The role of information aggregators in tax compliance

Abstract

Fifty-six percent of the US taxpaying population uses a paid tax preparer, but the effect of these tax preparation services on tax compliance is not well understood. Although governments conceal the algorithms they use to determine which taxpayers to audit, tax preparation firms with large client bases may be able to infer these algorithms and therefore offer strategic advice to taxpayers. This paper formalizes this role, using a simple asymmetric information model where agents can purchase information about the government’s enforcement rules. In a competitive market for tax preparation services, demand for tax preparers is selective and increases in taxpayer income. Moreover, the presence of tax preparers always reduces compliance when tax preparers have perfect information. Perhaps surprisingly, if the demand for strategic advice is high enough, the government can mitigate evasion by revealing full information about its audit rule. Alternatively, when the tax preparers have imperfect information about the audit rule, if the top income within an audit class is low enough, the government can utilize tax preparer firms to increase compliance by influencing the audit information available to these firms.

A Appendix

In this section, I will provide details about omitted proofs and some of the equations.

The following inequality is always true in this model:

$$\begin{aligned} 0<\underline{\beta }<\omega<y_{\omega }={\overline{\beta }}<{\overline{y}}=2\overline{\beta }-\underline{\beta }<m \end{aligned}$$

(30)

1.1 A.1 Agent’s problem

$EI_{dif}$ increases with income.

$$\begin{aligned}&EI_{dif}= {\left\{ \begin{array}{ll} 0 &{} y<\underline{\beta } \\ \frac{t \left( y-\underline{\beta }\right) ^2}{4 \left( \overline{\beta }-\underline{\beta }\right) } &{} \underline{\beta } \le y<\overline{\beta } \\ -\frac{t \left( 2 \overline{\beta }^2-\underline{\beta }^2-4 y \overline{\beta }+2 y \underline{\beta }+y^2\right) }{4 \left( \overline{\beta }-\underline{\beta }\right) } &{} \overline{\beta } \le y<{\overline{y}} \\ \frac{1}{2} t \left( \overline{\beta }-\underline{\beta }\right) &{} {\overline{y}} \le y \\ \end{array}\right. } \end{aligned}$$

(31)

$$\begin{aligned}&\frac{\partial EI_{dif}}{ \partial y}= {\left\{ \begin{array}{ll} 0 &{} y<\underline{\beta } \\ \frac{t \left( y-\underline{\beta }\right) }{2 \left( \overline{\beta }-\underline{\beta }\right) } &{} \underline{\beta }\le y<\overline{\beta } \\ \frac{t \left( 2 \overline{\beta }-\underline{\beta }-y\right) }{2 \left( \overline{\beta }-\underline{\beta }\right) } &{} \overline{\beta } \le y<{\overline{y}} \\ 0 &{} {\overline{y}}<y\\ \end{array}\right. } \end{aligned}$$

(32)

Derivatives in all intervals are positive since $\overline{\beta }>y>\underline{\beta }$.

1.2 A.2 Demand

$y_d(c)$ is continuous in c.

$$\begin{aligned} y_d(c)= {\left\{ \begin{array}{ll} y_{y}=\underline{\beta }+\frac{2\sqrt{ct(\overline{\beta }-\underline{\beta })}}{t} \quad &{}\text {if } \ 0\le c \le \frac{t}{4}(\overline{\beta }-\underline{\beta })\\ y_{{\overline{\beta }}}=2\overline{\beta }-\underline{\beta }-\frac{\sqrt{2t((\overline{\beta }-\underline{\beta })(2t(\overline{\beta }-\underline{\beta })-2c)}}{t} &{}\text {if } \ \frac{t}{4}(\overline{\beta }-\underline{\beta })\le c \le \frac{1}{2}({\overline{\beta }}-{\underline{\beta }})t\\ m &{}\text {if } \ c > \frac{1}{2}({\overline{\beta }}-{\underline{\beta }})t \end{array}\right. } \end{aligned}$$

(33)

It is obvious that each cutoff function is continuous in c. Then, showing that the cutoffs do not jump at the boundaries is enough. Plugging in c at each boundary point will yield the following:

$$\begin{aligned} y_d(c)= {\left\{ \begin{array}{ll} y_y=y_{\overline{\beta }}=\overline{\beta } &{}\text {if} \ c=\frac{t}{4}(\overline{\beta }-\underline{\beta })\\ y_{\overline{\beta }}={\overline{y}} &{}\text {if} \ c=\frac{1}{2}({\overline{\beta }}-{\underline{\beta }})t \end{array}\right. } \end{aligned}$$

(34)

1.2.1 A.2.1 Comparative statics

In this section, I will analyze the comparative statics of the demand. As the demand thresholds are determined where $EI_{dif}(y)-c=0$, implicit function theorem can be utilized to locally analyze how thresholds vary with respect to the parameters of the model.

Derivative of $EI_{dif}$ with respect to ${\underline{\beta }}$:

$$\begin{aligned} \frac{\partial EI_{dif}}{\partial {\underline{\beta }}}= {\left\{ \begin{array}{ll} 0 &{} y<\underline{\beta } \\ \frac{t \left( y-\underline{\beta }\right) \left( -2 \overline{\beta }+\underline{\beta }+y\right) }{4 \left( \overline{\beta }-\underline{\beta }\right) ^2} &{} \underline{\beta }\le y<\overline{\beta } \\ -\frac{t \left( 2 \overline{\beta }^2+\underline{\beta }^2-2 \overline{\beta } \left( \underline{\beta }+y\right) +y^2\right) }{4 \left( \overline{\beta }-\underline{\beta }\right) ^2} &{} \overline{\beta }\le y<{\overline{y}} \\ -\frac{t}{2} &{} {\overline{y}}<y \\ \end{array}\right. } \end{aligned}$$

(35)

