We now examine the equilibrium in the model. We begin by specifying a condition on parameter values for which, given that ${w}_{F}>{w}_{F}^{\ast}$, a market-clearing informal wage ${w}_{I}={w}_{I}^{\ast}$ below the formal wage *w*
_{
F
}
exists.

### Proposition 2

If ${A}_{I}<\left(1-{\varphi}_{d}\right){w}_{F}+\left|{W}_{I}^{i}\left({w}_{F}\right)\right|$ then there exists a unique wage ${w}_{I}^{\ast}\in \left(0,{w}_{F}\right)$ such that ${L}_{I}^{d}=\left|{W}_{I}^{i}\right|.$

The intercept on the informal labour demand curve must not be too high – for otherwise the condition that *w*
_{
I
} < *w*
_{
F
}
would be violated. Similarly, the condition imposes a lower bound on *w*
_{
F
}, while the probability *ϕ*
_{
d
}
that an informal employer will renege on paying the wage must not be too high.

Given that the condition in Proposition 2 holds, we can interpret the wage *w*
_{
I
} underlying Figure 2 as the equilibrium wage ${w}_{I}^{\ast}$, so that the figure represents the equilibrium. The comparative statics of the model, to which we now turn, can be understood intuitively by reference to the figure. We are primarily interested in the effect of variations of parameter values on the size and membership of the sets shown in the figure; but most of these effects work through their impacts on two endogenous variables, the cut-off value $\alpha =\hat{\alpha}$ of log ability and the informal wage rate *w*
_{
I
}. We therefore begin by specifying the comparative statics of these variables.

The parameters *w*
_{
F
}, *ϕ*
_{
c
} and *ϕ*
_{
d
} may be interpreted as, at least to some extent, policy variables. *w*
_{
F
}
may be interpreted as a minimum wage rate imposed by the government. *ϕ*
_{
c
}
is the probability that informal wages will be confiscated by the government, and this can be increased by greater effort in detection or a greater willingness to confiscate once detected. *ϕ*
_{
d
} is the probability that the employer will choose not to pay the informal wage. The government may be able to reduce this by strengthening contract enforcement of wage payment to informal workers. The parameters *A*
_{
F
} and $\stackrel{\u0304}{\alpha}$ may be interpreted as indicators of economic development. *A*
_{
F
}
is the intercept of the demand curve for formal wage labour, and we may assume that the demand for the goods produced by formal wage labour increases as the economy grows. $\stackrel{\u0304}{\alpha}$ is the upper bound of log ability. An increase in $\stackrel{\u0304}{\alpha}$ can be interpreted as a rise in the average level of ability in the economy. The remaining parameters, *ψ* and *A*
_{
I
} are less easily interpreted as policy tools or indicators of development.

### Lemma 4

In equilibrium, the comparative statics of the log ability cut-off $\hat{\alpha}$ and the informal wage rate *w*
_{
I
}
are given by

Log ability cut-off

$\frac{d\hat{\alpha}}{d{A}_{F}}<0;\phantom{\rule{2em}{0ex}}\frac{d\hat{\alpha}}{d{w}_{F}}>0;\phantom{\rule{2em}{0ex}}\frac{d\hat{\alpha}}{\mathrm{d\psi}}>0;\phantom{\rule{2em}{0ex}}\frac{d\hat{\alpha}}{d\stackrel{\u0304}{\alpha}}\gtrless 0;\phantom{\rule{2em}{0ex}}\frac{d\hat{\alpha}}{d{A}_{I}}=\frac{d\hat{\alpha}}{\mathrm{d\varphi}}=0;$

Informal wage rate

$\begin{array}{ccc}\frac{d{w}_{I}}{d{A}_{F}}& >0;& \phantom{\rule{3em}{0ex}}\frac{d{w}_{I}}{d{w}_{F}}<0;\phantom{\rule{2em}{0ex}}\frac{d{w}_{I}}{\mathrm{d\psi}}<0;\phantom{\rule{2em}{0ex}}\frac{d{w}_{I}}{d\overline{\alpha}}\gtrless 0;\\ \frac{d{w}_{I}}{d{A}_{I}}& >0;& \phantom{\rule{0.1em}{0ex}}\frac{d{w}_{I}}{d{\varphi}_{d}}>\frac{d{w}_{I}}{d{\varphi}_{c}}>0\text{.}\end{array}$

A higher demand for formal wage labour, as represented by *A*
_{
F
}, is associated with a lower cut-off log ability $\hat{\alpha}$, while a higher formal wage rate *w*
_{
F
} is associated with a higher cut-off ability. A higher value of *ψ*
represents a greater risk in self employment. At the margin, this causes the first preference of some voluntarily self employed workers to change to formal employment. The existence of relatively high ability among this group pushes up the cut-off value $\hat{\alpha}$ at which the available formal jobs are filled. The demand for informal wage workers and income risk in informal wage employment have no impact on the formal employment decision. Although we shall be able to say something below about the effects of varying $\stackrel{\u0304}{\alpha}$, we are not able to sign $d\hat{\alpha}/d\stackrel{\u0304}{\alpha}$ (or $d{w}_{I}/d\stackrel{\u0304}{\alpha}$).

