Informal unemployment and education

3.1 Matching

Manual and highly educated workers search for jobs in both a formal and an informal sector. For simplicity, we assume that only unemployed workers search for jobs. This is a simplification, i.e. we do not acknowledge that the connection to the labour market given by working in the formal or informal sector may bring about job opportunities not available while unemployed. The matching functions for the four categories of jobs are given by \({X_{l}^{j}} = \left ({v_{l}^{j}}\right)^{\frac {1}{2}} \left (\left ({\sigma _{l}^{j}}\right)^{\gamma }u_{l}\right)^{\frac {1}{2}}\), where \({X_{l}^{j}}\) is the sectorial matching rate, \({v_{l}^{j}}\) is the sectorial vacancy rate, and u _l is the unemployment rate and j=F,I and l=m,h. The rates are defined as the numbers relatively to the labour force of manual and highly educated workers, respectively. The exponents in the matching function are set to be equal to half in order to simplify the welfare analysis where we derive the optimal tax and punishment system when we have imposed the traditional Hosios condition. In that case, we can disregard congestion externalities as the elasticity of the expected duration of a vacancy is equal to the bargaining power of workers in a symmetric Nash bargaining situation.⁸

Workers allocate search effort optimally across the formal and the informal sector. A worker with educational level l will direct \({\sigma _{l}^{F}}\) units of search for a formal sector job and \({\sigma _{l}^{I}}\) units of search for an informal sector job. Thus, workers with different levels of education may differ in their allocation of search time across a formal and informal sector. Each worker’s total search intensity is, however, exogenously given and normalized to unity, i.e. \({\sigma _{l}^{F}}+{\sigma _{l}^{I}}=1,\;l={m,h}\). The parameter γ<1 captures the effectiveness of search falls with search effort, i.e. the first unit of search in one sector is more effective than the subsequent units of search. This could capture that different search methods are used when searching for a job in a market. The more time that is used in order to search in a market, the less efficient search methods have to be used. The transition rates into informal and formal sector employment for a particular worker i are \(\lambda _{li}^{I}=\left (\sigma _{li}^{I}\right)^{\gamma }\left ({\theta _{l}^{I}}\right)^{\frac {1}{2}}\) and \(\lambda _{li}^{F}=\left (1-\sigma _{li}^{I}\right)^{\gamma }\left ({\theta _{l}^{F}}\right)^{\frac {1}{2}}\), where \({\theta _{l}^{I}}={v_{l}^{I}}/\left (\left ({\sigma _{l}^{I}}\right)^{\gamma }u_{l}\right)\) and \({\theta _{l}^{F}}={v_{l}^{F}}/\left (\left (1-{\sigma _{l}^{I}}\right)^{\gamma }u_{l}\right)\) are labour market tightness, l=m,h, measured in effective search units. The rates at which vacant jobs become filled are \({q_{l}^{j}}=\left ({\theta _{l}^{j}}\right)^{-\frac {1}{2}},\;j={F,I},\;l={m,h}\).

3.2 Value functions

where r is the exogenous discount rate, \({w_{l}^{j}}\) is the sector wage, and s is the exogenous separation rate. R is a lump sum transfer that all individuals receive from the government which reflects that the government has some positive revenue requirements.⁹ The parameter a is the rate by which a worker is dying, and it captures that there is a constant flow of workers out of the labour market at each instant of time. Analogously, there is an equally sized flow of workers into the labour market each time period as people are born at the same rate. This keeps the population constant, normalized to unity, and enables us to look at the impact of various policies on educational attainment despite the fact that education is an irreversible investment.

where z is the payroll tax rate and \({y_{l}^{j}},\;j={F,I},\;l={m,h}\), is productivity. The parameter p is the auditing rate which captures the probability of being detected employing a worker in the informal sector and α is the associated firm punishment fee rate. Vacancy costs are indexed by factor k to the productivity in the sector and written \(k{y_{l}^{j}}\;,j={F,I},\;l={m,h}\).¹⁰ The concealment costs, κ _l , l=m,h, capture that it is costly to hide income from the tax authorities. The costs could, for example, capture what Kleven et al. (2011) refer to as third-party reporting. When there is third-party reporting of income, such as the firm reporting the wage payments directly to the tax authorities, this has to be agreed upon also by the worker, which is costly. These concealment costs could also be other direct costs associated with concealing evasion, as well as morality costs associated with evading taxes.

If firms hiring highly educated workers have a harder time concealing their activities than firms hiring manual workers, then κ _h >κ _m . This is the case if, for example, third-party reporting is more common for highly educated workers, or as assumed in Kleven et al. (2011), the marginal costs of evasion increase with the amount of income evaded. Although this is likely to be the case, we do not a priori impose any restriction on the values of κ _l , l=h,m.