Derivative of thresholds with respect to $\underline{\beta }$

$$\begin{aligned} \frac{\partial y_d(c)}{\partial {\underline{\beta }}}=-\frac{\frac{\partial EI_{dif}}{\partial {\underline{\beta }}}}{\frac{\partial EI_{dif}}{\partial y}}= {\left\{ \begin{array}{ll} \frac{2 \overline{\beta }-\underline{\beta }-y}{2 (\overline{\beta }- \underline{\beta })} &{}\underline{\beta }\le y<\overline{\beta } \\ \frac{1}{2}+\frac{(y-\overline{\beta })(y+\underline{\beta })}{2 \left( \overline{\beta }-\underline{\beta }\right) \left( 2 \overline{\beta }-\underline{\beta }-y\right) }&{} \overline{\beta }<y<{\overline{y}} \\ \end{array}\right. } \end{aligned}$$

(36)

Note that as $2\overline{\beta }-\underline{\beta }={\overline{y}}$, the thresholds always increase, when $\underline{\beta }$ increases. As demand decreases when the threshold increases, an increase in $\underline{\beta }$ has a negative effect on the demand.

Derivative of $EI_{dif}$ with respect to ${\overline{\beta }}$:

$$\begin{aligned} \frac{\partial EI_{dif}}{\partial {\overline{\beta }}}= {\left\{ \begin{array}{ll} 0 &{} y<\underline{\beta } \\ -\frac{t \left( y-\underline{\beta }\right) ^2}{4 \left( \overline{\beta }-\underline{\beta }\right) ^2} &{} \underline{\beta }\le y<\overline{\beta } \\ \frac{t \left( -2 \overline{\beta }^2+4 \overline{\beta } \underline{\beta }-\underline{\beta } \left( \underline{\beta }+2 y\right) +y^2\right) }{4 \left( \overline{\beta }-\underline{\beta }\right) ^2}&{} \overline{\beta }\le y<{\overline{y}} \\ \frac{t}{2} &{} {\overline{y}}<y \\ \end{array}\right. } \end{aligned}$$

(37)

Derivative of thresholds with respect to $\overline{\beta }$

$$\begin{aligned} \frac{\partial y_d(c)}{\partial {\overline{\beta }}}=-\frac{\frac{\partial EI_{dif}}{\partial {\overline{\beta }}}}{\frac{\partial EI_{dif}}{\partial y}}= {\left\{ \begin{array}{ll} \frac{y-\underline{\beta }}{2( \overline{\beta }-\underline{\beta })} &{} \underline{\beta }\le y<\overline{\beta } \\ \frac{1}{2}-\frac{(y-\overline{\beta })(y+\underline{\beta })}{2 \left( \overline{\beta }-\underline{\beta }\right) \left( 2 \overline{\beta }-\underline{\beta }-y\right) }&{} \overline{\beta }<y<{\overline{y}} \\ \end{array}\right. } \end{aligned}$$

(38)

The equation when the threshold falls in the interval $[\underline{\beta },\overline{\beta })$ is positive, which implies that as $\overline{\beta }$ increases the demand for preparation services should fall. However, if the threshold is in the interval $[\overline{\beta },{\overline{y}})$, the sign is ambiguous. If the threshold is lower than $\sqrt{2}\left( \overline{\beta }-\underline{\beta }\right) +\underline{\beta }$, an increase in $\overline{\beta }$ has a negative effect on demand, whereas, once the threshold exceeds this level, the effect switches signs and become positive. Derivative of $EI_{dif}$ with respect to t:

$$\begin{aligned} \frac{\partial EI_{dif}}{\partial t}= {\left\{ \begin{array}{ll} 0 &{} y<\underline{\beta } \\ \frac{\left( y-\underline{\beta }\right) ^2}{4 \left( \overline{\beta }-\underline{\beta }\right) } &{} \underline{\beta }\le y<\overline{\beta } \\ -\frac{2 \overline{\beta }^2-\underline{\beta }^2-4 y \overline{\beta }+2 y \underline{\beta }+y^2}{4 \overline{\beta }-4 \underline{\beta }} &{} \overline{\beta }\le y<{\overline{y}} \\ \frac{1}{2} \left( \overline{\beta }-\underline{\beta }\right) &{} {\overline{y}}<y \\ \end{array}\right. } \end{aligned}$$

(39)

Derivative of thresholds with respect to t

$$\begin{aligned} \frac{\partial y_d(c)}{\partial t}=-\frac{\frac{\partial EI_{dif}}{\partial t}}{\frac{\partial EI_{dif}}{\partial y}}= {\left\{ \begin{array}{ll} - \frac{y-\underline{\beta }}{2 t} &{} \underline{\beta }\le y<\overline{\beta } \\ \frac{-2 \overline{\beta }^2+\underline{\beta }^2+4 y \overline{\beta }-2 y \underline{\beta }-y^2}{2 t \left( -2 \overline{\beta }+\underline{\beta }+y\right) } &{} \overline{\beta }\le y<{\overline{y}} \\ \end{array}\right. } \end{aligned}$$

(40)

The first expression in the above function is negative; therefore, as the tax rate increases, demand for tax preparers also increases. The second expression is also negative. To see this, first, notice that the denominator of this expression is negative as $2\overline{\beta }-\underline{\beta }=\overline{\beta }>y$. Also, notice that the numerator increases in y; therefore, the lowest possible value the numerator can take is when $y=\overline{\beta }$. Plugging in this value boils the numerator down to $2(\overline{\beta }-\underline{\beta })^2>0$. Consequently, the demand for tax preparers unambiguously increases as the tax rate rises.

