Dynamic tax evasion with audits based on visible consumption

Abstract

We solve the problem of a representative agent who maximises the expected present utility of his intertemporal consumption under the assumption that an optimal fraction of his wealth is hidden to the tax authorities (we show conditions under which evasion is expedient). Evasion affects the capital dynamics in two ways: the growth rate of capital increases because some taxes are not paid, but when caught evading the consumer has to pay a fine (proportional to evasion). Consumption can be allocated between ordinary goods and so-called visible goods. The latter are used by the Government for targeting the audit, since they are considered like an indicator of consumer’s income. In fact, the probability of being caught is a function of the distance between the actual and the presumed consumption in visible goods. We find a closed form solution to the dynamic optimization problem and show how fiscal and audit parameters affect the optimal evasion and the optimal allocation between the two types of consumptions.

Appendices

Appendix

Optimization problem

Given the optimization problem (5) and the behaviour of the capital in (3) the corresponding Hamilton–Jacobi–Bellman equation is

$$\begin{aligned} 0= & {} \frac{\partial J_{t}\left( k_{t}\right) }{\partial t}-\rho J_{t}\left( k_{t}\right) +\frac{\partial J_{t}\left( k_{t}\right) }{\partial k_{t}}\left( 1-\tau \right) Ak_{t}\\&+\max _{c_{t},e_{t},g_{t}}\left[ \begin{array}{l} c_{t}+\beta \frac{\left( g_{t}-\underline{g}\right) ^{1-\delta }}{1-\delta }+\frac{\partial J_{t}\left( k_{t}\right) }{\partial k_{t}}\left( \tau e_{t}Ak_{t}-pc_{t}-g_{t}\right) \\ +\lambda _{t}\left( c_{t},e_{t}\right) \left( J_{t}\left( k_{t}-\eta e_{t}Ak_{t}\right) -J_{t}\left( k_{t}\right) \right) \end{array}\right] , \end{aligned}$$

where $J_{t}\left( k_{t}\right) e^{-\rho t}$ is the value function, whose boundary (transversality) condition is

$$\begin{aligned} \lim _{t\rightarrow \infty }J_{t}\left( \bullet \right) e^{-\rho t}=0 \end{aligned}$$

(13)

Given (4), the derivatives of $\lambda _{t}\left( c_{t},e_{t}\right)$ w.r.t consumption and evasion are

$$\begin{aligned}&\frac{\partial \lambda _{t}\left( c_{t},e_{t}\right) }{\partial c_{t}}=\lambda _{1}p,\\&\frac{\partial \lambda _{t}\left( c_{t},e_{t}\right) }{\partial e_{t}}=\lambda _{1}\alpha y_{t}\left( 1-\tau \right) . \end{aligned}$$

The Jacobian of the optimization problem, with respect to $c_{t}$, $e_{t}$ and $g_{t}$ is

$$\begin{aligned} \nabla _{c_{t},e_{t},g_{t}}J=\left[ \begin{array}{l} 1-\frac{\partial J_{t}\left( k_{t}\right) }{\partial k_{t}}p+p\lambda _{1}\left( J_{t}\left( k_{t}-\eta e_{t}Ak_{t}\right) -J_{t}\left( k_{t}\right) \right) \\ \frac{\partial J_{t}\left( k_{t}\right) }{\partial k_{t}}\tau Ak_{t}+\lambda _{1}\alpha y_{t}\left( 1-\tau \right) \left( J_{t}\left( k_{t}-\eta e_{t}Ak_{t}\right) -J_{t}\left( k_{t}\right) \right) \\ \quad +\lambda _{t}\left( c_{t},e_{t}\right) \frac{\partial J_{t}\left( k_{t}-\eta e_{t}Ak_{t}\right) }{\partial e_{t}}\\ \beta \left( g_{t}-\underline{g}\right) ^{-\delta }-\frac{\partial J_{t}\left( k_{t}\right) }{\partial k_{t}} \end{array}\right] , \end{aligned}$$

and the Hessian matrix is

$$\begin{aligned} \mathcal {H}_{c_{t},e_{t},g_{t}}J=\left[ \begin{array}{llllll} 0&{} \quad p\lambda _{1}\frac{\partial J_{t}\left( k_{t}-\eta e_{t}Ak_{t}\right) }{\partial e_{t}} &{} \quad 0\\ p\lambda _{1}\frac{\partial J_{t}\left( k_{t}-\eta e_{t}Ak_{t}\right) }{\partial e_{t}}\lambda _{1}&{}\quad 2\alpha y_{t}\left( 1-\tau \right) \frac{\partial J_{t}\left( k_{t}-\eta e_{t}Ak_{t}\right) }{\partial e_{t}}+\lambda _{t}\left( c_{t},e_{t}\right) \frac{\partial ^{2}J_{t}\left( k_{t}-\eta e_{t}Ak_{t}\right) }{\partial e_{t}^{2}} &{} \quad 0\\ 0 &{} \quad 0 &{} \quad -\delta \beta \left( g_{t}-\underline{g}\right) ^{-\delta -1} \end{array}\right] \end{aligned}$$

