Selfemployment in an equilibrium model of the labor market
 Jake Bradley^{1}Email author
Received: 1 February 2016
Accepted: 10 May 2016
Published: 10 June 2016
Abstract
Selfemployed workers account for between 8 and 30 % of participants in the labor markets of OECD countries (Blanchower, Selfemployment: more may not be better, 2004). This paper develops and estimates a general equilibrium model of the labor market that accounts for this sizable proportion. The model incorporates selfemployed workers, some of whom hire paid employees in the market. Employment rates and earnings distributions are determined endogenously and are estimated to match their empirical counterparts. The model is estimated using the British Household Panel Survey (BHPS). The model is able to estimate nonpecuniary amenities associated with employment in different labor market states, accounting for both different employment dynamics within state and the misreporting of earnings by selfemployed workers. Structural parameter estimates are then used to assess the impact of an increase in the generosity of unemployment benefits on the aggregate employment rate. Findings suggest that modeling the selfemployed, some of whom hire paid employees implies that small increases in unemployment benefits leads to an expansion in aggregate employment.
JEL Classification J21, J24, J28, J64
Keywords
1 Introduction
The proportion of total employment made up by the selfemployed in the UK rose steadily over the period 2000–2004. The number of selfemployed increased by 8.9 % compared with an increase of 0.1 % of paid employees. This growth was across gender, region, and industry (Lindsay and Macauley 2004). Over this period, the average selfemployment rate was at 11.5 % of total employment and this large proportion is not unique to the UK: across all OECD countries this proportion varied from 8 to 30 % (Blanchflower 2004). Overall employment, wage determination and dynamics across the two sectors are clearly intrinsically linked, especially when one also considers that in the UK one third of all the selfemployed hire at least one paid employee (Moralee 1998). Considering the size and importance of selfemployment, literature that incorporates it into a model of the labor market is relatively sparse.
This paper develops a searchtheoretic general equilibrium model of the labor market that incorporates selfemployed individuals. The selfemployed entrepreneurs are treated in a Schumpeterian way, as a source of innovation (Schumpeter 1934)^{1}. It is distinct from other labor market models of selfemployment in that after innovation, the selfemployed agent can work on their own, as an ownaccount worker, or begin hiring workers from a labor market with search frictions. The importance of incorporating the selfemployed into an equilibrium model of the labor market is seen by looking at simulations of labor market policy. Conventional wisdom would suggest that a rise in unemployment benefits would have an adverse effect on aggregate unemployment—workers require better job offers or better ideas to exit unemployment for paid or selfemployment respectively. Introducing the selfemployed as a source of job creation introduces a counterweight to this straightforward mechanism. If only sufficiently good ideas create employment opportunities, making the selfemployed more fastidious about which ideas to act on could create employment opportunities for other agents. This paper finds that for small increases in the generosity of unemployment benefits, aggregate employment increases. This is used as an illustrative example to show the importance of considering the selfemployed when implementing active labor market policy.
In the model, there are two types of agents, large private sector firms who vary in their productivity and ex ante homogeneous workers who are exposed to innovative ideas which arrive at an exogenous Poisson rate that is dependent on their labor market status. When an agent gets an idea, its quality is drawn from a known distribution and the agent decides whether to act on it. If they choose to start a business, then depending on the quality of the draw, they may commence attempting to hire workers as paid employees. Thus, paid employees are either hired by large firms or selfemployed recruiters. All workers are finitely lived and when in paid employment are exposed to an exogenous probability that they lose their job; all wage offers and job arrival rates to paid employment are determined endogenously. This rich setting allows for a multitude of avenues in which the two sectors are interlinked.
The model is structurally estimated using the British Household Panel Survey (BHPS). The identification strategy allows primitive productivity distributions and hiring behavior of both types of firm to be uncovered. The distribution of productivity amongst large firms is uncovered, as in Bontemps et al. (2000) by an inversion of the wage offer distribution; the productivity amongst the selfemployed is obtained directly from their earnings; and hiring behavior is estimated in order to match the distribution of firm size amongst large privately owned companies and small firms owned by selfemployed recruiters. Estimates suggest that the productivities of the two types of firms are very similar; however, selfemployed owned firms are responsible for a much smaller share of employment because they face much greater frictions in hiring paid employees. It is necessary to uncover output and hiring through the guise of an equilibrium model as there exist large limitations of data on the selfemployed. Data are specifically limited in information regarding the recruiting selfemployed. The BHPS is useful as it distinguishes the selfemployed between those who hire paid employees and those who do not. However, it does not identify the paid employees who are hired by recruiting selfemployed nor does it provide information on output. Therefore, in order to infer rates of hiring and production, the paper leans heavily on a model that provides a great deal of structure to the data.
There are models that embed selfemployment into a labor market equilibrium. However, there are none, to the author’s knowledge, that allow for the two distinct types of selfemployment discussed. Kumar and Schuetze (2007) develop a model of the labor market that incorporates selfemployment and assess the effect of changes to unemployment insurance and the minimum wage on labor market equilibrium. Selfemployed hire paid employees as in this paper, but unlike this model, there is no wage dispersion within sector nor are there any movements across sectors. Thus, they restrict a direct interaction between the two sectors. Narita (2014) and Margolis et al. (2014) allow for wage dispersion within sector and structurally estimate the parameters of their models using data from Brazil and Malaysia, respectively. However, unlike Kumar and Schuetze (2007) and this paper, the selfemployed are restricted to being ownaccount workers (they are restricted from hiring). Also, although they allow for more mobility than Kumar and Schuetze (2007), they still omit any direct transitions between paid and selfemployment.
Although not all concern themselves with selfemployment, perhaps the most similar papers methodologically are Meghir et al. (2015), Bradley et al. (2015), and Millán (2012). All introduce another sector into an equilibrium labor market setting allowing for a great deal of mobility between and across sectors, be it an informal, public, or selfemployed sector. Decisions made by workers depend not only on the wage they are offered but also future prospects associated with the sector. Millán (2012) suggests that selfemployment is used as a route out of unemployment. This paper does not consider this hypothesis, but workers may be encouraged to become selfemployed as they face the opportunity of starting a business which can grow. Only Kumar and Schuetze (2007) and this paper entertain the idea of the selfemployed as employers, and only this paper distinguishes between the selfemployed as ownaccount workers and recruiters.