Derivative of $EI_{dif}$ with respect to $({\overline{\beta }} -{\underline{\beta }})$: To do this, define ${\overline{\beta }}=\omega +\epsilon$ and ${\underline{\beta }}=\omega -\epsilon$ and take derivatives with respect to $\epsilon$

$$\begin{aligned} \frac{\partial EI_{dif}}{\partial \epsilon }= {\left\{ \begin{array}{ll} 0 &{} y<\underline{\beta } \\ \frac{t (\epsilon +y-\omega ) (\epsilon -y+\omega )}{8 \epsilon ^2} &{} \underline{\beta }\le y<\overline{\beta } \\ -\frac{t (\epsilon +y-\omega ) (\epsilon -y+\omega )}{8 \epsilon ^2} &{} \overline{\beta }\le y<{\overline{y}} \\ t &{} {\overline{y}}<y \\ \end{array}\right. } \end{aligned}$$

(41)

Derivative of thresholds with respect to $\epsilon$, where I substitute $\epsilon =\frac{\overline{\beta }-\underline{\beta }}{2}$ and $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial y_d(c)}{\partial \epsilon }=-\frac{\frac{\partial EI_{dif}}{\partial \epsilon }}{\frac{\partial EI_{dif}}{\partial y}}= {\left\{ \begin{array}{ll} \frac{y-\overline{\beta }}{\overline{\beta }-\underline{\beta }} &{} \underline{\beta }\le y<\overline{\beta } \\ \frac{\left( y-\overline{\beta }\right) \left( y-\underline{\beta }\right) }{\left( \overline{\beta }-\underline{\beta }\right) \left( -2 \overline{\beta }+\underline{\beta }+y\right) }&{} \overline{\beta }\le y<{\overline{y}} \\ \end{array}\right. } \end{aligned}$$

(42)

1.3 A.3 Ex post compliance

1.3.1 A.3.1 Evasion

Lemma 2:

Proof

As 1 explains, the two candidates for lowest possible change in evasion compared to the case where all taxpayers self-prepare are: (i) where all taxpayers hire a firm; (ii) with the smallest positive demand possible, i.e., where the demand cutoff is at $\overline{\beta }$.

Total reported income in (i) is

$$\begin{aligned} \begin{aligned} RI(\underline{\beta })&=\int _{0}^{\omega }ydy+\int _{\omega }^{m}\omega dy \\&=\frac{1}{8} \left( \overline{\beta }+\underline{\beta }\right) \left( -\overline{\beta }-\underline{\beta }+4 m\right) >0 \end{aligned} \end{aligned}$$

(43)

Total reported income in (ii) is

$$\begin{aligned} RI({\overline{y}})&=\int _0^{\underline{\beta }} y \, dy+\int _{\underline{\beta }}^{{\overline{y}}} r \, dy+\int _{{\overline{y}}}^m \omega \, dy\nonumber \\&=\frac{1}{2} \left( \overline{\beta } \left( m-\underline{\beta }\right) +m \underline{\beta }\right) \end{aligned}$$

(44)

$$\begin{aligned} RI({\overline{y}})-RI(\underline{\beta })&=\int _{\underline{\beta }}^{\omega } (y-r) \, dy+\int _{\omega }^{\overline{\beta }} (\omega -r) \, dy\nonumber \\&=\frac{1}{8} \left( \overline{\beta }-\underline{\beta }\right) ^2>0 \end{aligned}$$

(45)

Therefore, aggregate reported income when $y_d=\overline{\beta }$ is higher than the aggregate reported income when everyone hires a firm. $\square$

Proposition 2:

Proof

Case 1

$$\begin{aligned} RI(y_{\omega })&=\int _0^{\underline{\beta }} y \, dy+\int _{\underline{\beta }}^{\overline{\beta }} r \, dy+\int _{\overline{\beta }}^m \omega \, dy\nonumber \\&=\frac{1}{4} \left( -\overline{\beta }^2+2 m \overline{\beta }+\underline{\beta } \left( 2 m-\underline{\beta }\right) \right) \end{aligned}$$

(46)

$$\begin{aligned} RI(\underline{\beta })&=\int _{0}^{\omega }ydy+\int _{\omega }^{m}\omega dy \nonumber \\&=\frac{1}{8} \left( \overline{\beta }+\underline{\beta }\right) \left( -\overline{\beta }-\underline{\beta }+4 m\right) >0 \end{aligned}$$

(47)

$$\begin{aligned} RI(\underline{\beta })-RI(y_{\omega })&=\int _{\underline{\beta }}^{\omega } (y-r) \, dy+\int _{\omega }^{\overline{\beta }} (\omega -r) \, dy\nonumber \\&=\frac{1}{8} \left( \overline{\beta }-\underline{\beta }\right) ^2>0 \end{aligned}$$

(48)

Case 2

$$\begin{aligned} RI({\overline{y}})&=\int _0^{\underline{\beta }} y \, dy+\int _{\underline{\beta }}^{{\overline{y}}} r \, dy+\int _{{\overline{y}}}^m \omega \, dy\nonumber \\&=\frac{1}{2} \left( \overline{\beta } \left( m-\underline{\beta }\right) +m \underline{\beta }\right) \end{aligned}$$

(49)

$$\begin{aligned} RI(\underline{\beta })-RI({\overline{y}})&=\int _{\underline{\beta }}^{\omega } (y-r) \, dy+\int _{\omega }^{\overline{\beta }} (\omega -r) \, dy\nonumber \\&=-\frac{1}{8} \left( \overline{\beta }-\underline{\beta }\right) ^2<0 \end{aligned}$$

(50)

$\square$

1.3.2 A.3.2 Reported income comparative statics

Case 1 Everyone hires a firm, i.e., $y_d(c)=\underline{\beta }$

Reported income:

$$\begin{aligned} RI(\underline{\beta })= \int _{0}^{\omega }ydy+\int _{\omega }^{m}\omega dy= \omega m-\frac{\omega ^2}{2} \end{aligned}$$

(51)

In this case, none of the parameters have an effect on aggregate reported income.

Case 2 Only taxpayers with income above the first demand threshold, i.e., $y>y_{y}=\underline{\beta }+\frac{2\sqrt{ct(\overline{\beta }-\underline{\beta })}}{t},$ hire a firm.

