The model is given by
$$ \frac{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}= \left(\frac{{\theta_{l}^{F}}}{{\theta_{l}^{I}}}\right) \frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l},\;l=m,h. $$
(32)
where \(\psi _{l}=\frac {1+p\alpha +\kappa _{l}}{1+z}\)
$$ {\omega_{l}^{F}}={w_{l}^{F}}\left(1+z\right),=\frac{1}{2}{y_{l}^{F}} \left(1+k\frac{{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\;l=m,h, $$
(33)
$$ {\omega_{l}^{I}}={w_{l}^{I}}\left(1+p\alpha+\kappa_{l}\right)= \frac{1}{2}{y_{l}^{I}}\left(1+\frac{{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}k\right),\;l=m,h, $$
(34)
$$ k\left(r+s+a\right)\left({\theta_{l}^{F}}\right)^{\frac{1}{2}} = \frac{1}{2}\left(1-\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right),\;m=h,l, $$
(35)
$$ k\left(r+s+a\right)\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}= \frac{1}{2}\left(1-\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right),\;m=h,l. $$
(36)
Educational choice
$$ c(\hat{e})=\frac{k}{1+z}\left({y_{h}^{F}}o_{h}-{y_{m}^{F}}o_{m}\right), $$
(37)
Employment and unemployment
$$ {n_{l}^{F}}=\frac{{\lambda_{l}^{I}}}{s+a+{\lambda_{l}^{F}}+{\lambda_{l}^{I}}}, \;{n_{l}^{F}}=\frac{{\lambda_{l}^{I}}}{s+a+{\lambda_{l}^{F}}+{\lambda_{l}^{I}}},\;l=h,m, $$
(38)
$$ u_{l}=\frac{s+a}{s+a+{\lambda_{l}^{F}}+{\lambda_{l}^{I}}}, \;{u_{l}^{o}}=\frac{s+a+{\lambda_{l}^{I}}}{s+a+{\lambda_{l}^{F}}+{\lambda_{l}^{I}}},\;l=h,m. $$
(39)
Budget constraint in terms of producer wages using \({\omega _{m}^{F}}={w_{l}^{F}}(1+z)\) and \({\omega _{l}^{I}}={w_{l}^{I}}(1+p\alpha +\kappa _{l}),\;l={m,h},\) yields
$$ \frac{z\hat{e}{n_{m}^{F}}{w_{m}^{F}}}{1+z}+ \frac{p\alpha\hat{e}{n_{m}^{I}}{w_{m}^{I}}}{1+p\alpha+\kappa_{m}}+ \frac{z\left(1-\hat{e}\right){n_{h}^{F}}{w_{h}^{F}}}{1+z}+ \frac{p\alpha\left(1-\hat{e}\right){n_{m}^{I}}{w_{h}^{I}}}{1+p\alpha+\kappa_{h}}=R $$
(40)
where R is the exogenous revenue requirement.
Tightness relatively to search intensity
We show that \(\frac {{\theta _{t}^{F}}}{\left (1-{\sigma _{t}^{I}}\right)^{1-\gamma }}<\frac {{\theta _{x}^{F}}}{\left (1-{\sigma _{x}^{I}}\right)^{1-\gamma }}\) when κ
t
>κ
x
in the following way. Differentiating equations (35), (36), and (32) with respect to κ
l
gives around the equilibrium
$$\frac{d{\theta_{l}^{F}}}{d\kappa_{l}}=\frac{\frac{\left(1-\gamma\right)}{1-{\sigma_{l}^{I}}}\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\frac{1}{2}\left(1+\frac{k{\theta_{l}^{F}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\frac{1}{{\theta_{l}^{I}}}}{D_{l}}\left(1+p\alpha+\kappa_{l}\right)>0, $$
$$\frac{d{\theta_{l}^{I}}}{d\kappa_{l}}=-\frac{\frac{1}{2}\left(1+\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\frac{1}{{\theta_{l}^{F}}}\left(1-\gamma\right)k{\theta_{l}^{I}}\left({\sigma_{l}^{I}}\right)^{\gamma-2}}{D_{l}}\left(1+p\alpha+\kappa_{l}\right)<0, $$
$$\frac{d{\sigma_{l}^{I}}}{d\kappa_{l}}=-\frac{\frac{1}{2}\left(1+\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\frac{1}{{\theta_{l}^{F}}}\frac{1}{2}\left(1+\frac{k\theta^{I}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\frac{1}{{\theta_{l}^{I}}}}{D_{l}}\left(1+p\alpha+\kappa_{l}\right)<0, $$
where
$$\begin{array}{@{}rcl@{}} D_{l}&=&\frac{\left(1-\gamma\right)\frac{1}{{\sigma_{l}^{I}}}}{{\theta_{l}^{I}}{\theta_{l}^{F}}4\left(1-{\sigma_{l}^{I}}\right)}\left(\frac{\frac{1}{\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}}k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}+1\right)\left(1-\frac{k\theta^{I}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\left(1-{\sigma_{l}^{I}}\right)\\ &&+\left(\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}+1\right)\left(1-\frac{\frac{1}{\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}}k\theta^{I}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right) \end{array} $$
Now, differentiating \(\frac {{\theta _{l}^{F}}}{\left (1-{\sigma _{l}^{I}}\right)^{1-\gamma }}\) with respect to κ
l
gives
$${} {{\begin{aligned} \frac{d\frac{{\theta_{l}^{F}}}{\left(1-\sigma_{l}\right)^{1-\gamma}}}{d\kappa_{l}}=\frac{d{\theta_{l}^{F}}\left(1-\sigma_{l}\right)^{\gamma-1}}{d\kappa_{l}}={\theta_{l}^{F}}\left(1-\sigma_{l}\right)^{\gamma-1}\left(\left(\theta^{F}\right)^{-1}\frac{d{\theta_{l}^{F}}}{d\kappa_{l}}+\left(1-\gamma\right)\left(1-\sigma_{l}\right)^{-1}\frac{d\sigma^{I}}{d\kappa_{l}}\right)= \end{aligned}}} $$
$${} {{\begin{aligned} =-\frac{{\theta_{l}^{F}}\left(1-\sigma_{l}\right)^{\gamma-1}}{4D_{l}}\frac{\left(1-\gamma\right)}{\left(1-{\sigma_{l}^{I}}\right)}\left(1+\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\frac{1}{{\theta_{l}^{F}}{\theta_{l}^{I}}}\left(1-\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\left(1+p\alpha+\kappa_{l}\right)<0. \end{aligned}}} $$
Hence, if κ
t
>κ
x
, then \(\frac {{\theta _{t}^{F}}}{\left (1-\sigma _{t}\right)^{1-\gamma }}<\frac {{\theta _{x}^{F}}}{\left (1-\sigma _{x}\right)^{1-\gamma }}.\)
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
.
