Temporary Agency Work and Labor Misallocation

RAQUEL CARRASCO

ISMAEL GÁLVEZ-INIESTA

BELÉN JEREZ

Documento de Trabajo 2024/09

Octubre de 2024

fedea

Las opiniones recogidas en este documento son las de sus autores y no coinciden necesariamente con las de Fedea.

Raquel Carrasco^† Ismael Gálvez-Iniesta^‡ Belén Jerez^§

September 27, 2024

Abstract

The triangular employment relationship between outsourced workers, intermediary employers, and user firms has received limited theoretical attention. This paper seeks to address this gap by focusing on temporary agency work. We develop a two-period labor search model in which firms can create jobs directly or through a temporary work agency (TWA). In both instances, match quality depends on unobservable attributes of workers and job vacancies. The agency acts as a matchmaker, providing flexibility by spreading termination risks across firms and certifying assignment quality through worker screening. However, worker poaching by user firms reduces the returns on the agency’s investments in recruitment and screening. This hold-up problem leads to ineficient assignments, prolonged TWA employment spells, and insuficient job creation. As in the data, TWA employment is more relevant for less skilled workers. We also find that these workers are more likely to be trapped in ineficient assignments. The distortions afecting agency workers are more severe when wages in direct-hire jobs are set through Nash bargaining rather than directed search. Although the hold-up problem in our model can be solved with transfers from poaching clients to the agency, such transfers are illegal in most EU countries. The paper concludes by presenting simulations and using Spanish data to validate the theoretical framework.

JEL classification: J20, J42, D58, D61.

Keywords: temporary agency work, intermediation, search and matching, screening, eficiency.

^∗We thank Paul Beaudry, William Fuchs, Giovanni Gallipoli, Francisco González, David Green, Li Hao, Matthias Kredler, Joe Ostroy, Michele Pellizzari, Mike Peters, and Felix Wellschimied for their comments, as well as participants the 2023 North American Summer Meeting of the Econometric Society at UCLA, VI Spanish Macroeconomics Network Workshop, and II Micro Retreat and Internal Macro/Applied Labor Workshop at UC3M, and seminars at the Vancouver School of Economics and University of Waterloo. Financial support by MICIN/AEI/10.13039/501100011033, grants RTI2018-095231-B-I00, PID2019-107161GB-C3, CEX2021-001181-M and PID2022-142496OB-I00, and Comunidad de Madrid, grant EPUC3M11 (V PRICIT) and H2019/HUM-5891, is gratefully acknowledged. Belén Jerez thanks the Vancouver School of Economics at UBC for its kind hospitality while working on this project. Any errors are ours.

^†Department of Economics, Universidad Carlos III de Madrid. E-mail: rcarras@eco.uc3m.es

^‡Department of Applied Economics, Universitat de les Illes Balears. E-mail: i.galvez@uib.es

^§Department of Economics, Universidad Carlos III de Madrid. E-mail: mjerez@eco.uc3m.es

1 Introduction

The rise of employee outsourcing is reshaping the modern workplace. Firms increasingly contract tasks to external providers, seeking specialized labor at potentially lower costs. Temporary agency work is a growing form of outsourcing¹. A review of the empirical literature reveals the diverse roles played by temporary work agencies (TWAs). These agencies act as intermediaries, matching firms with temporary workers to address fluctuating demands and facilitate worker screening (Katz and Krueger, 1999; Neugart and Storrie, 2002; Houseman et al., 2003). Because TWAs specialize in recruitment, they can fill short positions by rotating workers across multiple clients, resulting in economies of scale and cost savings (Houseman and Polivka, 2000; Abraham and Taylor, 1996). For individuals, TWAs can provide access to work, particularly when unemployed or seeking flexible arrangements (Katz and Krueger, 2019). Agency work thus can be viewed as an innovation in intermediation aimed at reducing matching and informational frictions. This form of outsourcing has unique characteristics due to the temporary nature of the contracts and strict regulations imposed in many countries.

As the prevalence of outsourcing has increased and TWAs come under scrutiny, the debate surrounding their impact on labor market outcomes has intensified. Extensive empirical research has been conducted on the subject; see the survey in Autor (2009), as well as more recent work by Bilal and Lhuillier (2021), Spitze (2022), Drenik et al. (2023), Carrasco et al. (2024), and Bergeaud et al. (2024), among others. These studies reveal that while outsourcing afects a wide range of jobs, low-skilled workers are disproportionately impacted. On average, jobs that are outsourced pay less and have shorter duration than direct-hire positions. TWA jobs, in particular, can help individuals exit unemployment, but are associated with high reentry rates and persistence and do not necessarily lead to permanent employment².

This paper delves into the potential drawbacks of TWAs by analyzing the specific mechanisms through which they operate. Our analysis highlights the multifaceted role of TWAs as intermediaries, serving as a bridge between firms (“user firms”) and workers. This triangular employment relationship has received limited theoretical attention, as highlighted by Autor (2009). Our paper seeks to contribute to filling this gap, by developing a tractable general equilibrium model that examines how TWA employment afects worker outcomes.

¹Temporary agency work has grown significantly in developed countries since the 1990s. Globally, agency workers increased from 8.9 million in 2006 to 49 million in 2019. Europe leads this market, accounting for 39% of global revenues in 2019. See World Employment Confederation (2021).

²The evidence on this last point is mixed. See also Filomena and Picchio (2022).

Our model embeds a TWA sector into a two-period labor search framework. We consider a TWA that holds monopsony power over workers while charging firms competitive fees equal to the marginal product of labor. This setting is consistent with recent evidence of monopsonistic TWA behavior (see Drenik et al., 2023; Carrasco et al., 2024). There is a continuum of unemployed individuals and a larger mass of one-worker firms that are free to enter the market. Jobs can be created directly or through the TWA. In both cases, match quality depends on unobservable attributes of workers and jobs. In particular, firms fill both “stepping-stone” jobs with potential for permanent positions and “dead-end” jobs with no long-term prospects (e.g. contract work), in line with the literature categorization (Booth et al., 2002). We use the model to compare the outcomes for direct-hire workers with those placed through the TWA. In addition, we analyze how the model’s predictions difer across skill levels.

There are several diferences between the direct-hire and intermediated markets in the model. First, in the direct-hire market workers may not find jobs, while firms may struggle to fill vacancies. In contrast, there is no rationing on the TWA market, reflecting the intermediary’s superior matching technology (Yavas, 1994). Second, vacancies with no permanent employment prospects are more prevalent in the TWA market, where jobs are shorter on average (as in the data³). While it is not profitable for firms to fill these vacancies directly, the TWA can do so by rotating its workers across user firms. Finally, by screening workers, the TWA gathers more information about skills and qualifications, as well as reliability, which is especially important for low-skilled jobs. It then certifies assignment quality⁴. In the direct-hire market, however, firms do not screen workers.

There are also contractual diferences between TWA and direct-hire jobs. The former are temporary, lasting for one period, whereas the latter initially last for one probationary period, with the possibility of transitioning to a permanent position upon reaching a certain output threshold. TWA workers can also become permanent employees of the user firm once their contracts expire if they reach this threshold. In this scenario, the firm and the TWA engage in a Bertrand-style competition, and the firm poaches the worker ofering a wage that equals the THA’s maximum bid. Workers who are not poached either renew their TWA contracts or are dismissed.

³See Storrie (2007), and empirical work by Givord and Wilner (2015), Addison et al. (2019) and Carrasco et al. (2022) for France, Germany, and Spain, respectively. Critically, Carrasco et al. (2024) find that worker self-selection does not fully explain the higher unemployment risk for TWA workers compared to direct-hire temporary workers.

⁴García-Pérez and Muñoz-Bullón (2005b) and Carrasco et al. (2024) find evidence of screening using the same data source we use for calibration.

The intermediation process in our model can cause several distortions that negatively impact workers, especially those with lower skills. Because user firms have incentives to poach highperforming workers, the TWA’s recruitment investments are reduced and some agency workers are not assigned to their most productive roles. Such misallocation results in ineficient repeated TWA employment spells and insuficient overall job creation. Ineficient assignments are more common among less skilled workers, for two reasons. First, misallocating these workers is less costly (in terms of reduced TWA fees). Second, some of these costs can be ofset by lowering wages, given the workers’ limited prospects for direct-hire employment⁵. Nonetheless, distortions may also occur in highly skilled positions. Our benchmark model assumes a competitive direct-hire search market (Peters, 1991). The misallocation of TWA workers becomes more severe when direct-hire wages are instead set through Nash bargaining (Pissarides, 2000). In this model variant, workers’ payofs are reduced and the TWA’s monopsonistic power is strengthened due to the suboptimal functioning of the direct-hire market.

A model calibration sheds further light on our mechanism and provides empirical support for our theoretical framework. The calibration replicates key characteristics of the Spanish TWA sector, focusing on male workers with low and medium skills, the main demographic group employed by the sector. We target the sector’s job creation share, the positive average wage gap between direct-hire temporary workers and TWA workers, and average transition rates to diferent states for the two worker groups. Due to the distortions that arise in equilibrium, welfare gains are modest relative to a situation without intermediation. Also, the TWA’s contribution to net employment creation is positive but small. When diferent skill groups are compared, our simulations align with two key observations from Spanish data. First, transitions to permanent employment become more frequent as skills rise, with this efect being more pronounced among agency workers. Second, the wage gap between agency and direct-hire temporary workers widens.

⁵We model the labor market as segmented by occupational skill groups, where lower skills are associated with lower average productivity.

Allowing transfers from poaching firms to the TWA solves the hold-up problem in our model. Such transfers induce the TWA to enhance its recruitment investments and place its workers eficiently. Many European Union (EU) countries, however, prohibit or greatly restrict non-compete clauses in TWA contracts. Even in countries like the UK, where TWAs are allowed to charge a “transfer fee” if a worker is hired by the user firm, transfers are restricted (Storrie, 2007). The lesson is that, even though these policies aim to safeguard workers’ rights and promote job stability, they can have unexpected consequences. A more sophisticated approach to regulation may then be required. Policies that allow for transfers may help reduce the type of distortions we identify.

