The equilibrium can be characterised by reducing the many equations in terms of many endogenous variables specified above, into just one equation in terms of one endogenous variable, namely θ. Four conditions define the equilibrium: (1) firms enter freely, such that the value of maintaining a vacancy is zero, (2) matches in the wage sector are consummated if and only if it is in the interests of worker and firm to do so, (3) steady state flows into and out of unemployment, selfemployment, and wageemployment are equal, for each type, and (4) individuals only take jobs—in either self or wageemployment—that are worth their while, given their type and their ability y. To find the equilibrium, we first use Condition (4) to derive some cutoff levels of productivity y, above and below which workers will forego work in either wage or selfemployment. Secondly, we use Condition (3) to derive the steadystate level of unemployment in the model. Finally, we bring these components together by using Condition (1) and rewriting the asset equation for posting a vacancy.
5.1 Productivity cutoffs
As in Roy’s (1951) classic model of occupational choice, the jobs that individuals are willing to take depend on their selfemployment productivity, y, as well as their type, k=A,B. Defining \(X_{e}^{k}(y) \equiv V_{e}^{k}(y)V_{u}^{k}(y)\), workers will take wageemployment only if its flow value is greater than unemployment, such that \(X_{e}^{k}(y)>0\). Similarly, defining \(X_{s}^{k}(y) \equiv V_{s}^{k}(y)V_{u}^{k}(y)\), workers will participate in selfemployment only if \(X_{s}^{k}(y)>0\). This implies there may be some typespecific cutoff values of y, where \(X_{e}^{k}(y) = 0\) and \(X_{s}^{k}(y)=0\), above and below which certain jobs will not be taken.
In order to interpret these cutoffs, we need to know how workers value wage and selfemployment differently—relative to unemployment—according to their productivity. That is, we need to understand how \(X_{e}^{k}(y)\) and \(X_{s}^{k}(y)\) change in y. Rearranging the worker value functions along with the wage schedule and differentiating, we can express \(\frac {\partial X_{e}^{k}(y)}{\partial y}\) in terms of \(\frac {\partial X_{s}^{k}(y)}{\partial y}\), showing that these two differentials have (weakly) opposite signs.
$$ \left(r + q_{e} + \gamma \mathbbm{1}_{\left[X_{e}^{k}(y) > 0\right]} m^{k}(\theta)\right) \frac{\partial X_{e}^{k}(y)}{\partial y} =  \mathbbm{1}_{\left[X_{s}^{k}(y) > 0\right]}\alpha \gamma \frac{\partial X_{s}^{k}(y)}{\partial y} $$
(10)
It can further be shown that \(\frac {\partial X_{s}^{k}(y)}{\partial y}\) is always positive, while the sign of \(\frac {\partial X_{e}^{k}(y)}{\partial y}\) depends on whether \(X_{s}^{k}(y) > 0\) or \(X_{s}^{k}(y) \leq 0\).
$$ \frac{\partial X_{e}^{k}(y)}{\partial y}\left\{ \begin{array}{ll} < 0 & \text{if}\,\, X_{s}^{k}(y) > 0 \\ = 0 & \text{if}\,\, X_{s}^{k}(y) \leq 0 \\ \end{array}\right. $$
(11)
At the first potential cutoff where \(X_{s}^{k}(y) = 0\), unemployed individuals are just indifferent between accepting and rejecting a selfemployment job: we label the level of productivity at which this occurs y^{∗k}. Since \(X_{s}^{k}(y)\) slopes upwards, unemployment is more valuable than selfemployment when productivity is below y^{∗k}. This means workers with y<y^{∗k} would never accept selfemployment opportunities. Intuitively, if y is low, the returns to selfemployment are insufficient to tempt workers away from queueing for wage jobs, where they could achieve higher earnings in the future.^{Footnote 14}
There is also a potential cutoff—which we label y^{∗∗k}—where \(X_{e}^{k}(y) = 0\) and unemployed workers are just indifferent between accepting and rejecting wage job offers. Since \(X_{e}^{k}(y)\) is weakly downward sloping, unemployment is more valuable than wageemployment when productivity is above y^{∗∗k}. Workers with y>y^{∗∗k} will never accept wageemployment offers, as they are better off biding their time in unemployment and waiting for a selfemployment job, in which they have high productivity and therefore greater earnings.