Relative unemployment rates (Proposition 1)
Unemployment is increasing in concealment costs if ψ
l
>1. Hence, if κ
t
>κ
x
, then u
t
>u
x
if ψ
l
>1. We show that in the following way, u
t
>u
x
if and only if \(s/\left (s+{\lambda _{x}^{F}}+{\lambda _{x}^{I}}\right)<s/\left (s+{\lambda _{t}^{F}}+{\lambda _{t}^{I}}\right)\) if an only if \({\lambda _{t}^{F}}+{\lambda _{t}^{I}}<{\lambda _{x}^{F}}+{\lambda _{x}^{I}}\). Hence, the condition holds if
$${} {{\begin{aligned} \frac{d\left({\lambda_{l}^{F}}+{\lambda_{l}^{I}}\right)}{d\kappa_{l}}= \frac{d\left[\left(1-{\sigma_{l}^{I}}\right)^{\gamma}\left({\theta_{h}^{F}}\right)^{\frac{1}{2}}+ \left({\sigma_{l}^{I}}\right)^{\gamma}\left({\theta_{h}^{I}}\right)^{\frac{1}{2}}\right]}{d\kappa_{l}} \end{aligned}}} $$
$${} {{\begin{aligned} =\gamma\left(-\left(1-{\sigma_{l}^{I}}\right)^{\gamma-1} \left({\theta_{l}^{F}}\right)^{\frac{1}{2}}+\left({\sigma_{l}^{I}}\right)^{\gamma-1} \left({\theta_{l}^{I}}\right)^{\frac{1}{2}}\right)\frac{d{\sigma_{l}^{I}}}{d\kappa_{l}}+ \frac{1}{2}\left(\frac{\left(1-\sigma^{I}\right)^{\gamma}}{\left({\theta_{l}^{F}}\right)^{\frac{1}{2}}} \frac{d{\theta_{l}^{F}}}{d\kappa_{l}}+\frac{\left(\sigma^{I}\right)^{\gamma}}{\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}} \frac{d{\theta_{l}^{I}}}{d\kappa_{l}}\right)<0. \end{aligned}}} $$
We substitute for the derivatives and the first-order condition for search intensity to obtain the condition equal to
$${} {{\begin{aligned} =\gamma\left(\frac{1}{\psi_{l}\frac{{y_{l}^{F}}}{{y_{l}^{I}}} \left({\theta_{l}^{F}}\right)^{\frac{1}{2}}}-\frac{1}{\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}}\right) \left(1+\frac{k{\theta_{l}^{I}}}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\left(\sigma_{li}^{I}\right)^{1-\gamma}}\right) \left(1+\frac{k{\theta_{l}^{I}}}{\left(\sigma_{li}^{I}\right)^{1-\gamma}}\right) \end{aligned}}} $$
$${} {{\begin{aligned} +\left(1-\gamma\right)\frac{k{\theta_{l}^{I}}}{\left(\sigma_{li}^{I}\right)^{1-\gamma}} \left(\frac{1}{\left({\theta_{l}^{F}}\right)^{\frac{1}{2}}}\frac{1}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}} \left(\frac{1}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}}+\frac{k{\theta_{l}^{I}}}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l} \left(\sigma_{li}^{I}\right)^{1-\gamma}}\right)-\frac{1}{\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}}\left(1+ \frac{k{\theta_{l}^{I}}}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\left(\sigma_{li}^{I}\right)^{1-\gamma}}\right)\right) \end{aligned}}} $$
which is negative when y
F/y
F≥1 and ψ
l
>1, as then \({\theta _{l}^{F}}>{\theta _{l}^{I}}\) giving \(\frac {1}{\frac {{y_{l}^{F}}}{{y_{l}^{I}}}\psi _{l}}\frac {1}{\left ({\theta _{l}^{F}}\right)^{\frac {1}{2}}}< \frac {1}{\left ({\theta _{l}^{I}}\right)^{\frac {1}{2}}}\) and \(\left (\frac {1}{\frac {{y_{l}^{F}}}{{y_{l}^{I}}}\psi _{l}}+ \frac {1}{\frac {{y_{l}^{F}}}{{y_{l}^{I}}}\psi _{l}}\frac {k\theta ^{I}}{\left ({\sigma _{l}^{I}}\right)^{1-\gamma }}\right)< \left (1+\frac {1}{\frac {{y_{l}^{F}}}{{y_{l}^{I}}}\psi _{l}}\frac {k\theta ^{I}}{\left ({\sigma _{l}^{I}}\right)^{1-\gamma }}\right)\). Hence, unemployment increases with ψ
l
, and hence, u
t
>u
x
when \({y_{h}^{F}}/{y_{l}^{I}}\geq 1\) and κ
t
>κ
x
.
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
.
