Discrimination in a search and matching model with self-employment

2.1 Search models, informality, and self-employment

Typically, search models focus on workers finding formal wage-employment jobs, either from unemployment or from an existing wage job through on-the-job search. However, only allowing for formal wage-employment is insufficient for modelling labour markets in developing countries, where informality and self-employment are prevalent. A number of recent papers have attempted to address this shortcoming.

Some economists have used wage-posting models in the spirit of Burdett and Mortensen (1998)—where employers post the wages associated with particular vacancies—to model informality. Narita (2012), for example, constructs a model in which workers can match with either formal- or informal-sector firms, as well as working in self-employment (or remaining unemployed).² At the same time, firms are able to choose between the obligation to pay taxes in the formal sector and the risk of being fined in the informal sector. Narita’s model is also dynamic, in the sense that workers have the potential to learn about good business opportunities throughout their life cycle. When estimated using Brazilian data, the model not only successfully reproduces the composition of the Brazilian labour force, but also demonstrates that the chances of transitioning into self-employment are significantly higher for more experienced workers. While work experience is not an aspect of heterogeneity considered in this paper and we do not use the wage-posting framework, Narita’s model is a helpful counterpoint to the class of models on which we primarily draw.

More commonly, Mortensen and Pissarides’ (1994) random matching and bargaining framework has been used as the basis for search models that allow for self-employment or an informal sector. In one example, Saatchi and Temple (2009) build a model with an urban formal sector—in which firms and workers randomly match and then bargain over wages—that coexists with an urban informal sector—in which workers are self-employed and earn some fixed level of output. They also allow for an agricultural sector, from which potential workers can migrate. When calibrated to Mexican data, Saatchi and Temple’s model produces a reasonably sized informal sector in the presence of formal sector frictions. Given its relative simplicity, the model can also be used to examine analytically the effects of shocks to sector-specific productivity, worker bargaining power, and tax policies, without requiring numerical methods. However, Saatchi and Temple’s model does not allow for worker heterogeneity: this is a vital element of our modelling approach, since (1) we want to allow for the possibility that only some workers face discrimination and (2) we are interested in the resulting distribution of worker productivity and earnings in different sectors of the economy.

Our paper therefore draws most directly on the model by Albrecht et al. (2009) and its adaptation by Kerr (2012), which allows for both informality and worker heterogeneity within a random matching and bargaining framework. In Albrecht et al.’s (2009) paper, unemployed individuals may transition into the formal sector—where productivity is heterogeneous and subject to shocks that lead matches to end or wages to be renegotiated—or the informal sector—where productivity and earnings are homogeneous. Noting that earnings dispersion is higher in the informal sector in African labour markets, Kerr (2012) modifies Albrecht et al.’s (2009) model by assuming that workers are heterogeneous in their informal sector productivity instead, with earnings dispersion arising in both sectors because workers’ outside options (their informal sector welfare) during the formal sector wage bargain differ. This paper develops these models further by incorporating a further element of worker heterogeneity: their exposure to discrimination.

2.2 Search models and discrimination

Competitive labour market models, in which there are no frictions associated with the coming together of employers and employees, often predict that discriminatory outcomes cannot persist in equilibrium. As Becker (1957) describes, in a simple model in which there are two types of workers (‘Type-As’ and ‘Type-Bs’) and some subset of prejudiced firms that incur a utility cost from employing Type-Bs, there are two possibilities, neither of which lead to Type-Bs faring worse than Type-As. If the number of prejudiced firms is initially low, the market instantaneously and fully segregates, such that Type-Bs only work for unprejudiced firms. Even if the share of prejudiced firms is initially large, any Type-Bs working at prejudiced firms for lower wages can be attracted by expanding unprejudiced firms, who offer higher wages to exploit the arbitrage opportunity: prejudiced firms go out of business until these arbitrage opportunities no longer exist, and the market fully segregates.

As such, to allow for the possibility that discriminatory outcomes persist in equilibrium, one of the assumptions underpinning the competitive labour market model is typically dropped. For example, in models of ‘statistical discrimination’, the assumption that employers have perfect information about the true underlying ability of potential employees is relaxed. Employers instead base their hiring decisions on observable characteristics, which they believe proxy for ability, such as gender or race.³ Search and matching models, by contrast, depart from the competitive labour market model by adding frictions into the process through which employers and employees meet: this affords employers monopsony power and allows discriminatory outcomes to persist in equilibrium.

A number of papers—typically using the wage-posting framework—explore the specific conditions under which discriminatory outcomes can persist in search models in equilibrium. Black (1995), for example, constructs a model in which firms are heterogeneous in terms of both their entrepreneurial ability and their ‘taste for discrimination’, where unprejudiced firms are willing to employ both Type-A and Type-B workers, but prejudiced firms will only employ Type-Bs.⁴ In Black’s model, it is only those prejudiced firms with higher entrepreneurial ability that are sufficiently competitive to survive while indulging their taste for discrimination. Interestingly, even though only unprejudiced firms will actually employ Type-Bs, such firms still exploit the fact that Type-B workers have worse outside options when posting wages, driving Type-dependent earnings gaps in the model. In a related model, Sasaki (1999) shows that prejudice among potential co-workers—meaning that Type-A workers prefer to work with other Type-As rather than Type-Bs—can also drive discriminatory outcomes in equilibrium, as firms have to compensate Type-As for working alongside Type-Bs. Sasaki also shows that an increased taste for discrimination among Type-As (the extent of their unwillingness to work with Type-Bs) increases their wages. As such, Type-As have an incentive to ensure discrimination persists.

