Income tax evasion: tax elasticity, welfare, and revenue


This paper provides a general equilibrium model of income tax evasion. As functions of the share of income reported, the paper contributes an analytic derivation of the tax elasticity of taxable income, the welfare cost of the tax, and government revenue as a percent of output. It shows how an increase in the tax rate causes the tax elasticity and welfare cost to increase in magnitude by more than with zero evasion. Keeping constant the ratio of income tax revenue to output, as shown to be consistent with certain US evidence, a rising productivity of the goods sector induces less evasion and thereby allows tax rate reduction. The paper derives conditions for a stable share of income tax revenue in output with dependence upon the tax elasticity of reporting income. Examples are provided with less and more productive economies in terms of the tax elasticity of reported income, the welfare cost of taxation and the tax revenue as a percent of output, with sensitivity analysis with respect to leisure preference and goods productivity. Discussion focuses on how the tax evasion analysis may help explain such fiscal tax policy as the postwar US income tax rate reductions along with discussion of government fiscal multipliers. Fiscal policy with tax evasion included shows how tax rate reduction induces less tax evasion, a lower welfare cost of taxation, and makes for a stable income tax share of output.

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  1. 1.

    See for example this decision of reporting income in Allingham and Sandmo (1972); see Hansen and Sargent (2005) on certainty equivalence; and see Ehrlich (1973, 1996), Becker (1968), and Klepper and Nagin (1989) on penalties and enforcement.

  2. 2.

    This is the main point of Kopczuk (2005).

  3. 3.

    For example, Alstadsæter et al. (2019a) present evidence on the amount of income hidden from taxation internationally, within banks, on the basis of wealth held in previously undisclosed bank accounts.

  4. 4.

    See for example Weisbach (2003) on the distinction between tax avoidance and evasion, with for example tax expenditures being legal but evasive, so that a hard line between these activities is tenous.

  5. 5.

    On the financial intermediation approach to the production of such services using Cobb-Douglas technology, see for example Sealey and Lindley (1977), Clark (1984), Hancock (1985) and Degryse et al. (2009).

  6. 6.

    See Lucas’s (1990) explanation of the productivity differential between developed and less developed countries in terms of the wage rate.

  7. 7.

    Weisbach (2003, p. 9) expands on the welfare approach to avoidance and evasion: “To put the problem in a welfarist framework, we cannot assume pre-existing definitions of tax avoidance and evasion. Instead, we must determine which responses to taxation will be treated in various fashions based directly on the welfare consequences of such treatment”.

  8. 8.

    U.S. Bureau of Economic Analysis (BEA), Federal government current tax receipts: Personal current taxes; from the Federal Reserve Bank of St. Louis; defined in BEA Table 3.4. as federal and state income taxes for 98% of the total reported in 2018; federal income taxes are 78% of that total.

  9. 9.

    World Bank Development Indicators; World Bank ID: GC.TAX.TOTL.GD.ZS.

  10. 10.

    World Bank Development Indicators: World Bank ID: GC.TAX.YPKG.RV.ZS.

  11. 11.

    World Bank Development Indicators; World Bank ID: GC.TAX.YPKG.RV.ZS.

  12. 12.

    While stable since 1959, the US income tax share of GDP has decreased slightly since 1978.

  13. 13.

    The model is a special, simplified, case of the Gillman and Kejak (2014) economy that includes both human and physical capital, with endogenous growth.

  14. 14.

    The dividend income is assumed to be hidden by the bank; it can be made taxable which introduces a squared \(\tau\) term that complicates the presentation of results while keeping them qualitatively the same.

  15. 15.

    Berk and Green (2004) assume a similar but exogenous upward sloping intermediation cost function for mutual funds supply.

  16. 16.

    See Gillman and Kejak (2005) for a parallel condition to Eq. (11) in a monetary generalization of the Baumol (1952) condition; the marginal cost of avoiding the inflation tax through banking services equals the marginal benefit which is the inflation tax rate itself.

  17. 17.

    The marginal cost for Fig. 2b can be solved for \(\tau\) from \(1-a_{E}= \frac{q_{E}}{c}=A_{E}\left( \frac{\tau \kappa A_{E}}{w}\right) ^{\frac{ \kappa }{1-\kappa }},\) such that \(\tau =\frac{w\left( 1-a_{E}\right) ^{\frac{ 1-\kappa }{\kappa }}}{\kappa \left( A_{E}\right) ^{\frac{1}{1-\kappa }}};\) here \(\frac{1-\kappa }{\kappa }\) is the power coefficient on the quantity \(1-a_{E},\) so that marginal cost rises at an increasing rate if \(\kappa <0.5\).

