Bargaining with renegotiation in models with on-the-job search

Layer 1: Overview

This paper resolves a long-standing theoretical impasse in labor search models: how to model wage bargaining when workers search on the job (OJS) and the quit rate depends on the wage. Shimer (2006) showed that this wage-dependent turnover creates a potentially non-convex bargaining set, causing the Nash bargaining solution to break down and generating equilibrium multiplicity. Gottfries introduces renegotiation — wages are fixed under a contract that expires at a Poisson rate γ, after which a new wage is bargained — as the device that simultaneously restores uniqueness and nests the earlier models of Pissarides (1994), Mortensen (2003), and Shimer (2006) as limit cases.

The model is a continuous-time, frictional labor market with risk-neutral firms and workers. Unemployed workers receive job offers at rate λu; employed workers receive outside offers at rate λe; matches dissolve exogenously at rate δ. Wages are determined through non-cooperative alternating-offers bargaining in the spirit of Rubinstein (1982) and Binmore et al. (1986), with worker bargaining power β. The key innovation is that contracted wages last until renegotiation, which arrives at a Poisson rate γ(F), where F indexes match quality (and hence wage expectations about future renegotiations). As γ → ∞ (continuous renegotiation), the model converges to Pissarides (1994): values solve the Nash bargaining solution with perfectly transferable values, and worker turnover is independent of the current contracted wage. As γ → 0 (no renegotiation), the model converges to the unique equilibrium from Shimer (2006) and Mortensen (2003), with wages playing a strong role in retaining workers. Equilibrium uniqueness follows because renegotiation makes match types payoff-relevant — wage expectations about future negotiations differ across types, so the Nash product cannot be constant on the support, pinning down the initial condition for the wage differential equation.

The main mechanism is a turnover-retention channel that amplifies worker bargaining power. Because a higher wage reduces the quit rate, and marginal quits are bilaterally inefficient (the firm loses its profits when the worker leaves), agreeing on a higher wage partially recoup losses through longer match duration. This acts as an additional source of worker surplus share on top of the primitive bargaining power β. The strength of this channel is governed by θ — the expected fraction of the discounted match duration covered by a given contracted wage. Higher θ (less frequent renegotiation) means wages matter more for turnover and workers extract more surplus. Lower θ (more frequent renegotiation) attenuates the channel.

Calibrated to US labor market data — a 45% monthly job-finding rate (Shimer 2012), a 3.2% monthly job-to-job transition rate (Moscarini and Thomsson 2007), a 5% unemployment rate, a 5% annual discount rate, and targeting a labor share of 2/3 and a lognormal wage-offer distribution with scale parameter σ = 0.16 (Gottfries and Teulings 2017) and a mean-to-minimum wage ratio of 1.7 (Hornstein et al. 2007) — the model implies sharply different primitive bargaining powers depending on the assumed renegotiation frequency. Under continuous renegotiation (γ = ∞), the calibrated bargaining power of workers is β = 0.46. Under never-renegotiated wages (γ = 0), β = 0.02. The implication is that the correct inference about worker bargaining power from observed wage distributions is very sensitive to the assumed renegotiation regime.

For minimum wages, the paper proves that, holding firm entry and the reservation wage constant, any minimum wage increase raises the entire wage distribution in the sense of first-order stochastic dominance (Proposition 2). However, the extent of spillovers above the minimum wage depends critically on renegotiation frequency. In a high-commitment economy (low γ) versus a low-commitment economy (high γ) with identical pre-policy wage distributions, the high-commitment economy exhibits strictly larger wage spillovers throughout the support above the minimum (Proposition 3). The intuition is that a spike in the mass of workers at the minimum wage creates a strong incentive for firms to offer higher wages to reduce costly turnover — but this incentive only materializes when wages are sticky enough that turnover responds appreciably to them. With continuous renegotiation, the spillover vanishes entirely and only a mass point at the minimum wage remains. In the limit of no renegotiation, the model resembles the wage-posting model, which produces especially large spillovers by construction.

An extension endogenizes the contract length. Firms optimally choose the renegotiation frequency after observing the match type. Two regimes emerge: when worker bargaining power is sufficiently high or productivity rises quickly relative to profits, firms prefer continuous renegotiation; otherwise, an interior contract length strictly above zero is optimal, and firms with all the bargaining power prefer no renegotiation. This implies that the polar assumptions of full commitment or no commitment standard in the literature arise only as boundary cases.