Providing the lower productivity cutoff y^{∗k} exists, which is guaranteed under relatively mild assumptions about b, three types of equilibria may prevail (see Fig. 2). This depends on the upper productivity cutoff y^{∗∗k}. The existence of both productivity cutoffs is discussed in Appendix 4.
When z_{0} is large relative to b, all individuals are willing to take wage job offers, so the upper cutoff at y^{∗∗k} does not exist (Panel A). In this case, the lower cutoff at y^{∗k} simply divides individuals into those that would also be willing to work in selfemployment (y≥y^{∗k}) and those that would not (y<y^{∗k}).
If, however, b is large relative to z_{0}, the flow value of unemployment is so high that no individuals are willing to take wage job offers, again meaning the upper cutoff at y^{∗∗k} does not exist (Panel B). In this type of equilibrium, individuals with low productivity levels (y<y^{∗k}) never exit unemployment, because the returns to selfemployment are insufficient to tempt them away from simply receiving b each period. However, individuals with higher productivity (y≥y^{∗k}) enter selfemployment if such jobs arrive.
In the final case, where both cutoffs y^{∗k} and y^{∗∗k} exist, individuals can be divided into three groups (Panel C). Individuals with very low productivity only ever accept opportunities in wageemployment, while individuals with very high productivity accept only selfemployment opportunities. However, individuals with y^{∗k}≤y<y^{∗∗k}, accept both self and wageemployment jobs. These individuals do not fully specialise in either wage or selfemployment and may transition (via unemployment) between different sectors throughout their lifetime. In the analysis that follows in Section 6, we restrict the values of z_{0} and b to ensure the existence of both productivity cutoffs, such that an equilibrium of this type prevails.
Using the worker value functions and the wage schedule to solve for y^{∗k} in \(X_{s}^{k}\left (y^{*k}\right) = 0\), the lower cutoff can be expressed in terms of the model’s exogenous parameters and the matching function:
$$ y^{*k} = \frac{b(r + q_{e}) + \gamma m^{k}(\theta) z_{0}}{\left(r + q_{e} + \gamma m^{k}(\theta)\right)} $$
(12)
The worker value functions and the wage schedule can also be used to solve for y^{∗∗k} in \(X_{e}^{k}\left (y^{**k}\right) = 0\). Since the m^{k}(θ) and w^{k}(y) terms are eliminated, y^{∗∗k} can be expressed as a function of the exogenous parameters of the model, meaning that y^{∗∗A}=y^{∗∗B}, and the upper cutoff is unaffected by type. This is because workers with high selfemployment productivity are not affected by discrimination in the wage sector, even if they are TypeBs, as they forego wageemployment opportunities anyway.
