Staffing agencies and in-house bargaining

This paper asks whether a labor market with search-and-matching frictions and firms producing under decreasing returns to labor is better characterized by in-house hiring with intra-firm wage bargaining (Stole-Zwiebel) or by an alternative arrangement in which intermediaries — “staffing agencies” — search for and employ workers and then rent them to producing firms on a frictionless, perfectly competitive market. The paper’s second and central question is what happens when firms can choose their optimal combination of the two arrangements simultaneously.

The model is static. There are Z homogeneous firms with production function F(n) satisfying F’’(n) < 0, N homogeneous workers, and a standard concave constant-returns-to-scale matching function M = m(V, N). Firms can post vacancies, workers search, and Nash bargaining with worker bargaining weight β determines wages. The analysis is conducted with fully general production and matching functions throughout, deviating to specific functional forms (Cobb-Douglas matching, power production function F(n) = An^α) only when needed to illustrate a particular efficiency result. All main results hold for both directed and random search.

Comparing the two polar arrangements (Theorem 3.1). When all hiring is in-house (Stole-Zwiebel), equilibrium firm size n^SZ, aggregate employment Zn^SZ, labor market tightness θ^SZ, and the equilibrium wage w^SZ are all strictly higher than their counterparts under full staffing-agency employment (n^SA, Zn^SA, θ^SA, w^SA). The mechanism is that under in-house hiring with decreasing returns, a worker’s threat to leave raises the marginal product — and hence the wage — of remaining workers, giving workers additional bargaining leverage. Firms respond by over-employing in-house hires to dilute each worker’s marginal product and thus moderate wages. This over-employment raises vacancy posting and tightness, which in general equilibrium bids up wages despite each firm’s individual wage-moderation motive.

Efficiency (Theorem 3.2). Under the standard Hosios condition — worker bargaining weight β equals the elasticity η(θ) of the matching function with respect to vacancies — the staffing-agency equilibrium achieves the social planner’s optimum (θ^SA = θ*), while the in-house equilibrium posts strictly too many vacancies (θ^SZ > θ^SA = θ*). The in-house arrangement can be optimal for some β > η when workers’ bargaining power is sufficiently high (Theorem 3.3, proved for Cobb-Douglas matching and power production function), because the over-employment incentive then counteracts the externality from underprovision of vacancies.

The main result: staffing agencies dominate (Theorem 3.4). When firms choose their profit-maximizing combination of in-house hires n^SZ and rented staffers n^SA, the unique equilibrium has n^SZ = 0 and n^SA > 0 — firms use only staffers. The key is Lemma 3.1: renting one additional staffer reduces the wage paid to in-house workers by more than does hiring one additional in-house worker (formally, ∂w^SZ/∂n^SA < ∂w^SZ/∂n^SZ). This asymmetry arises because staffers cannot leave during intra-firm bargaining breakdowns — they remain regardless — so each additional staffer tightens the firm’s fallback position more effectively than an additional in-house hire. With continuous labor, any positive mass of in-house workers leaves residual scope for further wage moderation through staffers, so the firm always finds it profitable to convert the last in-house hire to a staffer. The corner solution n^SZ = 0 is thus the unique equilibrium. With discrete labor, a firm would be indifferent between exactly one and zero in-house workers.

The paper also notes that this staffing-agency arrangement is formally equivalent to the “labor packer” or intermediate-good setup widely used in applied macroeconomics (e.g., Gertler, Sala, and Trigari 2008) to avoid Stole-Zwiebel complications, providing a micro-foundation for that modeling convention.

Q: Why does in-house hiring with decreasing returns to labor generate higher wages and employment than the staffing-agency arrangement? A: Under decreasing returns, if a worker’s wage negotiation breaks down and the worker leaves, the marginal product of the remaining n−1 workers rises. This gives each in-house worker additional bargaining leverage beyond the standard β parameter. To counteract this, firms over-employ in-house hires to keep the marginal product low. In general equilibrium this raises tightness θ^SZ > θ^SA, which in turn raises wages w^SZ > w^SA even though each individual firm’s motive was wage moderation.

Q: What is the formal basis for the in-house wage equation, and what does it depend on? A: Following Stole and Zwiebel’s stability condition, the continuous-labor wage for a firm with n workers is w(n) = (1−β)b + n^(−1/β) ∫₀ⁿ z^((1−β)/β) F’(z) dz. The wage depends on the entire distribution of marginal products over [0, n], not merely on the marginal product at n. In the special case of a power production function, the integral yields an explicit power function in n.

Q: When is the staffing-agency equilibrium socially efficient? A: Under the standard Hosios condition β = η(θ*), the staffing-agency equilibrium attains exactly the planner’s tightness (θ^SA = θ*), because bargaining in staffing agencies is standard — the worker’s outside option does not affect other workers’ wages and so the usual efficiency characterization applies. The in-house equilibrium then strictly over-posts vacancies (θ^SZ > θ*).

Q: Can the in-house equilibrium ever be socially optimal? A: Yes, but only under specific parameter conditions. Theorem 3.3 shows that with Cobb-Douglas matching (η constant) and a power production function F(n) = An^α, there exists a threshold β̂ ∈ (η, 1) at which θ^SZ = θ*. The intuition is that strong worker bargaining power creates a vacancy-underprovision problem; the over-employment incentive under in-house hiring then partially corrects it. The functional form restriction is made for expositional convenience; the core logic is general.