First note that Eq. 33 states that $y_y \in [\underline{\beta },\overline{\beta }]$. As $\omega \in [\underline{\beta }, \overline{\beta }]$ is also the case, the analysis of reported income will be different whether $y_y>\omega$ or not.

Case 2a $y_y<\omega$ Note that from Eq. 33, one can see that to have $y_d(c)=y_y$, it should be the case that $0\le c \le \frac{1}{4}(\overline{\beta }-\underline{\beta })$. Moreover, $y_y<\omega$ is only possible if $0\le c \le \frac{1}{16}(\overline{\beta }-\underline{\beta })$ Reported income:

$$\begin{aligned} \begin{aligned} RI(y_{y_1})&= \int _0^{\underline{\beta }} y \, dy + \int _{\underline{\beta }}^{y_y} r \, dy +\int _{y_y}^{\omega } y \, dy +\int _{\omega }^m \omega \, dy\\&\frac{c \underline{\beta }}{t}-\frac{c \overline{\beta }}{t}+m \omega -\frac{\omega ^2}{2} \end{aligned} \end{aligned}$$

(52)

Taking the derivative with respect to $\underline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI(y_y)}{\partial \underline{\beta }} =\frac{c}{t}>0 \end{aligned}$$

(53)

Taking the derivative with respect to $\overline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI(y_y)}{\partial \overline{\beta }} =-\frac{c}{t}<0 \end{aligned}$$

(54)

Derivative of reported income with respect to $\epsilon =({\overline{\beta }} -{\underline{\beta }})$: To do this, define ${\overline{\beta }}=\omega +\epsilon$ and ${\underline{\beta }}=\omega -\epsilon$ and take derivatives with respect to $\epsilon$, and then substitute $\epsilon =(\overline{\beta }-\underline{\beta })$ and $\omega =\frac{1}{2} (\overline{\beta }+\underline{\beta })$

$$\begin{aligned} \frac{\partial RI(y_y)}{\partial {\epsilon }} =-\frac{c}{t}<0 \end{aligned}$$

(55)

Derivative with respect to t:

$$\begin{aligned} \frac{\partial RI(y_y)}{\partial t}=\frac{c \left( \overline{\beta }-\underline{\beta }\right) }{t^2}>0 \end{aligned}$$

(56)

Derivative with respect to c:

$$\begin{aligned} \frac{\partial RI(y_y)}{\partial c}=-\frac{\overline{\beta }-\underline{\beta }}{t}<0 \end{aligned}$$

(57)

Case 2b $y_y<\omega$ For similar reasons to Case 2a, $y_y>\omega$ is only possible if $\frac{1}{16}(\overline{\beta }-\underline{\beta })\le c \le \frac{1}{4}(\overline{\beta }-\underline{\beta })$ Reported income:

$$\begin{aligned} \begin{aligned} RI(y_{y_2})&= \int _0^{\underline{\beta }} y \, dy + \int _{\underline{\beta }}^{y_y} r \, dy+\int _{y_y}^m \omega \, dy\\&\frac{2 c \overline{\beta }+\underline{\beta } \left( 4 \sqrt{c t \left( \overline{\beta }-\underline{\beta }\right) }-2 c-2 t \omega \right) -4 \omega \sqrt{c t \left( \overline{\beta }-\underline{\beta }\right) }+t \underline{\beta }^2+2 m t \omega }{2 t} \end{aligned} \end{aligned}$$

(58)

Taking the derivative with respect to $\underline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI(y_{y_2})}{\partial \underline{\beta }} =\frac{5 \sqrt{c t \left( \overline{\beta }-\underline{\beta }\right) }-t \overline{\beta }+t \underline{\beta }-2 c}{2 t}>0 \end{aligned}$$

(59)

Taking the derivative with respect to $\overline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI(y_{y_2})}{\partial \overline{\beta }} =-\frac{\sqrt{c t \left( \overline{\beta }-\underline{\beta }\right) }-2 c}{2 t}<0 \end{aligned}$$

(60)

Derivative of reported income with respect to $\epsilon =({\overline{\beta }} -{\underline{\beta }})$: To do this, define ${\overline{\beta }}=\omega +\epsilon$ and ${\underline{\beta }}=\omega -\epsilon$ and take derivatives with respect to $\epsilon$, and then substitute $\epsilon =(\overline{\beta }-\underline{\beta })$ and $\omega =\frac{1}{2} (\overline{\beta }+\underline{\beta })$

$$\begin{aligned} \frac{\partial RI(y_{y_2})}{\partial {\epsilon }} =\frac{-6 \sqrt{c t \left( \overline{\beta }-\underline{\beta }\right) }+t \left( \overline{\beta }-\underline{\beta }\right) +4 c}{4 t}<0 \end{aligned}$$

(61)

Derivative with respect to t:

$$\begin{aligned} \frac{\partial RI(y_{y_2})}{\partial t}=\frac{\left( \overline{\beta }-\underline{\beta }\right) \left( \sqrt{c t \left( \overline{\beta }-\underline{\beta }\right) }-2 c\right) }{2 t^2}>0 \end{aligned}$$

(62)

Derivative with respect to c:

$$\begin{aligned} \frac{\partial RI(y_{y_2})}{\partial c}=-\frac{\left( \overline{\beta }-\underline{\beta }\right) \left( \sqrt{c t \left( \overline{\beta }-\underline{\beta }\right) }-2 c\right) }{2 c t}<0 \end{aligned}$$

(63)

Case 3 Only taxpayers with income above the first demand threshold, i.e., $y>y_{\overline{\beta }}=2\overline{\beta }-\underline{\beta }+\frac{\sqrt{2t((\overline{\beta }-\underline{\beta })(2t(\overline{\beta }-\underline{\beta })-2c)}}{t},$ hire a firm.

