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.

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Notes

  1. 1.

    In 2015, the European Commission launched their ambition to step up efforts to combat tax evasion and tax fraud in order to have a more fair and transparent taxation. The Tax Transparency Package presented in 2015 was followed by the Anti Tax Avoidance Package in 2016. The OECD has also increased their ambition to reduce tax evasion. The OECD initiated the Global Forum of Transparency and Exchange of Information for Tax Purposes in the early 2000s, and it now constitutes the largest tax organisation in the world including both OECD and non-OECD countries (see http://www.oecd.org/tax/transparency/).

  2. 2.

    For the 2007 survey, see http://ec.europa.eu/commfrontoffice/publicopinion/archives/ebs/ebs_284_en.pdf. For the 2013 survey, see http://ec.europa.eu/commfrontoffice/publicopinion/archives/ebs/ebs_402_en.pdf.

  3. 3.

    See Gordon and Li (2009), Kleven et al. (2011, 2016), Kleven (2014), Pomeranz (2015), and Bjorneby et al. (2017).

  4. 4.

    See Andreoni et al. (1998), Perry (2007), and Packard et al. (2012).

  5. 5.

    See Packard et al. (2012).

  6. 6.

    This holds in particular for the informal sector measures that capture the type of undeclared work modelled in this paper. Namely workers that, at a given point in time, work either in the formal or in the informal sector. Although this is the preferred measure to use according to Packard et al. (2012) when considering available measures for comparisons of the informal sector sizes across countries, it disregards under-declared work (usually referred to as envelope wages). Envelope wages are a phenomenon particularly prevalent in countries in eastern Europe.

  7. 7.

    Hazans (2011) also include Cyprus and Israel in the group of southern European countries. As our calibrations are based on mainly OECD data, Cyprus is left out from our group of southern European countries. Israel is left out as it is not a European country.

  8. 8.

    The seminal paper by Allingham and Sandmo (1972) and Srinivasan (1973), where under-reporting of income is modelled as a decision made under uncertainty, provides early theoretical analyses of tax evasion. Subsequent papers have since then enhanced the basic model of individual behaviour by, for example, incorporating endogenous labour supply decisions. See for example Sandmo (1981) for an early contribution of endogenous labour supply and under-reporting of income. Also equilibrium models with tax evasion have been developed, as, for example, the early study by Cremer and Gahvari (1993) and the studies by Tonin (2011) and Prado (2011).

  9. 9.

    See also Bosch and Esteban-Pretel (2012) for a model based on a similar set-up calibrated by use of flow data from Brazil. Also Ulyssea (2010) uses Brazilian data to calibrate a search model where firms need intermediate goods from both a formal and an informal sector in their production process of a final good.

  10. 10.

    There are also numerous empirical studies on issues of informality in low- and middle-income countries. See for example Günther and Launov (2012).

  11. 11.

    There are also papers based on search and matching models considering different aspects of tax evasion than what is done here. For example, while this paper considers workers that at a given point in time, either work in the formal or the informal sector, other studies are concerned with workers being partly employed in the formal sector and partly employed in the informal sector. See Kolm and Nielsen (2008) and Di Nola et al. (2017) for two papers on this issue. Other papers have looked at the impact of tax evasion on other outcomes, such as the impact of tax evasion on educational attainment (Kolm and Larsen 2016).

  12. 12.

    Workers being paid cash above their reported wage is an important aspect of informal sector work although not considered in this paper. See Kolm and Nielsen (2008), Williams (2009), and Di Nola et al. (2017) for papers on envelope wages. For policy purpose, however, it may be of importance to study the different types of informal jobs separately. Workers with no attachment to the formal sector may be in a weaker position than workers that only work in the informal sector as a complement to their formal sector job.

  13. 13.

    We view all workers not being employed as job searchers. Thus, we do not consider the impact on labour force participation in the model.

  14. 14.

    The government cannot exclude the informal sector workers when distributing the transfer as the government does not know who the informal sector workers are (if it did, it could just punish all of them).

  15. 15.

    They provide an explanation for why development may reduce tax evasion as development is often associated with employment growth, and thus more whistle-blowers.