In order to improve the transparency of the model, we disregard taxation, expected punishment, and concealment costs on the worker side. This is of no importance for the results.

Workers allocate their search between sectors to equalize the net marginal returns to search effort across the two sectors.

3.3 Wage determination

where we have imposed symmetry and the free entry condition, \({V_{l}^{j}}=0,\;j={F,I},\;l={m,h}.\)

An increase in tightness, \({\theta _{l}^{j}}\), makes it easier for an unemployed worker to find a job and at the same time harder for a firm to fill a vacancy. This improves the worker’s relative bargaining position, resulting in higher wage demands. An increase in search will instead increase the firm’s relative bargaining position. This is the case as firms will then find it easier to match with a new worker in case of no agreement. The improved bargaining position for firms moderates wage pressure.

3.4 Labour market tightness

By use of the equilibrium search allocation equation in (32), where \(\frac {{\theta _{l}^{I}}}{\left (\sigma _{li}^{I}\right)^{1-\gamma }} = \frac {{\theta _{l}^{F}}}{\left (1-\sigma _{li}^{I}\right)^{1-\gamma }} \frac {{y_{l}^{F}}}{{y_{l}^{I}}}\psi _{l}\), in (15), it becomes clear that the wedge, ψ _l , and productivity differences, \({y_{l}^{F}}/{y_{l}^{I}}\), are the crucial factors determining the size of the formal sector in relation to the informal sector.¹¹ In case productivity is the same in the formal and informal sectors, hence, \({y_{l}^{F}}/{y_{l}^{I}}=1\), then when ψ _l >1, and thus expected punishment plus concealment costs are higher than payroll taxes, informal sector producer wages are higher than formal sector producer wages. In this case, it is relatively more attractive for firms to enter the formal sector, which makes formal sector tightness exceed informal sector tightness. Hence, we obtain that \({\theta _{l}^{F}}>{\theta _{l}^{I}}\) and \({\sigma _{l}^{I}}<\frac {1}{2},\;l={m,h}\) if \(\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}>1\) and vice versa when \(\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}<1\). Notice that the formal sector exceeds the informal sector \({\theta _{l}^{F}}>{\theta _{l}^{I}}\) and \({\sigma _{l}^{I}}<\frac {1}{2},\;l={m,h}\) both if the wedge is equal to 1, ψ _l =1, and the formal sector is more productive than the informal sector, \({y_{l}^{F}}/{y_{l}^{I}}>1\), as well as if the formal and informal sectors are equally productive and the wedge is larger than 1, ψ _l >1.

As the formal sector exceeds the informal sector in size in most high-income countries, it is most realistic to consider the case when \(\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}>1\). This implies considering the situation when the expected punishment rate plus concealment costs exceed the tax rate, i.e. p α+κ _l >z, when both the formal and informal sectors are equally productive, which does not seem unrealistic given a broad interpretation of concealment costs. In fact, as discussed in the introduction, positive concealment costs κ _l >0 such that p α+κ _l >z could potentially explain the puzzle of why we observe a relatively small informal sector although we, at the same time, observe rather low audit rates and fairly modest fines, i.e. p α<z. In addition, when the productivity in the formal sector exceeds that of the informal sector, the formal sector is even more likely to exceed the informal sector in size. However, we do not a priori impose any restrictions on the size of ψ _l , p α, or κ _l when deriving the results in this paper. When discussing results that depend on the size of ψ _l , however, we focus the discussion on what we believe is the most realistic case.

In Fig. 5, we can use Eqs. (14) and (15) to derive relative tightness as a function of search intensity and illustrate this equation in a \(({\sigma _{l}^{I}},{\theta _{l}^{F}}/{\theta _{l}^{I}})\) diagram together with Eq. (11). Both equations have a negative slope, and the former curve will be flatter than the latter around the equilibrium insuring a stable equilibrium.¹² When the wedge increases, ψ l′>ψ _l (or \({y_{l}^{F}}/{y_{l}^{I}}\) increases), then the search intensity decreases for given relative labour market tightness, \({\theta _{l}^{F}}/{\theta _{l}^{I}}\); this reduction in search intensity increases \({\theta _{l}^{F}}/{\theta _{l}^{I}}\) and thereby \({\sigma _{l}^{I}}\) until a new equilibrium is reached. In Fig. 5, we have left out subscript l to ease exposition.

