Income tax evasion: tax elasticity, welfare, and revenue

Abstract

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.

Appendix: Equilibrium conditions and solution

The consumer problem is

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

(31)

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

$$\begin{aligned}&c_{t}:\frac{1}{c_{t}}-\lambda _{t}=0; \end{aligned}$$

(32)

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

(33)

$$\begin{aligned}&a_{E}:\lambda _{t}\left[ \left( 1-\tau \right) -\left( 1-p_{Et}\right) \right] =0; \end{aligned}$$

(34)

$$\begin{aligned}&d_{Et}:\lambda _{t}r_{Et}-\varphi _{t}=0; \end{aligned}$$

(35)

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

(36)

$$\begin{aligned} \varphi _{t}:w\left( 1-x_{t}\right) -d_{Et}=0. \end{aligned}$$

(37)

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

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

(38)

$$\begin{aligned}&l_{Et:}:w=\kappa p_{Et}A_{E}\left( \frac{l_{Et}}{d_{Et}}\right) ^{\left( \kappa -1\right) }; \end{aligned}$$

(39)

$$\begin{aligned}&d_{Et}:r_{Et}=\left( 1-\kappa \right) p_{Et}A_{E}\left( \frac{l_{Et}}{d_{Et}} \right) ^{\kappa }. \end{aligned}$$

(40)

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

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

(41)

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

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

(42)

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:

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

(43)

For leisure in turn the solution is

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

(44)

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

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

(45)

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

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

(46)

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

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

(47)

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

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

(48)

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

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

(49)

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

(50)

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 $\frac{wl_{E}}{wl} =\tau \kappa \left( 1-a_{E}\right) .$