Impact of higher punishment on sector allocation (Proposition 2)
Raising the audit rate p
l
or the punishment fee α increases the wedge, ψ
l
=(1+p
α+κ
l
)/(1+z). Differentiating Eqs. (35), (36), and (32) with respect to ψ
l
gives around the equilibrium
$$\frac{d\theta^{F}}{d\psi_{l}}=\frac{\frac{\left(1-\gamma\right)}{\left(1-{\sigma_{l}^{I}}\right)} \frac{1}{2}\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}} \left(1+\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right) \frac{1}{{\theta_{l}^{I}}}}{D_{l}}\frac{1}{\psi_{l}}>0 $$
$$\frac{d\theta^{I}}{d\psi_{l}}=\frac{-\frac{\left(1-\gamma\right)}{\left({\sigma_{l}^{I}}\right)} \frac{1}{2}\left(1+\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right) \frac{1}{{\theta_{l}^{F}}}\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}}{D_{l}}\frac{1}{\psi_{l}}>0 $$
$$\frac{d\sigma^{I}}{d\psi_{l}}=\frac{-\frac{1}{2}\frac{1}{{\theta_{l}^{F}}{\theta_{l}^{I}}} \left(1+\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\frac{1}{2} \left(1+\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)}{D_{l}}\frac{1}{\psi_{l}}<0, $$
where
$$\begin{array}{@{}rcl@{}} D_{l}&=&\frac{\left(1-\gamma\right)}{4{\theta_{l}^{F}}{\theta_{l}^{I}}}\frac{1}{\left(1-{\sigma_{l}^{I}}\right){\sigma_{l}^{I}}}\left\{ \left(1+\frac{k{\theta_{l}^{I}}}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\left(1-\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\left(1-\sigma^{I}\right)\right.\\ &&+\left.\sigma_{li}^{I}\left(1-\frac{k{\theta_{l}^{I}}}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\left(1+\frac{k{\theta_{l}^{I}}}{\left(\sigma_{li}^{I}\right)^{1-\gamma}}\right)\right\}, \end{array} $$
which is positive. Hence, as \(\lambda _{li}^{I}=\left (\sigma _{li}^{I}\right)\gamma \left ({\theta _{l}^{I}}\right)^{\frac {1}{2}}\) and \(\lambda _{li}^{F}=\left (1-\sigma _{li}^{I}\right)^{\gamma }\left ({\theta _{l}^{F}}\right)^{\frac {1}{2}},\) by inspection of Eq. (38), it follows that \(d{n_{l}^{F}}/d\psi _{l}>0,\;d{n_{l}^{I}}/d\psi _{l}<0,l={m,h}\). The impact on wages is then
$$ \frac{d{\omega_{l}^{F}}}{d\psi_{l}}=\frac{1}{2}{y_{l}^{F}}k\frac{\frac{{\theta_{l}^{F}}}{\left(1-\sigma_{li}^{I}\right)^{1-\gamma}}}{d\psi_{l}},\;l=m,h, $$
(41)
$$ \frac{d{\omega_{l}^{I}}}{d\psi_{l}}=\frac{1}{2}{y_{l}^{I}}k\frac{d\frac{{\theta_{l}^{I}}}{\left(\sigma_{li}^{I}\right)^{1-\gamma}}}{d\psi_{l}},\;l=m,h, $$
(42)
as
$${} \frac{d\frac{{\theta_{l}^{F}}}{\left(1-\sigma_{l}\right)^{1-\gamma}}}{d\psi_{l}}=\frac{d{\theta_{l}^{F}}\left(1-\sigma_{l}\right)^{\gamma-1}}{d\psi_{l}}={\theta_{l}^{F}}\left(1-\sigma_{l}\right)^{\gamma-1}\left(\frac{1}{{\theta_{l}^{F}}}\frac{d{\theta_{l}^{F}}}{d\psi_{l}}+\left(\!1-\gamma\right)\left(1-\sigma_{l}\right)^{-1}\!\frac{d\sigma^{I}}{dd\psi_{l}}\right)= $$
$${\kern24pt} =-\frac{\left(1-\sigma_{l}\right)^{\gamma-1}}{4D_{l}\psi_{l}}\frac{1-\gamma}{1-{\sigma_{l}^{I}}}\left(1+\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\frac{1}{{\theta_{l}^{I}}}\left(1-\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right)<0. $$
(43)
and
$$\frac{d\frac{{\theta_{l}^{I}}}{\sigma_{l}^{1-\gamma}}}{d\psi_{l}}=\frac{d{\theta_{l}^{I}}\left(\sigma_{l}\right)^{\gamma-1}}{d\psi_{l}}={\theta_{l}^{I}}\left(\sigma_{l}\right)^{\gamma-1}\left(\frac{1}{{\theta_{l}^{I}}}\frac{d{\theta_{l}^{I}}}{d\psi_{l}}-\left(1-\gamma\right)\left(\sigma_{l}\right)^{-1}\frac{d\sigma^{I}}{d\psi_{l}}\right)= $$
$$=-\frac{\left(\sigma_{l}\right)^{\gamma-1}}{4D_{l}\psi_{l}}\frac{1-\gamma}{{\sigma_{l}^{I}}}\left(1-\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\frac{1}{{\theta_{l}^{F}}}\left(1+\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}\right)<0. $$
Impact of higher punishment on unemployment rates (Proposition 3)
Raising the audit rate p or the punishment fee α increases the wedge, ψ
l
=(1+p
α+κ
l
)/(1+z). Differentiating Eq. (39) with respect to ψ
l
gives
$$\frac{du_{l}}{d\psi_{l}}=-\frac{s}{\left(s+{\lambda_{l}^{I}}+{\lambda_{l}^{F}}\right)^{2}}\left(\frac{d{\lambda_{l}^{F}}}{d\psi_{l}}+\frac{d{\lambda_{l}^{I}}}{d\psi_{l}}\right) $$
where
$$\frac{d{\lambda_{l}^{F}}}{d\psi_{l}}+\frac{d{\lambda_{l}^{I}}}{d\psi_{l}}=\frac{d\left(1-{\sigma_{l}^{I}}\right)^{\gamma}\left({\theta_{l}^{F}}\right)^{\frac{1}{2}}+d\left({\sigma_{l}^{I}}\right)^{\gamma}\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}}{d\psi_{l}} $$
$$\gamma\left(-\left(1-{\sigma_{l}^{I}}\right)^{\gamma-1}\left({\theta_{l}^{F}}\right)^{\frac{1}{2}}+\left({\sigma_{l}^{I}}\right)^{\gamma-1}\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}\right)\frac{d{\sigma_{l}^{I}}}{d\psi_{l}} $$
$$+\frac{1}{2}\left(\frac{1-{\sigma_{l}^{I}}}{\left({\theta_{l}^{F}}\right)^{\frac{1}{2}}}\frac{d{\theta_{l}^{F}}}{d\psi_{l}}+\frac{\left({\sigma_{l}^{I}}\right)^{\gamma}}{\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}}\frac{d{\theta_{l}^{I}}}{d\psi_{l}}\right) $$
Substituting for the derivatives and the first-order condition for search intensity, we obtain that d