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3.5 Education

When workers decide whether to acquire higher education or remain as manual workers, they compare the value of unemployment as an educated worker and the associated costs of higher education to the value of unemployment as a manual worker. Workers that find it optimal to acquire higher education view this as a once and for all investment in human capital, and it takes place as soon as the worker enters the labour market. As in most studies, we assume that education is costly but it takes no time.¹³ The cost of higher education depends on individual ability, e _i ∈[0,1], and is given by c(e _i ), where c ^′(e _i )<0 and c ^′′(e _i )>0.¹⁴

where \(o_{l}={\theta _{l}^{F}}/\left (1-{\sigma _{l}^{I}}\right)^{1-\gamma },\;l={h,m}.\) Equation (17) gives \(\hat {e}\) as a function of the endogenous variables \({\theta _{l}^{F}}\) and \({\sigma _{l}^{I}},\;l={m,h}.\) Workers with \(e\leq \hat {e}\) choose not to acquire education, whereas workers with \(e>\hat {e}\) acquire education. Hence, \(\hat {e}\) and \(1-\hat {e}\) constitute the manual and educated labour forces, respectively. The right-hand side of Eq. (17) is the expected income gain of attaining education. This gain needs to be positive in order for, at least some, workers to proceed to higher education. The fact that productivity is higher for highly educated workers, which gives rise to an educational wage premium, provides incentives for higher education. However, higher education may potentially also be associated with losses in expected income. For example, if concealment costs are higher for highly educated workers, i.e. κ _h >κ _m , relatively more attractive informal employment opportunities for manual workers will be foregone in case of higher education. This reduces the incentives for education.¹⁵

Clearly, in order to study the non-trivial case where at least some workers proceed to higher education, it is necessary to assume that there is a net gain in the expected income of higher education. Thus, we need to assume that productivity differences between manual and highly educated workers are sufficiently high, i.e. \({y_{h}^{F}}/{y_{m}^{F}}>o_{m}/o_{h}\). Moreover, to guarantee a non-trivial interior solution where at least some, but not all, individuals choose to acquire education, the individual with the highest ability faces a very low cost of education, more specifically c(1)=0, and the individual with the lowest ability faces a very high cost of education, i.e. l i m _e→0 c(e)=∞. See the Appendix for the proof of the existence of \(\hat {e}\in (0,1)\).

3.6 Employment and unemployment

A comparison of the unemployment rates for manual and highly educated workers requires assumptions about the size of the concealment costs. If concealment costs are higher for educated workers, i.e. κ _h >κ _m , the official unemployment rate is always lower for highly educated workers than for manual workers, i.e. \({u_{h}^{o}}<{u_{m}^{o}}\). This is also what is observed in data. However, if furthermore, \(\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}>1,\;l={h,m}\), and hence the informal sector is smaller than the formal sector, the actual unemployment rate is higher for the highly educated workers, u _h >u _m , i.e. in this case, manual workers have a lower actual unemployment rate than highly educated workers. The following Proposition summarizes the results.

4.1 Sector allocation

Although the results of fully financed punishment of informal activities in Propositions 2 and 3 hold irrespective of how the government budget restriction is affected, to stress the intuition, we present the results based on the standard case when an increase in p or α increases relative punishment ψ _l .¹⁹ The effects on the allocation of search and employment across the formal and the informal sector are summarized in the following Proposition.

More zealous enforcement will make informal work less attractive, inducing unemployed workers to reallocate their search effort towards the formal sector. In addition, when search is reallocated towards the formal sector, the wage bargaining position strengthens for firms in the formal sector whereas it falls for firms in the informal sector. The lower producer wages in the formal sector stimulate formal firms to open vacancies, while at the same time, informal firms are discouraged to open new vacancies as they now face higher producer wages. As a consequence that both vacancies and search effort are reallocated towards the formal sector, the formal sector employment rate increases at the expense of informal employment. These mechanisms can explain the empirical findings in Almeida and Carneiro (2012) who use data on inspections carried out in Brazil.