1.1 Related literature

The role of intermediaries in mitigating market frictions is extensively studied in the theoretical literature, which generally classifies them into two categories. Matching intermediaries, such as online platforms or public employment agencies, connect buyers and sellers without trading goods themselves (Rubinstein and Wolinsky, 1987; Gehrig, 1993; Yavas, 1994). In turn, inventory-holding intermediaries, like car dealerships, buy and resell goods, often certifying quality (Biglaiser, 1993). TWAs combine aspects of both types. They directly employ workers, addressing informational frictions by screening them, and matching frictions by connecting them to firms. Analyzing the impact of this type of intermediation on the labor market is challenging due to its hybrid nature and the dynamic triangular employment relationship involved.

Our work is related to two previous contributions. Autor (2001) analyzes a partial equilibrium model in which workers receive free training from competing TWAs. Based on US data, this study provides empirical evidence that TWAs screen workers and sell information about worker quality to user firms. Ciapanna (2011) studies the interaction between consulting firms, client companies, and consultants using a dynamic matching game. Similar to the TWA in our model, consulting firms ofer flexibility and information that improve matching outcomes. Ciapanna (2011) finds that, while intermediation can enhance welfare if agents are patient, it also sufers from hold-up problems and ineficiencies caused by strategic poaching, resulting in lower match quality and welfare compared to the optimal allocation. Her work difers from ours in that competition in the consulting industry prevents consultant misallocation. Also, the model is too complex to consider heterogeneous jobs (e.g. varying average duration or match quality) or to conduct comparative statics, as we do.

Our paper complements independent work by Bergeaud et al. (2024). Building on Drenik et al. (2023) and based on a rich French dataset, these authors show that the wage gap between direct-hire workers and TWA workers, while positive on average, exhibits significant cross-sectional variation, being negative in some cases. To rationalize this fact, they develop a quantitative search model where TWAs compete directly with firms in the hiring process while ofering them the option of hiring TWA services⁶. A cost-saving benefit, arising from faster hiring and human resource management outsourcing, leads to an over-reliance on TWAs which depresses the equilibrium wages of TWA workers. Our theoretical analysis shows that screening by a monopsonistic TWA can lead to additional ineficiencies beyond the wage gap, such as worker misallocation, excessive repetition of TWA employment spells, and low transition rates from TWA to permanent employment. Furthermore, a hold-up problem reduces TWA employment compared to the optimal allocation.

The paper is organized as follows. To motivate our theoretical framework, Section 2 briefly reviews Spain’s legal framework for TWAs, along with some descriptive facts regarding TWA workers and user firms. Sections 3 and 4 describe the model and characterize the equilibrium. Section 5 characterizes the optimal allocation and the distortions in the TWA sector, and Section 6 presents comparative statics results. Section 7 describes the calibration and the results of our simulations, and discusses empirical evidence that supports our theoretical framework. Section 8 includes concluding remarks. All proofs are in the Appendix.

2 TWAs in Spain

2.1 Legal framework

As in other EU countries, Spanish law requires that TWAs and their workers sign employment contracts, while the TWA signs a commercial contract with the user firm. Commercial contracts must include the reasons and duration of the TWA assignment, the job description, the qualifications required, the worker’s salary, the location, the work schedule, TWA fees, and collective bargaining agreements for both the TWA and the user firm. Workers may be hired for an indefinite period or a fixed term that coincides with the duration of the commercial contract. However, permanent employment contracts for TWA workers have historically been rare (Storrie, 2007; Countouris et al., 2016).

⁶Previous quantitative studies on outsourcing include Bilal and Lhuillier (2021), Spitze (2022) and Bostanci (2022).

Spanish legislation for regular fixed-term employment contracts applies to TWA contracts⁷. In particular, workers who continue to work for the user firm after completing their contract are considered permanent workers of the firm. Commercial contracts cannot contain non-compete clauses (prohibiting the user firm from hiring agency workers). TWAs must train their workers before they join the user firm, and workers cannot bear any training costs⁸. EU legislation mandates equal pay for TWA workers and direct-hire workers performing the same task at the user firm (EU Directive 2008/104/EC); however, there is evidence of non-compliance in diferent member states (e.g. Böheim and Cardoso, 2009; Bergeaud et al., 2024).

Although TWAs were authorized to operate in Spain with strict restrictions in 1994, the 2008 financial crisis resulted in an increasing deregulation of TWAs across the EU. In particular, current TWA sector collective agreements in Spain include a union commitment to avoid clauses that exclude TWA services.

2.2 Descriptive facts

This section provides an overview of the relevant features of TWAs in Spain, describing the characteristics of the workers employed through these agencies and the firms that use TWA services.

Spain has historically had one of the highest temporary employment rates in the OECD. At the start of the Great Recession, about one-third of salaried employees had temporary contracts. According to the Continuous Sample of Working Lives (Muestra Continua de Vidas Laborales,

⁷Until 2022, the maximum legal contract duration was 12 months (within 18 months) for a “casual contract”, and three years for a “contract for specific tasks”.

⁸Hiring TWA workers to complete hazardous tasks, replace workers on legal strike, or fill vacancies caused by recent layofs is illegal. User firms must comply with other basic worker rights regarding security and health at work. Unlike user firms’ employees, TWA workers receive neither holiday periods nor extraordinary payments, but proportional payments are included in their wages. See Carrasco et al. (2022) for further details.

MCVL hereafter), a dataset from Spanish Social Security records, TWA employment in 2019 represented about 1% of salaried jobs for workers aged 16 to 65 and 3% for males aged 20 to 45 with at most secondary education. Despite these modest shares, agency work plays a significant role in job creation and job destruction. In 2019, TWA contracts accounted for 13% of all contracts signed each month and 14% of all terminated contracts. All these shares have been growing since 2009.

Table A1 describes the worker composition by contract type in 2019. Temporary direct-hire and TWA workers have similar age distributions. Males are more predominant in TWA contracts (61%) compared to direct-hire temporary contracts (around 50%). This aligns with part-time work being less common among TWA workers (25%) than regular temporary workers (35%). Low-educated workers are also overrepresented in TWA jobs, with workers with at most secondary education accounting for 68% of agency contracts, compared to 59% for regular temporary contracts. There are also notable diferences in contract duration and skill types. Specifically, 97% of agency contracts correspond to medium-low and low-skilled occupations, compared to 73% of temporary direct-hire contracts. While TWA contracts account for 18% of all temporary contracts signed in 2019, this share is 36% low-skilled occupations. TWA contracts also tend to be shorter; 32% lasted only one day, and 84% lasted less than a month, compared to 18% and 57%, respectively, for regular temporary contracts.

Although the MCVL does not provide information about user firms, according to TWA statistics from the Spanish Social Security, agency workers mainly concentrate in manufacturing (27%), agriculture, forestry, and fishing (19%), accommodation and food services (15%), transportation and storage (14%), and wholesale and retail trade (11%). TWA contracts are particularly overrepresented in manufacturing, accounting for over 50% of all contracts in 2019, and in transportation and storage, where the share exceeds 40%. In 2017, 260 TWAs operated in Spain, with three agencies accounting for 42% and ten agencies for 67% of total revenues. These agencies employed 800 workers each month on average, according to the Spanish TWA association (ASEMPLEO).

Figure A1 illustrates the aggregate employment transition patterns by contract type among young, low-skilled males. Here, A represents agency contracts, D direct-hire temporary contracts, P permanent contracts, and U denotes unemployment. The figure displays the proportion of A, D, and P contracts that transition to states A, D, P, and U from month t to month t + 1 (in panels b, c, and d, respectively). Transitions are defined as contract changes, with the same or a diferent employer. For instance, contracts with the same employer accounted for 56% of AA transitions and 27% of DD transitions in 2019. Transitions to U are more frequent for A than for D contracts throughout the sample period (Panel b). For transitions to P the opposite is true (Panel a). The most significant diference is observed in transitions to with 30% of A contracts and around 1% of D contracts exiting to this state (Panel d). For transitions to D, the proportion is around 15-18% for D contracts and about half of that for A contracts (Panel c).

3 Model

In this section, we describe the two-period model economy. There is a measure one of unemployed workers, a larger mass of one-worker firms that can freely enter the economy, and a TWA that employs multiple workers. To keep things simple, agents are risk-neutral, there is no discounting, and the value of unemployment is set to zero. We omit time subscripts when this does not lead to confusion.

The temporary job market comprises direct-hire and TWA segments, respectively denoted by . Workers are ex-ante symmetric, and so are vacancies in each market segment. Yet ex-post match quality depends on unobservable characteristics of workers and vacancies. The distribution of workers and vacancy types is common knowledge. There is a continuum of worker types, distributed according to the c.d.f.

\[F (x) = 1 - \gamma + \gamma G (x),\tag{3.1}\]

where and G is with . That is, a mass of workers are of type 0, and the rest are distributed on with density . There are two types of jobs, , where match output is given by

\[y _ {j} (x) = \left\{ \begin{array}{l l} y _ {0} & \text { if } j = 0, \\ (1 + z x) y _ {0} & \text { if } j = 1, \end{array} \right.\tag{3.2}\]

where . Type-0 jobs then generate the same output regardless of the worker’s type, whereas type-1 jobs generate more output when higher worker types perform them (the percentage diference in productivity between type-x and type-0 workers being zx). We assume that jobs with output y₀ are destroyed after one period; , they correspond to short-term demands or have low average quality. However, if output exceeds , there is a second production stage. In this sense, type-0 jobs represent “dead ends”, while type-1 jobs represent potential “stepping stones”⁹.

The share of type-1 vacancies in market m is denoted by , where . Hence, agency jobs are shorter on average than direct-hire jobs, as in the data. As we shall show, although it is never profitable for firms to fill directly these shorter jobs, the TWA can do so by rotating its workers after each job, thus spreading termination risk across multiple firms (see Section 4.1). We also assume that , so agency jobs of type 1 are scarce relative to the mass of workers whose productivity at these jobs exceeds y₀.

The timing is as follows. Job creation takes place at the start of period 1 in the direct-hire and TWA markets. Once markets close, there is an initial production stage. Contract renewals and dismissals take place at the start of period 2, based on period-1 output. We describe these decisions in detail below. We focus on the interesting case where direct-hire and agency jobs coexist in equilibrium, so workers get the same expected payof in both markets and their participation decision corresponds to a mixed strategy.