u
l
/d
ψ
l
has the same sign as
$$\gamma\left(\frac{1}{\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}}\frac{1}{\left({\theta_{l}^{F}}\right)^{\frac{1}{2}}}-\frac{1}{\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}}\right)\left(1+\frac{1}{\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}}\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\left(1+\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right) $$
$$\begin{array}{@{}rcl@{}}{\kern20pt} &+&\left(1-\gamma\right)\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\left(\frac{1}{\left({\theta_{l}^{F}}\right)^{\frac{1}{2}}}\frac{1}{\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}}\left(\frac{1}{\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}}+\frac{1}{\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}}\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\right.\\ &-&\left.\frac{1}{\left({\theta_{l}^{I}}\right)^{\frac{1}{2}}}\left(1+\frac{1}{\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}}\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\right) \end{array} $$
Hence,
$$\frac{{du}_{l}}{d\psi_{l}}\lesseqgtr0\text{ if and only if } {\left({y_{l}^{F}}/{y_{l}^{I}}\right)\psi_{l}\lesseqgtr1.} $$
The impact on the official unemployment rate resulting from an increase in the audit rate or the punishment fee corresponds to
$$\frac{d{u_{l}^{o}}}{d\psi_{l}}=\frac{\left(s+{\lambda_{l}^{I}}+{\lambda_{l}^{F}}\right)\frac{d{\lambda_{l}^{I}}}{d\psi_{l}}-\left(s+{\lambda_{l}^{I}}\right)\left(\frac{d{\lambda_{l}^{F}}}{d\psi_{l}}+\frac{d{\lambda_{l}^{I}}}{d\psi_{l}}\right)}{\left(s+{\lambda_{l}^{I}}+{\lambda_{l}^{F}}\right)^{2}}=\frac{{\lambda_{l}^{F}}\frac{d{\lambda_{l}^{I}}}{d\psi_{l}}-\left(s+{\lambda_{l}^{I}}\right)\left(\frac{d{\lambda_{l}^{F}}}{d\psi_{l}}\right)}{\left(s+{\lambda_{l}^{I}}+{\lambda_{l}^{F}}\right)^{2}} $$
Impact of higher punishment on education (Propositions 4 and 5)
A closer examination of (37) reveals that changes in the audit rates or punishment rates affect the share of educated workers, \(1-\hat {e}\), through ψ
l
only, whereas changes in the tax rate, z, have a direct effect on \(1-\hat {e}\) in addition to the effects working through ψ
l
. Therefore, in order to consider the effects of a fully financed change in the punishment rates on the number of educated workers, we have to account for repercussions on \(1-\hat {e}\) following adjustments in the tax rate. However, let us first consider the impact on \(1-\hat {e}\) of a change in the tax and expected punishment separately:
$$\frac{\partial\left(1-\hat{e}\right)}{\partial\left(p\alpha\right)}|_{z}=-\frac{k}{c^{\prime}(e)\left(1+z\right)}\left({y_{h}^{F}}\frac{d\frac{{\theta_{h}^{F}}}{\left(1-{\sigma_{h}^{I}}\right)^{1-\gamma}}}{d\left(p\alpha\right)}-{y_{m}^{F}}\frac{d\frac{{\theta_{m}^{F}}}{\left(1-{\sigma_{m}^{I}}\right)^{1-\gamma}}}{d\left(p\alpha\right)}\right) $$
$$\frac{\partial\left(1-\hat{e}\right)}{\partial z}|_{p_{l}\alpha}=-\psi_{l}\frac{\partial\left(1-\hat{e}\right)}{\partial\left(p\alpha\right)}|_{z}+\frac{c\left(\hat{e}\right)}{c^{\prime}\left(\hat{e}\right)\left(1+z\right)} $$
Using Eq. (43), we obtain
$$ \frac{d\frac{k{\theta_{l}^{F}}}{\left(1-{\sigma_{l}^{I}}\right)^{1-\gamma}}}{d\left(p\alpha\right)}=-\frac{\frac{k{\theta_{l}^{I}}}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\left({\sigma_{l}^{I}}\right)^{1-\gamma}}}{\frac{\left(1+\frac{k{\theta_{l}^{I}}}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\left(1-\frac{k{\theta_{l}^{I}}}{\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)}{\left(1-\frac{k{\theta_{l}^{I}}}{\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\left({\sigma_{l}^{I}}\right)^{1-\gamma}}\right)\left(1+\frac{k{\theta_{l}^{I}}}{\left(\sigma_{li}^{I}\right)^{1-\gamma}}\right)}\frac{1-\sigma^{I}}{{\sigma_{l}^{I}}}+1}\frac{1}{\psi_{l}}\frac{1}{1+z},l=h,m, $$
(44)
whereby the educational impacts become
$$\frac{\partial\left(1-\hat{e}\right)}{\partial\left(p\alpha\right)}|_{z}=-\frac{k}{c^{\prime}(e)\left(1+z\right)^{2}}\left({y_{h}^{F}}\frac{{do}_{h}}{d\left(p\alpha\right)}-{y_{m}^{F}}\frac{{do}_{m}}{d\left(p\alpha\right)}\right) $$
$$\frac{\partial\left(1-\hat{e}\right)}{\partial z}|_{p_{l}\alpha}=-\psi_{l}\frac{\partial\left(1-\hat{e}\right)}{\partial\left(p\alpha\right)}|_{z}+\frac{c\left(\hat{e}\right)}{c^{\prime}\left(\hat{e}\right)\left(1+z\right)} $$
where \(o_{l}=\frac {1}{\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}}\frac {k{\theta _{l}^{I}}}{\left ({\sigma _{l}^{I}}\right)^{1-\gamma }}=\frac {k{\theta _{l}^{F}}}{\left (1-{\sigma _{l}^{I}}\right)^{1-\gamma }},l=h,m\) and
$$ \frac{{do}_{l}}{d\psi_{l}}=-\frac{\frac{1}{\psi_{l}}o_{l}}{\frac{\left(o_{l}+1\right)\left(1-\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\right)\psi_{l}o_{l}\right)}{\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}o_{l}+1\right)\left(1-o_{l}\right)}\frac{\left(1-{\sigma_{l}^{I}}\right)}{{\sigma_{l}^{I}}}+1}<0,\;l=h,m. $$