  18. 18.

    For Examples 1 and 2, \(\kappa =0.36\) and \(\alpha =1;\) for Example 1 \(w=0.094\) and \(A_{E}=0.53,\) while for Example 2 productivities are higher at \(w=1.41\) and \(A_{E}=1.03\).

  19. 19.

    See also Feldstein (1999) on the welfare cost of income taxation.

  20. 20.

    In related work, it is clarified how compensating only goods consumption is not a feasible equilibrium; see Gillman (2020a).

  21. 21.

    \(\underset{a_{E}\longrightarrow 0}{lim} \left( \frac{z}{w}\right) =\left( 1-\tau \kappa \right) ^{-\frac{\alpha }{ 1+\alpha }}+\tau \kappa -1\).

  22. 22.

    This result is similar to a parallel monetary literature in Lucas (2000) and Gillman (2020b) in which the welfare cost of the inflation tax in this case is the cost of bank time used in avoiding the inflation tax through exchange credit.

  23. 23.

    See Benk et al. (2005, 2008, 2010) for \(\kappa <0.5\) for inflation tax avoidance.

  24. 24.

    Starting at \(75\%\) of income reported, assume the steady tax rate reduction would yield \(80\%\) of reported income at the end of 60 years. Then, a simple average would be \(77.5\%\) and \(\dot{\Delta }\tau \simeq -2\left( \frac{ \frac{0.36}{1-0.36}\frac{1-0.775}{0.775}}{1-\frac{0.36}{1-0.36}\frac{1-0.775 }{0.775}}\right) =-\,0.39\) per year, or about a \(21\%\) reduction over 55 years.

  25. 25.

    Egger et al. (2013) use the same “Taxing Wages” OECD methodology for their study on the probability of corporate headquarter locations.

  26. 26.

    For a literature review on estimates of shadow economy sizes internationally, see for example, Schneider and Enste (2000).

  27. 27.

    For a lower tax rate of \(20\%,\) then increasing \(A_{E}\) by \(15\%\) to 0.612 would also yield a \(40\%\) evasion rate.

  28. 28.

    Using Appendix solution for output which equals consumption when \(z=0\) (see Eq. 48), \(y=c=0.040,\) and that \(w/y=\frac{\left( 0.094\right) }{0.040},\) then \(z/y=0.063\).

  29. 29.

    I am grateful for this suggestion by an anonymous referee.

  30. 30.

    Note that this includes labor income, profit and capital gains which is broader than the concept of the paper’s model which can be extended to include capital and the capital income tax with similar results.

  31. 31.

    “It is compiled from National Accounts Statistics (NAS), published annually by the Central Statistical Organization, Government of India and supplemented by Input–Output tables, Annual Survey of Industries & National Sample Survey Organizations (NSSO) surveys on employment & unemployment.” according to Krishna et al. (2018).” See: _worldklems2018_dkd.pdf?m=1528208436.

  32. 32.

    “Over 20 countries in the world, including five central and eastern European Member States and seven EU neighbouring countries, have introduced a so-called “flat tax” (initially the three Baltic countries in 1994-1995, followed since 2001 by a second wave of countries including Russia, Serbia, Ukraine, Slovakia, Georgia, Romania, the former Yugoslav Republic of Macedonia, Montenegro and Albania—see table). The “flat tax” concept is usually associated with the academic works of Hall and Rabushka (1983, (1985) who consider a single tax rate applied to both personal and corporate income beyond a given threshold or “basic allowance” (ECB Monthly Bulletin, September euro 2007, p. 81).”

  33. 33.

    Gorodnichenko et al. (2009), for example, examine flat tax reform, tax evasion and welfare in Russia; Azacis and Gillman (2010) show welfare effects of flat tax reform in the Baltics, including transition dynamics; and Holter et al. (2019), and Alstadsæter et al. (2019a) address Laffer curves, flat and progressive taxes, and inequality.

  34. 34.

    Galí et al. (2007) qualify their results by finding that the financing by tax increases needs to be more associated with an increase in debt than with an increase in current spending.