3.1 Direct-hire jobs

The search process in market d is competitive or directed (Peters, 1991; Moen, 1997; Acemoglu and Shimer, 1999). Firms first pay a vacancy posting cost κ. Then, they simultaneously post an ofer specifying the worker’s share of the match surplus. In doing so, they commit to paying a piece-rate wage for each unit produced. Workers observe all the ofers and choose among them, randomizing when indiferent. Firms and workers who choose a given ofer form a submarket where they meet randomly, since worker and vacancy types are not observable at this point¹⁰. Once a match is formed, ofers are enforced (there is no renegotiation). All agents have common rational beliefs about the mass of workers each ofer attracts in equilibrium.

⁹We set the productivity at type-0 jobs to y₀ to simplify the exposition. Our theoretical results hold also when these jobs have higher productivity; all we require is that type-0 jobs last for one period.

¹⁰Sattinger (1996, 2003) and Albrecht and Vroman (2002) introduce search frictions in the Roy (1951) model.

We index submarkets by the associated ofers and denote the mass of workers and vacancies in submarket by u and v. The mass of jobs created in this submarket is . The matching function M has standard properties: it is increasing, , and displays constant returns to scale, with and . The associated job-filling and job-finding probabilities are

\[\begin{array}{l} q (\theta) = \mathcal {M} (u, v) / v = \mathcal {M} (\theta^ {- 1}, 1), \\ q (\theta) \theta = \mathcal {M} (u, v) / u = \mathcal {M} (1, \theta), \end{array}\tag{3.3}\]

where is the number of vacancies per worker (or tightness) in submarket In particular, workers (firms) are more (less) likely to be matched when θ is higher.

Since workers and firms are ex-ante symmetric, it is direct to show that a single ofer is posted in equilibrium that satisfies the Hosios (1990) condition:

\[\phi = \eta (\theta) \equiv \frac {- q ^ {\prime} (\theta) \theta}{q (\theta)}.\tag{3.4}\]

This condition ensures that workers and firms receive a payof that is equal to their social contributions to match formation, so no ineficiencies arise in the direct-hire market (Peters, 1991). There, a single submarket is active, where the worker’s bargaining share equals the elasticity of the firm’s matching probability. As is standard, the elasticity is assumed non-decreasing. Hence, when is higher, workers are not only more likely to find jobs directly but also tend to have greate bargaining power.

Denote the total mass of unemployed workers and vacant jobs in market d by and respectively. The mass of direct-hire jobs created in period 1 is then , where

\[\theta = v _ {d} / u _ {d}.\tag{3.5}\]

After jobs are created in period 1, output is realized and the first production stage takes place. In period 2, contract renewals and dismissals of direct-hire workers are handled as follows. If output equals , the job is destroyed (at no cost), so workers reenter unemployment and firms exit the economy. If output exceeds , however, there is a second production stage with two possible outcomes. With probability λ, match productivity is scaled up by , as a result of learning and job-specific human capital accumulation, and the worker is promoted. With complementary probability, output remains constant, and the worker’s temporary contract is renewed. We interpret promotions as permanent job ofers, even though our model has only two periods.

Direct-hire workers then transition to three possible states in period 2, as shown in Table B1 (first row). They become unemployed with probability . Those who remain employed are promoted with probability λ; otherwise, their temporary contracts are renewed. These workers never transition to TWA jobs though. This is a reasonable simplifying assumption given the low observed frequency of these transitions (e.g. Givord and Wilner, 2015; Carrasco et al., 2024).

3.2 TWA jobs

Following Yavas (1994), we assume an intermediated market that is not subject to rationing, so the probability of finding/filling a TWA job is . The number of agency jobs then equals the number of agency workers:

\[v _ {a} = 1 - u _ {d}.\tag{3.6}\]

Due to its superior matching technology, more jobs are created when the TWA is in operation (compared to an environment without intermediation); i.e., total job creation is

\[q (\theta) (\theta) u _ {d} + 1 - u _ {d} > q (\theta) (\theta).\]

The TWA’s share in job creation is

\[s _ {a} = \frac {1 - u _ {d}}{q (\theta) \theta u _ {d} + 1 - u _ {d}} = \frac {1}{q (\theta) \theta \left(\frac {1}{v _ {a}} - 1\right) + 1}.\tag{3.7}\]

This share falls when the probability of finding a job directly, , increases, and when more workers choose market d over market a.

The TWA observes the type of vacancies its clients seek to fill and learns the types of its workers at a cost¹². It then signs an identical employment contract with each worker stipulating the wage, and a commercial contract with each user firm specifying the certified output and the associated fee. Both contracts last for one period. The TWA has no incentives to lie about the output it certifies. If it did, firms can renege on the contract and refuse to pay the fee¹³. Also, the screening and certification process cost is assumed to be low enough to be more profitable than assigning workers randomly to firms.

¹¹This keeps the model simple. Our results also hold when these probabilities are constant and higher than in market d.

¹²The screening process in Autor (2001) is more complex than in this paper because workers observe imperfect signals about their types.

The TWA’s non-labor cost depends on the mass of jobs it creates and includes matching, screening, contracting, and regulation compliance costs. This cost is denoted by , where c is strictly increasing, strictly convex, and , with . Also, lim , so it is not feasible for the TWA to hire all workers.

Once workers are assigned to user firms, there is a first production stage. As in the case of direct-hire jobs, firms want to continue production in period 2 if worker output exceeds (and exit the economy otherwise). In this case, worker productivity is scaled up by with probability and the worker gets a permanent job ofer from the user firm. That is, in our benchmark model firms treats TWA and direct-hire workers equally if they produce identical output in the second period. Whenever a TWA worker receives an ofer from the user firm, the firm and the TWA compete for the worker a la Bertrand, and the firm poaches the worker by ofering her a wage equal to the TWA’s maximum bid. The TWA may ofer a contract renewal to those workers who are not poached and assign these workers to a diferent client, whereas workers who are not assigned to (or poached by) a client reenter unemployment.

4 Search equilibrium

In this section, we describe the agents’ payofs in each market and characterize the equilibrium.

4.1 Payofs in market d

The expected bilateral surplus from creating a direct-hire job is

\[\mathcal {S} _ {d} = (1 - \pi_ {d} \gamma) y _ {0} + \pi_ {d} \gamma \int_ {0} ^ {1} [ y _ {1} (x) + (1 - \lambda) y _ {1} (x) + \lambda p y _ {1} (x) ] d G.\tag{4.1}\]

¹³With a longer time horizon, truthful revelation would follow from a reputational argument Biglaiser, 1993).

With probability , output equals , so there is a single production stage. With complementary probability, output is higher and there are two production stages. Recall that, in period 2, output is scaled up by p with probability λ and remains constant otherwise. Combining (3.2) and (4.1), and integrating by parts gives

\[S _ {d} / y _ {0} = 1 - \pi_ {d} \gamma + \pi_ {d} \gamma (2 + (p - 1) \lambda) \left(z (1 - \int_ {0} ^ {1} G (x) d x) + 1\right).\tag{4.2}\]

The bilateral surplus in market d increases with the share of jobs that last for two periods (as well as with because average productivity is higher in this case.

Workers find jobs directly with probability , and receive a share of the surplus, so their payof in market d is

\[U = q (\theta) \theta \eta (\theta) S _ {d}.\tag{4.3}\]

Likewise, firms hire a worker with probability after their vacancy posting cost is sunk, receiving the rest of the surplus. Due to free entry, firms break even in equilibrium¹⁴:

\[\kappa = q (\theta) (1 - \eta (\theta)) S _ {d}.\tag{4.4}\]

The equilibrium value of θ is given by (4.4), and (4.3) then determines U. As is standard in these models, when productivity is higher, direct-hire jobs are more abundant and the workers’ payof is higher. That is, θ and U increase with . These equilibrium variables do not depend on the terms of trade that prevail in market a. This model feature is consistent with the small observed TWA employment share. Our comparative statics analysis in turn focuses on how the TWA’s decisions are afected by changes in direct-hire employment prospects (as described by U).

An important remark is in order. In our model, firms would never fill directly the shorter jobs the TWA helps create. Workers only seek these jobs if their expected payof equals U. Since the implied bilateral surplus is lower than , the associated job-finding probability must be higher than . Hence, the implied job-filling probability is lower than , resulting in negative profits. In short, if (4.3) holds for these jobs then (4.4) is violated, meaning that these shorter jobs are not created in market d.

¹⁴We assume lim_θ→0 , so market d is active in equilibrium.

4.2 The TWA’s problem

Since user firms break even in equilibrium, TWA fees in each period equal worker output (the marginal product of labor). This is trivial if the worker is only employed during period 1. Take the case of a poached worker, who is employed in both periods. Bertrand’s competition between the TWA and the user firm implies that the entire surplus goes to the poached worker in period 2. The reason is that, for the TWA, the value of assigning the worker to another client equals the worker’s output. Given that the TWA cannot commit to not competing with the firm, its fee cannot exceed the worker’s output in period 1 to compensate for the fact that the worker is poached in period 2. Such a fee would imply negative profits for the user firm.

The TWA chooses the mass of workers it hires, and their wage, . In doing so, it takes as given the share of type-0 vacancies in market a, the worker type distribution in (3.1), and the workers’ equilibrium payof, U. Once it pays for screening costs, the TWA perfectly observes worker and vacancy types; it then chooses how to assign workers to clients in each period.

Since fees equal match output, in period 2 it is optimal to assign the most productive workers to type-1 vacancies, to maximize total fees. This need not be the case in period 1 though. To see why, suppose type-1 vacancies are filled with the most productive types in t = 1. If so, output exceeds in and the firm wants to continue production in t = 2. At this stage, however, it is optimal for the firm and the worker to sign a direct-hire contract to save on the TWA’s fee. In this scenario, the firm poaches the worker by ofering her a wage equal to her period-2 output (the TWA’s maximum bid).

The alternative strategy, to avoid poaching, is to fill type-1 vacancies with type-0 workers initially and assign more productive workers to type-0 jobs instead. Once type-0 jobs are destroyed, more productive workers are reassigned to type-1 jobs, and type-0 workers are fired. This strategy ensures that the TWA earns the fee in t = 2, as long as the worker does not get a permanent ofer from the user firm. The implied cost is a reduction in period-1 fees, due to lower assignment quality.