The consensus in the empirical literature is that there exists a substantial paid employment premium over being selfemployed. Hamilton (2000) finds that taking into account within sector earnings growth and without distinguishing between ownaccount selfemployed and recruiters, there is a 35 % differential between median earnings of a selfemployed individual and a paid employee with 10 years of experience in the USA. To rationalize workers’ career choices, “results suggest that the nonpecuniary benefits of selfemployment are substantial”. Critiquing the shortcomings of the existing literature Hamilton (2000) goes on to say “results presented here are of a reduced form [...] structural estimates of the compensating wage differential, for example, would require [...] the probability of observing particular employment and earnings sequences”. To my knowledge, this is the first paper that attempts to structurally estimate the nonpecuniary benefits of selfemployment while taking into account different employment options, including starting and growing one’s own business. A priori, it is not clear which sector has the preferable employment and earnings profile. On the one hand, those in paid employment are better exposed to other jobs in paid employment and can climb the job ladder thusly. However, the selfemployed are far more likely to become recruiters. If their firm successfully grows, so will their earnings. Consistent with the empirical literature, this paper too finds a large positive amenity associated with selfemployment.
The rest of the paper is structured as follows. Section 2 presents and solves the equilibrium model. Section 3 presents the data, which corrects the earnings of the selfemployed, attempts to validate assumptions made in the model, and explains the moments used for identification. Section 4 outlines the estimation protocol, the results, and the fit of the model. Section 5 runs counterfactual policy simulations to asses the effects of increasing unemployment benefit on aggregate employment, and finally, Section 6 concludes.
2 The model
2.1 The environment
Time is continuous, and at any point, there exists a unit mass of ex ante homogeneous workers and a mass N of private sector firms who are heterogeneous in their level of productivity. Workers and firms are riskneutral and discount the future at a rate r>0. Workers can be in one of three broad states: unemployment, paid employment, or selfemployment. They leave the labor market at an exogenous Poisson rate μ which is independent of their labor market state; they are replaced with new agents who are born into unemployment. Firms are infinitely lived^{2}.
Paid employment comes from two sources, large private sector firms and selfemployed individuals. Job offers arrive to unemployed agents from firms at a rate λ _{0} and paid employees in large firms at a rate λ _{1}=κ λ _{0}. κ is the relative search intensity of an employed worker compared to an unemployed one. Wages are drawn from a known distribution F(·). Job offers can also arrive from selfemployed recruiters at a rate \({\lambda _{0}^{s}}\) to the unemployed. For tractability, it is assumed that the recruiters direct all their search intensity to the unemployed, attempts will be made to justify said assumption in Section 3. These wages are drawn from a known distribution F ^{ s }(·). All job offer arrival rates and wage offer distributions are endogenous objects. A match is destroyed at a Poisson rate δ. Both firm and worker continue to exist, but the worker is reallocated to unemployment. If the worker exits the labor market, at a rate μ, the firm will continue to exist, but with one less worker. Agents are also exposed to the possibility of having an innovative idea. The rate agents get ideas follows a Poisson process which is dependent on one’s labor market state. Ideas arrive to unemployed agents at a rate η _{0} and to paid employees in large firms at a rate η _{1}. The quality (productivity) of the idea is drawn from an exogenous distribution Γ(·). Depending on the draw, this will allow workers to cross the market from unemployment or paid employment and become selfemployed. The reservation productivity of paid employees will depend on their current wage.
Selfemployment spells begin as an agent working on his own producing output according to the draw from Γ(·). There is a search friction in the matching process: hiring workers takes time. One can think of this as time required for vetting, preliminary training etc. So, as will transpire, some of the selfemployed will wish to hire but are time restricted and therefore hire at a suboptimal rate.
It is assumed that the selfemployed hire at a Poisson rate h ℓ, where h is exogenous and constant for all firms (independent of size and productivity) and ℓ is the size of the firm, can take any positive integer value. Thus, by construction, the growth of a firm adheres to Gibrat’s law ^{3}. The rate of total hiring is proportional to the firm size as the hiring process requires a certain amount of work. Larger firms can share this workload over a greater number of workers and hence the rate of hiring increases proportionally with the size. In a way akin to Coles and Mortensen (2011), the size of the firm ℓ will follow an endogenously determined Markov process.
Finally, the model is derived in a steadystate. The stocks of agents in each of the three states are constant over time as are the distributions of firm size, productivity amongst the selfemployed, and the distribution of wages amongst the paid employees. To solve the model, one needs to (i) derive the value functions of the three states, (ii) derive reservation levels for which workers change states, (iii) calculate the size of each state and derive the ergodic distributions of wages and productivity within states, and (iv) derive the profit maximizing wage policy of large private sector firms.
2.2 Value functions
Workers can be in one of five states: paid employment in a large firm; unemployment; paid employment, employed by a selfemployed recruiter; ownaccount selfemployment (working alone); and recruiting selfemployment (hiring paid employees). In all states, workers are maximizing their lifetime income discounted at a rate r. The following subsections derive the lifetime values for each state.
2.2.1 Paid employees
A paid employee working for a large firm has two sources of revenue flow from employment, a basic wage and a nonpecuniary amenity, which ex ante can be positive or negative; it is measured relative to being a selfemployed worker. As well as the revenue flow of income and amenity, the value function of a paid employee has the option value of transiting into other states, namely the option value of unemployment, higher value paid employment (employed by firm or a selfemployed agent), and selfemployment. At any point, there is a possibility that the agent exits the labor force. All future revenue flows are discounted at a rate r.
2.2.2 The selfemployed
All selfemployed agents start their selfemployment spell as an ownaccount worker. That is, they employ only themselves. If an agent aims to recruit workers, they will arrive at a rate h ℓ, where ℓ is the integer number of employees they already have. Workers will quit the firm at a rate (μ+δ), exiting to either unemployment or out of the labor force. Thus, the number of workers a recruiter employs will follow a Markov process. The only search friction that exists for a recruiter is that hiring is a timeconsuming process. For a firm with an integer number of employees ℓ, it takes an estimated time 1/h ℓ to hire a worker. Unlike the Burdett and Mortensen (1998) model, a higher wage neither increases the rate of recruitment nor retention. For ease of exposition, it is assumed that the selfemployed never exit selfemployment for unemployment.
Since the value of recruitment decreases with the wage rate, the optimal wage is independent of y and ℓ and solves the equality W ^{ s }(w ^{⋆})=U. An explicit solution will be given for w ^{⋆} in the next section.
For an individual not to set about recruiting, the option value of having one worker must be negative. A selfemployed individual with productivity y will aim to recruit workers if S _{ R }(y)≥S _{ O }(y). Given certain regularity conditions, to follow, there exists a threshold productivity level ψ _{1}, above which selfemployed agents intend to recruit.
2.3 Reservation strategies
A worker’s strategy can be characterized by a set of reservation values which depends on his current labor market state.