As in this paper, some economists have even tried to consider what discrimination might mean in search models that explicitly capture self-employment. Borjas and Bronars (1989), for example, build a model in which certain consumers derive dis-utility from buying goods from Type-B self-employed sellers (but not Type-As). However, since matches are random, consumers initially have imperfect information about the Type and price ‘quoted’ by self-employed sellers, and contacting other sellers is costly. This not only drives down the average earnings of self-employed Type-Bs, but also lowers the incentives for high-ability Type-Bs to become self-employed in the first place. However, while Borjas and Bronars’ (1989) model is crucial for understanding discrimination within self-employment, the approach taken in this paper is different, insofar as we focus on the effects that wage-sector discrimination has the self-employed.

This paper is closest to a series of studies that use a random matching and bargaining framework to try and empirically estimate the extent to which discrimination may drive race or gender earnings gaps observed in the labour market. In one example, Flabbi (2010) constructs a model in which certain firms incur some heterogeneous flow of dis-utility from hiring Type-Bs, while for workers, productivity, job arrival rates, and termination rates are all Type-specific.⁵ Estimating the model to understand gender labour market inequality in the USA, Flabbi shows that as many as half of firms may be prejudiced and that discrimination accounts for around two thirds of the gender earnings gap.

To our knowledge, however, no existing search models that incorporate discrimination also allow for both self- and wage-employment. This constitutes one key contribution that this paper aims to make. Additionally, rather than assuming firms are heterogeneous like much of the existing literature, we characterise discrimination using a simple stochastic process that directly affects the matching function. This appears to be a sufficient and tenable approach for understanding how discrimination in the wage sector affects the self-employed.

4.1 Background

The population of workers, of size N, is divided into two, such that some proportion π^A are Type-As (men) and some proportion π^B are Type-Bs (women). The latter are affected by discrimination. We assume that π^A=π and π^B=(1−π). Individuals are also heterogeneous in their self-employment ability, y∼Uniform(0,1). This ability distribution is the same for Type-As and Type-Bs.

To model discrimination, we assume that Type-Bs face the possibility of some extra shock that immediately destroys a match with a particular firm. This shock hits potential matches with probability λ. By modelling discrimination as an exogenous and stochastic feature of the matching process, we can maintain the assumption that firms are ex ante homogeneous, such that there is no inherent heterogeneity in firms’ preferences.

Our approach is therefore a departure from the models of Borjas and Bronars (1989), Black (1995), Sasaki (1999), and other search and matching models that allow for discrimination, as we do not attempt to endogenise the process through which prejudiced firms survive. The plausibility of this setup hinges on discrimination being widespread and similar in intensity across employers. Intuitively, λ can be understood as the proportion of ‘interviewers’ at each firm that are prejudiced and therefore unwilling to employ Type-B individuals. Direct evidence on this issue is somewhat scarce. Bowlus and Eckstein (2002) show that more than half of firms (56 percent) in the USA have a ‘dis-utility factor’ associated with employing black workers, but it is not clear these findings translate to gender discrimination in urban Ghana. Nevertheless, descriptive evidence indicates that gender gaps in terms of access to formal jobs and earnings persist across almost all industries in the Ghanaian economy (Baah-Boateng 2012). This suggests that λ>0 for at least some firms in each industry. Thus, although treating firms as homogenous is a strong assumption, modelling discrimination as an exogenous process in this way provides a tenable basis for our model.

4.2 Worker value functions

Both types of individuals can work in either wage- or self-employment, queuing for jobs in each sector in unemployment. Following Mortensen and Pissarides (1994), workers are assumed to be risk neutral, have infinite lifespans, and discount the future at rate r.

Despite what the transition matrices for urban Ghana show, we do not allow for direct transitions between self- and wage-employment, as doing so adds substantial complication to the model. Moreover such transitions are still captured by workers ability to move between self- and wage-employed jobs through unemployment.¹²

4.3 Firm value functions

4.4 Wage determination

The differences in wages between Type-As and Type-Bs enter through the unemployment term, \(V_{u}^{k} (y)\). Conditional on y, this is lower for Type-Bs because their probability of a successful match that takes them out of unemployment into wage-employment is reduced by discrimination.

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.

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.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.

5.2 Steady-state employment flows

5.3 The free-entry condition

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 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.

