Underground activities and labour market performance

Abstract

We build a general equilibrium search and matching model with an informal sector. We consider the impact of traditional policy instruments discussed in the tax evasion literature, such as changes in the tax and punishment system and the employment protection legislation, as well as the impact of concealment costs, on labour market outcomes. The model is calibrated to and simulated on the northern and southern European countries, where countries in the south have significantly higher informal sectors than countries in the north. We conclude that differences in tax and punishment systems cannot explain the observed difference. Instead, we find that stricter employment protection legislation in southern Europe, as well as the higher tax morale and more extensive use of third-party reporting in northern Europe, are potential candidates for explaining the difference.

Appendix

In this Appendix, we derive the impact on labour market performance of higher concealment costs, expected tax and punishment rates as well as higher employment protection.

1.1 Impact of concealment costs when $\mu =0$:

We consider the equilibrium when $\mu =0,$ which is given by Eqs. (16), (18) and then Eq. (17) when $\mu =0$, that is

$$\begin{aligned} 2\left( r+s\right) k\left( \theta ^{F}\right) ^{\eta }=y-\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }} \end{aligned}$$

(20)

We differentiate the three equations in $\sigma$, $\theta ^{F},$$\theta ^{I}$ and $\kappa$ to obtain around the equilibrium and for $\psi >1$:

$$\begin{aligned} \frac{\mathrm{d}\sigma }{\mathrm{d}\kappa }=\frac{\eta \left( y+\frac{\left( 1-\eta \right) }{\eta }\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\right) }{D}\left( \theta ^{F}\right) ^{-1}\frac{1}{\psi } \frac{\mathrm{d}\psi }{\mathrm{d}\kappa }<0, \end{aligned}$$

$$\begin{aligned} \frac{\mathrm{d}\theta ^{F}}{\mathrm{d}\kappa }= & {} \frac{\frac{\left( 1-\gamma \right) }{1-\sigma } \frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}}{-D}\frac{1}{\psi } \frac{\mathrm{d}\psi }{\mathrm{d}\kappa }>0,\\ \frac{\mathrm{d}\theta ^{I}}{\mathrm{d}\kappa }= & {} \frac{\frac{\left( 1-\gamma \right) }{\sigma } \frac{k\theta ^{I}}{\sigma ^{1-\gamma }}}{\eta \left( y+\frac{\left( 1-\eta \right) }{\eta }\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) }\theta ^{I}\frac{\mathrm{d}\sigma }{\mathrm{d}\kappa }<0, \end{aligned}$$

where

$$\begin{aligned} D= & {} -\frac{1-\gamma }{\sigma }\left( \left( \frac{1}{1-\sigma }-F\right) \left( y-\frac{k \theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\right) \eta +\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\left( 1-F\right) \right) \\&\quad \left( \theta ^{F}\right) ^{-1}<0, \end{aligned}$$

where $F=\frac{\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}}{\left( \eta \left( y-\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) +\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) }<1$ and as $\left( y-\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\right) >0$ from Eq. (17). The impact on unemployment is then:

$$\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}\kappa }=\frac{-s}{\left( s+\lambda ^{F}+\lambda ^{I}\right) ^{2}}\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\kappa }+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\kappa }\right) . \end{aligned}$$

Unemployment therefore increases if $\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\kappa }+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\kappa }\right) <0$. We derive this sum to be

$$\begin{aligned} \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\kappa }+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\kappa }= & {} \sigma _{i}^{\gamma } \left( \theta ^{I}\right) ^{\left( 1-\eta \right) }\frac{1}{\sigma }\left( \left( 1-\frac{\left( \theta ^{I}\right) ^{\eta }}{\left( \theta ^{F}\right) ^{\eta }}\frac{1}{\psi }\right) \gamma \frac{\mathrm{d}\sigma }{\mathrm{d}\kappa }-\frac{ \left( 1-\gamma \right) \left( 1-\eta \right) }{-D\psi ^{2}\theta ^{F}} \frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right. \\&\quad \left. \left( \frac{\left( \psi y+ \frac{\left( 1-\eta \right) }{\eta }\frac{k\theta ^{I}}{\sigma ^{1-\gamma }} \right) }{\left( y+\frac{\left( 1-\eta \right) }{\eta }\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) }-\frac{\left( \theta ^{I}\right) ^{\eta }}{ \left( \theta ^{F}\right) ^{\eta }} \frac{1}{\psi }\right) \frac{\mathrm{d}\psi }{\mathrm{d}\kappa }\right) \lesseqgtr 0, \end{aligned}$$

for $\psi \gtreqless 1.$ And hence unemployment increases (decreases/unaffected) with concealment costs, $\kappa$ when for $\psi \gtreqless 1.$