Hence, given monotonicity of the value functions, ψ and ϕ are reciprocals of one another. Similarly, the wage that makes an unemployed agent indifferent between continuing unemployment and being employed by a large firm is ϕ _{0} and solves the equality W(ϕ _{0})=U, and the productivity of an idea that makes an unemployed agent indifferent between continuing unemployment and being selfemployed at that productivity is ψ _{0} and solves the equality S(ψ _{0})=U. Clearly, ψ(ϕ _{0})=ψ _{0}.
Solutions for ψ _{0}, ϕ _{0}, and an ODE defining ϕ(y) are provided in “Solving for reservation strategies” in the Appendix.
2.4 Steady state
The model is derived in a steady state. A steady state is defined as a constant share of agents in each labor market state, and the distributions of wages amongst the paid employees, the productivity amongst the selfemployed, and the distribution of employment size amongst recruiters are all stationary.
The steady state is defined by Eqs. (18) through (21). The sum of all agents in the economy is equal to unity. The flow into unemployment equals the outflow. The flow out of selfemployment (paid employment) below a productivity y (wage w) is equal to the flow into selfemployment (paid employment) below a productivity y (wage w). As well as this, the distribution of labor force size amongst selfemployed recruiters that is dictated by a Markov process has reached its ergodic distribution. This last object is denoted as Σ(ℓ); it is the measure of recruiters and those who intend to recruit hiring an integer ℓ workers.
Equations (18), (19), (20), and (21) are solved simultaneously for the endogenous objects \(({N_{e}^{s}}, N_{u}, {N_{e}^{f}} G(\phi (y)), N_{s} \Gamma _{s}(y))\). These coupled with the distribution Σ(ℓ) define the steady state allocation of agents. The solution for these objects is provided in “Solving for the steady state” in the Appendix.
2.5 Private sector firms
Private sector firms are large and infinitely lived. The law of large numbers is employed, and they are modeled following Bontemps et al. (2000). This paper considers the familiar equilibrium where firms post wages and commit to those wages for their lifetime. The higher the wage a firm posts, the larger the firm (fewer quits and more hires); this is at the cost of making less profit per worker. As pointed out by Coles (2001), this is not a dynamically consistent wage posting strategy without commitment on wages. To see this, imagine a firm has grown to its steady state size. Rather than posting its optimal wage w, it will be strictly better off to pay its workers’ reservation wage ϕ _{0}. Over a period d t→0, no workers will quit and the firm will make strictly greater profits. Coles (2001) identifies a dynamically consistent equilibrium without relying on commitment. Interestingly, the wages posted by selfemployed recruiters are a dynamically consistent strategy, without having to rely on commitment. It is the belief of the author that despite theoretical issues regarding dynamic consistency, the tractability of Bontemps et al. (2000) makes it very well suited to modeling the behavior of large firms.
Thus, if wages are increasing in productivity, then one can infer that F(w)=Γ ^{ f }(y(w)), where the relationship y(w) is given by (24) and Γ ^{ f }(·) is the cumulative distribution of productivity amongst large private sector firms. Thus, the distribution of wage offers in the private sector F(w) can be retrieved if we know the distribution of productivity across firms.
2.6 Equilibrium characterization
Given exogenous parameters r,a,b,μ,δ,η _{0},η _{1},h,p(y),κ,Γ(·),Γ ^{ f }(·),N,h ^{ f }, an equilibrium is characterized by the following conditions: (i) agents behave optimally in their career decisions, solutions to ϕ _{0}, ψ _{0}, ψ _{1}, and ϕ(y); (ii) the economy is in steadystate, the inflow from a given state equals the outflow, solutions to Γ _{ s }(·), G(·), N _{ u }, \({N_{e}^{f}}\), \({N_{e}^{s}}\), N _{ s }, and Σ(ℓ); (iii) large private sector firms and selfemployed recruiters offer their workers’ wages optimally, w(p) and w ^{⋆}; and (iv) when firms behavior is aggregated the wage offer distribution F(·) and the offer arrival rates λ _{0}, \({\lambda ^{s}_{0}}\), and λ _{1} are determined. Attention is restricted to a specific class of equilibria where selfemployed individuals exist, some of whom recruit paid employees. To guarantee this equilibrium, the exogenous parameter space needs to be constrained.
Assumption 1.
h∈(0,r+2μ+δ)
Assumption 2.
Either p ^{′}(y)≥0 and p ^{′′}(y)>0 or \(p^{\prime }(y) > \frac {r+ 2 \mu + \delta  h}{r + \mu } \) and p ^{′′}(y)≥0.
Proposition.
Given Assumptions 1 and 2, an equilibrium will exist with selfemployed recruiters.
The intuition is as follows. These assumptions are needed for two reasons. Firstly, Assumption 1 guarantees the nonnegativity of the value function of a recruiter; see (8). Given this, Assumption 2 guarantees that for a sufficiently good idea, a selfemployed agent will actively seek to hire workers.
Proof.
The above is positive if Assumption 1 holds. Recruiters will exist if, for some y, the value of intending to recruit, S _{ R }(y), is greater than being an ownaccount selfemployed indefinitely, S _{ O }(y). S _{ O }(y) is a linear function of y, and S _{ R }(y) is a linear function of p(y). If p ^{′}(y)≥0 and p ^{′′}(y)>0, the first part of Assumption 2. Clearly, for sufficiently large y, S _{ R }(y)>S _{ O }(y) and given Assumption 1, recruiters will exist in equilibrium.
If p(y) is linear and increasing in y, the second part of Assumption 2. Then both S _{ O }(y) and S _{ R }(y) are linear and increasing in y. Thus, for sufficienlty large y, recruiter will exist if \(S_{R}^{\prime }(y) > S_{O}^{\prime }(y)\). This is the case, given \(p^{\prime }(y)>\frac {r+ 2 \mu + \delta  h}{r + \mu } \), contained in Assumption 2. Q.E.D.
Identification result: A sufficient condition for the existence of recruiters is \(p^{\prime } (y) >\frac {r + 2\mu + \delta }{r+ \mu }\)
Class of equilibrium: the equilibrium is restricted to one where selfemployed recruiters exist. Clearly, some selfemployed will be ownaccount workers. These can fall into two categories either they are ownaccount workers, but because of the frictions present in the market, as yet, they have been unable to hire anyone. Or, they are ownaccount workers with no intention to recruit. For the latter type of agent to exist, it is required that S _{0}(y)>S _{ R }(y) for some y and because \(S_{R}^{\prime }(y) > S_{O}^{\prime }(y)\) for all y, this is equivalent to S _{ O }(ψ _{0})>S _{ R }(ψ _{0}).
Since one cannot restrict attention to either class as neither can be invalidated by data, any simulation of the model needs to repeatedly check in what equilibrium it is in. One where all selfemployed intend to recruit or one where some selfemployed intend to stay ownaccount indefinitely.
3 Data
The model described is estimated using the British Household Panel Survey (BHPS). It is identified using transition rates across labor market states, the earnings of paid employees and selfemployed workers and the distribution of firm size.
Exogenous parameters are estimated by simulated method of moments. The principle of the estimation technique is to find values of the structural parameters that minimize a function of the difference between a chosen set of moments from the data and data simulated with these values of the structural parameters.
The role of the rest of the data section are twofold. Firstly, it aims to inform the reader about the data that is used to estimate the model. Secondly, it attempts to find empirical support for predictions of the model as well as assumptions made for tractability.
3.1 The sample
The data used in the analysis are taken from the BHPS, a longitudinal dataset of British households. Data were first collected in 1991, but attention is restricted to five waves covering the period from 2004 to 2008. The sample comprises of prime age (21–60) white male lowskilled workers. Lowskilled is defined as not having obtained Alevel qualifications. These are the highest qualifications available for students aged 18 in the UK, before they enter higher education. A worker is an individual who is never inactive in the period looked at: if out of work, they declare themselves to be actively seeking work. The hourly earnings distribution is adjusted by treating the bottom and top 2.5 % of the distribution as missing, hopefully ridding the sample of erroneously reported earnings.
Composition
Paid employment  Ownaccount  Recruiter  