Below we characterize the TWA’s optimal assignment given and . There are user firms with type-1 vacancies, filled with the most productive worker types in . These are types , where i.e.,

\[G (x ^ {*}) = 1 - \frac {\pi_ {a}}{\gamma}.\tag{4.5}\]

Lower types in are dismissed in

Let us describe how each worker type is assigned in . If type x is assigned to a type-1 vacancy, the TWA earns in , and nothing in (as the worker is poached). If is initially assigned to a type-0 vacancy and then reassigned to a type-1 vacancy, the TWA gets in , and with probability 1 − λ it gets in (and nothing otherwise). The second strategy is optimal if

\[y _ {1} (x) - w _ {a} < y _ {0} - w _ {a} + (1 - \lambda) (y _ {1} (x) - w _ {a}).\tag{4.6}\]

The first strategy is optimal if the inequality in (4.6) is reversed, and the TWA is indiferent between both strategies if

\[\lambda (y _ {1} (x) - w _ {a}) = y _ {0} - w _ {a}.\tag{4.7}\]

Since increases with , the optimal assignment is characterized by a threshold xˆ. Types above (below) xˆ are initially assigned to type-1 (type-0) vacancies. This is intuitive, as the cost of deviating from the eficient (output maximizing) assignment is greater when x is higher. The threshold is interior if the value of x that solves (4.7) lies in . If this value exceeds 1 then if it is lower than then . Using (3.2), we may then write

\[\hat {x} = \min \left\{1, \max \left\{x ^ {*}, \frac {1}{z} \left(\frac {1}{\lambda} - \frac {\left(\frac {1}{\lambda} - 1\right) w _ {a}}{y _ {0}} - 1\right) \right\} \right\}.\tag{4.8}\]

Proposition 1. TWA workers of type are assigned to type-1 vacancies in period 1 and poached in period 2, where is given by (4.8). Other types are initially assigned to type-0 vacancies. They are then reassigned to type-1 vacancies in period 2 , and dismissed otherwise.

In our model, infrequent promotions to permanent employment are associated with greater misallocation. When the promotion probability λ approaches one, we reach the static scenario and the TWA’s assignment is eficient because its margin in period 2 is zero regardless of its strategy. Instead, when λ approaches zero, the distortion is maximal, and all type-1 jobs are initially filled with type-0 workers. For intermediate values of λ, “top types” in the range (ˆx, 1] are assigned eficiently, but “intermediate types” in are not. Thus similar types (close to xˆ) face very diferent outcomes.

Corollary 2. If λ is close to 1 then , so the TWA’s assignment is eficient. is close to 0 then , and all type-1 vacancies are filled with type-0 workers in period 1.

Given the threshold xˆ, the TWA wage satisfies the workers’ participation constraint:

\[\begin{array}{l} U = w _ {a} + (1 - \lambda) \gamma (G (\hat {x}) - G (x ^ {*})) w _ {a} + \lambda \int_ {x ^ {*}} ^ {\hat {x}} p y _ {1} (x) \gamma d G + \int_ {\hat {x}} ^ {1} [ (1 - \lambda) y _ {1} (x) + \lambda p y _ {1} (x) ] \gamma d G \\ = w _ {a} [ 1 + (1 - \lambda) \gamma (G (\hat {x}) - G (x ^ {*})) ] + \lambda \gamma p y _ {0} \int_ {x ^ {*}} ^ {1} (1 + z x) d G + (1 - \lambda) \gamma y _ {0} \int_ {\hat {x}} ^ {1} (1 + z x) d G, \end{array}\tag{4.9}\]

where the last equality in (4.9) follows from (3.2), after rearranging. Condition (4.9) ensures that workers get the same expected payof in markets a and d. Although all TWA workers earn wage in period 1, their period-2 earnings depend on their type. Lower types in become unemployed, with zero income. Top types in (ˆx, 1] are poached by firms, so their wage is with probability , and with probability λ. Intermediate types in renew their TWA contracts with probability ; with complementary probability, they get a permanent job that pays

The TWA chooses its labor demand to maximize its total profits given xˆ and . The associated first-order condition is

\[c ^ {\prime} (v _ {a}) + U = S _ {a} (\hat {x}).\tag{4.10}\]

The left-hand side of (4.10) is the marginal cost of hiring a TWA worker, which equals the marginal non-labor cost plus the worker’s expected payof. In turn, represents average per-worker fees:

\[\begin{array}{r l} & S _ {a} (\hat {x}) = \int_ {\hat {x}} ^ {1} y _ {1} (x) \gamma d G + (1 - \gamma + \gamma G (\hat {x})) y _ {0} + (1 - \lambda) \int_ {x ^ {*}} ^ {\hat {x}} y _ {1} (x) \gamma d G \\ & \qquad = \gamma y _ {0} \int_ {x ^ {*}} ^ {1} (1 + z x) d G - \lambda \gamma y _ {0} \int_ {x ^ {*}} ^ {\hat {x}} (1 + z x) d G + (1 - \gamma + \gamma G (\hat {x})) y _ {0}, \end{array}\tag{4.11}\]

where the last equality follows from (3.1)–(3.2). Since types in (ˆx, 1] are initially assigned to type-1 jobs, the associated fee is in t = 1, and zero in t = 2. Types in are assigned to type-0 jobs, so the fee in is . If , the TWA ofers the worker a contract renewal, which is accepted as long as the worker is not promoted by the user firm. Hence, the fee in is with probability , and zero otherwise. If , the worker is fired in

Transition rates for TWA workers are displayed in Table B1 (second row). Given that only type-0 jobs are destroyed in , TWA workers enter unemployment with probability Since workers who remain employed get promoted with probability transitions to permanent employment occur with probability . The TWA retains workers of type who do not get a permanent job, so repeated TWA spells occur with probability . In turn, TWA workers transition to regular temporary employment with probability 1 − λ if The share of workers who experience this transition is .

4.3 Equilibrium

An equilibrium with intermediation is defined as follows.

Definition 1. A search equilibrium with intermediation is a vector that satisfies equations (3.5), (3.6), (4.3), (4.4), and (4.8)–(4.10).

The system of equilibrium equations is recursive and has a unique solution. As noted, equation (4.4) gives and (4.3) then determines U. Given U, (4.8) and (4.9) determine and , and (4.10) gives . Finally, (3.5) and (3.6) determine and .

Market a is active in equilibrium if the TWA’s marginal cost is low enough and the workers’ direct-hire prospects are suficiently poor (low U). Formally, by equation (4.10), , as the cost function c is convex. In what follows, we assume that these conditions hold. Proposition 3. A unique equilibrium with exists for suficiently high κ and ， where satisfies (4.5).

We now compare the transition rates of TWA workers and their direct-hire counterparts. If the share of type-0 (dead-end) vacancies in market a is suficiently high relative to market the former workers are more likely to reenter unemployment than the latter. They are also less likely to transition to regular temporary and permanent jobs (see Table B1). This scenario aligns with the evidence reported in Figure A1 (see also Givord and Wilner, 2015). In our benchmark model, average transition rates to permanent employment are lower for TWA workers because they are less likely to stay employed , and all workers who keep their jobs face the same promotion chances (regardless of productivity and type of contract).

Proposition 4. If , TWA workers are more likely on average to reenter unemployment than their direct-hire counterparts and less likely to transition to regular temporary and permanent jobs.

Consider the TWA’s efect on net job creation. Although one of the roles of the TWA is to speed up job creation, under the conditions in Proposition 4, job destruction is also greater when the TWA operates. It is direct to show that when direct-hire positions are hard to find, net job creation (period-2 employment) is higher nonetheless. However, when the chances of finding directhire jobs are good, intermediation can reduce net job creation if TWA contracts have a suficiently low average duration (compared to direct-hire temporary contracts). The Appendix shows that this is consistent with market a being active.

Proposition 5. If , total job destruction in period 2 is higher with intermediation than without it, whereas net job creation is higher if , and lower otherwise.

The sign of the average wage gap between direct-hire temporary workers and TWA workers is ambiguous in the model. This is intuitive. In equilibrium workers are indiferent between markets a and d, and the associated wage gap ought to reflect this indiference. Even though their employment spells are shorter , workers who participate in market a have easier access to employment. They also earn higher wages when they transition to permanent employment. Recall that, while direct-hire workers receive a piece-rate wage, the wage of poached TWA workers equals their output by Bertrand competition. If the benefits of working as a TWA worker outweigh the higher unemployment risk, these workers accept lower wages in equilibrium. The calibration in Section 7 replicates the positive average wage gap between direct-hire temporary workers and TWA workers observed in our dataset. Given that transitions to permanent jobs are infrequent, this positive gap indicates that the TWA’s positive job creation efect outweighs its negative job

destruction efect.

5 Welfare analysis

In this section, we characterize a constrained eficient allocation.

Trivially, an eficient TWA assignment involves placing the most productive workers in type-1 jobs in period 1. That is, worker types in are assigned to type-1 jobs, and lower types are assigned to type-0 jobs. In period 2, the former workers enter permanent employment with probability λ, and the latter reenter unemployment. The average output of TWA workers under the eficient assignment is

\[\begin{array}{r l} & {\bar {S} _ {a} (x ^ {*}) \equiv \int_ {x ^ {*}} ^ {1} \Bigl (y _ {1} (x) + (1 - \lambda) y _ {1} (x) + \lambda p y _ {1} (x) \Bigr) \gamma d G + \Bigl (1 - \gamma + \gamma G (x ^ {*}) \Bigr) y _ {0}} \\ & {\qquad = \gamma y _ {0} \Bigl (2 + \lambda (p - 1) \Bigr) \int_ {x ^ {*}} ^ {1} (1 + z x) d G + (1 - \gamma + \gamma G (x ^ {*})) y _ {0}.} \end{array}\tag{5.1}\]

The diference between equations (5.1) and (4.11) is that includes the output of all workers in period 2, while only includes the output of the workers that the TWA retains (total TWA fees). Hence,

At a constrained eficient allocation, total output net to job creation costs is maximal. That is, and solve the following problem:

\[\begin{array}{r l} {W ^ {*}} & {= \max _ {\theta , v _ {d}, v _ {a} \in \mathbf {R} _ {+} ^ {2} \times [ 0, 1 ]} q (\theta) \theta S _ {d} (1 - v _ {a}) + \bar {S} _ {a} (x ^ {*}) v _ {a} - \kappa v _ {d} - c (v _ {a})} \\ {s. t.} & \\ & {v _ {d} = \theta (1 - v _ {a}),} \end{array}\]

since . The first and second terms in the objective function give the total output of direct-hire and TWA workers, respectively. The third and fourth terms are the associated job creation costs. The optimal allocation satisfies (since . Substituting the constraint in the objective function yields

\[{W ^ {*}} {= \max _ {\theta , v _ {a} \in \mathbf {R} _ {+} ^ {2}} \theta (q (\theta) S _ {d} - \kappa) (1 - v _ {a}) + \bar {S} _ {a} (x ^ {*}) v _ {a} - c (v _ {a}).}\]