The impact on observable unemployment is

$$\begin{aligned} \frac{\mathrm{d}u^{o}}{\mathrm{d}\kappa }=\frac{\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\kappa }\left( \lambda ^{F}+\lambda ^{I} \right) -s\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\kappa }\right) }{\left( s+\lambda ^{F}+\lambda ^{I} \right) ^{2}}<0, \end{aligned}$$

as

$$\begin{aligned} \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\kappa }= & {} \left( 1-\sigma _{i}\right) ^{\gamma } \left( \theta ^{F}\right) ^{\left( 1-\eta \right) }\left( -\frac{\gamma }{1-\sigma } \frac{\mathrm{d}\sigma }{\mathrm{d}\kappa }+\frac{\left( 1-\eta \right) }{\theta ^{F}}\frac{\mathrm{d}\theta ^{F}}{\mathrm{d}\kappa }\right) >0 \\ \frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\kappa }= & {} \sigma _{i}^{\gamma }\left( \theta ^{I}\right) ^{\left( 1-\eta \right) }\left( \frac{\gamma }{\sigma }\frac{\mathrm{d}\sigma }{\mathrm{d}\kappa }+\frac{\left( 1-\eta \right) }{\theta ^{I}}\frac{\mathrm{d}\theta ^{I}}{\mathrm{d}\kappa }\right) <0. \end{aligned}$$

The impact on relative employment is:

$$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}\kappa }=\frac{\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\kappa }\left( \lambda ^{F} \right) -\lambda ^{I}\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\kappa }\right) }{\left( \lambda ^{F}+ \lambda ^{I}\right) ^{2}}<0. \end{aligned}$$

1.2 Impact of expected tax and punishment rates when $\mu =0$:

We again consider the equilibrium when $\mu =0,$ which is given by Eqs. (16), (20) and (18).

We differentiate the three equations in $\sigma$, $\theta ^{F},$$\theta ^{I}$ and $l=\frac{1+p\alpha }{1-p\delta },\frac{1}{\phi ^{F}}$, to obtain around the equilibrium and for $\psi >1$:

$$\begin{aligned} \frac{\mathrm{d}\sigma }{\mathrm{d}l}= & {} \frac{\eta \left( y+\frac{\left( 1-\eta \right) }{\eta }\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\right) }{D} \left( \theta ^{F}\right) ^{-1}\frac{1}{\psi }\frac{\mathrm{d}\psi }{\mathrm{d}l}<0, \\ \frac{\mathrm{d}\theta ^{F}}{\mathrm{d}l}= & {} \frac{\frac{\left( 1-\gamma \right) }{1-\sigma }\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}}{-D} \frac{1}{\psi }\frac{\mathrm{d}\psi }{\mathrm{d}l}>0, \\ \frac{\mathrm{d}\theta ^{I}}{\mathrm{d}l}= & {} \frac{\frac{\left( 1-\gamma \right) }{\sigma }\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}}{\eta \left( y+\frac{\left( 1-\eta \right) }{\eta } \frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) }\theta ^{I}\frac{\mathrm{d}\sigma }{\mathrm{d}l}<0, \end{aligned}$$

where

$$\begin{aligned} D= & {} -\frac{1-\gamma }{\sigma }\left( \left( \frac{1}{1-\sigma }-F\right) \left( y-\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\right) \eta +\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\left( 1-F\right) \right) \\&\quad \left( \theta ^{F}\right) ^{-1}<0, \end{aligned}$$

where $F=\frac{\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}}{\left( \eta \left( y-\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) +\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) }<1$ and as $\left( y-\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\right) >0$ from Eq. (17). The impact on unemployment is then:

$$\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}l}=\frac{-s}{\left( s+\lambda ^{F}+\lambda ^{I}\right) ^{2}}\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}l}+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}l}\right) . \end{aligned}$$