(45)
For existence of an interior solution for education, we need \({y_{h}^{F}}o_{h}-{y_{m}^{F}}o_{m}>0\). Hence, education increases if \({y_{h}^{F}}\frac {{do}_{h}}{d\psi _{h}}-{y_{m}^{F}}\frac {{do}_{m}}{d\psi _{m}}>0\). As \(\frac {{do}_{l}}{d\psi _{l}},\;l={h,m}\) is negative, and \({y_{h}^{F}}/{y_{m}^{F}}>o_{m/}o_{h}\), then for existence of an interior solution for \(\hat {e}\), if
$$ \left\vert \frac{{do}_{m}}{d\psi_{m}}\right\vert /\left\vert \frac{{do}_{h}}{d\psi_{h}}\right\vert >{y_{h}^{F}}/{y_{m}^{F}}>o_{m}/o_{h}. $$
(46)
then education increases with p
α. Consider the case where κ
h
>κ
m
. As ψ
l
increases with κ
l
, then for such a solution to exist, we need that \(\left \vert \frac {{do}_{l}}{d\psi _{l}}\right \vert,\;l={m,h}\) is decreasing in concealment costs whereby \(\left \vert \frac {{do}_{m}}{d\psi _{m}}\right \vert >\left \vert \frac {{do}_{h}}{d\psi _{h}}\right \vert \). We first show that that is the case. Multiply the numerator and denominator by ψ
l
to obtain
$$\left|\frac{{do}_{l}}{d\psi_{l}}\right|=\frac{o_{l}}{\frac{\left(o_{l}+1\right)\left(1-\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\right)\psi_{l}o_{l}\right)}{\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}o_{l}+\frac{1}{\psi_{l}}\right)\left(1-o_{l}\right)}\frac{\left(1-{\sigma_{l}^{I}}\right)}{{\sigma_{l}^{I}}}+\psi_{l}},\;l=h,m. $$
Substituting for the tightness equations, \(1-\frac {k{\theta _{l}^{F}}}{\left (1-{\sigma _{l}^{I}}\right)^{1-\gamma }}= 1-o_{l}=k\left (r+s+a\right)\left ({\theta _{l}^{F}}\right)^{\frac {1}{2}}2\) and \(1-\frac {k{\theta _{l}^{I}}}{\left ({\sigma _{l}^{I}}\right)^{1-\gamma }}= 1-\left ({y_{l}^{F}}/{y_{l}^{I}}\right)\psi _{l}o_{l}=k\left (r+s+a\right)\left ({\theta _{l}^{I}}\right)^{\frac {1}{2}}2\) and use the fact that \(\frac {1-{\sigma _{l}^{I}}}{{\sigma _{l}^{I}}}=\left (\frac {{\theta _{l}^{F}}}{{\theta _{l}^{I}}}\right)^{\frac {1}{1-\gamma }} \left (\frac {{y_{l}^{F}}}{{y_{l}^{I}}}\psi _{l}\right)^{\frac {1}{1-\gamma }}\) according to the search equation to obtain
$$ \left|\frac{{do}_{l}}{d\psi_{l}}\right|=\frac{o_{l}}{A_{l}\left(\frac{{\theta_{l}^{F}}}{{\theta_{l}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\right)^{\frac{1}{1-\gamma}}+\psi_{l}},\;l=h,m, $$
(47)
where \(A_{l}=\frac {\left (o_{l}+1\right)}{\left (\frac {{y_{l}^{F}}}{{y_{l}^{I}}}o_{l}+\frac {1}{\psi _{l}}\right)}.\) Differentiating (47), ψ
l
, we obtain the following expression for \(\frac {d\left |\frac {{do}_{l}}{d\psi _{l}}\right |}{d\psi _{l}}\):
$${} {{\begin{aligned} \frac{\frac{{do}_{l}}{d\psi_{l}}\left(\!\!A_{l}\left(\!\frac{{\theta_{l}^{F}}}{{\theta_{l}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}\left(\!\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\right)^{\frac{1}{1-\gamma}}\!+\psi_{l}\right)-o_{l}\left(\!\left(\!\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\right)^{\frac{1}{1-\gamma}}\left(\!\left(\!\frac{{\theta_{l}^{F}}}{{\theta_{l}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}\left(\frac{{dA}_{l}}{d\psi_{l}}+\frac{1}{\psi_{l}}\frac{A_{l}}{1-\gamma}\right)+A_{l}\frac{d\left(\frac{{\theta_{l}^{F}}}{{\theta_{l}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}}{d\psi_{l}}\!\right)+1\right)}{\left(A_{l}\left(\frac{{\theta_{l}^{F}}}{{\theta_{l}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}\right)^{\frac{1}{1-\gamma}}+\psi_{l}\right)^{2}}<0, \end{aligned}}} $$
(48)
as substituting for \(\frac {{do}_{l}}{d\psi _{l}}\) using the expression from Eq. (45) gives
$${} {{\begin{aligned} \frac{{dA}_{l}}{d\psi_{l}}=\frac{\frac{{do}_{l}}{d\psi_{l}}\left(\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}-1\right)o_{l}+\frac{1}{\psi_{l}}-1\right)+\frac{o_{l}+1}{{\psi_{l}^{2}}}}{\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}o_{l}+\frac{1}{\psi_{l}}\right)^{2}}=\frac{\frac{1}{\psi_{l}}o_{l}\left(1-\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}-1\right)o_{l}\right)+\frac{o_{l}+1}{{\psi_{l}^{2}}}\frac{\left(o_{l}+1\right)\left(1-\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\right)\psi_{l}o_{l}\right)}{\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}o_{l}+1\right)\left(1-o_{l}\right)}\frac{\left(1-{\sigma_{l}^{I}}\right)}{{\sigma_{l}^{I}}}+\frac{1}{{\psi_{l}^{2}}}}{\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}o_{l}+\frac{1}{\psi_{l}}\right)^{2}\left(\frac{\left(o_{l}+1\right)\left(1-\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\right)\psi_{l}o_{l}\right)}{\left(\frac{{y_{l}^{F}}}{{y_{l}^{I}}}\psi_{l}o_{l}+1\right)\left(1-o_{l}\right)}\frac{\left(1-{\sigma_{l}^{I}}\right)}{{\sigma_{l}^{I}}}+1\right)}>0, \end{aligned}}} $$
for \(\frac {{y_{l}^{F}}}{{y_{l}^{I}}}-1<1\) (sufficient condition) and from the equilibrium equations we have \(d\left ({\theta _{h}^{F}}/{\theta _{h}^{I}}\right)/d\psi _{l}>0\) and d
o
l
/d
ψ
l
<0.