$$ y^{**k} = \frac{z_{0}\left(\alpha + q_{s} + r\right)  b\left(r + q_{s}\right)}{\alpha} $$
(13)
5.2 Steadystate employment flows
The flows between unemployment, selfemployment, and wageemployment, for Type k=A,B workers depend on their productivity relative to the cutoffs, y^{∗k} and y^{∗∗k}. Individuals with low productivity (y<y^{∗k}) will never take selfemployment jobs, so the only relevant flow for them is between wageemployment and unemployment. By contrast, individuals with high productivity (y>y^{∗∗k}) will never take wageemployment, so they only move between selfemployment and unemployment. It is only individuals with middling ability (y^{∗k}≤y<y^{∗∗k}) that are willing to become both wage and selfemployed, and therefore flow from unemployment to self and wageemployment (and back). In a steady state, these employment flows must balance so that the proportion of time that a Typek, ability y individual is unemployed (u^{k}(y)), selfemployed \(\left (n_{s}^{k} (y)\right)\), or wageemployed \(\left (n_{e}^{k} (y)\right)\) remains unchanged and adds up to 1. The proportion of time spent by Typek individuals with ability y in unemployment, selfemployment, and wageemployment can be written:
$$ u^{k}(y) =\left\{ \begin{array}{ll} \frac{q_{e}}{m^{k}(\theta) + q_{e}} & \text{if}\,\, y < y^{*k} \\ \frac{q_{e}}{m^{k}(\theta) + q_{e} + \alpha \frac{q_{e}}{q_{s}}} & \text{if} \,\,y^{*k} \leq y < y^{**k} \\ \frac{q_{s}}{\alpha + q_{s}} & \text{if}\,\, y \geq y^{**k} \\ \end{array}\right. $$
(14)
$$ n_{s}^{k}(y) =\left\{ \begin{array}{ll} 0 & \text{if}\,\, y < y^{*k} \\ \frac{\alpha \frac{q_{e}}{q_{s}}}{m^{k}(\theta) + q_{e} + \alpha \frac{q_{e}}{q_{s}}} & \text{if}\,\, y^{*k} \leq y < y^{**k} \\ \frac{\alpha}{\alpha + q_{s}} & \text{if}\,\, y \geq y^{**k} \\ \end{array}\right. $$
(15)
$$ n_{e}^{k}(y) =\left\{ \begin{array}{ll} \frac{m^{k}(\theta)}{m^{k}(\theta) + q_{e}} & \text{if}\,\, y < y^{*k} \\ \frac{m^{k}(\theta)}{m^{k}(\theta) + q_{e} + \alpha \frac{q_{e}}{q_{s}}} & \text{if} \,\,y^{*k} \leq y < y^{**k} \\ 0 & \text{if} \,\,y \geq y^{**k} \\ \end{array}\right. $$
(16)
The total number of unemployed Typeks, which is needed to pin down θ, can be written as the product of the average proportion of time that a Typek individual spends in unemployment (E[u^{k}(y)]) and the total number of Tykeks in the population^{Footnote 15}. The total number of unemployed individuals (U) is then simply the sum of U^{A} and U^{B}. The Typespecific numbers of wageemployed and selfemployed people \(\left (N_{e}^{k}\right)\) and \(\left (N_{s}^{k}\right)\) and the total numbers of wageemployed and selfemployed people (N_{e}) and (N_{s}) can be calculated in the same way.
$$ U^{k} = N \pi^{k} \mathrm{E}\left[u^{k}(y)\right] = N \pi^{k} \left[ \int_{0}^{y^{*k}} u^{k}(y) f(y) dy + \int_{y^{*k}}^{y^{{*{*k}}}} u^{k}(y) f(y) dy + \int_{y^{{*{*k}}}}^{1} u^{k}(y) f(y) dy \right] $$
(17)
5.3 The freeentry condition
To find the model’s equilibrium, we write a single equation in which the only unknown is labour market tightness, θ. We do this by rewriting the asset equation for posting a vacancy in Eq. (7), assuming that the freeentry of firms drives Π_{v} down to 0, and incorporating all the information on occupational choice and steadystate employment flows from Sections 5.1 and 5.2. Setting Π_{v}=0 and noting that the max operators are redundant because firms will not accept matches that do not provide them with a positive profit, we can write:
$$ c = \frac{m^{A}(\theta)}{\theta} \frac{U^{A}}{U^{A} + U^{B}} \mathrm{E} \left[\Pi_{e}^{A} (y)\right] + \frac{m^{B}(\theta)}{\theta} \frac{U^{B}}{U^{A} + U^{B}} \mathrm{E} \left[\Pi_{e}^{B} (y)\right] $$
(18)
The expectations operators in Eq. (18) are taken over all the possible values of y, separating out TypeAs and TypeBs, with which a firm could match. Since firms can only match with unemployed workers, we apply Bayes’ Law to adjust the productivity distribution for the entire population^{Footnote 16}. The underlying productivity distribution for the full population, f(y), is the same for TypeAs and TypeBs, but the productivity distribution for unemployed individuals, \(h_{u}^{k} (y)\), differs according to type, because TypeAs and TypeBs do not have equal chances of being in unemployment.