From Eq. 33, one can see that to have $y_d(c)=y_y$, it should be the case that $\frac{1}{4}(\overline{\beta }-\underline{\beta })\le c \le \frac{1}{2}(\overline{\beta }-\underline{\beta }).$

Reported income:

$$\begin{aligned} \begin{aligned}&RI(y_{\overline{\beta }})= \int _0^{\underline{\beta }} y \, dy+\int _{\underline{\beta }}^{y_2} r \, dy+\int _{y_2}^m \omega \, dy\\&\frac{2 \left( c \underline{\beta }-\overline{\beta } \left( t \underline{\beta }+c+2 t \omega \right) +t \omega \left( \underline{\beta }+m\right) \right) +\left( 2 \omega -2 \overline{\beta }\right) \sqrt{2 t \left( \overline{\beta }-\underline{\beta }\right) \left( t \left( \overline{\beta }-\underline{\beta }\right) -2 c\right) }+3 t \overline{\beta }^2}{2 t}\\ \end{aligned} \end{aligned}$$

(64)

Taking the derivative with respect to $\underline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI(y_{\overline{\beta }})}{\partial \underline{\beta }} =\frac{1}{2} \left( -\left( \overline{\beta }-\underline{\beta }\right) +\frac{\sqrt{2} \left( \overline{\beta }-\underline{\beta }\right) \left( t \left( \overline{\beta }-\underline{\beta }\right) -c\right) }{\sqrt{t \left( \overline{\beta }-\underline{\beta }\right) \left( t \left( \overline{\beta }-\underline{\beta }\right) -2 c\right) }}+\frac{2 c}{t}\right) >0 \end{aligned}$$

(65)

Taking the derivative with respect to $\overline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned}&\frac{\partial RI(y_{\overline{\beta }})}{\partial \overline{\beta }} =2 \left( \overline{\beta }-\underline{\beta }\right) -\frac{\left( \overline{\beta }-\underline{\beta }\right) \left( 3 t \left( \overline{\beta }-\underline{\beta }\right) -5 c\right) }{\sqrt{2 t \left( \overline{\beta }-\underline{\beta }\right) \left( t \left( \overline{\beta }-\underline{\beta }\right) -2 c\right) }}-\frac{c}{t} \end{aligned}$$

(66)

$$\begin{aligned}&\frac{\partial RI(y_{\overline{\beta }})}{\partial \overline{\beta }}= {\left\{ \begin{array}{ll} >0 &{}\text {if } \frac{1}{4}t(\overline{\beta }-\underline{\beta })\le c \le (\sqrt{2}-1)t(\overline{\beta }-\underline{\beta }) \\ <0 &{}\text {if } (\sqrt{2}-1)t(\overline{\beta }-\underline{\beta })\le c \le \frac{1}{2}t(\overline{\beta }-\underline{\beta }) \end{array}\right. } \end{aligned}$$

(67)

Derivative of reported income with respect to $\epsilon =({\overline{\beta }} -{\underline{\beta }})$: To do this, define ${\overline{\beta }}=\omega +\epsilon$ and ${\underline{\beta }}=\omega -\epsilon$ and take derivatives with respect to $\epsilon$, and then substitute $\epsilon =(\overline{\beta }-\underline{\beta })$ and $\omega =\frac{1}{2} (\overline{\beta }+\underline{\beta })$

$$\begin{aligned} \frac{\partial RI(y_{\overline{\beta }})}{\partial {\epsilon }} =\frac{5}{2} \left( \overline{\beta }-\underline{\beta }\right) +\frac{\left( \overline{\beta }-\underline{\beta }\right) \left( 3 c-2 t \left( \overline{\beta }-\underline{\beta }\right) \right) }{\sqrt{\frac{1}{2} t \left( \overline{\beta }-\underline{\beta }\right) \left( t \overline{\beta }-t \underline{\beta }-2 c\right) }}-\frac{2 c}{t}<0 \end{aligned}$$

(68)

Derivative with respect to t

$$\begin{aligned} \frac{\partial RI(y_{\overline{\beta }})}{\partial t}=-\frac{c \left( \overline{\beta }-\underline{\beta }\right) \left( \sqrt{2} \sqrt{t \left( \underline{\beta }-\overline{\beta }\right) \left( -t \overline{\beta }+t \underline{\beta }+2 c\right) }-2 t \overline{\beta }+2 t \underline{\beta }+4 c\right) }{2 t^2 \left( t \overline{\beta }-t \underline{\beta }-2 c\right) }<0 \end{aligned}$$

(69)

Derivative with respect to c:

$$\begin{aligned} \frac{\partial RI(y_r)RI(y_{\overline{\beta }})}{\partial c}=\frac{\left( \overline{\beta }-\underline{\beta }\right) \left( \sqrt{2} \sqrt{t \left( \underline{\beta }-\overline{\beta }\right) \left( -t \overline{\beta }+t \underline{\beta }+2 c\right) }-2 t \overline{\beta }+2 t \underline{\beta }+4 c\right) }{2 t \left( t \overline{\beta }-t \underline{\beta }-2 c\right) }>0 \end{aligned}$$

(70)

Case 4 Only taxpayers with income above ${\overline{y}}$, i.e., $y_d(c)={\overline{y}}.$

Again, from Eq. 33, one can see that to have $y_d(c)={\overline{y}}$, it should be the case that $c=\frac{1}{2}(\overline{\beta }-\underline{\beta }).$

Reported income:

$$\begin{aligned} \begin{aligned} RI({\overline{y}})&= \int _0^{\underline{\beta }} y \, dy+\int _{\underline{\beta }}^{{\overline{y}}} r \, dy+\int _{{\overline{y}}}^m \omega \, dy\\&\overline{\beta }^2-\frac{\underline{\beta }^2}{2}+\omega \left( m-2 \overline{\beta }+\underline{\beta }\right) \end{aligned} \end{aligned}$$