Hence, as \(\frac {d\left \vert \frac {{do}_{l}}{d\psi _{l}}\right \vert }{d\psi _{l}}<0\), then \(\frac {d\left \vert \frac {{do}_{l}}{d\psi _{l}}\right \vert }{d\kappa _{l}}<0\), so when κ
h
>κ
m
, then \(\left |\frac {{do}_{m}}{d\psi _{m}}\right |>\left |\frac {{do}_{h}}{d\psi _{h}}\right |\). We observe that \(\frac {d\left \vert \frac {{do}_{l}}{d\psi _{l}}\right \vert }{d\psi _{l}}<0\) both because the numerator decreases with ψ
l
and the denominator increases with ψ
l
. Rewriting the expression determining the sign of \(\frac {\partial \left (1-\hat {e}\right)}{\partial \left (p\alpha \right)}|_{z}\), Eq. (46) as
$$\frac{\frac{o_{m}}{A_{m}\left(\frac{{\theta_{m}^{F}}}{{\theta_{m}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}\left(\frac{{y_{m}^{F}}}{{y_{m}^{I}}}\psi_{m}\right)^{\frac{1}{1-\gamma}}+\psi_{m}}}{\frac{o_{h}}{A_{h}\left(\frac{{\theta_{h}^{F}}}{{\theta_{h}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}\left(\frac{{y_{h}^{F}}}{{y_{h}^{I}}}\psi_{h}\right)^{\frac{1}{1-\gamma}}+\psi_{h}}}=g\left(\kappa_{h},\kappa_{m}\right)o_{m/}o_{h}>{y_{h}^{F}}/{y_{m}^{F}}>o_{m/}o_{h}, $$
where
$$g\left(\kappa_{h},\kappa_{m}\right)=\frac{D_{\frac{{do}_{h}}{d\psi_{h}}}}{D_{\frac{{do}_{m}}{d\psi_{m}}}}=\frac{A_{h}\left(\frac{{\theta_{h}^{F}}}{{\theta_{h}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}\left(\frac{{y_{h}^{F}}}{{y_{h}^{I}}}\psi_{h}\right)^{\frac{1}{1-\gamma}}+\psi_{h}}{A_{m}\left(\frac{{\theta_{m}^{F}}}{{\theta_{m}^{I}}}\right)^{\frac{1}{1-\gamma}-\frac{1}{2}}\left(\frac{{y_{m}^{F}}}{{y_{m}^{I}}}\psi_{m}\right)^{\frac{1}{1-\gamma}}+\psi_{m}}>1, $$
when κ
h
>κ
m
and \(\frac {{y_{h}^{F}}}{{y_{h}^{I}}}\geq \frac {{y_{m}^{F}}}{{y_{m}^{I}}}\) (or equivalently \(\frac {{y_{h}^{F}}}{{y_{m}^{F}}}\geq \frac {{y_{h}^{I}}}{{y_{m}^{I}}}\)) as the denominator of \(\left \vert \frac {{do}_{l}}{d\psi _{l}}\right \vert \) increases with ψ
l
. We conclude that if \(\frac {{y_{h}^{F}}}{{y_{m}^{F}}}\in \left [\frac {o_{m}}{o_{h}},g\left (\kappa _{h},\kappa _{m}\right)\frac {o_{m}}{o_{h}}\right ]\) education increases with p
α and when \(\frac {{y_{h}^{F}}}{{y_{m}^{F}}}\in \left [g\left (\kappa _{h},\kappa _{m}\right)\frac {o_{m}}{o_{h}},\infty \right ]\) education falls with p
α.
Impact of higher punishment on unemployment (Proposition 6)
Raising the audit rate p or the punishment fee α increases the wedge, ψ
l
=(1+p
l
α+κ
l
)/(1+z). Differentiating total unemployment with respect to ψ
l
gives
$$\frac{{dU}_{TOT}}{d\psi_{l}}=\frac{d\hat{e}}{d\psi_{l}}\left(u_{m}-u_{h}\right)+\hat{e}\frac{{du}_{m}}{d\psi_{l}}+(1-\hat{e})\frac{{du}_{h} }{d\psi_{l}} $$
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.
Total official unemployment changes according to
$$\frac{{dU}_{TOT}^{o}}{d\psi_{l}}=\frac{d\hat{e}}{d\psi_{l}}\left({u_{m}^{o}}-{u_{h}^{o}}\right)+\hat{e}\frac{d{u_{m}^{o}}}{d\psi_{l}}+(1-\hat{e})\frac{d{u_{h}^{o}}}{d\psi_{l}}<0, $$
where the last two terms are negative and therefore when κ
h
>κ
m
and \(\frac {{y_{h}^{F}}}{{y_{h}^{I}}}\psi _{h}\geq \frac {{y_{m}^{F}}}{{y_{m}^{I}}}\psi _{m}\geq 1 \frac {{dU}_{TOT}^{o}}{d\psi _{l}}<0\) when \(\frac {d\hat {e}}{d\psi _{l}}\leq 0,\) as \(\left ({u_{m}^{o}}-{u_{h}^{o}}\right)>0\). When \(\frac {d\hat {e}}{d\psi _{l}}\leq 0\), the sign of \(\frac {{dU}_{TOT}^{o}}{d\psi _{l}}\) is ambiguous.
Socially optimal solution for \({\theta _{m}^{F}},{\theta _{m}^{I}},{\theta _{h}^{F}},{\theta _{h}^{I}},{\sigma _{m}^{I}},{\sigma _{h}^{I}},\hat {e.}\)
For simplicity, we here let \({y_{l}^{F}}={y_{l}^{I}},\;l={h,m}.\) We make use of a utilitarian welfare function, which is obtained by adding all individuals’ steady state flow values of welfare and let r+a=r
a
. This accounts for that both the formal and the informal economy generate welfare in the economy. The social welfare function is written as
$$W=\hat{e}\tilde{W}_{m}+\int_{\hat{e}}^{1}\tilde{W}_{h}de, $$
where
$$\tilde{W}_{m}=u_{m}r_{a}U_{m}+\sum_{j=F,I}{n_{m}^{j}}r_{a}{E_{m}^{j}}+\sum_{j=F,I}{n_{m}^{j}}r_{a}{J_{m}^{j}}+\sum_{j=F,I}{v_{m}^{j}}r_{a}{V_{m}^{j}}+{n_{m}^{I}}r_{a}J_{m}^{law}, $$
$$\tilde{W}_{h}=u_{h}r_{a}U_{h}+\sum_{j=F,I}{n_{h}^{j}}r_{a}{E_{h}^{j}}+\sum_{j=F,I}{n_{h}^{j}}r_{a}{J_{h}^{j}}+\sum_{j=F,I}{v_{h}^{j}}r_{a}{V_{h}^{j}}+{n_{h}^{I}}r_{a}J_{h}^{law}-c(e). $$
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.