4.2 Unemployment rates

As became clear in Proposition 2, employment in the formal sector increases at the expense of employment in the informal sector following more severe punishment of the informal sector. While this is somewhat expected, it is a priori not clear what would happen to the unemployment rates. We have the following results:

The actual unemployment rates increase with more severe punishment of informal work if \(\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}>1\). The reason for this is that the large concealment costs discourage workers from searching, and firms from opening vacancies, in the informal sector. Increased punishment of the informal sector will encourage further reallocation of search and workers away from the informal sector, where relatively efficient search methods are used, towards the formal sector. Total search efficiency then falls, inducing unemployment to increase. The fact that search becomes less efficient when reallocated towards the formal sector also has an impact on unemployment working through wage formation and tightness. As search is reallocated towards the formal sector, the wage demand is moderated in the formal sector and exaggerated in the informal sector. As the efficiency of search in the formal sector increases by less than the efficiency of search in the informal sector is reduced, the informal sector wage push will dominate the formal sector wage moderation. Thus, the incentives to open up a vacancy in the formal sector subsides the disincentives to open up a vacancy in the informal sector; the formal sector tightness will increase by less than the informal sector tightness falls when \(\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}>1\). The opposite holds if \(\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}<1\). In this case, too much search and too many firms are allocated into the informal sector as there is a relative cost advantage of producing underground. Total search efficiency would then improve when the government tries to combat the informal sector. The official unemployment rate always falls with more harsh punishment of informal activities as workers to a larger extent become formally employed. In this unemployment measure, workers in the informal sector were counted as unemployed to start with.

4.3 Education

From (17), it is clear that more severe relative punishment of the informal sector affects the number of educated workers as such policy increases ψ _l . This effect is further reinforced if the tax rate is reduced in order to assure a balanced government budget as the increase in ψ _l is reinforced by a reduction in z. However, a reduced payroll tax rate will also have a direct effect on the stock of educated workers. More specifically, a reduction in the tax rate, z, for a given wedge, will increase the number of educated workers. This follows as taxation is more harmful to high income earners, and consequently, a tax reduction will improve the income relatively more for high income earners. However, before considering repercussions working through the budget constraint, let us first consider the impact of a more zealous enforcement policy on education, for a given tax rate. We have the following results:

The impact of a more zealous enforcement policy on educational attainment depends on how attractive underground work is to manual and educated workers. When concealment costs are higher for highly educated workers, more zealous enforcement policies tend to induce more workers to educate themselves. This follows as κ _h >κ _m implies that manual workers to a larger extent face informal labour market opportunities. Therefore, more zealous enforcement policies, which make it less attractive to work in the informal sector, will be more harmful to manual workers. This effect may, however, be counteracted by the fact that highly educated workers have higher productivity and therefore earn higher wages. As also informal activities are highly productive for these workers, this implies that more harsh punishment, in this perspective, is more harmful for the highly educated worker. Thus, even if highly educated workers face less informal employment opportunities, these opportunities are more profitable. This reduces educational incentives.

Which of the two effects dominate will thus depend on how sizeable the differences in informal employment opportunities and productivity are. If underground employment opportunities in an economy foremost are available to manual workers, more harsh punishment of underground activities will push more workers into education, thus increasing the stock of educated workers in the economy. However, if these employment opportunities to a large extent also are available for highly educated workers, harder punishment will harm highly educated workers more as these opportunities are more profitable to productive workers. This leads to less workers educating themselves.

Note that Proposition 4 only provides the sufficient conditions for when the educational stock increases and when it falls with more harsh punishment of the informal sector without considering the financing of the reform. Provided that we are located on the positively sloped side of the Laffer curve, we can conclude the following:

This simply follows as taxation as a direct effect is more harmful for high income earners, and consequently, a tax reduction, in order to maintain a balanced government budget, will be more beneficial for high income earners, thus encouraging educational attainments.

4.4 Unemployment

From Propositions 3, 4, and 5, it follows that more severe punishment of the informal sector potentially increases the total number of unemployed workers. If the formal sector is larger than the informal sector, the unemployment rates for both manual and highly educated workers are augmented. Moreover, if informal employment opportunities to a significantly larger extent are available for manual workers, more workers will attain higher education when informal activities are punished more severely. This tends to increase total unemployment as the actual unemployment rate, including informal work, is higher for highly educated workers. Also, recall that this reallocation effect is reinforced if we are located on the positively sloped side of the Laffer curve. Thus, in this case, total unemployment increases both because the unemployment rates for all workers increase and because workers are reallocated towards the sector where the unemployment rate is highest. More generally, the Proposition summarizes the result:

5.1 Optimal punishment policy

The welfare analysis above indicates that it may be optimal to punish tax-evading activities carried out by manual workers more severely than those carried out by highly educated workers. For example, if concealment costs are higher for highly educated workers, a punishment policy with ψ _m =ψ _h =1 is only possible if the manual workers to a larger extent than highly educated workers face punishment of informal activities. That is, p α has to be set relatively higher for manual workers if κ _m <κ _h in order to induce ψ _m =ψ _h =1.

This raises the question of whether it is possible or not to target the punishment fees and audit rates towards the sector employing manual vs highly educated workers. While governments potentially could, and in fact do,²³ target their audits to specific sectors, i.e. allowing for p _m to differ from p _h , this is less likely the case for the fee rates.