The first-order conditions with respect to and are, respectively,

\[q (\theta^ {*}) (1 - \eta (\theta^ {*})) S _ {d} = \kappa ,\tag{5.2}\]

\[\theta^ {*} (q (\theta^ {*}) S _ {d} - k) \geq \bar {S} _ {a} (x ^ {*}) - c ^ {\prime} (v _ {a} ^ {*}), \text { with equality if } v _ {a} ^ {*} > 0.\tag{5.3}\]

Equation (5.2) gives , and coincides with the equilibrium condition in (4.4), meaning that this margin is not distorted. We can then write (5.3) as

\[c ^ {\prime} (v _ {a} ^ {*}) + q (\theta^ {*}) \theta^ {*} \eta (\theta^ {*}) S _ {d} \geq \bar {S} _ {a} (x ^ {*}), \text { with equality if } v _ {a} ^ {*} > 0.\tag{5.4}\]

Since , the assumptions in Proposition 3 imply

\[c ^ {\prime} (0) + q (\theta^ {*}) \theta^ {*} \eta (\theta^ {*}) S _ {d} < S _ {a} (\hat {x}) < \bar {S} _ {a} (x ^ {*}).\]

Comparing (5.4) with the equilibrium condition in (4.10), it follows that , since c is convex. Thus the equilibrium allocation is ineficient.

Proposition 6. In equilibrium, job creation in market a is ineficiently low: . Moreover, if , the TWA’s assignment is ineficient and the average output of TWA workers is ineficiently low. In contrast, when the search process in market d is competitive, no distortions arise in this market:

The hold-up problem implies that the intermediary’s social benefit is greater than its private benefit (see also Ciapanna, 2011). The TWA’s failure to account for the impact of its assignment on the future output of poached workers results in ineficient assignments and insuficient TWA job creation. Crucially, distortions are more severe when the workers’ direct-hire employment prospects are poorer (U is lower).

Proposition 7. There exist values U and such that:

(i) then xˆ = 1, so all TWA workers are misallocated;

(ii) then falls with U , reducing the misallocation of TWA workers;

(iii) then , so the TWA’s assignment is eficient.

Additionally, if then fal ls with reducing the extensive margin distortion. If then does not depend on U.

TWA workers also experience distorted transitions as a result of their misallocation. In particular, the equilibrium features repeated TWA spells for worker types in , which do not occur at the optimal allocation (see the last row of Table B1). Furthermore, the likelihood of transitioning to regular temporary work decreases as misallocation becomes more severe. By contrast, transitions to permanent employment and unemployment are not distorted in our benchmark model¹⁵. In Section 7, we calibrate a model variant in which the probability of being promoted to a permanent job increases with the time the worker has been working at the firm (i.e., due to increased on-thejob learning and human capital accumulation). In this variant, the transition rate to permanent employment is distorted whenever a TWA worker is misallocated in period 1. In Section 7, we use simulations to illustrate this additional ineficiency.

Adding a clause in commercial contracts that requires firms that poach workers to pay a fee to the TWA would eliminate the distortions, thereby solving the hold-up problem.

Proposition 8. The constrained eficient allocation can be implemented as an equilibrium if commercial contracts specify an additional contingent fee equal to when a worker is poached, where is worker output in period 2.

5.1 Random search and Nash bargaining

Consider an alternative model where the wages of direct-hire workers are determined through generalized Nash bargaining after they are matched to firms. Denote the workers’ exogenous bargaining weight by . The definition of equilibrium is the same, except that replaces in equations (4.3)–(4.4). Since each match constitutes a bilateral monopoly (with a single firm/worker), the Hosios rule does not generally hold and the outcome in market d is ineficient (Pissarides, 2000).

Welfare is lower in this model, for two reasons. First, if the Hosios rule does not hold, job creation in market is ineficient . Second, the fact that search externalities are not internalized in this market reduces the workers’ payof, relaxing the participation constraint of

¹⁵This need not be the case with more than two periods.

TWA workers and strengthening the TWA’s monopsony power. This exacerbates the distortions in market a, where is now higher (Proposition 7). The comparative static results in the next section also hold in this alternative model.

6 Comparative statics

This section reports comparative statics results.

We first analyze how the model’s predictions difer across skill levels. To this aim, we assume that the temporary job market is segmented by occupational skill groups, with lower skills associated with lower overall productivity. To analyze how equilibrium outcomes difer among skill groups, we thus examine equilibria for diferent average productivity levels.

Consider a reduction in , which lowers output proportionally in all matches within the market. Due to this output reduction, workers have a harder time finding direct-hire jobs and earn lower wages at those jobs (θ and U fall). Proposition 9 shows that a higher share of TWA workers are misallocated in this scenario (xˆ increases¹⁶). The reasons are twofold. First, when workers are less productive, misallocation is less costly for the TWA (in terms of reduced fees). Second, some of the costs can be passed on to workers by reducing their wages because they have poorer outside options (lower reservation utilities). If is suficiently low, all TWA workers are misallocated. Above a threshold value for , intermediate (but not types are misallocated. In this range, the share of misallocated workers falls with . This intensive-margin distortion may or may not disappear as rises; in some cases, it will persist no matter how high is.

Proposition 9. There exists such that if Otherwise, . If then decreases with . There exists a scalar K such that for al l y₀ if . Otherwise, for y₀ suficiently high.

Proposition 10 describes how a reduction in afects other endogenous variables. It also shows that an increase in the firms’ vacancy posting cost, κ, has the same qualitative efects.

Proposition 10. y₀ decreases and/or κ increases, θ, U, and and xˆ increases when

¹⁶Note that changes in do not afect the optimal threshold in (4.5).

interior. The efect on and is generally ambiguous, but it is positive if

As y₀ falls, agency workers earn less. So do their direct-hire counterparts, whose wages are a piece-rate of their output. Yet falls more than . The efect on TWA job creation is generally ambiguous, as two opposing forces are at work. The reduction in lowers the TWA’s fees and thus its marginal benefit, while the fall in reduces its marginal cost. The second efect dominates if permanent ofers are not very likely (low λ) and productivity diferences within a skill group are small (low z). In this scenario, TWA jobs are more prevalent among less skilled workers.

When the proportion of type-1 jobs falls, average worker productivity is also reduced and workers face higher unemployment risk. Specifically, if and fall proportionally, all workers are more likely to lose their jobs after the first period. Yet the efect is stronger for direct-hire temporary workers (see Table B1). Although transitions to unemployment are exogenous in our environment, Section 7 presents evidence consistent with these implications. The efects on market d are analogous to those described in Proposition 10 (θ and U fall due to a reduction in . The efects on market a are generally ambiguous, but we explore them numerically in Section 7. As we will show, our simulations are in line with the theoretical findings in Proposition 10.

We have already noted that there is greater misallocation when transitions to permanent employment are uncommon. The equilibrium efects of a reduction in λ are similar to those in Proposition 10. The efect on TWA job creation is generally ambiguous, but it is positive if the misallocation of agency workers is not too severe.

Proposition 11. If λ falls, θ, U, and and xˆ increases when interior. The efect on and is ambiguous in general, but it is positive if and xˆ is close to .

As expected, a reduction in the TWA’s operating costs (e.g. due to the deregulation or innovations in the intermediation technology) increases TWA employment and reduces regular temporary employment. The other variables are not afected in this simple model.

Proposition 12. A downward shift in raises and and reduces but does not afect other variables.

7 Numerical calibration

In this section, we introduce a model variation to examine the impact of diferential promotion probabilities for workers joining a firm in diferent periods. Specifically, we assume that workers who join in period 1 are twice as likely to be promoted to permanent positions compared to those who join in period 2 (whose firm-specific human capital accumulation is lower). That is, we set the associated promotion probabilities equal to λ and , respectively, where is an exogenous parameter. As shown in Table B2, all average transition rates are now distorted when the threshold xˆ deviates from its optimal value, except for those leading to unemployment. We will further examine these distortions through simulations.

The agency’s cost function is quadratic, , and the distribution G is uniform. We assume an urn-ball matching process in the direct-hire market:

\[q (\theta) = \frac {1 - e ^ {- \theta}}{\theta}.\]

This micro-founded process captures the notion of directed job search (Peters, 1991). The worker’s share of the surplus in direct-hire jobs is then given by , where is increasing, with lim and . This means that direct-hire workers receive a larger share of the surplus when market d is tighter.

We select the model parameters to replicate the characteristics of the Spanish TWA sector, focusing on male temporary workers with low and medium skills, the primary demographic group employed by this sector (Carrasco et al., 2022). Our analysis uses data from 2005-2019 obtained from the Spanish Social Security registers (MCVL).

Table 1 presents the benchmark parameter values used in our numerical simulations. We begin by setting the values of parameters externally. We normalize the value of to 1. Let A, D, P, and U denote temporary agency and direct-hire employment, permanent employment, and unemployment, respectively. We set to match the average transition rate into unemployment for agency workers (AU), . This rate is 0.075 in the data, resulting in a relatively high value of . We set to match the average wage gap between permanent and temporary direct-hire

Table 1: Calibration of Model Parameters

workers, which in our sample is 34.8%.

We internally calibrate the remaining parameters to jointly match the following data moments in the model’s equilibrium: the TWA sector’s share in job creation, the wage gap between direct-hire temporary workers and agency workers, and the corresponding transition rates for both categories of workers (DU, DP, AP, and AD)¹⁷. About 20% of newly created jobs are agency jobs. The average wages of agency workers are approximately 5.5% lower than those of direct-hire temporary workers. In the model, the wage gap is calculated as the diference between the TWA wage and the average direct-hire wage in period 1. Regarding transition rates, since the empirical unemployment hazard is higher for TWA workers, we need that . Table 1 confirms the model’s ability to reproduce the observed average transition rates. The observed rates are consistent with the theoretical results of our benchmark model (Proposition 4). Specifically, transitions to direct-hire permanent (temporary) employment are less likely for TWA workers than for direct-hire temporary workers.