Q: What is Lemma 3.1 and why is it the key to the main result? A: Lemma 3.1 states that, given any positive number of in-house hires n^SZ > 0, renting one additional staffer reduces the wage paid to in-house workers by more than does hiring one additional in-house worker: ∂w^SZ/∂n^SA < ∂w^SZ/∂n^SZ. This is proved by showing the relevant integral in the difference (∂w^SZ/∂n^SA − ∂w^SZ/∂n^SZ) is negative for n^SZ > 0 given F’’ < 0.

Q: Why does renting an additional staffer moderate in-house wages more than hiring an in-house worker? A: An in-house worker who is hired can, in principle, leave during a bargaining breakdown, triggering renegotiation all the way down to zero in-house workers and driving the firm’s fallback to zero profit. A rented staffer cannot leave; at minimum, all rented staffers remain in production regardless of in-house bargaining outcomes. Each additional staffer thus raises the firm’s floor payoff in bargaining by more than an additional in-house hire does, generating stronger wage moderation.

Q: Why does Theorem 3.4 produce a corner solution rather than an interior mix? A: Because labor is treated as a continuous input, any strictly positive mass n^SZ > 0 of in-house workers leaves the marginal in-house worker with positive bargaining leverage through the threat-to-leave mechanism. The firm can always improve its bargaining position by converting that marginal in-house worker to a staffer. This margin is present no matter how small n^SZ is, so the only equilibrium is n^SZ = 0. In discrete labor the firm would be indifferent between exactly one and zero in-house hires.

Q: What happens to labor market tightness when both arrangements coexist? A: In the mixed equilibrium the tightnesses for in-house and staffer jobs must be equal in equilibrium (θ^SZ = θ^SA). If one tightness were higher, workers would prefer that job type (higher wage and higher probability of finding it), but firms would reduce vacancy posting there (costlier to fill), automatically equalizing tightness. This equilibration occurs even though in equilibrium vacancy posting for in-house jobs goes to zero.

Q: Do the results require directed search or hold under random search as well? A: The results hold under both directed and random search. Appendix 3.C establishes that with random search and a single pooled matching function M = m(V^SZ + V^SA, N), the unique equilibrium also features n^SZ = 0. The directed-search assumption is made without loss of generality.

Q: What does this paper imply for the applied macroeconomics literature’s “labor packer” modeling convention? A: The paper provides a formal micro-foundation for the labor-packer or intermediate-good approach used in New Keynesian DSGE models (e.g., Gertler, Sala, and Trigari 2008) to sidestep Stole-Zwiebel bargaining. In that literature, a “wholesale firm” or “packer” searches for workers and sells their services to final-goods firms under perfect competition — formally identical to the staffing-agency arrangement in this paper. Theorem 3.4 shows this arrangement is the unique equilibrium outcome of rational firm choice, so the shortcut is not merely convenient but theoretically grounded.

Q: What does the empirical literature say about wage differentials between in-house and agency workers? A: Drenik et al. (2023), using Argentine administrative data linking temp agencies to user firms, estimate a significant wage premium for in-house hires relative to temp workers. This is consistent with the paper’s theoretical prediction that w^SZ > w^SA in the polar-case comparison (Theorem 3.1), though the paper itself presents no empirical estimation.

Q: What are the implications of the staffing-agency arrangement for measured labor shares? A: The paper notes that costs for staffers typically appear in firm accounts as intermediate input costs rather than labor costs. A shift from in-house hires to staffers therefore reduces measured labor costs and, because it also reduces value added (by more than the labor-cost reduction), lowers the measured labor share at the firm even when actual labor input and output are unchanged.

Q: What future extensions do the authors identify as priorities? A: The authors flag three main extensions: (i) heterogeneous workers and firms, which could generate predictions about which firms use each hiring mode; (ii) worker effort/loyalty differences between in-house and agency workers that could make in-house hiring attractive ex post; and (iii) a frictional rental market for staffers or heterogeneous tasks within the firm, where insufficient staffer supply in certain sub-markets could restore a role for in-house hiring.

Staffing agency (in this paper’s sense): An intermediary that posts vacancies on the frictional labor market, employs workers through standard Nash bargaining, and rents those workers one-for-one to producing firms on a frictionless, perfectly competitive market. The staffing agency is separated from the firm’s production decisions; its search activity has constant returns to scale.

In-house hiring with Stole-Zwiebel bargaining: A market arrangement in which the producing firm itself posts vacancies, employs workers, and conducts intra-firm Nash bargaining. Under decreasing returns to labor, the bargaining outcome for worker i depends on the firm’s payoff if that worker left, which in turn depends on wages paid to the remaining n−1 workers — generating a system of interdependent bargaining problems captured by the differential equation w(n) = (1−β)b + n^(−1/β) ∫₀ⁿ z^((1−β)/β) F’(z) dz.

Wage-moderation incentive (over-employment): Under in-house hiring, a firm has an incentive to hire more workers than a social planner would recommend, because additional workers reduce each worker’s marginal product and hence the wage the firm must pay. This incentive is present because decreasing returns mean a departing worker raises the marginal product of remaining workers, giving each in-house worker leverage over the firm.

Differential wage-moderation effect (Lemma 3.1): The finding that, given any positive mass of in-house hires, renting one additional staffer reduces in-house wages by more than hiring one additional in-house worker (∂w^SZ/∂n^SA < ∂w^SZ/∂n^SZ). The asymmetry arises because staffers cannot leave during intra-firm bargaining breakdowns, so they provide a more effective floor to the firm’s fallback payoff.

Hosios condition (as applied here): The standard efficiency condition β = η(θ), where β is the worker’s Nash bargaining weight and η(θ) is the elasticity of the job-offer arrival rate with respect to tightness. When this condition holds, the staffing-agency equilibrium is socially optimal (θ^SA = θ*) and the in-house equilibrium is inefficient (θ^SZ > θ*).