the same number of individuals will evade taxes in the presence and absence of a tax buyout, so that we can ignore auditing resources in the definition of the government’s budgetary constraint.

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I am extremely grateful for constructive suggestions by two anonymous referees; and helpful comments by Mathew Rablen, Nadine Riedel, Dirk Schindler, and participants of the CESifo area conference on Public Sector Economics in München, the Shadow 2011 conference in Münster, the meeting of the Association for Public Economic Theory in Lisbon and a seminar in Frankfurt.

## Appendix 1

### Appendix 1

### 1.1 (a) Strict concavity of expected utility EU in gross income *Y*

Using \(\tilde{{f}}=\frac{{f}}{({t}-{r})^\alpha }\) and \(\tilde{{t}}:=1-( {{t}-{r}})( {1+\tilde{{f}}})>0\), the total derivatives of expected utility EU as defined in Eq. (2.2) with respect to income *Y* are found to be:

where EU\(_\mathrm{VV}\) is defined in (2.4). The other derivatives are given by:

Substituting (2.4), (7.3) and (7.4) into (7.2) and simplifying, we obtain:

Note that the signs of (7.1) and (7.5) are independent of the magnitudes of *B* and *r*. Therefore, expected utility EU increases with gross income *Y* at a decreasing rate, irrespective of whether a tax buyout is utilised or not.

### 1.2 (b) Expected utility and expected tax payments

We have EU\(_\mathrm{VV} <\) 0 from the second-order condition, \(t > r\), and \(\partial {S}/\partial {B}>\) 0 because otherwise the government could reduce *B*, thereby raise expected payments per capita *S* and make individuals better off (see the derivation below Eq. (3.1)). For \(\Omega :={p}-( {1-{p}})\tilde{{f}}=1-( {1-{p}})(1+\tilde{{f}})>0\) (see the discussion below Eq. (2.3)), we can define a difference \(A\):

Collecting common terms yields:

Substituting in accordance with (2.4)–(2.6) and rearranging we obtain:

Collecting common terms and simplifying, we find:

### 1.3 (c) Derivation of Fig. 1

Suppose \(\alpha \) = 0. The definition of \({Y}_3 ,{V}^{*} ( {{Y}_3\vert {a}=1})={Y}_1 ( {{t}-{r}})\), and the assumption of an interior solution imply that \({V}^{*} ( {{Y}_3\vert {a}=1})<{Y}_3 ( {{t}-{r}})\) and \({Y}_1 <{Y}_3\) hold. The line \(B^{u}\) is defined by \(\mathrm{EU}( {{Y};{ V}^{*} ( {{Y}\vert {a}=1})})\), the line *W* by \(\mathrm{EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=0})})\), and the line \(B^\mathrm{c}\) by \(\mathrm{EU}( {Y};{V} = Y_{1}(t-r); {a}=1)\). Expected utility levels \(\mathrm{EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=0})})\) and \(\mathrm{EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=1})})\) are the same at an income level \(Y_{1} = B/r\) according to Proposition 1a. Therefore, the lines \(B^{u}\) and *W* cross at the income level \(Y_{1}\). Furthermore, \(\mathrm{EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=0})})<\mathrm{EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=1})})\) at any income \(Y> Y_{1}\) according to Proposition 1b. This implies that the line \(B^{u}\) lies above *W* at \(Y = Y_{3}\). Since \({V}^{*} ( {{Y}_3 \vert {a}=1})={Y}_1 ( {{t}-{r}})\), we have EU\(( {{Y};{V}={Y}_1 ( {{t}-{r}});{a}=1})=\mathrm{EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=1})})\) for \(Y = Y_{3}\) and given levels of *B* and *r*. Hence, the lines \(B^{u}\) and \(B^\mathrm{c}\) coincide at an income level \(Y_{3}\). Finally, \(B^\mathrm{c}\) lies below *W* at \(Y = Y_{1}\) because quasi-voluntary payments exceed the optimal unconstrained level \({V}^{*} ( {{Y}_1 \vert {a}=0})\). Evaluating \(\mathrm{EU}( {{Y};{V}={Y}_1 ( {{t}-{r}});{a}=1})\) at \({Y}={Y}_1 ={B}/{r}\), yields \(\hbox {EU}( {{Y};{V}={Y}_1 ( {{t}-{r}});{ a}=1;{B}={Y}_1 {r}})={pu}( {{Y}_1 ( {1-{t}})})+( {1-{p}}){u}( {{Y}_1 ( {1-{t}})})\). Expected utility \(\mathrm{EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=0})})\) calculated at an income \(Y_{1}\) will equal \(\mathrm{EU}( {{Y};{V}={Y}_1 ( {{t}-{r}});{a}=1;{B}={Y}_1 {r}})\) if \({V}^{*} ( {{Y}_1 \vert {a}=1})={Y}_1 ( {{t}-{r}})\) holds. Since the optimal unconstrained payment \({V}^{*}\) at an income \(Y_{1}\) is lower than \({Y}_1 ( {{t}-{r}})\), given tax evasion, \(\mathrm{EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=0})})>\mathrm{EU}( {{Y};{V}={Y}_1 ( {{t}-{r}});{a}=1;{B}={Y}_1 {r}})\) holds at \(Y = Y_{1}\).