Table 2 presents the key equilibrium outcomes and welfare results of our model simulation. Regarding job creation, 18.8% of workers secure their initial employment through the TWA, while the rest search for jobs directly. In equilibrium, 76.6% of workers find jobs in the direct-hire market, where the job-finding probability is 0.94. Hence, resorting to the TWA increases the job-finding rate by 6 percentage points in this model economy. Nonetheless, in line with Proposition 6, the optimal allocation comprises a higher proportion of TWA jobs (57.2%), a lower proportion of direct-hire jobs (40.4%), and higher job creation, with 97.6% of workers securing employment (Column 3). Comparing the equilibrium outcomes with and without intermediation is also important (Columns 1 and 2). While more direct-hire jobs would be available without the TWA, fewer workers would find employment (94.3 vs 95.4%). Recall that the TWA’s decisions do not afect the outcomes of direct-hire workers. The latter have identical equilibrium wages and job-finding rates with and without the TWA.

¹⁷As noted, DA rates are low in the data and zero in the model.

Table 2: Equilibrium and Welfare

Regarding job destruction, 5.2% of the workers reenter unemployment after the first period in equilibrium. Because regular jobs are more common, they contribute more to job destruction even if TWA jobs have higher job destruction rates. Notably, job destruction reaches a higher level (6.3%) in the optimal allocation because of the increased prevalence of TWA jobs (recall that transition rates to unemployment are the same in both allocations). Despite this, net job creation is higher in the optimum, as the positive impact on job creation prevails. The optimal net job creation rate is 91.3%, compared to 90.2% in equilibrium and 89.6% without the TWA. Thus the

TWA’s contribution to net job creation at equilibrium is only 0.6%, but almost triples to 1.7% in the optimal allocation.

Consider transitions to diferent employment states. The repeated TWA spells that occur in equilibrium do not take place at the optimal allocation, which makes transitions to permanent employment more likely for agency workers.

The distortions in the TWA sector reduce welfare (output net of job creation costs) by approximately 5.5%. Specifically, the TWA sector generates a welfare gain of 1.7 percentage points compared with a scenario without intermediation. If the distortions in this sector are eliminated, the welfare gains from intermediation more than quadruple to 7.26%.

Figure 1 depicts the expected utility that diferent worker types obtain in the direct-hire and TWA markets. As anticipated, the presence of the TWA sector benefits some worker types while harming others. The diference between the two curves represents the gain or loss each type experiences due to TWA intermediation. Notably, a small group of top-type workers (0.942, 1] consistently benefit from intermediation due to eficient job matching and increased bargaining power.

Worker types in (0.047, 0.942), who are assigned ineficiently in the TWA sector in the first period, face a more nuanced situation. The most productive among them do worse in this sector, whereas the less productive benefit from TWA. This is intuitive. All these types earn the same TWA wage, regardless of productivity (unlike their direct-hire counterparts). This leads to a crosssubsidization efect, where more productive workers subsidize less productive ones.

Low-type workers in (0, 0.047) also fare worse in the TWA sector, where they are always assigned to dead-end jobs and face a higher risk of termination. Additionally, part of their production surplus is captured by the TWA as fees. By contrast, the least productive (type-0) workers benefit from the TWA. Although they are always fired after one period, they earn the same wage as other TWA workers who are more productive, efectively receiving a subsidy from them.

Figure 1: Expected utility of a type-x worker: direct-hire vs TWA market

7.1 Diferences by skills and testable implications

Figures 2 and 3 illustrate the efect of a proportional change in and , which we interpret as a variation in occupational skills. Recall that the efect of this change is ambiguous in our benchmark model.

Lower values of and indicate lower average worker productivity and increased unemployment risk (as the share of dead-end jobs in the TWA and direct-hire markets is higher). As noted in Section 6, the gap in unemployment hazard rates between TWA and direct-hire temporary workers (AU/DU hazard ratio) narrows, meaning that the increase in the hazard is smaller for TWA workers (bottom left-hand panel of Figure 3). Although our model assumes exogenous hazard rates to unemployment, these findings are consistent with the empirical evidence we present below.

In the calibrated model, the fraction of TWA workers who are ineficiently assigned to type-0 jobs raises as the share of such jobs increases in both markets (i.e., lower and . This fraction is given by , so it equals one when is suficiently low (see to the upper left panel of Figure 2). As expected, due to the reduction in average job productivity, welfare falls in both the optimal and equilibrium allocations (see the lower right panel). Yet it is noteworthy that, due to greater misallocation in the TWA sector, the percentage of welfare loss in equilibrium is higher when workers have lower skills. So is the percentage diference in the TWA’s contribution to net job creation between the equilibrium and optimal allocations, relative to an economy without intermediation (lower left panel).

In sum, through this exercise, the model provides three testable implications regarding the relationship across skill categories between (i) the share of jobs created by the TWA sector, (ii) transitions out of temporary jobs, and (iii) the wage gap between direct-hire temps and agency workers. Below we validate our model predictions using Spanish data from the MCVL. We use the information on whether individuals work through a TWA and their assigned occupational codes as proxies for job skill requirements. For each skill category, we calculate the share of jobs created through a TWA and the average wage gap between direct-hire temps and agency workers. Moreover, using the panel dimension of the data, we compute average transition rates from temporary employment to diferent labor market states for both types of workers within each skill category. While our occupational classification difers from the EU-LFS, it maintains a hierarchical structure. Specifically, workers are grouped into five primary occupational categories based on their entry-level wages¹⁸.

Skills and TWA share According to the calibrated model, the TWA sector creates a higher share of jobs when workers are less skilled (see Figure 2, lower left panel). Given that these workers have lower chances of finding employment directly, more people turn to the TWA is larger because θ is lower). As a result, the number of agency (direct-hire) jobs created increases (falls), pushing up the TWA’s employment share (lower left panel of Figure 2). As shown in Column (1) of Table 3, the model prediction aligns with the data. Evidence from the European Labour Force Survey further supports this prediction (see Table A2 in the Appendix). In most European countries, the proportion of TWA employment is highest among low-skilled workers and generally decreases as job skill requirements increase.

¹⁸Very high skilled: 1 - Engineers, college graduates, and senior managers; High-skilled: Technical engineers and graduate assistants, 3 - Administrative and technical managers; Medium-high skilled: Non-graduate assistants, 5 - Administrative oficers, 6 - Subordinates; Medium-low skilled: Administrative assistants, First, and second class oficers, 9 - Third class oficers; Low-skilled: Labourers.

Notes: The figure shows the changes in each key variable after a proportional change in and when remains the same as in the baseline calibration (1.057). The vertical dashed line corresponds to the value of in the baseline calibration (0.925). The net job creation (NJC) gap is % diference in NJC relative to an economy without intermediation. The welfare (W) loss is computed as % diference in W relative to the optimal allocation.

Skills and transition hazards We have already noted that while unemployment hazards decrease as workers’ skills increase, the positive gap in these hazards between TWA and direct-hire temporary workers widens in the model. Columns 1 and 2 of Table A3 compare job destruction rates, defined as the proportion of workers transitioning into unemployment, across the five primary occupational groups in our sample. Across all occupational groups, TWA workers consistently experience higher job destruction rates than direct hires. Moreover, these rates generally decrease as skill levels increase. However, the ratio of the unemployment hazards between TWA workers and direct-hire temps is more pronounced in the two highest-skill groups. Specifically, this ratio is approximately 1.4 for the two lower-skill groups, whereas it rises to 1.5 and 1.8 for the two higher-skill groups (Column 2 of Table 3).¹⁹

The model also predicts that workers with higher skills are more likely to transition to permanent employment (upper left panel of Figure 3). In the data, transition rates to permanent employment among TWA increase with skill levels, mirroring the model’s predictions. A similar trend is observed for direct-hire temporary workers up to medium-low skill levels (see Column 4 of Table A3). Importantly, as predicted by the model, the ratio of the probability transition between TWA workers and direct-hire counterparts increases with skill levels (upper left-hand panel of Figure 3). Specifically, Column 3 of Table 3 shows that the ratio of transition rates (AP/DP) is approximately 0.6 for very low-skilled workers and increases to 2.7 for very high-skilled workers. This is intuitive since the gains from screening should be larger for the latter workers.

Finally, the probability of TWA reemployment is higher for more skilled workers in the model (bottom right-hand panel of Figure 3). This result is also consistent with the data in Table 3.

Skills and the wage gap More skilled workers earn higher wages in the model because they are more productive (see the top right-hand panel of Figure 2). Critically, however, the positive wage gap between direct-hire temps and TWA workers widens as and rise.

There are opposing forces at play, which afect the wage gap. On the one hand, the surplus from job creation increases in the direct-hire market due to higher average job quality, so workers have higher chances of finding jobs directly. At the same time, the positive gap in unemployment risk between TWA jobs and direct-hire jobs widens. This means that the advantages of faster job placement through the TWA become less important, while the downside of higher unemployment risk becomes more significant. Both changes make the TWA market less attractive, tightening the participation constraint of agency workers and reducing the wage gap. On the other hand, when workers have higher skills, TWA work can be more valuable as a stepping stone to permanent positions. Misallocation of TWA workers is less severe in this case, making these transitions relatively more likely for them. Furthermore, when TWA workers transition to permanent jobs, their wage increase is larger (because it reflects their entire output, whereas directly hired temps receive a share of their output). These improved prospects make TWA work more attractive, relaxing the participation constraint of TWA workers and increasing the wage gap.

¹⁹A similar analysis, conditioned on covariates such as gender, years of education, and age, produces comparable results. Specifically, the job destruction rate remains relatively stable across the three lowest occupational categories. In contrast, for the two highest-skilled groups, including controls narrows the gap in job destruction rates between TWA and direct-hire temporary jobs. Nevertheless, the hazard ratio (AU/DU) remains higher for these higher-skilled groups compared to the lower end of the skills distribution.

According to our simulations, as workers’ skills increase, the benefits of transitioning to permanent employment outweigh the downsides of temporary agency work. Thus the wages that satisfy the participation constraint of TWA workers rise more slowly than those of direct-hire temps, and the wage gap widens. This prediction is in line with the data. Column (4) of Table 3 shows that the observed wage gap is more pronounced in high-skilled occupations, with direct-hire temps earning, on average, 20% more. The gap narrows to around 10% for medium-skilled jobs and is almost negligible for low-skilled jobs.