Unemployment therefore increases if $\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}l}+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}l}\right) <0$. We derive this sum to be

$$\begin{aligned} \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}l}+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}l}= & {} \frac{\left( \theta ^{I}\right) ^{1-\eta }}{\sigma ^{1-\gamma }}\left( \left( 1-\frac{\left( \theta ^{I}\right) ^{\eta }}{\left( \theta ^{F}\right) ^{\eta }} \frac{1}{\psi }\right) \gamma \frac{\mathrm{d}\sigma }{\mathrm{d}l}-\frac{\left( 1-\gamma \right) \left( 1-\eta \right) }{-D\psi ^{2}\theta ^{F}}\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right. \\&\quad \left. \left( \frac{\left( \psi y+\frac{\left( 1-\eta \right) }{\eta }\frac{k \theta ^{I}}{\sigma ^{1-\gamma }}\right) }{\left( y+\frac{\left( 1-\eta \right) }{\eta }\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) }- \frac{\left( \theta ^{I}\right) ^{\eta }}{\left( \theta ^{F}\right) ^{\eta }} \frac{1}{\psi }\right) \frac{\mathrm{d}\psi }{\mathrm{d}l}\right) \lesseqgtr 0, \end{aligned}$$

for $\psi \gtreqless 1.$ And hence unemployment increases with expected auditing $l=\frac{1+p\alpha }{1-p\delta }$ and punishment rates and decreases with taxes, $l=1/\phi ^{F}$, when $\psi \gtreqless 1.$

The impact on observable unemployment is then

$$\begin{aligned} \frac{\mathrm{d}u^{o}}{\mathrm{d}l}=\frac{\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}l}\left( \lambda ^{F}+\lambda ^{I}\right) -s \left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}l}\right) }{\left( s+\lambda ^{F}+\lambda ^{I}\right) ^{2}}<0, \end{aligned}$$

as

$$\begin{aligned} \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}l}= & {} \left( 1-\sigma _{i}\right) ^{\gamma }\left( \theta ^{F}\right) ^ {\left( 1-\eta \right) }\left( -\frac{\gamma }{1-\sigma }\frac{\mathrm{d}\sigma }{\mathrm{d}l}+\frac{ \left( 1-\eta \right) }{\theta ^{F}}\frac{\mathrm{d}\theta ^{F}}{\mathrm{d}l}\right) >0 \\ \frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}l}= & {} \sigma _{i}^{\gamma }\left( \theta ^{I}\right) ^{ \left( 1-\eta \right) }\left( \frac{\gamma }{\sigma }\frac{\mathrm{d}\sigma }{\mathrm{d}l}+ \frac{\left( 1-\eta \right) }{\theta ^{I}}\frac{\mathrm{d}\theta ^{I}}{\mathrm{d}l}\right) <0. \end{aligned}$$

The impact on relative employment is

$$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}l}=\frac{\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}l}\left( \lambda ^{F}\right) -\lambda ^{I} \left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}l}\right) }{\left( \lambda ^{F}+\lambda ^{I}\right) ^{2}}<0. \end{aligned}$$

1.3 Higher employment protection, higher $\mu .$

We differentiate Eqs (16)–(18) with respect to $\sigma$, $\theta ^{F},$$\theta ^{I}$ and $\mu$ to obtain around the equilibrium:

$$\begin{aligned} \frac{\mathrm{d}\sigma }{\mathrm{d}\mu }= & {} \frac{1}{-D}\left( \theta ^{F}\right) ^{-1}>0, \\ \frac{\mathrm{d}\theta ^{F}}{\mathrm{d}\mu }= & {} \frac{\frac{\left( 1-\gamma \right) }{\sigma }\left( \frac{1}{\left( 1-\sigma \right) }-\frac{\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}}{ \eta \left( y+\frac{\left( 1-\eta \right) }{\eta }\frac{k\theta ^{I}}{\sigma ^{1-\gamma }} \right) }\right) }{D}<0, \\ \frac{\mathrm{d}\theta ^{I}}{\mathrm{d}\mu }= & {} \frac{\frac{\left( 1-\gamma \right) }{ \sigma }\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}}{\eta \left( y+\frac{\left( 1-\eta \right) }{\eta }\frac{k\theta ^{I}}{\sigma ^{1-\gamma }} \right) }\theta ^{I}\frac{\mathrm{d}\sigma }{\mathrm{d}\kappa }>0, \end{aligned}$$