To prove that the intersection of the lines \(B^\mathrm{c}\) and *W* is unique, implying that \(Y_{1} < Y_{2} < Y_{3}\), we furthermore have to show that all lines are increasing and strictly concave in *Y*. In Appendix 1a we have already done so for \(B^{u}\) and \(W\). Since, furthermore, \(\hbox {EU}( {{Y};{V}={Y}_1 ( {{t}-{s}});{ a}=1})\) does not depend on *Y* via *V*, its derivatives are simply the partial derivatives of \(\hbox {EU}( {{Y};{V}^{*} ( {{Y}\vert {a}=0})})\) with respect to *Y*, \(\partial \mathrm{EU}/\partial {Y}\), as derived in (7.1) and (7.3). Therefore, the line \(B^\mathrm{c}\) is also increasing and strictly concave in *Y*.

### 1.4 (d) Derivative \(\partial {Y_{2}}/\partial {Y_{1}}\) in Eq. (4.4)

The income \(Y_{2}\) is defined by (4.2), which, for ease of exposition, we will slightly rewrite:

The impact of a rise in \(Y_{1}\) on \(Y_{2}\) is given by \(\frac{\partial {Y}_2 }{\partial {Y}_1 }=-\frac{{\partial {Z}}/{\partial {Y}_1 }}{{\partial {Z}}/{\partial {Y}_2 }}\). Furthermore, we have:

From the first-order condition (2.3) we know that:

At an income \(Y_{2}\), an individual cannot make the optimal payment \({V}^{*} ( {{Y}_2 \vert {a}=1})\) but has to pay more in order to avoid detection. Since expected utility is strictly concave in *V* (cf. (2.4)), the derivative (2.3) evaluated at \({Y}_1 ( {{t}-{r}})>{V}^{*} ( {{Y}_2 \vert {a}=1})\) is negative. Therefore, the term in square brackets in (7.11) is positive and \(\partial {Z}/\partial {Y_{1}} >\) 0 applies, irrespective of whether the repercussion of a change in \(Y_{1}\) on *B* is taken into account or not, since \(B = Y_{1}r\).

To derive \(\partial {Z}/\partial {Y_{2}} <\) 0, we note that the line \(B^\mathrm{c}\) crosses *W* only once in Fig. 1 and does so from below (see Appendix 1b). Therefore, the line \(B^\mathrm{c}\) has a larger slope than *W* at \(Y_{2}\), and \(\partial {Z}/\partial {Y_{2}} <\) 0 must hold. Combining this result with \(\partial {Z}/\partial {Y_{1}} >\) 0 implies that \(\partial {Y_{2}}/\partial {Y_{1}}>\) 0.

### 1.5 (e) Difference \({T}_{W} ( {{Y}_2 })-{T}_{{B}^\mathrm{c}} ( {{Y}_2 })\) in Eq. (4.4)

Note that \(Y_{2}\) will only exist if \({V}^{*} ( {{Y}_2 \vert {a}=1})<{Y}_1 ( {{t}-{r}})\). If tax evasion remains undetected, the tax payment by an individual who accepts the tax buyout will be higher than by someone who declines the offer; implying that \({Y}_2 -{V}^{*} ( {{Y}_2 \vert {a}=1})>{Y}_2 -{Y}_1 ( {{t}-{r}})-B\) holds. Inspection of Eq. (4.2), which defines \(Y_{2}\), then clarifies that \(Y_{2}\) will only exist if the income when caught evading taxes is greater for someone who has accepted the tax buyout scheme than for an individual who has declined the offer. Therefore, \({x}^\mathrm{c}( {W})={Y}_2 ( {1-{t}( {1+{f}})})+{V}^{*} ( {{Y}_2 \vert {a}=1}){f}<{Y}_2 ( {1-( {{t}+{r}})( {1+{f}})})+{Y}_1 ( {{t}-{r}}){f}-{B}={x}^\mathrm{c}( {B})\) and \({x}^\mathrm{n}( {B})<{x}^\mathrm{n}( {W})\) hold, where \(B(W)\) indicates that the buyout offer has been accepted (declined). Since \(Y_{2}\) is defined such that \({pu}( {{x}^\mathrm{n}( {W})})+( {1-{p}}){u}( {{x}^\mathrm{c}( {W})})={pu}( {{x}^\mathrm{n}( {B})})+( {1-{p}}){u}( {{x}^\mathrm{c}( {B})})\) holds, expected income \({px}^\mathrm{n}( {B})+( {1-{p}}){x}^\mathrm{c}( {B})\) must be less than expected income \({px}^\mathrm{n}( {W})+( {1-{p}}){x}^\mathrm{c}( {W})\), given strict risk aversion. Since expected income can also be expressed as \({Y}_2 -{T}_{W} ( {{Y}_2 })={Y}_2 -( {{px}^{n}( {W})+( {1-{p}}){x}^\mathrm{c}({W})})\hbox { and } Y_{2} -{T}_{{B}^\mathrm{c}} ({{Y}_2})={Y}_2 -( {{px}^\mathrm{n}( {W})+( {1-{p}}){x}^\mathrm{c}( {W})})\), expected tax payments in this constrained case must exceed the payments made if the individual does not accept the tax buyout offer, implying that \(T_{W}(Y_{2})- T_{{B}^\mathrm{c}}(Y_{2})\) is negative.

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Goerke, L. Income tax buyouts and income tax evasion.
*Int Tax Public Finance* **22**, 120–143 (2015). https://doi.org/10.1007/s10797-013-9302-z

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DOI: https://doi.org/10.1007/s10797-013-9302-z