Employment share  82.8 %  12.6 %  4.6 % 
Mean hours worked per week  40.39  43.34  51.68 
Raw data  
Mean earnings  10.69  9.34  11.14 
Standard dev. of earnings  3.88  5.81  10.44 
Adjusted data  
Median earnings  9.92  13.20  14.93 
Mean earnings  10.69  15.71  18.72 
Standard dev. of earnings  3.88  9.76  17.55 
There are a number of points worth noting from Table 1. There is a nonnegligible share of recruiters, consistent with the findings of Moralee (1998), with approximately one third of selfemployed agents hiring paid employees. The selfemployed work longer hours than their paid employee counterparts and the recruiting selfemployed work significantly more still. There is a clear ordering in the second moment of the earnings distribution across the three labor market states irrespective of whether one looks at the raw or the adjusted data. The ordering of the first moment is ambiguous however; the preferred specification that will be used in estimation is the adjusted data. Looking at the large implied differences, the importance of correcting for the misreporting of earnings is evident. The ranking of the second moment of the earnings distribution is not targeted in the estimation, but will still be replicated by the theoretical model.
Transition matrix: 2004–2008
Unemployment  Small paid  Large paid  Ownaccount  Recruiter  

employment  employment  
Unemployment  –  0.0009  0.0899  0.0066  0.0019 
Paid employment (s)  0.0160  0.0160  0.0841  0.0053  0.0027 
Paid employment (l)  0.0034  0.0002  0.0079  0.0009  0.0000 
Ownaccount  0.0011  0.0000  0.0056  –  0.0027 
Recruiter  0.0007  0.0000  0.0043  0.0108  – 
The rates in bold are those that the model is capable of replicating, some of which the estimator will target, and those in plain text are ones that the model is unable to generate. The model is unable to generate a number of classes of transitions. One is any transition into selfemployment where the worker instantly becomes a recruiter. This is inconsistent with a labor market with search frictions, as one cannot hire individuals instantaneously. The second is a simplifying assumption made for tractability: once in selfemployment an individual is allowed little labor market mobility. Of the transitional moments omitted, the rate at which a recruiter transits to being an ownaccount worker is the most glaring. In a given month there is a 1 % chance of a recruiter losing its workforce and becoming an ownaccount worker. This model is not able to generate this, in order to keep the result that a recruiter’s value is proportional to the number of employees it has (Eq. (7)), which helps the tractability of the model. Also, while 1 % seems large, in fact, from Table 1 just 4.6 % of employed individuals are recruiters so this translates into very few transitions missed. Finally, as in models of wage posting, firm heterogeneity and on the job search like Bontemps et al. (2000), there is a onetoone relation between firm size and offered wage. In an identical way, this model can therefore not rationalize why a paid employee would take a wage cut and move to a smaller firm.
The parameter h is identified using the transition rate from ownaccount selfemployed to becoming a recruiter. Since the employee size distribution of selfemployed recruiters is governed by μ, δ and h, this could be an alternative source of identification, conditional on μ and δ. The firm size distribution amongst private sector employers is used to identify h ^{ f }, the volume of vacancies a single firm posts. The selfemployed are asked how many people they employ; the paid employees are asked how many other people are in their place of work. These numbers are grouped into size bins and from these it is difficult to infer the size distribution of private sector firms. Therefore, purely for comparison, data on firm size from the Office of National Statistics (ONS) in the midpoint of the sample period is used.
Private sector firm size distribution 2006
Firm size  No. of firms  Percent 