Notes: The figure shows the change in each transition after a proportional change in and when remains the same as in the baseline calibration (1.057). The vertical dashed line corresponds to the value of in the baseline calibration (0.925). The dashed black line in the left panels displays the ratio of each transition probability between TWA workers and direct-hire temporary workers, with values corresponding to the right-hand axis.

Table 3: Data for model predictions

Notes: Sample of workers aged 16 to 55 for the years 2015-2019. Source: MCVL

7.2 The role of screening

A relevant question is whether the observed diferences in transition rates of agency workers and direct-hire temps can be replicated in a scenario in which the TWA enhances matchmaking between workers and firms, but does not screen workers. In the absence of screening, the TWA’s assignment is necessarily random. In this case, the transition rates of TWA workers are not distorted; i.e., they are the same in the equilibrium and optimal allocations. This is because the planner and the TWA are constrained by the same random matching technology.

Whereas transition rates to unemployment are as before, there are no repeated TWA spells in this setting. Jobs with output are destroyed after one period, while jobs with higher output always lead to a transition to direct-hire employment (after the TWA and the firm compete for the worker). Transition rates from TWA to permanent (and temporary) direct-hire jobs are obtained by replacing with in the expressions that describe the corresponding rates for direct-hire workers (in Table B2). Thus the gap in the transition rate to as described by the hazard ratio AP/DP, equals . This implies that a proportional change in and does not afect this gap. The same is true about transitions to D. Consequently, this alternative model cannot explain the observed diferences in transition rates between TWA workers and direct-hire temps.

8 Final remarks

Despite their rapid growth over the past decades in many developed economies, temporary agency employment arrangements have received limited theoretical attention. We view this paper as a useful building block for additional research.

A TWA has two major functions in our general equilibrium model. The first is to screen workers and certify assignment quality, similarly to Autor (2001). The second is to facilitate matching between workers and firms relative to the direct search process (see also Bergeaud et al., 2024). Accordingly, individuals who do not use the TWA have lower chances of finding a job, as documented in the empirical literature (e.g. García-Pérez and Muñoz-Bullón, 2005a; Storrie, 2007; Autor, 2009; Voss et al., 2013). In line with the data, we also assume that TWA contracts have a lower average duration than direct-hire temporary contracts (abstracting away from the factors that generate these diferences).

We deviate from existing work by considering a TWA that holds monopsony power over workers while charging competitive fees to firms equal to the marginal product of labor. The rationale for this assumption is that, as TWAs specialize in recruitment, they are able to achieve economies of scale that give them market power. Moreover, unlike firms that hire workers directly, TWAs may extract more surplus from their labor pool, where workers compete directly for jobs. Recent empiri cal work finds evidence of monopsonistic behavior by TWAs in the low-skilled labor market (Drenik et al., 2023; Carrasco et al., 2024). In our model, such behavior can lead to labor misallocation. We analyze the implications of these distortions on worker outcomes, employment, and welfare, and examine how TWA jobs diferentially afect diferent skill groups. Furthermore, we calibrate the model and use simulations to derive several testable predictions backed up by data from Spain. Interestingly, some key empirical facts cannot be rationalized in a counterfactual scenario where the TWA does not screen workers.

Our model is highly stylized and therefore easy to interpret, though this comes at the cost of imposing strong simplifying assumptions. First, there are two periods, though additional distortions may arise with a longer horizon. Extending the model in this direction is key to analyzing how TWA employment afects the worker’s labor market prospects in the medium and long run. Second, we assume that workers are ex-ante symmetric and equally likely to work in TWA jobs. However, unobserved worker heterogeneity is relevant to explain the high persistence of this type of employment in the data (Carrasco et al., 2022). Workers who self-selected into TWA employment may be more averse to unemployment, resulting in a lower labor supply elasticity that reinforces TWA monopsonistic behavior. While we assume a single agency, the degree of monopsonistic power in the TWA industry and its efects on equilibrium outcomes should be investigated. Third, endogenizing firm behavior is important. An ongoing debate exists on the reasons why firms employ TWA workers and the impact that this has on productivity (e.g. Autor, 2001, 2003; Hirsch and Mueller, 2012; Bergeaud et al., 2024; Beneito et al., 2024). Extending the model to incorporate some of the above features will help identify winners and losers, quantify welfare efects, and examine the efects of regulations on TWA workers’ outcomes. We leave these issues for future research.

References

1. Abraham, K. G. and Taylor, S. K. (1996). Firms’ Use of Outside Contractors: Theory and Evidence. Journal of Labor Economics, 14(3):394–424.
2. Acemoglu, D. and Shimer, R. (1999). Holdups and Eficiency with Search Frictions. International Economic Review, 40(4):827–49.
3. Addison, J. T., Teixeira, P., Grunau, P., and Bellmann, L. (2019). Worker Representation and Temporary Employment in Germany: The Deployment and Extent of Fixed-Term Contracts and Temporary Agency Work. Journal of Participation and Employee Ownership, 2(1):24–26.
4. Albrecht, J. and Vroman, S. (2002). A Matching Model with Endogenous Skill Requirements. International Economic Review, 43(1):283–305.
5. Autor, D. H. (2001). Why Do Temporary Help Firms Provide Free General Skills Training? The Quarterly Journal of Economics, 116(4):1409–1448.
6. Autor, D. H. (2003). Outsourcing at Will: The Contribution of Unjust Dismissal Doctrine to the Growth of Employment Outsourcing. Journal of Labor Economics, 21(1):1–42.
7. Autor, D. H. (2009). Studies of Labor Market Intermediation. NBER Books. National Bureau of Economic Research.
8. Beneito, P., Garcia-Vega, M., Vicente-Chirivella, O., and Wilemme, G. (2024). Robots and firms’ labour search: The role of temporary work agencies. Technical report, University of Nottingham, GEP.
9. Bergeaud, A., Cahuc, P., Malgouyres, C., Signorelli, S., and Zuber, T. (2024). The wage of temporary agency workers. CEP Discussion Papers dp2014, Centre for Economic Performance, LSE.
10. Biglaiser, G. (1993). Middlemen as Experts. RAND Journal of Economics, 24(2):212–223.
11. Bilal, A. and Lhuillier, H. (2021). Outsourcing, Inequality and Aggregate Output. NBER Working Paper Series 29348, National Bureau of Economic Research.
12. Böheim, R. and Cardoso, A. R. (2009). Temporary Help Services Employment in Portugal, 1995- 2000. In Studies of Labor Market Intermediation, NBER Chapters, pages 309–334. National Bureau of Economic Research, Inc.
13. Booth, A. L., Francesconi, M., and Frank, J. (2002). Temporary Jobs: Stepping Stones Or Dead Ends? Economic Journal, 112(480):189–213.
14. Bostanci, G. (2022). Productivity Gains from Labor Outsourcing: The Role of Trade Secrets. Mimeo.
15. Carrasco, R., Gálvez-Iniesta, I., and Jerez, B. (2022). Do Temporary Help Agencies Help? Temporary Employment Transitions for Low-Skilled Workers. Working papers in Economics 34756, Universidad Carlos III de Madrid.
16. Carrasco, R., Gálvez-Iniesta, I., and Jerez, B. (2024). Do Temporary Help Agencies Help? Temporary Employment Transitions for Low-Skilled Workers. Labour Economics, 90:102586.
17. Ciapanna, E. (2011). Directed Matching with Endogenous Markov Probability: Clients or Competitors? RAND Journal of Economics, 42(1):92–120.
18. Countouris, N., Deakin, S., Freedland, M., Koukiadaki, A., and Prassl, J. (2016). Report on Temporary Employment Agencies and Temporary Agency Work. Technical report, International Labor Ofice: Geneva.
19. Drenik, A., Jager, S., Plotkin, P., and Schoefer, B. (2023). Paying Outsourced Labor: Direct Evidence from Linked Temp Agency-Worker-Client Data. The Review of Economics and Statistics, 105(1):206–216.
20. Filomena, M. and Picchio, M. (2022). Are Temporary Jobs Stepping Stones or Dead Ends? A Meta-analytical Review of the Literature. International Journal of Manpower, 43(9):60–74.
21. García-Pérez, J. I. and Muñoz-Bullón, F. (2005a). Are Temporary Help Agencies Changing Mobility Patterns in the Spanish Labour Market? Spanish Economic Review, 7(1):43–65.
22. García-Pérez, J. I. and Muñoz-Bullón, F. (2005b). Temporary Help Agencies and Occupational Mobility. Oxford Bulletin of Economics and Statistics, 67(2):163–180.
23. Gehrig, T. (1993). Intermediation in Search Markets. Journal of Economics & Management Strategy, 2(1):97–120.
24. Givord, P. and Wilner, L. (2015). When Does the Stepping-Stone Work? Fixed-Term Contracts Versus Temporary Agency Work in Changing Economic Conditions. Journal of Applied Econometrics, 30(5):787–805.
25. Hirsch, B. and Mueller, S. (2012). The Productivity Efect of Temporary Agency Work: Evidence from German Panel Data. Economic Journal, 122(562):216–235.
26. Hosios, A. J. (1990). On the Eficiency of Matching and Related Models of Search and Unemployment. Review of Economic Studies, 57(2):279–98.
27. Houseman, S. N., Kalleberg, A. L., and Erickcek, G. A. (2003). The Role of Temporary Agency Employment in Tight Labor Markets. ILR Review, 57(1):105–127.
28. Houseman, S. N. and Polivka, A. E. (2000). The Implications of Flexible Stafing Arrangements for Job Stability. In On the Job: Is Long-Term Employment a Thing of the Past?, pages 427–462. New York: Russell Sage Foundation.
29. Katz, L. F. and Krueger, A. B. (1999). The High-Pressure U.S. Labor Market of the 1990s. Brookings Papers on Economic Activity, 30(1):1–88.
30. Katz, L. F. and Krueger, A. B. (2019). The Rise and Nature of Alternative Work Arrangements in the United States, 1995–2015. ILR Review, 72(2):382–416.
31. Moen, E. R. (1997). Competitive search equilibrium. Journal of Political Economy, 105:385–411.
32. Neugart, M. and Storrie, D. (2002). Temporary Work Agencies and Equilibrium Unemployment. Working Papers in Economics 83, University of Gothenburg.
33. Peters, M. (1991). Ex ante price ofers in matching games non-steady states. Econometrica, pages 1425–1454.
34. Pissarides, C. A. (2000). Equilibrium Unemployment Theory, 2nd Edition. MIT Press Books. The MIT Press.
35. Roy, A. D. (1951). Some Thoughts On The Distribution Of Earnings. Oxford Economic Papers, 3(2):135–146.
36. Rubinstein, A. and Wolinsky, A. (1987). Middlemen. The Quarterly Journal of Economics, 102(3):581–593.
37. Sattinger, M. (1996). Search and Discrimination. Labour Economics, 3(2):143–167.
38. Sattinger, M. (2003). A Search Version of the Roy Model. Discussion Papers 03-08, University at Albany, SUNY, Department of Economics.
39. Spitze, S. (2022). The Equilibrium Efects of Domestic Outsourcing. Mimeo.
40. Storrie, D. (2007). Temporary Agency Work in the European Union‚ÄîEconomic Rationale and Equal Treatment. Flexibility and Stability in Working Life, 1:103–122.
41. Voss, E., Vitols, K., Farvaque, N., Broughton, A., Behling, F., Dota, F., Leonardi, S., and Naedenoen, F. (2013). The Role of Temporary Agency Work and Labour Market Transitions in Europe: Institutional Frameworks, Empirical Evidence, Good Practice and the Impact of Social Dialogue. Technical report, Hamburg: Eurociett/UNI Europa Project.
42. World Employment Confederation (2021). Economic Report 2021. Technical report.
43. Yavas, A. (1994). Middlemen in Bilateral Search Markets. Journal of Labor Economics, 12(3):406– 429.