(71)

Taking the derivative with respect to $\underline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI({\overline{y}})}{\partial \underline{\beta }} =\frac{1}{2} \left( \overline{\beta }-\underline{\beta }\right) >0 \end{aligned}$$

(72)

Taking the derivative with respect to $\overline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI({\overline{y}})}{\partial \overline{\beta }} =\overline{\beta }-\underline{\beta }>0 \end{aligned}$$

(73)

Derivative of reported income with respect to $\epsilon =({\overline{\beta }} -{\underline{\beta }})$: To do this, define ${\overline{\beta }}=\omega +\epsilon$ and ${\underline{\beta }}=\omega -\epsilon$ and take derivatives with respect to $\epsilon$, and then substitute $\epsilon =(\overline{\beta }-\underline{\beta })$ and $\omega =\frac{1}{2} (\overline{\beta }+\underline{\beta })$

$$\begin{aligned} \frac{\partial RI(y_{{\overline{y}}})}{\partial {\epsilon }} =\frac{1}{2} \left( \overline{\beta }-\underline{\beta }\right) >0 \end{aligned}$$

(74)

Derivative with respect to t:

$$\begin{aligned} \frac{\partial RI({\overline{y}})}{\partial t}=0 \end{aligned}$$

(75)

Derivative with respect to c:

$$\begin{aligned} \frac{\partial RI({\overline{y}})}{\partial c}=0 \end{aligned}$$

(76)

Case 5 No one hires the firm, i.e., $y_d(c)=m$.

Reported income:

$$\begin{aligned} \begin{aligned} RI(m)&= \int _0^{\underline{\beta }} y \, dy + \int _{\underline{\beta }}^{{\overline{y}}} r \, dy + \int _{{\overline{y}}}^m \overline{\beta } \, dy \\&\quad -\overline{\beta }^2-\frac{\underline{\beta }^2}{2}+\overline{\beta } \left( \underline{\beta }+m\right) \end{aligned} \end{aligned}$$

(77)

Taking the derivative with respect to $\underline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI(m)}{\partial \overline{\beta }} =\overline{\beta }-\underline{\beta }>0 \end{aligned}$$

(78)

Taking the derivative with respect to $\overline{\beta }$ and then substituting $\omega =\frac{\overline{\beta }+\underline{\beta }}{2}$

$$\begin{aligned} \frac{\partial RI(m)}{\partial \underline{\beta }} =m-2 \overline{\beta }+\underline{\beta }>0 \end{aligned}$$

(79)

Derivative of reported income with respect to $\epsilon =({\overline{\beta }} -{\underline{\beta }})$:

$$\begin{aligned} \frac{\partial RI(y_m}{\partial {\epsilon }} =0 \end{aligned}$$

(80)

Derivative with respect to t:

$$\begin{aligned} \frac{\partial RI(m)}{\partial t}=0 \end{aligned}$$

(81)

Derivative with respect to c:

$$\begin{aligned} \frac{\partial RI(m)}{\partial c}=0 \end{aligned}$$

(82)

1.4 A.4 Effect of tax preparation firms when tax preparation costs are increasing in income

Demand cutoff schedule is as follows:

If $c<\frac{t(\overline{\beta }-\underline{\beta })}{4\overline{\beta }}$

$$\begin{aligned} D(c)=\left[ \frac{2 c \overline{\beta }+\underline{\beta } (t-2 c)+2 \sqrt{c \left( \overline{\beta }-\underline{\beta }\right) \left( c \overline{\beta }+\underline{\beta } (t-c)\right) }}{t},\frac{t \left( \overline{\beta }-\underline{\beta }\right) }{2 c}\right] \end{aligned}$$

(83)

If $\frac{t(\overline{\beta }-\underline{\beta })}{4\overline{\beta }}<c<\frac{t(\overline{\beta }-\underline{\beta })}{4\overline{\beta }-2\underline{\beta }}$

$$\begin{aligned} D(c)=\left[ \frac{-4 c \overline{\beta }+4 c \underline{\beta }-\sqrt{\left( 4 c \overline{\beta }-4 c \underline{\beta }-4 t \overline{\beta }+2 t \underline{\beta }\right) ^2-4 t \left( 2 t \overline{\beta }^2-t \underline{\beta }^2\right) }+4 t \overline{\beta }-2 t \underline{\beta }}{2 t},\frac{t \left( \overline{\beta }-\underline{\beta }\right) }{2 c}\right] \end{aligned}$$

(84)

If $\frac{t(\overline{\beta }-\underline{\beta })}{4\overline{\beta }-2\underline{\beta }}<c<\frac{t \left( {\overline{y}}-\sqrt{2 \overline{\beta }^2-\underline{\beta }^2}\right) }{2 \left( \overline{\beta }-\underline{\beta }\right) }$

$$\begin{aligned} \begin{aligned} D(c)=\left[ {\overline{y}}-\frac{4 c( \overline{\beta }- \underline{\beta })+\sqrt{\left( 4 c \overline{\beta }-4 c \underline{\beta }-4 t \overline{\beta }+2 t \underline{\beta }\right) ^2-4 t \left( 2 t \overline{\beta }^2-t \underline{\beta }^2\right) }}{2 t}, \right. \\ \left. {\overline{y}}-\frac{4 c( \overline{\beta }- \underline{\beta })-\sqrt{\left( 4 c \overline{\beta }-4 c \underline{\beta }-4 t \overline{\beta }+2 t \underline{\beta }\right) ^2-4 t \left( 2 t \overline{\beta }^2-t \underline{\beta }^2\right) }}{2 t}\right] \end{aligned} \end{aligned}$$

(85)

Where the first interval spans the part of region 1, all of region 2, and part of region 3. The second interval spans parts of regions 2 and 3, and the third interval only spans parts of second region 2.