Total  2,084,495  100 
0–4  1,391,960  66.78 
5–9  317,745  15.24 
10–19  178,820  8.58 
20–49  120,870  5.80 
50–99  41,905  2.01 
100–249  23,100  1.11 
250–499  6740  0.32 
500–999  2440  0.12 
1000 or more  915  0.04 
Selfemployed recruiter’s size distribution 2004–08
Firm size  No. of firms  Percent 

Total  151  100 
1–2  65  43.05 
3–9  65  43.05 
10–24  15  9.93 
25–49  1  0.66 
50–99  0  0 
100–199  1  0.66 
200–499  1  0.66 
500–999  1  0.66 
1000 or more  2  1.32 
Weighted regressions by industry classification
Nonunemployment exit rate  Mean (log) wage  

Coefficient  \(\underset {\left (0.0008 \right)}{0.033}\)  \(\underset {\left (0.016 \right)}{0.84}\) 
Constant  \(\underset {\left (0.00005 \right)}{0.014}\)  \(\underset {\left (0.00092 \right)}{2.4}\) 
All parameters are statistically significant to any conventional significance level. When the explanatory variable equals zero, that implies there are no selfemployed recruiters in that onedigit industry classification. If the explanatory variable equals one, it means that all employed individuals in that industry are selfemployed recruiters. If one interprets the former as an instance where the probability a paid employee is hired by a selfemployed recruiter is zero and in the latter the probability equals one, the coefficients have a straightforward interpretation. Simply, they are the difference in mobility and earnings associated with paid employment, given one is hired by a private sector firm or a selfemployed recruiter. With this in mind, the results are supportive of the restrictions imposed on model. Using the parameter estimates from the mobility regression, a linear projection would imply that a paid employee would have zero chance, assuming nonnegative probability, of exiting his current job for any other employment state but unemployment. The earnings regression suggests that a paid employee hired by a selfemployed recruiter will on average earn 84 % less than one hired by a large private sector firm. This paper does not aim to suggest that these restrictions are true, rather they are not too important and seem to be borne out by the data.
4 Estimation
4.1 The estimation protocol
The model is estimated using a simulated generalized method of moments estimator (SGMM). The estimation is performed over a number of steps. The reason a multistepped estimation is implemented is because it makes clear the source of identification for the moments. It also aids the estimation in not allowing the constraints put on the parameter space in Section 2.6 to be violated.
Some endogenous parameters can be computed without solving the model. The endogenous parameters F(·),ψ _{0} and ϕ _{0} are all fixed according to their empirical counterparts. F(·) is directly observable as the wage distribution for those who have transited into paid employment straight from unemployment. Note, strictly speaking this also includes those hired by the selfemployed. To keep these to a minimum, only those transiting to a sector with less than 10 % of employment made up by recruiters are considered^{6}. ψ _{0} and ϕ _{0} are the minimum observed earnings amongst the ownaccount selfemployed and ϕ _{0} the infimum of the support of F(·). The nonpecuniary amenities a and b are treated as free parameters so to equalize Eqs. (29) and (30) in the Appendix. After all other parameters are estimated, Γ ^{ f }(·) is computed so F(·) is rationalized according to Eq. (24).
It proves simpler to also treat λ _{0} and λ _{1} as temporary exogenous parameters in the estimation and uncover the underlying exogenous parameters ex post of estimation. The total contacts that large private sector firms make is driven by a combination of the number of firms N and the contact per firm h ^{ f }. While matching the number of contacts the relative size of these two objects will be set to best match the aggregate firm size distribution—including the selfemployed recruiters.
Estimation works as follows. In the first step, the transition rates are exactly identified and the following vector of parameters are estimated (δ,h,λ _{0},λ _{1},η _{0},η _{1},μ). Conditional on these, and the parameters fixed ex ante the selfemployed productivity parameters are estimated to match the earnings of ownaccount and recruiting selfemployed, they are (β,m _{ y },s _{ y }). These two steps are continuously iterated on until all parameter estimates are stable. Finally, as discussed in the final step after the parameters have converged, endogenous parameters are rationalized by their exogenous primitives and the aggregate firm size distribution is fitted with its empirical counterpart. Each step is described in more detail below.
4.1.1 Stage 1: transition parameters
In this stage, the remaining transition rates, reported in “Transition rate moment conditions” in the Appendix, are matched. They are the monthly rate at which individuals transit from paid employment in a firm of any size to unemployment, unemployment to paid employment, unemployment to becoming an ownaccount worker, paid employment to another paid employer, paid employment to becoming an ownaccount worker, and from an ownaccount selfemployed worker to a recruiter.
These moment conditions are reported in “Transition rate moment conditions” in the Appendix 6and are exactly identified by the parameters (δ,h,λ _{0},λ _{1},η _{0},η _{1},μ); λ _{0} and λ _{1} are endogenous to the model and are rationalized in Section 4.1.3.
4.1.2 Stage 2: selfemployed earnings
The distribution of earnings among ownaccount workers is given by the solution to the set of steady state equations in Section 2.4. The earnings distribution and productivity distribution are equivalent as ownaccount workers earn their output. Earnings for recruiters differ by their level of productivity and their size. The distribution of profits is calculated by summing over the measure at each size distribution, Σ(ℓ). Deciles from the data are matched with deciles from the simulated model using the mean and variance of log productivity, m _{ y } and s _{ y } as well as the parameter β which describes the increased profitability associated with hiring. The deciles of the two earnings distributions are fitted using an equally weighted matrix, following the criterion in Altonji and Segal (1996). The authors show that when moment conditions are based on relatively few observations, an equally weighted matrix often performs better than an optimally weighted one.
After the exactlyidentified first stage has fitted the transition rates, conditional on these estimates, the overidentified second stage fits the earnings of ownaccount workers and recruiters according to m _{ y }, s _{ y }, and β. These parameters are updated and stage one is repeated; this is done until all the estimated parameters have converged. It is found that this multistepped procedure performs better in fitting the data than a singlestep protocol.
4.1.3 Stage 3: ex post calculations
κ, the degree to which paid employees are exposed to private sector firms relative to the unemployed, is given by \(\kappa = \frac {\lambda _{1}}{\lambda _{0}}\). The endogenous parameters λ _{0} and λ _{1} are estimated in Section 4.1.1.
The distribution of productivity amongst large private sector firms is identified nonparametrically. Equation (24) is computed so the productivity of a firm paying w is known and given by the relation y(w). Then, since this is an increasing function (verified by repeated simulation) F(w)=Γ ^{ f }(y(w)), where F(w) is nonparametrically estimated before the first step.
Paid employees are asked the size of their employer, put into the same size bins as Table 4. The firm size distribution of selfemployed owned firms depends on μ and h and has already been determined. In order to fit the number of workers in firms of certain size or less, the model is only able to adjust the firm size distribution of large private sector firms, who are responsible for the majority of paid employment. This distribution ℓ _{ f }(w) is given by Eq. (22). The only parameter that is left undetermined is h ^{ f }, the number of contacts each firm makes. The distribution of firm size employing paid employees’ is matched by minimizing the KolmogorovSmirnov distance between the theoretical and empirically observed distribution. Thus, all that is left to estimate is the measure of firms in the economy N. This is set to equalize the accounting identity, given by Eq. (25).
4.2 Results
Parameter estimates
δ  η _{0}  η _{1}  μ 
\(\underset {\left (0.0003 \right)}{0.0034}\)  \(\underset {\left (0.0002 \right)}{0.0066}\)  \(\underset {\left (0.00002 \right)}{0.0011}\)  \(\underset {\left (0.0002 \right)}{0.0107}\) 
a  b  
\(\underset {\left (0.77 \right)}{6.53}\)  \(\underset {\left (1.42 \right)}{11.50}\)  
h  h ^{ f }  N  κ 
\(\underset {\left (0.0001 \right)}{0.0070}\)  \(\underset {\left (0.1069 \right)}{2.7305}\)  \(\underset {\left (0.0005 \right)}{0.0138}\)  \(\underset {\left (0.0119 \right)}{0.4075}\) 
m _{ y }  s _{ y }  β  
\(\underset {\left (0.0927 \right)}{2.4796}\)  \(\underset {\left (0.0242 \right)}{0.2689}\)  \(\underset {\left (0.0315 \right)}{1.6740}\) 
The transitional parameters given in the first section of Table 6 are all monthly Poisson rates. At first glance, h seems fairly small; 35 % of new ownaccount workers who aim to recruit will hire someone in their first 5 years. However, because hires are made at a rate h ℓ, the frequency of hires increases as the number of employees grows, 88 % (98.5 %) of selfemployed recruiters with five (ten) workers will hire another worker in the next 5 years.
The rate at which individuals receive innovative ideas is also infrequent. If one spends the majority of their lives in paid employment, it is highly likely that they will never have a single idea. The rate at which agents are exposed to ideas is six times higher when they are unemployed compared to when they are in paid employment. When an idea does arrive, it is drawn from a lognormal distribution with the mean and standard deviation of the natural log of productivity as given by m _{ y } and s _{ y } in Table 6.
Amenities presented in the second panel of Table 6 are measured in pounds per hour. The value of leisure b is negative and large, meaning for lowskilled male workers, there is a large stigma associated with unemployment^{7}. The nonpecuniary amenity associated with selfemployment is £6.53 per hour. This is commonly referred to in the literature as the benefit associated with “being your own boss”. Comparing this with Table 1 reveals this as 49 % (44 %) of the adjusted median wage of ownaccount workers (recruiters).
The parameter κ has a slightly different interpretation as it ordinarily would have. It is the ratio of job offers that paid employees in large firms receive from other large firms compared with job offers received by unemployed agents from large firms. Since the unemployed are also exposed to job offers from selfemployed recruiters which the paid employees are not, the estimate of κ is inflated in comparison with other canonical models. Estimates of N and h ^{ f } suggest that the mass of private sector firms is equivalent to approximately 1.4 % of the total active members of the labor market and that in total, the firms are in contact with 3.8 % of all agents active in the labor market in a given month.
β is the factor by which production increases when an ownaccount worker begins to recruit. Recall, when an agent recruits, he steps down from production and acts as a managerial overseer. Since β is given by 1.674, a recruiting selfemployed will make less after hiring their first worker than they were previously. However, losses are recouped as soon as they hire their second employee.
4.3 The fit
To review, moments that were specifically targeted were a selection of transition rates, information on firm sizes, and deciles of the earnings distributions of selfemployed workers.
Fit of deciles of the earnings distribution for ownaccount workers
Decile  Theoretical moment  Empirical moment 