$$ h_{u}^{k} (y) = \frac{u^{k}(y) f (y)}{\left(U^{k}/{N \pi^{k}}\right)} $$
(19)
Using the asset equations for filled jobs and the relevant wage schedules, we can therefore write the freeentry condition without the expectations operators as:
$$ \begin{aligned} c =& {\frac{m(\theta)}{\theta}} \frac{(1\gamma)}{(r + q_{e})} \left[ {\frac{U^{A}}{U^{A} + U^{B}}} \left[\int_{0}^{{y^{*A}}}\left(z_{0}  r {V_{u}^{A} (y)}\right) \frac{{u^{A}(y)} f(y)}{\left(U^{A}/{N \pi}\right)}dy \right. \right.\\ &+ \left. \int_{{y^{*A}}}^{y^{**A}}\left(z_{0}  r {V_{u}^{A} (y)}\right) \frac{{u^{A}(y)} f(y)}{\left(U^{A}/{N \pi}\right)}dy \right] \\ & + {\frac{U^{B}}{U^{A} + U^{B}}} (1  \lambda) \left[ \int_{0}^{{y^{*B}}}\left(z_{0}  r {V_{u}^{B} (y)}\right) \frac{{u^{B}(y)} f(y)}{\left(U^{B}/{N (1\pi)}\right)}dy \right.\\ & +\left.\left. \int_{{y^{*B}}}^{y^{**B}}\left(z_{0}  r {V_{u}^{B} (y)}\right) \frac{{u^{B}(y)} f(y)}{\left(U^{B}/{N (1\pi)}\right)}dy \right] \right] \end{aligned} $$
(20)
The only unknowns besides θ in Eq. (20)—aside from the unemployment rates and levels and the productivity cutoffs covered in the previous sections—are the unemployment flow value terms, \(rV_{u}^{A} (y)\) and \(rV_{u}^{B} (y)\). These will differ for individuals either side of y^{∗k}. For workers with y^{∗k}≤y<y^{∗∗k}, for whom both self and wageemployment are an option, the worker value functions can be rearranged for:
$$ r V^{k}_{u} (y) =\frac{b(r + q_{s})(r + q_{e}) + \alpha y (r + q_{e}) + \gamma m^{k}(\theta) z_{0} (r + q_{s})}{(r + q_{s})(r + q_{e}) + \alpha (r + q_{e}) + \gamma m^{k}(\theta) (r + q_{s})} $$
(21)
Similarly, for workers with y<y^{∗k}, who take only wageemployment opportunities, we can write:
$$ r V^{k}_{u} (y) =\frac{b(r + q_{e}) + \gamma m^{k}(\theta) z_{0}}{(r + q_{e}) + \gamma m^{k}(\theta)} $$
(22)
As such, Eq. (20) is an expression with only one unknown: θ. By imposing a functional form on the matching function and finding values for the exogenous parameters, we can solve this equation using numerical methods.
As shown in Appendix 5, the following standard conditions for the matching function (m(θ)) guarantee the existence of the equilibrium:

1.
m(θ) is increasing in θ.

2.
\(\frac {m(\theta)}{\theta }\) is decreasing in θ.

3.
\({\lim }_{\theta \rightarrow 0} m(\theta) = 0\) and \({\lim }_{\theta \rightarrow \infty } m(\theta) = \infty \).

4.
\({\lim }_{\theta \rightarrow 0} \frac {m(\theta)}{\theta } = \infty \) and \({\lim }_{\theta \rightarrow \infty } \frac {m(\theta)}{\theta } = 0\).
As in other models that incorporate worker heterogeneity into the Mortensen and Pissarides (1994) framework, the uniqueness of the equilibrium cannot be guaranteed analytically (Chéron et al. 2011; Albrecht et al. 2017). We therefore follow Albrecht et al. (2009) and numerically verify that the righthandside of the freeentry condition in Eq. (20) is monotonically decreasing for all values of θ in the simulations that follow.