Provided that we are located on the positively sloped side of the Laffer curves, we can conclude that

The result in Proposition 7 follows straightforwardly from the first-order conditions when evaluated at the p _m and p _h which induces ψ _m =ψ _h =1. The first term on the right-hand side of Eqs. (28) and (29) then disappears as the distortions in search and allocation of tightness across the formal and the informal sector are fully eliminated. In this case, there are no other distortions present except that too few workers have chosen to educate themselves. Recall that this is a consequence that taxation harms high income earners relatively more. This distortion can, however, be corrected for by increasing the audits in the manual sector and reducing them in the sector for highly educated workers, which is captured by the right-hand side in (30) and (31). As informal sector work for manual workers becomes less attractive when the government increases the number of audits, manual workers are encouraged to acquire higher education. Similarly, less audits in the highly educated sector further encourages workers to acquire higher education.

If concealment costs are higher in the sector employing highly educated workers, i.e. κ _h >κ _m , there are even further incentives for the government to focus their audits on the manual sector. This follows as high concealment costs work as a self-regulating punishment of informal sector activities. Thus, if concealment costs are higher in the sector employing highly educated workers, this sector will be in less need of audits as concealment costs will do part of the job of limiting the size of the informal sector.

Moreover it follows that

When deciding on the optimal audit rates, the government faces a trade-off between two distortions and it is never optimal to fully eliminate one of them. When the stock of educated workers is at its socially optimal level, there is an inefficient allocation of search and jobs across the formal and informal sectors. Welfare then improves as the stock of educated workers is reduced below its socially optimal level as this will only be a second-order effect in comparison to the improved welfare following a more efficient sectorial allocation.

5.2 Optimal punishment policy when concealment costs are high

In deriving the optimal audit rates in the previous section, it was implicitly assumed that audit rates could be chosen freely without restrictions. For example, according to Proposition 7, the audit rates should be chosen such that \(p_{m}^{\ast }>p_{h}^{\ast }\) so as to get \(\psi _{h}^{\ast }<1<\psi _{m}^{\ast }\). However, this is only possible if concealment costs are not too high. If, for example, κ _h >z, then ψ _h >1 even when p _h is very small. Replacing the first-order condition in (29) with the appropriate Kuhn-Tucker conditions, \(\frac {dW}{{dp}_{h}}+\mu =0\), p _h ≥0, and μ p _h =0, where μ is the Lagrange multiplier for the constraint p _h ≥0, then suggests that the audit rate in the sector should be set as low as possible when κ _h >z. Concealment costs are simply high enough to self-regulate the size of the informal sector facing highly educated workers, and there is no need for additional audits of this sector.²⁴

Taking off in real-world observations from high-income economies, this may not be an unrealistic scenario. Evidence indicates that manual workers, or workers with a lower level of formal education, to a substantially larger degree face informal employment opportunities compared to highly educated workers. Pedersen and Smith (1998) using comprehensive survey data find that almost half of the informal sector activities in Denmark is carried out within the construction sector. They also find that around 70 % of the total hours performed in the informal sector is carried out within the service sector or construction sector.

Potential explanations for why manual, in contrast to highly educated, workers engage in informal activities are that manual workers to a larger extent work in industries which handle cash payments or are to a lesser extent subject to third-party reporting. Firms and workers in industries dealing with cash payments, or which to a lesser extent are subject to third-party reporting, will find it easier, and thus less costly, to conceal their tax evasion. Taking this at face value implies that concealment costs for highly educated workers, κ _h , could be very large. If κ _h is assumed to approach infinity, informal employment opportunities facing highly educated workers will become infinitely small, leading to that basically no firms will post informal sector vacancies to highly educated workers and none of the highly educated workers will allocate search effort into the informal sector. All the results derived in Propositions 1 to 6 account for this special case, including the now clear-cut result that higher punishment fees, or a general increase in the audit rate, encourage more workers to educate themselves. This follows as less workers will remain as manual workers as the foregone informal employment opportunities when attaining education has become less attractive. Moreover, the socially optimal audit rate is again being determined by an audit rate which implies that \(p_{m}^{\ast }\) is set large enough so as to get \(\psi _{m}^{\ast }>1,\) although not high enough to induce an efficient stock of educated workers.