  35. 35.

    See Stokey and Rebelo (1995), Turnovsky (2000) and Azacis and Gillman (2010) on flat tax change with endogenous growth; Auerbach (2002), for example, discusses the “dynamic scoring” of the 2001 US Tax Act that considers changes in the economy’s growth rate.

  36. 36.

    Slemrod (2018) discusses features of the 2017 Act that broaden the tax base on the basis of some long-standing, well-known, issues leftover from previous tax reforms.

  37. 37.

    Piketty et al. (2018, p. 600) analyze many US tax acts and note in contrast, for example, that “The 2013 tax reform has partly reverted the long-run decline in top tax rates”.

  38. 38.

    Eliminating the 10% investment tax credit was estimated by the US Congressional Joint Tax Committee to be able to raise revenue sufficient to pay for a seven percent decrease in the corporate tax rate from 46 to 39%; the elimination of this credit became part of the Reagan Treasury proposal and the final Act; see McLure et al. (1987) for other details.

  39. 39.

    Holter et al. (2019), for example, find that income tax rate reduction lowers tax revenue depending in degree on the nature of progressivity of the code; Trabandt and Uhlig (2011) estimate US and European Laffer curves and calculate the percent of labor and capital income tax cuts that are self-financing.


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I acknowledge support from the University of Missouri Hayek Professorship endowment fund and am indebted to the anonymous referees of this journal, and appreciate related discussion with Michal Kejak and Tamas Csabafi.

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Correspondence to Max Gillman.

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Appendix: Equilibrium conditions and solution

Appendix: Equilibrium conditions and solution

The consumer problem is

$$\begin{aligned}&\underset{c_{t},x_{t},a_{Et},d_{Et}}{Max}\ln c_{t}+\alpha \ln x_{t} \nonumber \\&+\,\lambda _{t}\left[ a_{Et}w\left( 1-x_{t}\right) \left( 1-\tau \right) +\left( 1-a_{Et}\right) w\left( 1-x_{t}\right) \left( 1-p_{Et}\right) +r_{Et}d_{Et}+\Gamma _{t}+z-c_{t}\right] \nonumber \\&+\,\varphi _{t}\left[ w\left( 1-x_{t}\right) -d_{Et}\right] , \end{aligned}$$

with equilibrium conditions when the Lagrangian multipliers are binding as follows:

$$\begin{aligned}&c_{t}:\frac{1}{c_{t}}-\lambda _{t}=0; \end{aligned}$$
$$\begin{aligned}&x_{t}:\frac{\alpha }{x_{t}}-\lambda _{t}\left[ a_{E}w\left( 1-\tau \right) +\left( 1-a_{Et}\right) w\left( 1-p_{Et}\right) \right] -\varphi _{t}w=0; \end{aligned}$$
$$\begin{aligned}&a_{E}:\lambda _{t}\left[ \left( 1-\tau \right) -\left( 1-p_{Et}\right) \right] =0; \end{aligned}$$
$$\begin{aligned}&d_{Et}:\lambda _{t}r_{Et}-\varphi _{t}=0; \end{aligned}$$
$$\begin{aligned}&\lambda _{t}:a_{Et}w\left( 1-x_{t}\right) \left( 1-\tau \right) +\left( 1-a_{Et}\right) w\left( 1-x_{t}\right) \left( 1-p_{Et}\right) +r_{Et}d_{Et}+\Gamma _{t}+z-c_{t}=0; \end{aligned}$$
$$\begin{aligned} \varphi _{t}:w\left( 1-x_{t}\right) -d_{Et}=0. \end{aligned}$$

Only for the case when \(\kappa =1\) would there lack a unique and well-defined equilibrium with tax evasion. Given \(\kappa \in \left[ 0,1\right) ,\) and \(A_{E}\in R_{+},\) then \(a_{Et}\in \left( 0,1\right]\).