  16. 16.

    See Packard et al. (2012) for a discussion.

  17. 17.

    This holds for many definitions of informal activities such as activities that are legal but are not declared to tax authorities or social security institutions (EC 2007; Hjsgaard et al. 2017). However, other definitions imply that there could be an informal sector although no taxes need to be paid. For example, definitions based on activities carried out without following the employment protection legislation.

  18. 18.

    See Fialová and Schneider (2011) and Hazans (2011).

  19. 19.

    See Packard et al. (2012).

  20. 20.

    Countries in central and eastern Europe also have fairly large informal sectors. However, the undeclared work in these countries seems to a larger extent be derived from under-declared work (usually referred to as envelope wages). See Packard et al. (2012) and Hazans (2011). This type of informal employment is not captured in our model or in the measure of informal sector employment we are using in this section.

  21. 21.

    See Feige (2016) for a paper discussing problems with these estimates of the informal sector.

  22. 22.

    See Packard et al. (2012) for a discussion of the three most commonly used methods of how to define an informal employee; firm size criterion, social contribution criterion, and contract criterion. The contract criterion is considered the preferred proxy for measuring informal employees based on reliability when it comes to comparisons across countries, and the fact that it is less ambiguous and significantly more observable than the other criterions.

  23. 23.

    The measure derived for Italy stems from 2006, whereas the rest of the countries derive from 2008/2009. Excluding Italy will increase the difference in the size of the informal sector between the regions, as will including Israel and Cyprus in accordance with the division between South and North in Hazans (2011).

  24. 24.

    The OECD calculates country specific measures for a tax wedge by calculating the combined central and sub-central government income taxes paid plus the employee and employer social security contribution taxes paid as a share of labour costs defined as gross wage earnings plus employer social security contributions. In our model, this corresponds to the following measure: \(\hbox {TW}=(z+t)/(1+z)\) where TW denotes the OECD wedge. Using this measure, we can derive the measure for the tax wedge presented in Sect. 2.3 as \((1+z)/(1-t)=1/(1-\hbox {TW})\). See http://stats.oecd.org/index.aspx?DataSetCode=TABLE_I5

  25. 25.

    See Torgler (2011a).

  26. 26.

    The Worldwide Governance Indicators (WGI) are a research data set summarising the views on the quality of governance provided by a large number of enterprise, citizen and expert survey respondents in industrial and developing countries. These data are gathered from a number of survey institutes, think tanks, non-governmental organisations, international organisations, and private sector firms. The index averages capturing government effectiveness in a country for the period 2006–2015 is 2.00 in North and 0.74 in South, where the estimates ranges from approximately \(-2.5\) (weak) to 2.5 (strong) governance performance.

  27. 27.

    The indexes for the rule of law, and regulatory quality shows the same pattern.

  28. 28.

    See Gertler and Trigari (2009) for a review of values used in other studies.

  29. 29.

    http://www.oecd.org/els/emp/oecdindicatorsofemploymentprotection.htm.

  30. 30.

    This is in line with the result from the field experiment in Denmark by Kleven et al. (2011) who concluded that third-party reporting was a very efficient way to reduce tax evasion. They found that employees almost never evaded taxes, whereas self-employed, where the cost of evasion can be considered to be substantially lower due to the absence of third-party reporting, did. Although, this model does not contain self-employed, it illustrates the potential efficiency of the policy instrument.

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Acknowledgements

We want to thank participants at the Shadow 2017 Conference in Warsaw, seminar participants at the School of Economics and Business, Norwegian University of Life Sciences, as well as Diana Hornshøj Jensen and Katia Usova for excellent research assistance.

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Correspondence to Birthe Larsen.

Appendix

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.

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:

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

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

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

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

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

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Kolm, A., Larsen, B. Underground activities and labour market performance. Int Tax Public Finance 26, 41–70 (2019). https://doi.org/10.1007/s10797-018-9505-4

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Keywords

  • Informal sector
  • Search and matching
  • Unemployment
  • Concealment costs
  • Employment protection legislation
  • Tax evasion

JEL Classification

  • H26
  • J64
  • E26