5.3 Multiple equilibria

Again, consider the case when the government can target the audits towards the sectors for manual and highly educated workers. That ψ _l =(1+p _l α+κ _l )/(1+z) can be obtained both through high tax and enforcement rates and through low tax and enforcement rates raises the issue of multiple equilibria.²⁵ The relationship between the punishment rates and the tax rate in each sector can then be written as \(p_{l}\alpha =\bar {\psi }_{l}(1+z)-(1+\kappa _{l})\) where the wedge in each sector is set to some fixed value \(\bar {\psi }_{l}\). A 1-unit increase in z followed by an increase in p _l α by \(\bar {\psi _{l}}\) units maintains the relative punishment rate given by \(\bar {\psi }_{l}\).

From the government budget constraint in (20), it is clear that any revenue requirement could then be reaped through such simultaneous increases in z and p _l α, if it was not for adjustments in the stock of educated workers. This follows as the tax base in terms of producer wages, \({\omega _{l}^{j}}\), and employment rates, \({n_{l}^{j}}\), only depends on ψ _l whereas education falls with higher taxes for given wedges. The fact that the tax base falls with higher taxation through reduced incentives for education opens up for the possibility of two equilibria where the government can collect the same revenue although at different levels of tax and enforcement rates.

In the high tax and enforcement economy, very few workers may choose to educate themselves which reduces the tax base and thus induces modest tax revenues even though tax rates are high (negatively sloped side of the Laffer curve). In the low tax and enforcement economy, in contrast, many workers find it worthwhile to educate themselves which induces a large tax base which enables the government to obtain equally high revenues despite low tax rates (positively sloped side of the Laffer curve). This scenario is potentially possible in our model.

The scenario is shown graphically in Fig. 6 in terms of a Laffer curve with tax revenues, R, on the vertical axis whereas the horizontal axis captures the tax rate, z, for given wedges, \(\bar {\psi }_{l}\). As the only tax base adjustment taking place is with regard to education, the direct effect is fairly strong indicating that revenues always tend to increase with z (the filled curve). However, the direct effect becomes less strong as taxation becomes heavier because there are fewer highly educated workers to tax. On the other hand, the response in terms of the number of workers acquiring higher education is stronger when z is low. This is captured by the convex cost function for education. Including auditing costs into the government budget constraint clearly tends to increase the likelihood of being in a situation where an increase in tax rate, and increases in p _l α so as to keep \(\bar {\psi }_{l}\) constant, no longer increases government revenues, thus also increasing the likelihood of multiple equilibria (dotted line in Fig. 6).

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In the case of two equilibria, the low tax and enforcement equilibria is preferable from a welfare point of view. As was seen in Section 5.1, it was optimal to correct for the distortion in terms of that too few workers did choose to educate themselves. However, it was not optimal to fully correct for this distortion, leaving the educated stock below a socially optimal value, since distortions on the labour market then became inefficiently high. Clearly, the high tax and enforcement economy worsens the problem by inducing an educational stock which is, for the same wedge, even further away from what is socially optimal. This leads the government to use the wedges to push the economy further away from an efficient labour market outcome in order to correct for this additional distortion in education. Both the labour market distortions and the educational distortion is in this economy larger at the socially optimal wedges than at the socially optimal wedges chosen under the low tax and enforcement economy.

8.1 Tightness relatively to search intensity

Hence, if κ _t >κ _x , then \(\frac {{\theta _{t}^{F}}}{\left (1-\sigma _{t}\right)^{1-\gamma }}<\frac {{\theta _{x}^{F}}}{\left (1-\sigma _{x}\right)^{1-\gamma }}.\)

8.2 Existence of \(\hat {e}\in \left (0,1\right)\)

Consider the educational Eq. (37). For a non-trivial solution, there needs to be a net gain in expected income of higher education. Thus, \({y_{h}^{F}}/{y_{m}^{F}}>o_{m}/o_{h}\). Moreover, to guarantee a non-trivial interior solution where at least some, but not all, individuals choose to acquire education, the individual with the highest ability faces a very low cost of education, more specifically c(1)=0, and the individual with the lowest ability faces a very high cost of education, i.e. \({\lim }_{e\rightarrow 0}c(e)=\infty \).

In the case κ _h ≤κ _m , then o _m /o _h <1, and hence, \({y_{h}^{F}}/{y_{m}^{F}}>o_{m}/o_{h}\) holds as \({y_{h}^{F}}>{y_{m}^{F}}\). If educated workers face higher concealment costs than manual workers κ _h >κ _m , then we need to assume that the productivity gain of education is large enough to assure that \({y_{h}^{F}}/{y_{m}^{F}}>o_{m}/o_{h}\) holds, which is possible as the right-hand side is independent of y _l .