10 %  7.3556  6.1111 
20 %  9.0002  7.8725 
30 %  10.4178  9.7059 
40 %  11.8070  11.4559 
50 %  13.2911  13.1993 
60 %  14.9602  15.6732 
70 %  17.0026  18.2312 
80 %  19.7461  21.1564 
90 %  24.2895  27.0048 
Fit of deciles of the earnings distribution for recruiting selfemployed
Decile  Theoretical moment  Empirical moment 

10 %  1.5080  2.8799 
20 %  4.5268  5.6617 
30 %  7.9110  8.6274 
40 %  9.6422  10.6302 
50 %  13.6360  14.9321 
60 %  18.0819  18.0795 
70 %  22.4150  21.5686 
80 %  29.7773  24.2646 
90 %  45.3309  48.5292 
Proportion of paid employees in firms of given size or less
Firm size  Theoretical moment  Empirical moment 

2  0.0312  0.0387 
9  0.0358  0.1916 
24  0.1198  0.3288 
49  0.3122  0.4765 
99  0.5656  0.5883 
199  0.9031  0.6956 
499  1.0000  0.8429 
999  1.0000  0.9234 
5 Counterfactual policy simulation
To illustrate the importance of explicitly modeling the selfemployed, specifically as a source of job creation, this section looks at the endogenous employment response as a consequence of a change in unemployment benefit. It turns out, in the simulations, for small increases in unemployment benefit, aggregate employment increases. But underlying this, aggregate employment shift is a large reallocation of workers: growth in paid employees in large private sector firms at the expense of small selfemployed owned firms and a shift in the composition of the selfemployed, who are now operating with better ideas, but are far less likely to take on workers. This result is in stark contrast to a typical one sector model of the labor market.
In a prototypical single sector model of this kind, the employment response is straightforward. Unemployment benefit increases the value of unemployment which in turn means that workers need higher wages to leave for paid employment. Thus, fewer firms can afford to employ workers and the unemployment exit rate falls, and with a constant employment exit rate, unemployment will unambiguously increase. This mechanism is confused somewhat with the introduction of a second sector, with differing transition rates across sector, the unemployment rate is the solution to a more complicated set of flow equations, and the net effect is ambiguous. The inclusion of the recruiting selfemployed confounds the issue yet further, as now there is a positive employment externality of one sector on another. On the one hand, if more people are unemployed, with the rate of ideas approximately six times as large in unemployment compared with employment, in aggregate, one would expect more ideas. Thus, agents only act on very good ideas, and perhaps, this leads to more recruiting selfemployed and hence more job creation. Conversely, an increase in unemployment benefits will make it more expensive for the recruiting selfemployed to hire workers, and therefore perhaps, fewer workers will be hired and selfemployment will be less desirable.
The exact specification of the policy is to change the value of b. Recall, that the estimated value of b is negative and can be thought of as the stigma associated with unemployment net of any existing unemployment benefit. In these simulations, a series of increases from zero to £3 per hour are considered. To put this in some context, assuming a 40h week, the maximum increase in benefit considered is equivalent to £120 per week. At the time of writing, a typical over 25yearold claimant would expect to get £73.10 per week, so the maximum amount considered represents a fairly large expansion in the degree of generosity. The practicalities of the simulation are similar to the estimation, with two exceptions. In the estimation, a and b were treated as free parameters, now these are fixed and ϕ _{0} and ψ _{0} are solved explicitly. Similarly, the wage offer distribution F(·) and the offer arrival rates (λ _{0},λ _{1}) are backed out from the productivity distribution of firms and the parameters governing the number of firms and contact rate per firm (N,h ^{ f }).
The fall in the level of recruiting selfemployed means that there are more workers to be hired by large firms, both directly through the lack of recruiters and crucially indirectly through the lack of paid employees in small selfemployed owned firms. This feedback effect is so large that for small increases in benefit, there is an expansion in aggregate employment. This is not necessarily the only mechanism that could generate such a phenomenon, but it highlights that ignoring the selfemployed who constitute such a large part of the aggregate economy may lead to misjudgments in active labor market policy.
6 Conclusions
This paper builds an equilibrium model of the labor market with frictions in which agents endogenously locate on either side of the market, as a paid employee, or a recruiting selfemployed individual. The model is able to replicate differential features of the earnings distributions of agents in different labor market states.
Using British data, the model is estimated and the career options of the selfemployed are critically assessed. Underreporting of earnings is taken into account as in Hurst et al. (2014), as are future employment and earnings profiles and any nonpecuniary amenity associated with either state. The estimated parameters are used in a counterfactual policy exercise that examines the effects of an increase in the generosity of unemployment benefits. Including the selfemployed in the model yields an interesting result, that for low levels of benefit, an increase will be associated with an expansion in aggregate employment. A prediction that stands in stark contrast to typical one sector structural models of the labor market.
There has been a recent surge in the literature that incorporates selfemployment into models of the labor market: Narita (2014), Margolis et al. (2014), and Millán (2012). As a result, there is a deeper understanding of the puzzle outlined by Hamilton (2000), of why agents choose selfemployment at all. By distinguishing between ownaccount workers and recruiters and giving the selfemployed the option to develop a firm, this paper goes further still. Improvements to the precision of estimates could be made from increases to the size of the data. However, finding data with the necessary information regarding employment spells and earnings and whether an individual is a recruiter could be a challenge.