The bank problem with the production function for \(q_{Et}\) substituted in from Eq. (7) and equilibrium conditions are given by

$$\begin{aligned}&\max _{l_{Et},d_{Et}}\Pi _{Et}=p_{Et}A_{E}\left( l_{Et}\right) ^{\kappa }\left( d_{Et}\right) ^{1-\kappa }-w_{t}l_{Et}-r_{Et}d_{Et}; \end{aligned}$$
$$\begin{aligned}&l_{Et:}:w=\kappa p_{Et}A_{E}\left( \frac{l_{Et}}{d_{Et}}\right) ^{\left( \kappa -1\right) }; \end{aligned}$$
$$\begin{aligned}&d_{Et}:r_{Et}=\left( 1-\kappa \right) p_{Et}A_{E}\left( \frac{l_{Et}}{d_{Et}} \right) ^{\kappa }. \end{aligned}$$

In equilibrium the time subscripts can be dropped. Equations (34) and (40) imply that \(\tau =p_{E}\) and \(r_{E}=\left( 1-\kappa \right) \tau A_{E}\left( \frac{l_{E}}{d_{E}}\right) ^{\kappa },\) such that Eq. (13) results above with \(r_{E}=\tau \left( 1-\kappa \right) \left( 1-a_{E}\right) .\) Substituting in that \(\Gamma _{t}=a_{E}\tau wl,\) that \(d_{E}=wl\) and that \(\tau =p_{E},\) the budget constraint of Eq. (36 ) writes as

$$\begin{aligned} c=wl\left( 1-\tau \right) +r_{E}wl+a_{E}\tau wl+z=wl\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] +z. \end{aligned}$$

The second main equation is the marginal rate of substitution between goods and leisure, which from Eq. (10), or the first-order conditions above, allows solving for leisure in terms of consumption as

$$\begin{aligned} x=\frac{\alpha c}{w\left( 1-\tau +r_{E}\right) }. \end{aligned}$$

We have the solutions for \(r_{E}=\tau \left( 1-\kappa \right) \left( 1-a_{E}\right)\) and \(a_{E}=1-A_{E}\left( \frac{p_{E}\kappa A_{E}}{w}\right) ^{\frac{\kappa }{1-\kappa }}\) from the bank equilibrium Eqs. (13) and (15) above, as well as from “Appendix” conditions 38, 39 and 40. Given the \(r_{E}\) and \(a_{E}\) solutions, Eqs. 41 and 42 are in two variables with two unknowns such that both c and x may be solved:

$$\begin{aligned} c= & {} wl\left[ 1-\tau \left( 1-a_{Et}\right) +r_{E}\right] +z=w\left( 1-x\right) \left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] +z; \nonumber \\ c= & {} w\left[ 1-\frac{\alpha c}{w\left( 1-\tau +r_{E}\right) }\right] \left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] +z; \nonumber \\ c= & {} \frac{w\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+\frac{z}{w}\right] }{ 1+\alpha \frac{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] }{\left( 1-\tau +r_{E}\right) }}=\frac{w\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+ \frac{z}{w}\right] }{1+\alpha \left( 1+\frac{a_{E}\tau }{\left( 1-\tau +r_{E}\right) }\right) }. \end{aligned}$$

For leisure in turn the solution is

$$\begin{aligned} x= & {} \frac{\alpha c}{w\left( 1-\tau +r_{E}\right) }=\frac{\alpha w\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+\frac{z}{w}\right] }{w\left( 1-\tau +r_{E}\right) \left[ 1+\alpha \left( 1+\frac{a_{E}\tau }{\left( 1-\tau +r_{E}\right) }\right) \right] }; \nonumber \\= & {} \left( \frac{\alpha }{1+\alpha }\right) \frac{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+\frac{z}{w}\right] }{1-\tau +r_{E}+\left( \frac{\alpha }{1+\alpha }\right) \tau a_{E}}. \end{aligned}$$

The leisure solution gives the solution for labor using the time constraint of \(l=1-x:\)

$$\begin{aligned} l=1-\frac{\alpha \left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+\frac{z}{w} \right] }{\left( 1-\tau +r_{E}\right) \left\{ 1+\alpha \frac{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] }{\left( 1-\tau +r_{E}\right) }\right\} } =1-\left( \frac{\alpha }{1+\alpha }\right) \frac{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+\frac{z}{w}\right] }{1-\tau +r_{E}+\left( \frac{\alpha }{1+\alpha }\right) \tau a_{E}}. \end{aligned}$$