8.3 Relative unemployment rates (Proposition 1)

The official unemployment rate facing t workers is higher than the official unemployment rate facing x workers; \({u_{t}^{o}}>{u_{x}^{o}}\) if and only if \(\left (s+{\lambda _{t}^{I}}\right)/\left (s+{\lambda _{t}^{F}}+{\lambda _{t}^{I}}\right)> \left (s+{\lambda _{x}^{I}}\right)/\left (s+{\lambda _{x}^{F}}+{\lambda _{x}^{I}}\right)\). This holds if an only if \({\lambda _{x}^{F}}\left (s+{\lambda _{t}^{I}}\right)>{\lambda _{t}^{F}}\left (s+{\lambda _{x}^{I}}\right),\) which is true when \({\lambda _{x}^{F}}>{\lambda _{t}^{F}}\) and \({\lambda _{x}^{I}}>{\lambda _{t}^{I}},\) that is, when κ _t >κ _x .

8.4 Impact of higher punishment on sector allocation (Proposition 2)

8.5 Impact of higher punishment on unemployment rates (Proposition 3)

8.6 Impact of higher punishment on education (Propositions 4 and 5)

8.7 Impact of higher punishment on unemployment (Proposition 6)

The last two terms are positive (≤0) when \(\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}\) is larger than one (≤1). The first term is positive if \(\frac {{y_{h}^{F}}}{{y_{m}^{F}}}\in \left [\frac {o_{m}}{o_{h}},g\left (\kappa _{h},\kappa _{m}\right)\frac {o_{m}}{o_{h}}\right ]\) where g(κ _h ,κ _m )>1 if when κ _h >κ _m and \(\frac {{y_{h}^{F}}}{{y_{h}^{I}}}\psi _{h}\geq \frac {{y_{m}^{F}}}{{y_{m}^{I}}}\psi _{m}\geq 1\) as then (u _m −u _h )<(=)0 and \(\frac {d\hat {e}}{d\psi _{l}}<0\). However, when \(\frac {{y_{l}^{F}}}{{y_{l}^{I}}}\psi _{l}<1\) and κ _h >κ _m , then (u _m −u _h )>0, and in case \(\frac {d\hat {e}}{d\psi _{l}}<0\), then unemployment falls, \(\frac {{dU}_{TOT}}{d\psi _{l}}<0\). If, \(\frac {{y_{h}^{F}}}{{y_{m}^{F}}}\in \left [g\left (\kappa _{h},\kappa _{m}\right)\frac {o_{m}}{o_{h}},\infty \right ]\), then \(\frac {d\hat {e}}{d\psi _{l}}>0\) and \(\frac {{dU}_{TOT}}{d\psi _{l}}\) has an ambiguous sign.

8.8 Socially optimal solution for \({\theta _{m}^{F}},{\theta _{m}^{I}},{\theta _{h}^{F}},{\theta _{h}^{I}},{\sigma _{m}^{I}},{\sigma _{h}^{I}},\hat {e.}\)

We assume that firms are owned by renters who do not work. This explains the presence of \(\sum _{j={F,I}}{n_{m}^{j}}r_{a}{J_{m}^{j}}+\sum _{j={F,I}}{v_{m}^{j}}r_{a}{V_{m}^{j}}\) and \(\sum _{j={F,I}}{n_{h}^{j}}r_{a}{J_{h}^{j}}+\sum _{j={F,I}}{v_{h}^{j}}r_{a}{V_{h}^{j}}\) in the welfare function. Moreover, we assume that the concealment costs for tax evasion-facing firms are payments to “lawyers” who only engage in concealing taxable income for other firms. The welfare function therefore includes \({n_{m}^{I}}r_{a}J_{m}^{\text {law}}={n_{m}^{I}}{w_{m}^{I}}\kappa _{m}\) and \({n_{h}^{I}}r_{a}J_{h}^{\text {law}}={n_{h}^{I}}{w_{h}^{I}}\kappa _{h}\). This assumption enables us to disregard from the waste associated with tax evasion if firms only pay these expenses to nobody.

