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, self-employment, and wage-employment are equal, for each type, and (4) individuals only take jobs—in either self- or wage-employment—that are worth their while, given their type and their ability y. To find the equilibrium, we first use Condition (4) to derive some cut-off levels of productivity y, above and below which workers will forego work in either wage- or self-employment. Secondly, we use Condition (3) to derive the steady-state 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 cut-offs
As in Roy’s (1951) classic model of occupational choice, the jobs that individuals are willing to take depend on their self-employment 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 wage-employment 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 self-employment only if \(X_{s}^{k}(y)>0\). This implies there may be some type-specific cut-off 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 cut-offs, we need to know how workers value wage- and self-employment 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 cut-off where \(X_{s}^{k}(y) = 0\), unemployed individuals are just indifferent between accepting and rejecting a self-employment 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 self-employment when productivity is below y∗k. This means workers with y<y∗k would never accept self-employment opportunities. Intuitively, if y is low, the returns to self-employment 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 cut-off—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 wage-employment when productivity is above y∗∗k. Workers with y>y∗∗k will never accept wage-employment offers, as they are better off biding their time in unemployment and waiting for a self-employment job, in which they have high productivity and therefore greater earnings.
Providing the lower productivity cut-off 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 cut-off y∗∗k. The existence of both productivity cut-offs is discussed in Appendix 4.
When z0 is large relative to b, all individuals are willing to take wage job offers, so the upper cut-off at y∗∗k does not exist (Panel A). In this case, the lower cut-off at y∗k simply divides individuals into those that would also be willing to work in self-employment (y≥y∗k) and those that would not (y<y∗k).
If, however, b is large relative to z0, the flow value of unemployment is so high that no individuals are willing to take wage job offers, again meaning the upper cut-off 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 self-employment are insufficient to tempt them away from simply receiving b each period. However, individuals with higher productivity (y≥y∗k) enter self-employment if such jobs arrive.
In the final case, where both cut-offs 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 wage-employment, while individuals with very high productivity accept only self-employment opportunities. However, individuals with y∗k≤y<y∗∗k, accept both self- and wage-employment jobs. These individuals do not fully specialise in either wage- or self-employment and may transition (via unemployment) between different sectors throughout their lifetime. In the analysis that follows in Section 6, we restrict the values of z0 and b to ensure the existence of both productivity cut-offs, 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 cut-off 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 mk(θ) and wk(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 cut-off is unaffected by type. This is because workers with high self-employment productivity are not affected by discrimination in the wage sector, even if they are Type-Bs, as they forego wage-employment opportunities anyway.
$$ y^{**k} = \frac{z_{0}\left(\alpha + q_{s} + r\right) - b\left(r + q_{s}\right)}{\alpha} $$
(13)
5.2 Steady-state employment flows
The flows between unemployment, self-employment, and wage-employment, for Type- k=A,B workers depend on their productivity relative to the cut-offs, y∗k and y∗∗k. Individuals with low productivity (y<y∗k) will never take self-employment jobs, so the only relevant flow for them is between wage-employment and unemployment. By contrast, individuals with high productivity (y>y∗∗k) will never take wage-employment, so they only move between self-employment and unemployment. It is only individuals with middling ability (y∗k≤y<y∗∗k) that are willing to become both wage- and self-employed, and therefore flow from unemployment to self- and wage-employment (and back). In a steady state, these employment flows must balance so that the proportion of time that a Type-k, ability y individual is unemployed (uk(y)), self-employed \(\left (n_{s}^{k} (y)\right)\), or wage-employed \(\left (n_{e}^{k} (y)\right)\) remains unchanged and adds up to 1. The proportion of time spent by Type-k individuals with ability y in unemployment, self-employment, and wage-employment 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 Type-ks, which is needed to pin down θ, can be written as the product of the average proportion of time that a Type-k individual spends in unemployment (E[uk(y)]) and the total number of Tyke-ks in the populationFootnote 15. The total number of unemployed individuals (U) is then simply the sum of UA and UB. The Type-specific numbers of wage-employed and self-employed people \(\left (N_{e}^{k}\right)\) and \(\left (N_{s}^{k}\right)\) and the total numbers of wage-employed and self-employed people (Ne) and (Ns) 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 free-entry 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 free-entry of firms drives Πv down to 0, and incorporating all the information on occupational choice and steady-state employment flows from Sections 5.1 and 5.2. Setting Πv=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 Type-As and Type-Bs, 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 populationFootnote 16. The underlying productivity distribution for the full population, f(y), is the same for Type-As and Type-Bs, but the productivity distribution for unemployed individuals, \(h_{u}^{k} (y)\), differs according to type, because Type-As and Type-Bs 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 free-entry 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 cut-offs 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 wage-employment 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 wage-employment 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 right-hand-side of the free-entry condition in Eq. (20) is monotonically decreasing for all values of θ in the simulations that follow.