The solution for time in banking \(l_{E}\) in turn comes from the solution for \(l_{E}/d_{E}\) in Eq. (12), or from “Appendix” bank conditions. Given that \(d_{E}=wl\) and that l is solved, \(l_{E}\) is found as

$$\begin{aligned} l_{E}=\tau \kappa \left( 1-a_{E}\right) l=\tau \kappa \left( 1-a_{E}\right) \left( 1-\left( \frac{\alpha }{1+\alpha }\right) \frac{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+\frac{z}{w}\right] }{1-\tau +r_{E}+\frac{\alpha \tau a_{E}}{1+\alpha }}\right) . \end{aligned}$$

Goods output is given by \(y=wl_{G}.\) Since \(l_{G}=1-x-l_{E},\) and from Eq. (16) that \(w\left( \frac{\tau \kappa A_{E}}{w}\right) ^{ \frac{1}{1-\kappa }}=\tau \kappa \left( 1-a_{E}\right) ,\) goods output follows as

$$\begin{aligned} y= & {} w\left[ 1-\left( 1-\frac{\alpha }{1+\alpha }\frac{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+\frac{z}{w}\right] }{1-\tau +r_{E}+\frac{\alpha \tau a_{E}}{1+\alpha }}\right) \right] \nonumber \\&-w\left[ \tau \kappa \left( 1-a_{E}\right) \left( 1-\frac{\alpha }{ 1+\alpha }\frac{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}+\frac{z}{w}\right] }{1-\tau +r_{E}+\frac{\alpha \tau a_{E}}{1+\alpha }}\right) \right] \nonumber \\= & {} \frac{w\left( 1-\tau +r_{E}-\alpha \frac{z}{w}\right) }{1+\alpha -\frac{ a_{E}\tau }{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] }}. \end{aligned}$$

The final part of the equilibrium is to confirm that the social resource constraint is respected by which \(y+z=c.\) Substituting in the above solution for y and adding z,  and rewriting, it results that

$$\begin{aligned} y+z= & {} c; \nonumber \\ \frac{w\left( 1-\tau +r_{E}-\alpha \frac{z}{w}\right) }{1+\alpha -\frac{ a_{E}\tau }{\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] }}+z= & {} \frac{w \left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] +z}{1+\frac{\alpha \left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] }{\left[ 1-\tau +r_{E}\right] }}= \frac{w\left[ 1-\tau \left( 1-a_{E}\right) +r_{E}\right] +z}{1+\alpha \left( 1+\frac{\tau a_{E}}{1-\tau +r_{E}}\right) }. \end{aligned}$$

Consider output and consumption with \(z=0,\) and note that with \(r_{E}=\tau \left( 1-\kappa \right) \left( 1-a_{E}\right)\)

$$\begin{aligned} 1-\tau \left( 1-a_{E}\right) +r_{E}= & {} 1-\tau \kappa \left( 1-a_{E}\right) ; \end{aligned}$$
$$\begin{aligned} 1-\tau +r_{E}= & {} 1-\tau \left[ a_{E}+\kappa \left( 1-a_{E}\right) \right] . \end{aligned}$$

Output and consumption equal the permanent income of the (Becker 1965) full value of time \(w\cdot 1\) minus the value of time used up in evasion through bank labor per unit of income wl,  which is \(w\tau \kappa \left( 1-a_{E}\right)\), as factored by \(\frac{1}{1+\alpha \left( 1+\frac{\tau a_{E} }{1-\tau +r_{E}}\right) }\), which is the fraction of permanent income consumed. The latter fraction rises as \(\tau\) rises from zero. The tax \(\tau\) causes less permanent income \(w\left[ 1-\tau \kappa \left( 1-a_{E}\right) \right]\) and effectively more leisure preference such that a higher fraction of a smaller permanent income is consumed. This is the effect of the tax distortion with tax evasion. Without tax evasion (\(a_{E}=1)\), as \(\tau\) increases, permanent income falls by more and the fraction of permanent income consumed rises by more such that output and consumption fall by more. The lesser distortion with tax evasion results despite that fact that tax evasion wastes resources as a fraction of income equal to the value of bank time per unit of income, or \(\frac{wl_{E}}{wl} =\tau \kappa \left( 1-a_{E}\right) .\)

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Gillman, M. Income tax evasion: tax elasticity, welfare, and revenue. Int Tax Public Finance (2020).

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  • Optimal evasion
  • Tax law
  • Welfare
  • Tax elasticity
  • Revenue
  • Productivity
  • Development

JEL Classification

  • E13
  • H21
  • H26
  • H30
  • H68
  • K34
  • K42
  • O11