Appendices

A Data Appendix

Table A1: Composition of workers by contract type, 2019

Notes: Sample of salaried workers aged 16 to 65 (295,293 workers). We pool all contracts signed in 2019. Source: MCVL.

Panel a. To permanent contract Panel c. To direct-hire temp contract

Panel b. To unemployment Panel d. To THA contract

Notes: Sample of low-educated male workers aged 20-44. Source: MCVL, 2005-2019.

Table A2: TWA share in salaried employment

Notes: We pool all salaried workers aged 16 to 65 for 2011-2019, and calculate the share of workers employed by temporary work agencies in each major occupational group across diferent countries. Occupations are categorized into four main groups based on skill requirements, using 1-digit level information (eight codes): Very high skilled: 1-Managers and 2-Professionals; High skilled: 3 Technicians and Associate Professionals Medium-skilled: 4 - Clerical support, 5 - Service and sales, 6 - skilled agricultural, forestry and fishery, 7 - craft and related trades, 8 - Plant and Machinery Operators and assemblers; Low-skilled: 9 - Elementary occupations. See https://ilostat.ilo.org/resources/concepts-and-definitions/classification-occupation/. Source: European Labour Force Survey.

Table A3: Data for model predictions

Notes: Sample of workers aged 16 to 55 for the years 2015-2019. We calculate the ratio of the number of agency and direct-hire temporary contracts ending in U, P, D, and A to the total number of contracts of each type. Source: MCVL.

B Theory Appendix

B.1 Worker transition rates

Table B1: Benchmark economy

Table B2: Model economy with variable and

Notes: A, D and P stand for TWA, direct-hire temporary and permanent jobs; U stands for unemployment. Both tables display transition rates in period 2 for A and D contracts at the equilibrium and optimal allocations

B.2 Proofs

Proof of Proposition 3. We have assumed that Assume where Since is non-decreasing, and is continuous and decreasing with lim and lim_θ→ then there exist unique values of θ and U that solve (4.3) and (4.4). Suppose , and take the case where . Combining (4.8) and (4.9), and integrating by parts, determines the value of xˆ as a function of U:

\[\begin{array}{r l} & {\frac {U}{\gamma y _ {0}} = [ 1 - \lambda (1 + z \hat {x}) ] \left(\frac {1}{\gamma (1 - \lambda)} + G (\hat {x}) - G (x ^ {*})\right) + \lambda p [ 1 + z - (1 + z x ^ {*}) G (x ^ {*}) - z \int_ {x ^ {*}} ^ {1} G (x) d x ]} \\ & {+ (1 - \lambda) [ 1 + z - (1 + z \hat {x}) G (\hat {x}) - z \int_ {\hat {x}} ^ {1} G (x) d x ].} \end{array}\tag{B.1}\]

This value is unique because the right-hand side of (B.1) is a decreasing function of i.e., the derivative of this term with respect to xˆ is

\[- \lambda z \left(\frac {1}{\gamma (1 - \lambda)} + G (\hat {x}) - G (x ^ {*})\right) - z \hat {x} g (\hat {x}) < 0.\tag{B.2}\]

The same is (trivially) true for other equilibrium variables, whether or not is interior. It remains to check that agency profits are non-negative for satisfying (4.10). Since is increasing, this is so provided . Since the eficient assignment is always feasible for the agency, , and it sufices to show that . As κ approaches implies that θ and U go to zero. Hence, by continuity, if then for κ close to □

Proposition 4 follows trivially from the expressions in Table B1.

Proof of Proposition 5. Total job creation in is in equilibrium. Without market , it would be lower: . Total job destruction in is + with intermediation, and without it. Since , the gap between the last two expressions is positive (job destruction is higher with intermediation):

\[v _ {a} (1 - \pi_ {a} - q (\theta) (\theta) (1 - \pi_ {d} \gamma)) > 0.\]

Finally, job creation net of job destruction (period-2 employment) is with intermediation, and without it, the diference being

The suficient condition in Proposition 5 holds trivially when , since θ does not depend on . Let us now show that a value of close to zero is compatible with the conditions in Proposition 3, so . If , all agency jobs are of type 0, and trivially. Denote the associated agency surplus by , and the threshold for κ implied by the argument in the proof of Proposition 3 by . Then and . For marginally higher values of , the agency surplus is necessarily higher: . The argument in the proof of Proposition 3 implies that the bound on denoted by , is now less stringent. It thus follows that and , where ϵ is suficiently low. Given that increases when κ is lower, the suficient condition in Proposition 5 then holds for all and ϵ suficiently low. □

Proof of Proposition 7. The value of xˆ that solves (B.1) increases when U falls, by (B.2). Thus the statement about the intensive margin follows from (4.8). Consider the extensive margin. Given that , (4.3) and (4.10) imply

\[c ^ {\prime} (v _ {a}) = S _ {a} (\hat {x}) - q (\theta^ {*}) \theta^ {*} \eta (\theta^ {*}) S _ {d}.\]

Hence, by (5.4),

\[c ^ {\prime} (v _ {a} ^ {*}) - c (v _ {a}) = \bar {S} _ {a} (x ^ {*}) - S _ {a} (\hat {x}).\]

If then does not afect . If , however, increases when U falls because xˆ increases. Thus is higher, and, since c is convex, so is □

Proof of Proposition 8. If a worker of type is assigned to a type-1 job in , the agency’s expected total fee is now

\[y _ {1} (x) - w _ {a} + (1 - \lambda) t _ {1} (x),\tag{B.3}\]

where is the amount charged to a user firm that poaches the worker. If (B.3) always exceeds the left-hand side of (4.6). □

Proof of Proposition 9. Combining (4.2) and (4.3), and rearranging, yields

\[\frac {U}{\gamma y _ {0}} = q (\theta) \theta \eta (\theta) \left(\frac {1}{\gamma} + \pi_ {d} [ (2 + (p - 1) \lambda) (z (1 - \int_ {0} ^ {1} G (x) d x) + 1) - 1 ]\right),\tag{B.4}\]

where θ is determined by (4.4). Suppose . The higher θ, the higher , and the lower xˆ, by (B.1)–(B.4). Since lim , there then exists θ such that for For large (rather than low) values of there are two possibilities. Define . Let N denote the right-hand side of (B.1) when

\[N = (1 - \lambda (1 + z x ^ {*})) \left(\frac {1}{\gamma (1 - \lambda)}\right) + (\lambda (p - 1) - 1) (1 + z - (1 + z x ^ {*}) G (x ^ {*}) - z \int_ {x ^ {*}} ^ {1} G (x) d x).\]

If lim , there then exists such that for . Otherwise, for all . Hence, . The result then follows as (4.2) and (4.4) imply that increases with , and goes to zero/infinity when does. □

Proof of Proposition 10. The previous proof shows that θ falls when falls. Since and fall, by (4.3) and (B.4) so do U and . There are two possibilities. First, is unafected by the change in . Since falls, by (4.9) so does . The THA’s profit maximization condition in (4.10) can be written as

\[\begin{array}{l} \frac {c ^ {\prime} (v _ {a}) + U}{\gamma y _ {0}} = \int_ {x ^ {*}} ^ {1} (1 + z x) d G + \left(\frac {1}{\gamma} - 1 + G (\hat {x})\right) \\ - \lambda \left(G (\hat {x}) - G (x ^ {*}) + z [ \hat {x} G (\hat {x}) - x ^ {*} G (x ^ {*}) ] - \int_ {x ^ {*}} ^ {\hat {x}} z G (x) d x\right), \end{array}\tag{B.5}\]

after integrating by parts. Since U falls by more than γy₀, (B.5) implies that and thus increase. By (3.6), falls. The second possibility is that . By Proposition 9, xˆ rises. By (4.8), so does . The derivative of the right-hand side of (B.5) with respect to xˆ is . So, if , right-hand side of (B.5) increases, and (again) so does . When k increases, the proof is similar (and simpler). □

Proof of Proposition 11. A similar argument to that in the previous proof implies that θ, U , and fall if λ falls. When , the positive efect on xˆ also follows from similar arguments. Given this, (4.8) implies that falls. Consider the last statement in the proposition. Suppose is unafected by the change in the case where is again similar. The derivative of the right-hand side of (B.5) with respect to is

\[- [ G (\hat {x}) - G (x ^ {*}) + z (\hat {x} G (\hat {x}) - x ^ {*} (G (x ^ {*})) - \int_ {x ^ {*}} ^ {\hat {x}} z G (x) d x ] < 0.\]

If xˆ is close to , this term negligible, so the right-hand side of (B.5) is essentially unchanged, and the same argument in the previous proof implies that increases. □