By making use of the asset equations, imposing the flow equilibrium conditions as well as the government budget restriction in (40), and considering the case of no discounting, i.e. r+a→0, we can write the welfare function as
$$ W=\hat{e}W_{m}+\int_{\hat{e}}^{1}W_{h}de, $$
(49)
$$ W_{m}=\left(1-u_{m}\right)y_{m}-u_{m}{ky}_{m}\Theta_{m}, $$
(50)
$$ W_{h}=\left(1-u_{h}\right)y_{h}-u_{h}{ky}_{h}\Theta_{h}-c(e), $$
(51)
where \(\Theta _{l}=\left (1-{\sigma _{l}^{I}}\right)^{\gamma }{\theta _{l}^{F}}+\left ({\sigma _{l}^{I}}\right)^{\gamma }{\theta _{l}^{I}},~l={m,h}\). This welfare measure is analogous to the welfare measure described in, for example, Pissarides (2000) as it includes aggregate production minus total vacancy costs, i.e. note that \(u_{l}\Theta _{l}k=\left ({v_{l}^{F}}+{v_{l}^{I}}\right)k,~l={m,h}\). With the assumption of risk neutral individuals, we ignore distributional issues and hence wages will not feature in the welfare function. We have to find the socially optimal choice of audit rates for the sector employing manual workers and the sector employing highly educated workers; the welfare function in (49)–(51) is maximized by choice of p
m
and p
h
subject to the market reactions given by (32), (35), (36), (37), and (39) and the government budget restriction in (40). This yields the following first-order conditions:
$$ \frac{dW}{{dp}_{m}}=\hat{e}\frac{{dW}_{m}}{d\psi_{m}}\frac{d\psi_{m}}{{dp}_{m}}+\frac{dW}{d\left(1-e\right)} \frac{d\left(1-e\right)}{{dp}_{m}}=0, $$
(52)
$$ \frac{dW}{{dp}_{h}}=\left(1-\hat{e}\right)\frac{{dW}_{h}}{d\psi_{h}}\frac{d\psi_{h}}{{dp}_{h}}+ \frac{dW}{d\left(1-e\right)}\frac{d(1-e)}{{dp}_{h}}=0 $$
(53)
where \(\frac {{dW}_{l}}{d\psi _{l}}=\left [\sum _{j={F,I}}\frac {{dW}_{l}}{d{\theta _{l}^{j}}} \frac {d{\theta _{l}^{j}}}{d\psi _{l}}+\frac {{dW}_{l}}{d{\sigma _{l}^{I}}}\frac {d{\sigma _{l}^{I}}}{d\psi _{l}}\right ],\;j={m,h}.\) Evaluating the first-order conditions at the levels of p
m
and p
h
ensuring that ψ
m
=ψ
h
=1 turns out to be very convenient and gives
$$ \frac{dW}{{dp}_{m}}\mid{}_{\psi_{m}=1}=\frac{dW}{d\left(1-\hat{e}\right)}\frac{d\left(1-\hat{e}\right)}{{dp}_{m}}>0, $$
(54)
$$ \frac{dW}{{dp}_{h}}\mid{}_{\psi_{h}=1}=\frac{dW}{d\left(1-\hat{e}\right)}\frac{d\left(1-\hat{e}\right)}{{dp}_{h}}<0 $$
(55)
Let us first derive the socially optimal choice of tightness, search, and stock of educated workers by maximizing the welfare function in (49)–(51) with respect to \({\theta _{m}^{F}}\), \({\theta _{m}^{I}}\), \({\theta _{h}^{F}}\), \({\theta _{h}^{I}}\), \({\sigma _{m}^{I}}\), \({\sigma _{h}^{I}}\), and \(\hat {e}\). The socially optimal solution is solved from the following seven conditions:
$$ \left(\sigma_{l}^{I\ast}\right)^{\left(\gamma-1\right)}-\left(1-\sigma_{l}^{I\ast}\right)^{\gamma-1}=0,~\rightarrow\ \sigma_{l}^{I\ast}=\frac{1}{2},\;l=m,h, $$
(56)
$$ -sk\left(\theta_{l}^{\ast I}\right)^{\frac{1}{2}}+\frac{1}{2}\left[1-\frac{k\theta_{l}^{*I}}{\left(\frac{1}{2}\right)^{1-\gamma}}\right]=0,\;l=m,h, $$
(57)
$$ \left(y_{h}-y_{m}\right)\frac{k\theta_{l}^{\ast I}}{\left(\frac{1}{2}\right)^{1-\gamma}}-c\left(\hat{e}^{*}\right)=0. $$
(58)
From the first-order conditions for tightness in the formal and informal sector for manual and highly educated workers, i.e. \(\frac {\partial W}{{\partial \theta _{l}^{I}}}=0,\;\frac {\partial W}{{\partial \theta _{l}^{F}}}=0,\;l={F,I},\) we get the following conditions: \(2sk\left ({\theta _{l}^{I}}\right)^{\frac {1}{2}}=u_{l}\left [1+k\Theta _{l}\right ]\) and \(2sk\left ({\theta _{l}^{F}}\right)^{\frac {1}{2}}=u_{l}\;l={m,h},\) which gives \({\theta _{l}^{F}}={\theta _{l}^{I}}\). Substitute \({\theta _{l}^{F}}={\theta _{l}^{I}}\) into the first-order condition for search effort, \(\frac {\partial W}{{\partial \sigma _{l}^{I}}}=0\), and the following condition determines the social optimal level of search: \(\left ({\sigma _{m}^{I}}\right)^{\gamma -1}-\left (1-{\sigma _{m}^{I}}\right)^{\gamma -1}=0\). This yields \({\sigma _{l}^{I}}=\frac {1}{2},\;l={m,h}\). Substitute \({\sigma _{l}^{I}}=\frac {1}{2},\;l={m,h}\) into \(2sk\left ({\theta _{l}^{I}}\right)=u_{l}\left [1+k\Theta _{l}\right ]\) and \(2sk\left ({\theta _{l}^{F}}\right)^{\frac {1}{2}}=u_{l}\left [1+k\Theta _{l}\right ],\;l={m,h},\) which