A higher demand for formal wage labour exerts upward pressure on the informal wage, whereas a higher formal wage has the opposite effect on the informal wage because it reduces formal labour demand. Greater risk in self employment causes a substitution effect from involuntary self employment to informal wage employment, with a negative impact on *w*
_{
I
}. A higher demand for informal labour is associated with a higher wage *w*
_{
I
}. Greater risk in informal wage employment causes a substitution of informal wage workers into involuntary self employment, exerting a positive impact on *w*
_{
I
}. However, this impact is greater for *ϕ*
_{
d
}
than for *ϕ*
_{
c
}: a marginal increase in the risk that the employer will renege on payment has a greater effect than the same marginal increase in the risk that the government will prevent both the worker and the employer from receiving informal income. Both of these increases in risk make informal wage employment less attractive to the worker, reducing informal labour supply and therefore increasing *w*
_{
I
}. But, as we have explained, the latter effect has no effect on labour demand, whereas the former increases labour demand, and so pushes *w*
_{
I
} up further.

Using Lemma 4 and a standard stability condition on the relative slopes of labour demand and supply, we obtain the comparative statics of employment.

### Proposition 3

In a stable equilibrium, the comparative statics of employment in each occupation are:

Voluntary self employment

$\begin{array}{c}\frac{d\left|{W}_{S}\right|}{\mathrm{d\psi}}\phantom{\rule{0.7em}{0ex}}<0;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{S}\right|}{d\overline{\alpha}}>0;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{S}\right|}{d{w}_{F}}<0;\\ \frac{d\left|{W}_{S}\right|}{d{A}_{F}}\phantom{\rule{0.7em}{0ex}}=\frac{d\left|{W}_{S}\right|}{d{A}_{I}}=\frac{d\left|{W}_{S}\right|}{\mathrm{d\varphi}}=0;\end{array}$

Formal wage employment

$\begin{array}{c}\frac{d\left|{W}_{F}^{r}\right|}{d{A}_{F}}\phantom{\rule{0.7em}{0ex}}=1;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{F}^{r}\right|}{d{w}_{F}}=-1;\\ \frac{d\left|{W}_{S}\right|}{d{A}_{F}}\phantom{\rule{0.7em}{0ex}}=\frac{d\left|{W}_{S}\right|}{d{A}_{I}}=\frac{d\left|{W}_{S}\right|}{\mathrm{d\varphi}}=\frac{d\left|{W}_{S}\right|}{\mathrm{d\psi}}=\frac{d\left|{W}_{S}\right|}{d\overline{\alpha}}=0;\end{array}$

Informal wage employment

$\begin{array}{c}\frac{d\left|{W}_{I}^{i}\right|}{d{A}_{F}}\phantom{\rule{0.7em}{0ex}}<0;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{I}^{i}\right|}{d{A}_{I}}>0;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{I}^{i}\right|}{d{\varphi}_{c}}<0;\\ \frac{d\left|{W}_{I}^{i}\right|}{d{\varphi}_{d}}\phantom{\rule{0.7em}{0ex}}>0;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{I}^{i}\right|}{\mathrm{d\psi}}>0;\\ \frac{d\left|{W}_{I}^{i}\right|}{d\overline{\alpha}}\phantom{\rule{0.7em}{0ex}}\gtrless 0\iff \frac{d{w}_{I}}{d\overline{\alpha}}\lessgtr 0;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{I}^{i}\right|}{d{w}_{F}}>0;\end{array}$

Involuntary self employment

$\begin{array}{c}\frac{d\left|{W}_{S}^{i}\right|}{d{A}_{F}}\phantom{\rule{0.7em}{0ex}}\gtrless 0\iff \frac{d\left|{W}_{I}^{i}\right|}{d{A}_{F}}\lessgtr -1;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{S}^{i}\right|}{d{A}_{I}}<0;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{S}^{i}\right|}{d{\varphi}_{c}}>0;\\ \frac{d\left|{W}_{S}^{i}\right|}{d{\varphi}_{d}}\phantom{\rule{0.7em}{0ex}}<0;\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{S}^{i}\right|}{\mathrm{d\psi}}\gtrless 0\iff \left|\frac{d\left|{W}_{S}\right|}{\mathrm{d\psi}}\right|\gtrless \frac{d\left|{W}_{I}^{i}\right|}{\mathrm{d\psi}};\phantom{\rule{2em}{0ex}}\frac{d\left|{W}_{S}^{i}\right|}{d\overline{\alpha}}\gtrless 0;\\ \frac{d\left|{W}_{S}^{i}\right|}{d{w}_{F}}\phantom{\rule{0.7em}{0ex}}\gtrless 0\iff \frac{d\left|{W}_{I}^{i}\right|}{d{w}_{F}}\lessgtr 1-\frac{d\left|{W}_{S}\right|}{d{w}_{F}}.\end{array}$