7.1 Varying z ₀

Higher levels of productivity in wage-employment—which can be seen as a proxy for economic growth—draw both Type-As and Type-Bs into wage-employment and out of self-employment. This is clear in the numerical simulations shown in Figure 5 in Appendix , yet it also emerges that \(\left. \frac {\partial N_{e}^{k}}{\partial z_{0}}\right |_{\theta = \theta ^{*}} < 0\) and \(\left. \frac {\partial N_{s}^{k}}{\partial z_{0}}\right |_{\theta = \theta ^{*}} > 0\). This results from the fact that both of the productivity cut-offs are increasing in z₀.

Increasing z₀ thus produces two countervailing effects. On the one hand, the resulting rise in y^∗k reduces the proportion of individuals who are willing to work in both self- and wage-employment: this places upward pressure on unemployment because more low-productivity individuals rule out becoming self-employed. On the other hand, the resulting rise in y^∗∗k increases the proportion of people who are willing to work in both self- and wage-employment: this suppresses unemployment because more high-productivity individuals consider becoming wage-employed as well as self-employed.

The balance of these two countervailing effects depends on m^k(θ) and α, such that discrimination makes it more likely that a rise in z₀ will increase unemployment for Type-Bs. A higher level of α boosts the upward pressure that the resulting rise in y^∗k has on unemployment while also reducing the downward pressure from the resulting rise in y^∗∗k. Intuitively, if self-employment jobs are easier to find, the increase in z₀ will induce fewer high-productivity to consider wage-employment as well as self-employment. The converse is true for m^k(θ). In particular, if it is easier to match with a wage job, high-productivity workers are more likely to consider wage-employment alongside self-employment and some relatively high-productivity workers start excluding the option of self-employment altogether. This is, however, less likely for Type-Bs if discrimination is more widespread as, with a higher λ, m^B(θ) will be lower. Thus, discrimination increases the likelihood that unemployment among Type-Bs will rise when the economy grows.

Since m^k(θ) enters the terms before \(\left. \frac {\partial y^{*k}}{\partial z_{0}}\right |_{\theta = \theta ^{*}}\) and \(\left. \frac {\partial y^{**k}}{\partial z_{0}}\right |_{\theta = \theta ^{*}}\) positively, the extent of discrimination, λ enters negatively. As such, discrimination may potentially slow the distribution of the proceeds of growth.

7.2 Varying b

The likelihood that increasing b leads to a rise in wage-employment (and a decline in self-employment) is therefore higher when m^k(θ) is lower. For Type-Bs, this happens when discrimination is more widespread (λ is higher). As such, unemployment benefits may offer a way of shifting employment from self-employment to the wage sector, especially for Type-Bs.

The effects of changing b on average ability and earnings in each sector are similarly ambiguous, although—as the simulations show—an interesting wedge emerges between earnings and ability in wage-employment. This arises because unemployment becomes a more tenable outside option, affording wage workers more bargaining power.

7.3 Varying α

While Section 6 considers how altering the matching function for wage-employment—by introducing discrimination—alters Type-As’ and Type-Bs’ labour market outcomes, there is a supplementary question over the impact of changing the arrival rate for opportunities in self-employment. As the simulations in Figure 7 in Appendix show, the number of individuals in self-employment rises at the expense of wage-employment if α is increased. These changes are almost entirely driven by the top of the productivity distribution. Since \(\left. \frac {\partial y^{*k}}{\partial \alpha }\right |_{\theta = \theta ^{*}} = 0\), there are only very small changes in y^∗k, brought about by the adjustment in labour market tightness θ. The upper cut-off, by contrast, declines relatively sharply with higher levels of α. When self-employment opportunities come along more frequently, higher-productivity workers of both types are more willing to forego wage-employment and pursue a job in which they have comparative advantage.

Correspondingly, there are moderate declines in average ability and earnings in all sectors of the economy, as α is increased. As the the upper cut-off falls, more individuals with lower productivity are drawn into self-employment. At the same time, it is those individuals with the highest ability among those who were previously willing to take both self- and wage-employment, that start to ignore the option of becoming wage-employed, with higher levels of α. Indeed, the effects of changing α are analogous to the effects of changing λ (as in Section 6), but the mechanism operates around the higher productivity cut-off, rather than the lower one.

13.1 Existence of y ^∗k

Since all the terms on the right-hand side of Eq. (29) are non-negative by construction, it is clear that the condition that y^∗k≥0 is met trivially.

By comparing the numerator and the denominator in Eq. (29), we can see y^∗k ≤1 is guaranteed by simply imposing that b≤ 1. Moreover, since 0≤z₀ ≤1, this will hold if b≤z₀.

13.2 Existence of y ^∗∗k

To guarantee the existence of the upper cut-off, we derive two conditions which rule out the equilibria shown in panels A and B of Fig. 2 (see Section 5.1).

This condition restricts the values of z₀ (relative to b), since otherwise the surplus and hence earnings from matches in wage-employment are too large for any individuals to forego wage-employed jobs. The condition also places a lower bound on α.

Thus, \(X_{e}^{k}\left (y^{*k}\right) > 0\) if z₀>b.

13.3 Simulation conditions

These are sufficient to guarantee the existence of both cut-offs, y^∗k and y^∗∗k.