Finally, it is worth stating that there are other salient features regarding selfemployment that have been overlooked in this analysis. Amongst others, poignant factors include cross employment state heterogeneity of workers, family structure, and financial constraints. Future research aimed at incorporating ex ante worker heterogeneity in order to explain the differences in composition between paid employees and the selfemployed could be extremely fruitful. In a metaanalysis of recent research, Parker (2004) (Table 3.3 page 104) suggests overwhelming crosscountry evidence that the selfemployed are older, better educated, have more labor market experience, and are wealthier than paid employees. Another research agenda that the author believes deserves particular focus is to incorporate asset accumulation into this type of model and examine the impacts of financial constraints on selfemployment. There is considerable empirical evidence suggesting that financial constraints play an important role in an individual’s decision to become selfemployed; for a UK context, see for example Blanchflower and Oswald (1998), Cowling and Mitchell (1997), and Black et al. (1996). Incorporating these features into the theoretical model and estimating the model, if possible, with a larger dataset could prove to be very fruitful future research projects.
7 Endnotes
^{1}For a comprehensive discussion of the history of economic thought regarding entrepreneurship, see Hébert and Link (1988).
^{2}Firms are assumed to be infinitely lived to keep features of the Bontemps et al. (2000) model. Haltiwanger et al. (2013) finds for larger firms in the USA—those with more than 500 employees, there is less than a 1 % chance that a firm will go out of business in a given year. Implying that large firms exist on average for longer than 100 years, considerably longer than a typical agent’s tenure in the labor market.
^{3}The empirical evidence on Gibrat’s law in relation to firm growth is mixed. For a summary of the literature, see Santarelli et al. (2006).
^{4} \(\mathbb {E}_{F} \max \left [ W(x) W(w), 0\right ] = \int \max \left [ W(x)  W(w), 0 \right ] dF(x)\) and \(\mathbb {E}_{\Gamma } \max \left [ S(z) W(w), 0 \right ] = \int \max \left [ S(z)  W(w), 0 \right ] d \Gamma (z)\).
^{5}Industry classification are based on The Standard Industrial Classification of economic activities 1992. They can be found by visiting url: http://www.ons.gov.uk/ons/guidemethod/classifications/ archivedstandardclassifications/ ukstandardindustrialclassification1992sic92/index.html.
^{6}This corresponds to omitting those who gained employment in sectors with onedigit industry code zero, six, or eight. That is, those employed in agriculture, forestry, and fishing; finance, insurance, and real estate; and certain services.
^{7}Interestingly, in a similar multisector model, Meghir et al. (2015) also estimate a large negative flow value for lowskilled unemployed workers in Sao Paulo, Brazil. Recall that b is treated as a free parameter to ensure Eq. (32) is satisfied. Thus, one reason the estimate of b is so small is that the value of selfemployment for a given productivity is underestimated because it contains a large option value of paid employment, as argued by Millán (2012).
^{8}For a comprehensive assessment of this phenomenon, see Levitt and List (2011)
8 Appendix
8.1 Solving for reservation strategies
However, if agents always intend to recruit, the solution is given by S(ψ _{0})=U, where ψ _{0}>ψ _{1}; this value for ψ _{0} is denoted as \(\check {\psi }_{0}\). The explicit solution for \(\check {\psi }_{0}\) depends on the parameterization of p(y).
with the initial condition ϕ(ψ _{0})=ϕ _{0}.
8.2 Solving for the steady state
where, \(\gamma _{s}(y) = \frac {d}{dy} \left \{\Gamma _{s}(y) \right \}\).
where \( \mathbb {I}_{\left \{y \geq \psi _{1} \right \}} \) is an indicator function taking the value one if {y≥ψ _{1}} is satisfied and zero otherwise.
Solving for the steady state is fairly cumbersome and requires an iterative solution. Initially, s(ℓ) is computed by simulating the Markov process. The outer loop iterates around the measure of potential recruiters \(N_{s} \overline {\Gamma }(\psi _{1})\) and the inner loop around the measure of unemployed. An initial guess is made regarding the number of potential recruiters, from which Σ(ℓ) is calculated (Eq. (36)) and so is \({N_{e}^{s}}\) (Eq. (37)). In the inner loop, the ODE (34) with the initial condition Γ _{ s }(ψ _{0})=0 is solved, where \({N_{e}^{f}} G(\phi (y))\) is imputed using Eq. (35). Then, N _{ u } is updated according to Eq. (18). Once the procedure converges to a solution, it goes to the outer loop. The number of potential recruiters has only been used to determine the measure Σ(ℓ) (and \({N_{e}^{s}}\)); it is updated using the solution to the differential equation in the inner loop. The whole process is iterated on until stable.
8.3 Misreporting of earnings by selfemployed
The data on income that is relied upon in the estimation are collected from survey data. While there are no clear incentives to lie about one’s income in a survey, where there would be, to say, tax authorities, there exists a literature that suggests people answer or behave differently when being studied. This is commonly known as the Hawthorne effect ^{8}. To calculate the degree of misreporting, this subsection follows the methodology proposed by Hurst et al. (2014). Relying on consumption and income data, which are obtained from the Expenditure and Food Survey (EFS), information is obtained about total personal weekly consumption, gross weekly income, the employment status of the individual, and a variety of demographic information. The sample is restricted to males in employment.
There are four identifying assumptions that allows the uncovering of the degree of misreporting. The income and expenditure relationship is governed by the loglinear Engel curve. Selfemployed agents systematically misreport their earnings by a factor κ _{ s }, it need not be assumed that κ _{ s }≤1 and paid employees provide an unbiased reporting of their income. Finally, both the selfemployed and paid employees provide unbiased reports of total expenditure.
Fraction of under reported income by selfemployed
IV  OLS  