where

$$\begin{aligned} D= & {} -\frac{1-\gamma }{\sigma }\left( \left( \frac{1}{1-\sigma }-F\right) \left( y-\mu -\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\right) \eta +\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }} \left( 1-F\right) \right) \\&\quad \left( \theta ^{F}\right) ^{-1}<0, \end{aligned}$$

where $F=\frac{\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}}{\left( \eta \left( y-\frac{k \theta ^{I}}{\sigma ^{1-\gamma }}\right) +\frac{k\theta ^{I}}{\sigma ^{1-\gamma }}\right) }<1$ and as $\left( y-\mu -\frac{k\theta ^{F}}{\left( 1-\sigma \right) ^{1-\gamma }}\right) >0$ from Eq. (17). The impact on unemployment is then:

$$\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}\mu }=\frac{-s}{\left( s+\lambda ^{F}+\lambda ^{I}\right) ^{2}} \left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\mu }+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\mu }\right) . \end{aligned}$$

The impact on unemployment therefore has the opposite sign of $\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\mu }+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\mu }\right)$ where

$$\begin{aligned} \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\mu }+\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\mu }= & {} \frac{1}{-D}\left( \theta ^{F}\right) ^ {-1}\sigma _{i}^{\gamma -1}\left( \theta ^{I}\right) ^{\left( 1-\eta \right) }\left( \left( \gamma +\left( 1-\eta \right) \left( 1-\gamma \right) F\right) \left( 1-\frac{1}{\psi }\frac{\left( \theta ^{I}\right) ^{\eta }}{\left( \theta ^{F}\right) ^{\eta }}\right) \right. \\&\quad \left. -\frac{1}{\psi }\frac{\left( \theta ^{I}\right) ^{\eta }}{\left( \theta ^{F}\right) ^ {\eta }}\frac{\left( 1-\eta \right) \left( 1-\gamma \right) }{\sigma }\left( 1-F\right) \right) \end{aligned}$$

where the sign is negative for $\psi \le 1$. The impact on observable unemployment is

$$\begin{aligned} \frac{\mathrm{d}u^{o}}{\mathrm{d}\mu }=\frac{\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\mu }\left( \lambda ^{F}+\lambda ^{I} \right) -s\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\mu }\right) }{\left( s+\lambda ^{F}+\lambda ^{I} \right) ^{2}}>0, \end{aligned}$$

as

$$\begin{aligned} \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\mu }= & {} \left( 1-\sigma _{i}\right) ^{\gamma }\left( \theta ^{F} \right) ^{\left( 1-\eta \right) }\left( -\frac{\gamma }{1-\sigma }\frac{\mathrm{d}\sigma }{\mathrm{d}\mu }+ \frac{\left( 1-\eta \right) }{\theta ^{F}}\frac{\mathrm{d}\theta ^{F}}{\mathrm{d}\mu }\right) <0 \\ \frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\mu }= & {} \sigma _{i}^{\gamma }\left( \theta ^{I}\right) ^{ \left( 1-\eta \right) }\left( \frac{\gamma }{\sigma }\frac{\mathrm{d}\sigma }{\mathrm{d}\mu }+\frac{\left( 1-\eta \right) }{\theta ^{I}}\frac{\mathrm{d}\theta ^{I}}{\mathrm{d}\mu }\right) >0. \end{aligned}$$

The impact on relative employment is

$$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}\mu }=\frac{\frac{\mathrm{d}\lambda ^{I}}{\mathrm{d}\mu }\left( \lambda ^{F}\right) -\lambda ^{I}\left( \frac{\mathrm{d}\lambda ^{F}}{\mathrm{d}\mu }\right) }{\left( \lambda ^{F}+\lambda ^{I}\right) ^{2}}>0. \end{aligned}$$