1.5 A.5 Effect of tax preparation firms when firms have imperfect information

The ex post income of a taxpayer, denoted by $EP_{self}$, who reports income based on her optimal reporting strategy, $r_{self}$, is as follows:

$$\begin{aligned} EP_{self}= {\left\{ \begin{array}{ll} y(1-t) &{}\text {if } \ y\le \underline{\beta } \\ \frac{(y-tr_{self})}{2}+\frac{(y-tr_{self}-(y-r_{self})t(1+\pi ))}{2} &{}\text {if } \ \underline{\beta } \le y \le \overline{\beta }\\ y-tr_{self} &{}\text {if } \quad \overline{\beta } \le y \le 2\overline{\beta }-\underline{\beta }\\ y-{\overline{\beta }} t &{}\text {if } \ y\ge 2\overline{\beta }-\underline{\beta } \end{array}\right. } \end{aligned}$$

(86)

Substituting $r_{self}$ simplifies $EP_{self}$ as follows:

$$\begin{aligned} EP_{self}= {\left\{ \begin{array}{ll} y(1-t) &{}\text {if } \ 0 \le y \le \overline{\beta } \\ y-t\frac{y+\underline{\beta }}{2} &{}\text {if } \ \overline{\beta } \le y \le 2\overline{\beta }-\underline{\beta }\\ y-{\overline{\beta }} t &{}\text {if } \ y\ge 2\overline{\beta }-\underline{\beta } \end{array}\right. } \end{aligned}$$

(87)

Similarly, replacing $[\underline{\beta },\overline{\beta }]$ by $[\underline{\beta }_f,\overline{\beta }_f]$ in Eq. 87 yields the ex post income of a taxpayer who hires a firm, denoted by $EP_{firm}$, can be defined as follows:

$$\begin{aligned} EP_{firm}= {\left\{ \begin{array}{ll} y(1-t) &{}\text {if } \ \underline{\beta }_f \le y \le \overline{\beta }_f \\ y-t\frac{y+\underline{\beta }_f}{2} &{}\text {if } \ \overline{\beta }_f \le y \le 2\overline{\beta }_f-\underline{\beta }_f\\ y-{\overline{\beta }_f} t &{}\text {if } \ y\ge 2\overline{\beta }_f-\underline{\beta }_f \end{array}\right. } \end{aligned}$$

(88)

The ex post excess benefit of hiring a firm is then, as follows:

$$\begin{aligned} EP_{dif}= {\left\{ \begin{array}{ll} 0 \quad &{}\text {if } \ 0 \le y \le \overline{\beta }_f \\ t y-\frac{1}{2} t \left( \underline{\beta }_{f}+y\right) \quad &{}\text {if } \ \overline{\beta }_f \le y \le \overline{\beta }\\ \frac{1}{2} t \left( \underline{\beta }+y\right) -\frac{1}{2} t \left( \underline{\beta }_{f}+y\right) \quad &{}\text {if } \ \overline{\beta } \le y \le 2\overline{\beta }_f-\underline{\beta }_f\\ \frac{1}{2} t \left( \underline{\beta }+y\right) -t \overline{\beta }_{f} \quad &{}\text {if } \ 2\overline{\beta }_f-\underline{\beta }_f \le y \le 2\overline{\beta }-\underline{\beta }\\ t (\overline{\beta }-\overline{\beta }_{f}) \quad &{}\text {if } \ y\ge 2\overline{\beta }-\underline{\beta } \end{array}\right. } \end{aligned}$$

(89)

Note that assuming that the taxpayer knows the exact ex post excess benefit of hiring a firm implies that the she also knows the belief spread of the firm. To avoid such an interpretation, I will define the average additional benefit for a taxpayer based on her reporting strategy and true income.

First, consider an individual whose income is below $\underline{\beta }$: this individual always reports her true income, regardless of the preparation method; hence, she does not need to hire a firm. Next, consider a taxpayer whose income is somewhere in the interval $[\underline{\beta },{\overline{y}}]$. The average ex post excess benefit of this individual is characterized by amassing ex post excess benefit from hiring a firm in the interval $[\underline{\beta },{\overline{y}}]$ and dividing by the interval itself.

$$\begin{aligned} \begin{aligned}&\frac{\displaystyle \int \limits _{\overline{\beta }_f}^{\overline{\beta }}\left( t y-\frac{1}{2} t \left( \underline{\beta }_{f}+y\right) \right) dy+\int \limits _{\overline{\beta }}^{2\overline{\beta }_f-\underline{\beta }_f} \left( \frac{1}{2} t \left( \underline{\beta }+y\right) -\frac{1}{2} t \left( \underline{\beta }_{f}+y\right) \right) dy}{(2 \overline{\beta }- \underline{\beta })- \underline{\beta }}\\&\qquad +\frac{\displaystyle \int \limits _{2\overline{\beta }_f-\underline{\beta }_f}^{2\overline{\beta }-\underline{\beta }}\left( \frac{1}{2} t \left( \underline{\beta }+y\right) -t \overline{\beta }_{f}\right) dy}{(2 \overline{\beta }- \underline{\beta })- \underline{\beta }}\\&\quad =\frac{t \left( -2 \overline{\beta } \left( \underline{\beta }+4 \overline{\beta }_{f}\right) +\overline{\beta }_{f} \left( 4 \underline{\beta }-2 \underline{\beta }_{f}\right) +5 \overline{\beta }^2-\underline{\beta }^2+3 \overline{\beta }_{f}^2+\underline{\beta }_{f}^2\right) }{8 \left( \overline{\beta }-\underline{\beta }\right) } \end{aligned} \end{aligned}$$

(90)