In order to solve these FOCs ($\nabla _{c_{t},e_{t},g_{t}}J=0$), a functional form for the guess function must be guessed. Since the utility function is linear, also the value function should be linear in capital. Thus, we assume³

$$\begin{aligned} J_{t}\left( k_{t}\right) =Fk_{t}+G, \end{aligned}$$

where F and G are constant that must solve the HJB equation. With this functional form the Jacobian and the Hessian become

$$\begin{aligned}&\nabla _{c_{t},e_{t},g_{t}}J=\left[ \begin{array}{l} 1-Fp-p\lambda _{1}\eta e_{t}Ak_{t}F\\ F\tau Ak_{t}-\lambda _{1}\alpha y_{t}\left( 1-\tau \right) \eta e_{t}Ak_{t}F-\lambda _{t}\left( c_{t},e_{t}\right) F\eta Ak_{t}\\ \beta \left( g_{t}-\underline{g}\right) ^{-\delta }-F \end{array}\right] ,\\&\mathcal {H}_{c_{t},e_{t},g_{t}}J=\left[ \begin{array}{lll} 0 &{}\quad -p\lambda _{1}\eta Ak_{t}F &{}\quad 0\\ -p\lambda _{1}\eta Ak_{t}F &{}\quad -\lambda _{1}2\alpha y_{t}\left( 1-\tau \right) \eta Ak_{t}F &{}\quad 0\\ 0 &{}\quad 0 &{}\quad -\delta \beta \left( g_{t}-\underline{g}\right) ^{-\delta -1} \end{array}\right] , \end{aligned}$$

where, for any positive F, we see that the stationary point is a saddle.

If this guess value function is substituted into the FOCs, the optimal evasion and consumption are obtained as functions of the constant F:

$$\begin{aligned}&e_{t}^{*}y_{t}=\frac{1-pF}{pF}\frac{1}{\lambda _{1}\eta },\\&pc_{t}^{*}=\alpha \left( 1-\tau \right) y_{t}+\frac{\tau -\eta \lambda _{0}-2\alpha \left( 1-\tau \right) \frac{1-pF}{pF}}{\eta \lambda _{1}}\\&g_{t}^{*}=\underline{g}+\left( \frac{F}{\beta }\right) ^{-\frac{1}{\delta }}. \end{aligned}$$

Now, $c_{t}^{*}$, $e_{t}^{*}$ and $g_{t}^{*}$ are substituted into the HJB equation in order to find G and F. The HJB becomes

$$\begin{aligned} 0= & {} -\rho Fk_{t}-\rho G+\left( F\left( 1-\alpha \right) +\frac{\alpha }{p}\right) \left( 1-\tau \right) y_{t}+\frac{\tau -\eta \lambda _{0}}{\eta \lambda _{1}}\left( \frac{1}{p}-F\right) \\&-\frac{\alpha \left( 1-\tau \right) }{\eta \lambda _{1}}\frac{\left( 1-pF\right) ^{2}}{p^{2}F}+\frac{\delta }{1-\delta }F^{1-\frac{1}{\delta }}\beta ^{\frac{1}{\delta }}-F\underline{g}, \end{aligned}$$

which can be split into two equations, one which contains $k_{t}$ and one which does not:

$$\begin{aligned} 0= & {} -\rho Fk_{t}+\left( F\left( 1-\alpha \right) +\frac{\alpha }{p}\right) \left( 1-\tau \right) Ak_{t}\\ 0= & {} -\rho G+\frac{\tau -\eta \lambda _{0}}{\eta \lambda _{1}}\left( \frac{1}{p}-F\right) -\frac{\alpha \left( 1-\tau \right) }{\eta \lambda _{1}}\frac{\left( 1-pF\right) ^{2}}{p^{2}F}+\frac{\delta }{1-\delta }F^{1-\frac{1}{\delta }}\beta ^{\frac{1}{\delta }}-F\underline{g}, \end{aligned}$$

from the first equation the value of F can be found:

$$\begin{aligned} F=\frac{\alpha }{p}\frac{\left( 1-\tau \right) A}{\rho -\left( 1-\alpha \right) \left( 1-\tau \right) A}, \end{aligned}$$

while the value of G is obtained from the second equation (but it is immaterial to our purposes). After substituting this value of F into the FOCs, the optimal values $c_{t}^{*}$, $e_{t}^{*}$ and $g_{t}^{*}$ in Proposition (1) are obtained.