where \(S_{l}=\left (\left ({\sigma _{l}^{I}}\right)^{\gamma -2}+\left (1-{\sigma _{l}^{I}}\right)^{\gamma -2}\right),\;l={m,h}, \;{\Delta _{l}^{I}}=-\frac {1}{2}\left (sk\left ({\theta _{l}^{I}}\right)^{-\frac {1}{2}}+k\left ({\sigma _{l}^{I}}\right)^{\gamma -1}\right), \;l={m,h}\) and \(\;{\Delta _{l}^{F}}=-\frac {1}{2}\left (sk\left ({\theta _{l}^{F}}\right)^{-\frac {1}{2}}+ k\left (1-{\sigma _{l}^{I}}\right)^{\gamma -1}\right),\;l={m,h}\). Therefore, \(H_{1}=\left (\gamma -1\right)\left (\left ({\sigma _{m}^{I}}\right)^{\gamma -2}+\left (1-{\sigma _{m}^{I}}\right)^{\gamma -2}\right)<0\) and the principal minors alternate in sign, for all variable values, i.e. \(H_{2}=-\left (\gamma -1\right)\left (\left ({\sigma _{m}^{I}}\right)^{\gamma -2}+ \left (1-{\sigma _{m}^{I}}\right)^{\gamma -2}\right){\Delta _{m}^{I}}>0,\ldots,H_{7}=\left (\gamma -1\right) \left (\left ({\sigma _{m}^{I}}\right)^{\gamma -2}+\left (1-{\sigma _{m}^{I}}\right)^{\gamma -2}\right) {\Delta _{m}^{I}}{\Delta _{m}^{F}}\left (\gamma -1\right)\left (\left ({\sigma _{h}^{I}}\right)^{\gamma -2}+ \left (1-{\sigma _{h}^{I}}\right)^{\gamma -2}\right){\Delta _{h}^{I}}{\Delta _{h}^{F}}c^{\prime }\left (\hat {e}^{\ast }\right)<0\), by which we have a global maximum.

8.9 Optimal does not induce the socially efficient stock of education (Corollary 8)

Evaluating (52) and (53) at \({p_{m}^{e}}\) and \({p_{h}^{e}}\) such that the socially optimal level of education is reached, i.e. \(\frac {dW}{d\left (1-e\right)}=0\). From Proposition 7, this requires that \({\psi _{m}^{e}}>1>{\psi _{h}^{e}}.\) This yields \(\frac {dW}{{dp}_{m}}\mid _{{\psi _{m}^{e}}>1}=\hat {e}\left [\frac {dW}{d{\theta _{m}^{F}}} \frac {d{\theta _{m}^{F}}}{d\psi _{m}}+\frac {dW}{d{\theta _{m}^{I}}}\frac {d{\theta _{m}^{I}}}{d\psi _{m}}+ \frac {dW}{d{\sigma _{m}^{I}}}\frac {d{\sigma _{m}^{I}}}{d\psi _{m}}\right ]\frac {d\psi _{m}}{{dp}_{m}}\) and \(\frac {dW}{{dp}_{h}}\mid _{{\psi _{h}^{e}}<1}=\left (1-\hat {e}\right)\left [\frac {dW}{d{\theta _{h}^{F}}} \frac {d{\theta _{h}^{F}}}{d\psi _{h}}+\frac {dW}{d{\theta _{h}^{I}}}\frac {d{\theta _{h}^{I}}}{d\psi _{h}}+ \frac {dW}{d{\sigma _{h}^{I}}}\frac {d{\sigma _{h}^{I}}}{d\psi _{h}}\right ]\frac {d\psi _{h}}{{dp}_{h}}\). From the derivations of the socially optimal solution for \({\theta _{m}^{F}},{\theta _{m}^{I}},{\theta _{h}^{F}},{\theta _{h}^{I}},{\sigma _{m}^{I}},{\sigma _{h}^{I}}\), it follows that \(\frac {dW}{d{\theta _{l}^{F}}}\mid _{\psi _{l}>1}<0, \frac {dW}{d{\theta _{l}^{I}}}\mid _{\psi _{l}>1}>0,\frac {dW}{{d{\sigma _{l}^{I}}}}\mid _{\psi _{l}>1}>0\) and \(\frac {dW}{d{\theta _{l}^{F}}}\mid _{\psi _{l}<1}>0,\;\frac {dW}{d{\theta _{l}^{I}}}\mid _{\psi _{l}<1}<0,\frac {{dW}_{l}}{d{\sigma _{l}^{I}}}\mid _{\psi _{l}<1}<0\) as the welfare function is maximized at ψ _l =1, i.e., \(\frac {dW}{d{\theta _{l}^{F}}}\mid _{\psi _{l}=1}=\frac {{dW}_{l}}{d{\theta _{l}^{I}}}\mid _{\psi _{l}=1}= \frac {{dW}_{l}}{d\sigma l}\mid _{\psi _{l}=1}=0\). It then follows that \(\frac {dW}{{dp}_{m}}\) and \(\frac {dW}{{dp}_{h}}\mid _{{\psi _{h}^{e}}<1}>0\).

8.10 Optimal punishment policy including auditing costs