Similarly, the average ex post excess benefit of hiring a firm for a taxpayer with income in the remaining interval, i.e., $y\in [2\overline{\beta }-\underline{\beta },m]$, where m is the top income level as defined before, is as follows:

$$\begin{aligned} \begin{aligned}&\frac{\displaystyle \int \limits _{2\overline{\beta }-\underline{\beta }}^{m}\left( t (\overline{\beta }-\overline{\beta }_{f})\right) dy}{m-(2 \overline{\beta }- \underline{\beta })}\\&\quad = t (\overline{\beta }-\overline{\beta }_{f}) \end{aligned} \end{aligned}$$

(91)

For analytical simplicity, assume that $\underline{\beta }_f=\frac{1}{4} \left( \overline{\beta }+3 \underline{\beta }\right)$, i.e., is halfway between $\underline{\beta }$ and $\omega$, and $\overline{\beta }_f=\frac{1}{4} \left( \underline{\beta }+3 \overline{\beta }\right)$, i.e., is halfway between $\overline{\beta }$ and $\omega$. Then, the average ex post benefit function can be written as follows:

$$\begin{aligned} AEP_{dif}= {\left\{ \begin{array}{ll} 0 \quad &{}\text {if } \ y\le \underline{\beta } \\ \frac{3}{64} t \left( \overline{\beta }-\underline{\beta }\right) \quad &{}\text {if } \ \underline{\beta }\le y \le 2\overline{\beta }-\underline{\beta }\\ \frac{1}{4} t \left( \overline{\beta }-\underline{\beta }\right) \quad &{}\text {if } \ y\ge 2\overline{\beta }-\underline{\beta } \end{array}\right. } \end{aligned}$$

(92)

The interpretation of the above equation is that the agent knows that hiring a tax preparer firm will have some benefit, but she is not sure exactly what that benefit is as she does not know what the firm knows. By assuming a variant of rational expectations, the agent expects to receive an average benefit from hiring a firm derived by using the ex post incomes. A taxpayer will hire a firm if her relevant $AEP_{dif}>c$, where c is the marginal cost of tax preparation services. The demand for tax preparation services in this case can be characterized by the following equation:

$$\begin{aligned} D(c)= {\left\{ \begin{array}{ll} {[}{\underline{\beta }},m] &{}\text {if } \ 0< c \le \frac{3}{64} t \left( {\overline{\beta }}-{\underline{\beta }}\right) \\ {[}2{\overline{\beta }}-{\underline{\beta }},m] &{}\text {if } \ \frac{3}{64} t \left( {\overline{\beta }}-{\underline{\beta }}\right) < c \le \frac{1}{4} t \left( {\overline{\beta }}-{\underline{\beta }}\right) \end{array}\right. } \end{aligned}$$

(93)

1.5.1 A.5.1 Effect of a tax preparation firm when the tax preparer market is a monopoly

The monopoly demand cutoff schedule as a function of the fee, f, the monopolist charges, is as follows:

$$\begin{aligned} y_d(f)= {\left\{ \begin{array}{ll} y_{y}=\underline{\beta }+\frac{2\sqrt{ft(\overline{\beta }-\underline{\beta })}}{t} \quad &{}\text {if } \ 0\le f \le \frac{t}{4}(\overline{\beta }-\underline{\beta })\\ y_{{\overline{\beta }}}=2\overline{\beta }-\underline{\beta }-\frac{\sqrt{2t((\overline{\beta }-\underline{\beta })(2t(\overline{\beta }-\underline{\beta })-2f)}}{t} &{}\text {if } \ \frac{t}{4}(\overline{\beta }-\underline{\beta })\le f \le \frac{1}{2}({\overline{\beta }}-{\underline{\beta }})t\\ m &{}\text {if } \ f> \frac{1}{2}({\overline{\beta }}-{\underline{\beta }})t \end{array}\right. } \end{aligned}$$

(94)

Given a demand cutoff $y_d(f)$, the firm maximizes the following profit function:

$$\begin{aligned} (m-y_d (f))(f-c) \end{aligned}$$

(95)

The maximization problem then yields the following fee schedule:

$$\begin{aligned} f(y_y)=&\frac{6 c \overline{\beta }-6 c \underline{\beta }+\sqrt{t} \left( \underline{\beta }-m\right) \sqrt{12 c \overline{\beta }-2 \underline{\beta } (6 c+m t)+t \underline{\beta }^2+m^2 t}-2 m t \underline{\beta }+t \underline{\beta }^2+m^2 t}{18 \left( \overline{\beta }-\underline{\beta }\right) } \end{aligned}$$

(96)

$$\begin{aligned} f(y_{\overline{\beta }})=&\frac{\sqrt{t^2 \left( -2 \overline{\beta }+\underline{\beta }+m\right) ^2 \left( 10 \overline{\beta }^2+7 \underline{\beta }^2+2 m \underline{\beta }-4 \overline{\beta } \left( 4 \underline{\beta }+m\right) +m^2\right) }}{18 \left( \overline{\beta }-\underline{\beta }\right) }\nonumber \\&\frac{t \left( -5 \underline{\beta }^2-2 \overline{\beta } \left( \overline{\beta }+2 m\right) +2 \underline{\beta } \left( 4 \overline{\beta }+m\right) +m^2\right) }{18 \left( \overline{\beta }-\underline{\beta }\right) } \end{aligned}$$

(97)

$$\begin{aligned} f({\overline{y}})=&\frac{1}{2} t \left( \overline{\beta }-\underline{\beta }\right) \end{aligned}$$

(98)

1.5.2 A.5.2 Risk-averse agents

Changing the risk preference of agents changes self-preparation expected income more than it affects anything else in the model. In cases where an agent chooses to underreport her income, imposing risk-averse behavior will decrease the level of expected income under self-preparation more than firm preparation. This is because agents are exposed to the risk of being audited in the former case, whereas there is no risk of audit in the latter. This implies that given a fixed marginal cost, demand for tax preparation services will be higher in the model with risk-averse agents. However, it is not obvious whether this change in assumptions will always lead to a higher evasion level.