As we might expect, voluntary self employment $\left|{W}_{S}\right|$ is decreasing in the riskiness *ψ* of self employment income and in the formal wage rate *w*
_{
F
}, but increasing in the average ability level in the labour force, as represented by $\overline{\alpha}$, while formal wage employment $\left|{W}_{F}^{r}\right|$ is increasing in the demand for formal labour, as given by the intercept *A*
_{
F
}, and decreasing in *w*
_{
F
}. More interesting results are obtained, however, for the other two employment types.

Informal wage employment $\left|{W}_{I}^{i}\right|$ is decreasing in the demand for formal labour, and it is increasing in the demand for informal wage labour and in the riskiness *ψ* of self employment. Also, because a higher *w*
_{
F
}
reduces $\left|{W}_{F}^{r}\right|$, $\left|{W}_{I}^{i}\right|$ is increasing in *w*
_{
F
}. However, greater risk *ψ* of not being paid an informal wage can affect $\left|{W}_{I}^{i}\right|$ in either direction, depending on the source of the risk. A higher risk *ϕ*
_{
c
} that the government will confiscate the informal earnings of both the employer and employees in an informal firm results in a smaller $\left|{W}_{I}^{i}\right|$. But when the higher risk is in the form *ϕ*
_{
d
}
– that an employer is more likely to renege on payment – a greater $\left|{W}_{I}^{i}\right|$ results. This is because an increase in *ϕ*
_{
d
}
both depresses the informal wage, and increases the demand for informal wage labour by making it more elastic. The two effects conflict, but, assuming a stable market equilibrium in the informal labour market, the latter effect dominates. This suggests that the strengthening of detection by the authorities of informal wage work, but *improvement* of the contractual rights of informal workers to be paid their wages, would lead to a reduction in the amount of informal wage work. Finally, a higher level of average ability, as given by $\overline{\alpha}$, affects $\left|{W}_{I}^{i}\right|$ with the opposite sign to its effect on the informal wage *w*
_{
I
}
– but this effect can go in either direction. Thus, an economy that is more developed in this sense *may* have more informal wage work. It is possible to show that $d{w}_{i}/d\overline{\alpha}$ is positive for sufficiently large $\overline{\alpha}$ and negative for $\overline{\alpha}$ sufficiently small. So, for small $\overline{\alpha}$, informal wage employment is increasing in the average level of ability, but this relationship reverses at higher levels of $\overline{\alpha}$.

Although involuntary self employment $\left|{W}_{S}^{i}\right|$ is found to be decreasing in the demand for informal wage labour, as represented by *A*
_{
I
}, it may be increasing or decreasing in the demand for formal labour, *A*
_{
F
}. This is because a rise in *A*
_{
F
}
reduces the proportion of workers who do not achieve their first choice, but also depresses *w*
_{
I
}. If the latter effect is sufficiently small, involuntary self employment is increasing in *A*
_{
F
}. Thus, if economic development is associated with a greater demand for the output of formal wage work, it may be found that it is also associated with more involuntary self employment. Similarly, the effect on $\left|{W}_{S}^{i}\right|$ of a higher *w*
_{
F
} can be of either sign. Also, the effects on $\left|{W}_{S}^{i}\right|$ of greater risk are not clear-cut. Greater risk *ψ* with respect to the earnings of the self employed can affect $\left|{W}_{S}^{i}\right|$ in either direction: there is a direct effect, discouraging involuntary self employment, but there is also an indirect effect which, by discouraging *voluntary* self employment, increases the supply of workers to involuntary self employment (and informal wage employment). The net effect can go either way, the underlying condition being that specified in the proposition. For similar reasons, a higher level of average ability, as represented by $\overline{\alpha}$ can affect $\left|{W}_{S}^{i}\right|$ in either direction. The effect on $\left|{W}_{S}^{i}\right|$ of an increase in the risks, *ϕ*
_{
c
}
and *ϕ*
_{
d
}, associated with informal wage employment is the opposite in sign to the effect of these risks on informal wage employment. But, since $d\left|{W}_{S}\right|/\mathrm{d\varphi}=d\left|{W}_{F}^{r}\right|/\mathrm{d\varphi}=0$, variation of $\left|{W}_{S}^{i}\right|$ in this case is exactly compensated by variation of the opposite sign in $\left|{W}_{I}^{i}\right|$.