β  \(\underset {\left (0.184 \right)}{0.913}\)  \(\underset {\left (0.049\right)}{0.368}\) 
γ  \(\underset {\left (0.117\right)}{0.474}\)  \(\underset {\left (0.090\right)}{0.264}\) 
κ _{ s }  \(\underset {\left (0.065\right)}{0.595}\)  \(\underset {\left (0.124\right)}{0.489}\) 
Standard errors in parentheses are obtained using a bootstrap procedure, redrawing the sample with repetition 500 times. The estimation is based upon earnings for the selfemployed which have been adjusted by the factor κ _{ s }.
8.4 Transition rate moment conditions

Unemployment to paid employment: \(1  \exp ( \lambda _{0}  {\lambda _{0}^{s}})\)

Unemployment to ownaccount selfemployment: \(1  \exp \left ( \eta _{0} \overline {\Gamma }(\psi _{0})) \right)\)

Paid employment (in large firm) to unemployment: 1− exp(−δ)

Paid employment (any) to unemployment: \(1  \exp ( \frac {{N^{s}_{e}}}{{N^{s}_{e}} + {N^{f}_{e}}} (\mu + \delta)  \frac {{N^{f}_{e}}}{{N^{s}_{e}} + {N^{f}_{e}}} \delta)\)

Paid employment to another paid employer: \(1  \exp ( \frac {{N_{e}^{f}} \lambda _{1}}{{N_{e}^{f}} + {N_{e}^{s}}} \int _{\phi _{0}}^{\infty } \overline {F} (x) d G(x))\)

Paid employment to ownaccount selfemployment: \(1  \exp ( \frac {{N_{e}^{f}} \eta _{1}}{{N_{e}^{f}} + {N_{e}^{s}}} \int _{\psi _{0}}^{\infty } \overline {\Gamma } (x) d G(\phi (x)))\)

Ownaccount selfemployment to recruiter: \(1  \exp ( h \overline {\Gamma }_{s}(\psi _{1}))\)
Declarations
Acknowledgements
The author would like to thank two anonymous referees and the editor of this journal, Pierre Cahuc for his insightful and constructive comments. I would also like to thank Jim Albrecht, Etienne Lalé, Fabien PostelVinay, Robert Shimer, Hélène Turon, Ludo Visschers, and Susan Vroman as well as seminar audiences at the Barcelona Graduate School of Economics, University of Bristol, The Search and Matching Conference 2014 at the University of Edinburgh, The International Association for Applied Econometrics 2014 at Queen Mary College, University of London, and The Bank of England for all their helpful insights. All errors are, of course, my own. Email: jb683@cam.ac.uk. Responsible Editor: Pierre Cahuc
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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