yields the four equations in (57) determining \({\theta _{m}^{F}},{\theta _{m}^{I}},{\theta _{h}^{F}},\) and \({\theta _{h}^{I}}\) in equilibrium. The socially optimal educational stock is determined by \(\partial W/\partial \left (1-\hat {e}\right)=W_{h} \left (\hat {e}\right)-W_{m}=y_{h}\left [1-u_{h} \left [1+k\Theta _{h}\right ]\right ]-y_{m}\left [1-u_{m}y_{m}\left [1+k\Theta _{m}\right ]\right ]-c\left (\hat {e}\right)=0\). Now use the equations determining the optimal levels of tightness, \(2sk\left ({\theta _{l}^{I}}\right)^{\frac {1}{2}}=u_{l}\) \right] and \(2sk\left ({\theta _{l}^{F}}\right)^{\frac {1}{2}}=u_{l}\left [1+k\Theta _{l}\right ],\;l={m,h},\) and the equation for the optimal educational level given by (58). To show that we have a global maximum, we differentiate W with respect to \({\sigma _{l}^{I}}\), \({\theta _{l}^{I}}\), \({\theta _{l}^{F}},l={m,h}\) and \(1-\hat {e}\) to obtain
$$\left(\sigma_{l}^{I\ast}\right)^{\gamma-1}-\left(1-\sigma_{l}^{I\ast}\right)^{\gamma-1}=0,\;l=m,h, $$
$$-sk\left(\theta_{l}^{\ast I}\right)^{\frac{1}{2}}+\frac{1}{2}\left[1-\frac{k\theta_{l}^{*I}}{\sigma^{1-\gamma}}\right]=0,\;l=m,h, $$
$$-sk\left(\theta_{l}^{\ast F}\right)^{\frac{1}{2}}+\frac{1}{2}\left[1-\frac{k\theta_{l}^{*F}}{\left(1-\sigma\right)^{1-\gamma}}\right]=0,\;l=m,h, $$
$$\left(y_{h}\frac{k\theta_{h}^{\ast I}}{\left({\sigma_{h}^{I}}\right)^{1-\gamma}}-y_{m}\frac{k\theta_{m}^{\ast I}}{\left({\sigma_{m}^{I}}\right)^{1-\gamma}}\right)-c\left(\hat{e}^{\ast}\right)=0. $$
The associated Hessian matrix is then
$$\begin{array}{@{}rcl@{}} \left|\begin{array}{ccccccc} \left(\gamma-1\right)S_{m} & 0 & 0 & 0 & 0 & 0 & 0\\ -\frac{k{\theta_{m}^{I}}}{2\left({\sigma_{m}^{I}}\right)^{2-\gamma}}\left(\gamma-1\right) & {\Delta_{m}^{I}} & 0 & 0 & 0 & 0 & 0\\ \frac{k{\theta_{m}^{F}}}{2\left(1-{\sigma_{m}^{I}}\right)^{2-\gamma}} & 0 & {\Delta_{m}^{F}} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \left(\gamma-1\right)S_{h} & 0 & 0 & 0\\ 0 & 0 & 0 & -\frac{k{\theta_{h}^{I}}}{2\left({\sigma_{h}^{I}}\right)^{2-\gamma}}\left(\gamma-1\right) & {\triangle_{h}^{I}} & 0 & 0\\ 0 & 0 & 0 & \frac{k\theta_{h}}{2\left(1-{\sigma_{h}^{I}}\right)^{2-\gamma}}\left(\gamma-1\right) & 0 & {\triangle_{h}^{F}} & 0\\ -y_{m}\left(\gamma-1\right)\frac{k\theta_{m}^{\ast I}}{\left({\sigma_{m}^{I}}\right)^{2-\gamma}} & -y_{m}k\left({\sigma_{m}^{I}}\right)^{\gamma-1} & 0 & \left(\gamma-1\right)y_{h}\frac{k\theta_{h}^{\ast I}}{\left({\sigma_{h}^{I}}\right)^{2-\gamma}} & y_{h}k\left({\sigma_{h}^{I}}\right)^{\gamma-1} & 0 & c'\left(\hat{e}^{\ast}\right) \end{array}\right| \end{array} $$
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.
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 \(\frac {dW}{{dp}_{h}}\mid _{{\psi _{h}^{e}}<1}>0\).
Optimal punishment policy including auditing costs
The government budget constraint with auditing costs, φ(p), is \(\frac {z\hat {e}{n_{m}^{F}}{w_{m}^{F}}}{1+z}+ \frac {p\alpha \hat {e}{n_{m}^{I}}{w_{m}^{I}}}{1+p\alpha +\kappa _{m}}+ \frac {z\left (1-\hat {e}\right){n_{h}^{F}}{w_{h}^{F}}}{1+z}+ \frac {p\alpha \left (1-\hat {e}\right){n_{m}^{I}}{w_{h}^{I}}}{1+p\alpha +\kappa _{h}}-\varphi (p)=R\), where p is the total intensity of audits, p=p
m
+p
h
. Adding costs of auditing has no impact on the positive analyses. The welfare function, however, is equal to \(W=\hat {e}W_{m}+\int _{\hat {e}}^{1}W_{h}de-\varphi (p),\) with first-order conditions for optimal audit rates:
$$\frac{dW}{{dp}_{m}}=\hat{e}\frac{{dW}_{m}}{d\psi_{m}}\frac{d\psi_{m}}{{dp}_{m}}+\frac{dW}{d\left(1-e\right)}\frac{d\left(1-e\right)}{{dp}_{m}}-\varphi^{\prime}(p)=0, $$
$$\frac{dW}{{dp}_{h}}=\left(1-\hat{e}\right)\frac{{dW}_{h}}{d\psi_{h}}\frac{d\psi_{h}}{{dp}_{h}}+\frac{dW}{d\left(1-e\right)}\frac{d\left(1-e\right)}{{dp}_{h}}-\varphi^{\prime}(p)=0, $$
where \(\frac {{dW}_{l}}{d\psi _{l}}=\sum _{j={F,I}}\frac {{dW}_{l}}{d{\theta _{l}^{j}}} \frac {d{\theta _{l}^{j}}}{d\psi _{l}}+\frac {{dW}_{l}}{d{\sigma _{l}^{I}}}\frac {d{\sigma _{l}^{j}}}{d\psi _{l}},\;j={m,h}.\) The optimal level of audits is reduced in both sectors. The result from Proposition 7, that welfare is maximized when the government to a larger extent targets its audits to the manual sector, i.e. p
m
>p
h
if κ
h
≥κ
m
, will still hold.