Layer 2: Deep Dive

What is the core theoretical problem this paper addresses?

Shimer (2006) demonstrated that when a worker’s quit rate depends on the contracted wage, the bargaining set can become non-convex, violating a key condition for the Nash bargaining solution. He proposed a non-cooperative alternating-offers bargaining game but showed that it produces a continuum of equilibria. The existing literature responded either by removing bargaining (wage posting, all bargaining power to firms) or by making turnover independent of the wage (counteroffers by the incumbent firm). Gottfries provides a solution that preserves both bargaining and wage-dependent turnover by introducing renegotiation.

How does renegotiation restore equilibrium uniqueness?

Without renegotiation and with homogeneous productivities (as in Shimer 2006), the match type F is not payoff-relevant: only the current contracted wage matters, so the Nash product is constant on the wage support and any wage in that support is a potential equilibrium outcome. With renegotiation, each type F is associated with a distinct expected future wage (wage expectation), which is payoff-relevant because it governs future turnover. Different types therefore face different Nash products, and the product cannot be constant across types. This forces the Nash product to be increasing to the left of the bargaining outcome and decreasing to the right for each type, providing a unique interior maximum and a unique initial condition w(0) = max{βx(0) + (1−β)wr, wmin}. The paper also shows that alternative refinements — large-friction limits or the case where λe = 0 — yield the same unique equilibrium.

How does the model nest Pissarides (1994) and Mortensen (2003)?

As γ → ∞ (continuous renegotiation, θ → 0), the contracted wage becomes irrelevant because future wages are renegotiated almost immediately. The worker’s quit decision is then independent of the current wage, so values solve the standard Nash bargaining solution with perfectly transferable values, exactly as in Pissarides (1994). As γ → 0 (no renegotiation, θ → 1), the wage lasts the full duration of the match, turnover responds maximally to wages, and the equilibrium values correspond to Mortensen (2003, Section 4.3.4) with a unique initial condition (rather than the multiplicity in Shimer 2006). Intermediate values of γ correspond to no prior model.

What is the mechanism by which workers receive a share of surplus exceeding their bargaining power β?

When a worker bargains for a higher wage, she reduces her quit probability. Marginal quits are bilaterally inefficient because the firm loses its profits when the worker leaves to a marginally better job (even though the transition is socially efficient once the new employer’s value is counted). The reduction in inefficient separations increases the joint match surplus. Formally, the extra surplus share comes from the term λe · [w’(F)/(δ+ρ+λe(1−F))] · [(δ+ρ+λe(1−F))/(δ+ρ+γ(F)+λe(1−F))] · Π(F,w(F)), which is the density of incoming offers per unit wage increase multiplied by the fraction of the match duration covered by the contracted wage, multiplied by the profit level lost at each marginal quit. This term is zero when γ → ∞ (continuous renegotiation) and is largest when γ = 0 (no renegotiation).

What is θ and what role does it play?

θ is defined for the homogeneous-productivity case as the expected fraction of the expected discounted match duration that an agreed wage remains in force. It captures the marginal relative importance of the current contracted wage versus the wage expectation (which governs future renegotiated wages). θ = 1 corresponds to no renegotiation (the contracted wage lasts the whole match), θ → 0 corresponds to continuous renegotiation. The renegotiation rate is γ(F) = [(1−θ)/θ] · (δ+ρ+λe(1−F)). A small increase in the wage by w’(F)dF decreases turnover by θ dF in the homogeneous case, so θ directly scales the turnover-retention channel and hence workers’ effective surplus share.

What does the calibration reveal about the relationship between renegotiation assumptions and inferred bargaining power?

Holding transition rates fixed (λu = 0.45, λe = 0.181, δ = 0.024 per month) and targeting a 2/3 labor share and a lognormal wage-offer distribution (σ = 0.16, mean-min ratio 1.7), the calibrated worker bargaining power β is 0.46 under continuous renegotiation (γ = ∞) and only 0.02 under no renegotiation (γ = 0). The calibrated productivity distribution also differs markedly: no-renegotiation requires a much fatter right tail in firm productivities to match the same wage distribution because the labor share falls sharply in the upper tail when bargaining power is low and wages are infrequent renegotiated.

What does the paper prove about minimum wage spillovers?

Proposition 2 proves that, holding firm entry constant and adjusting unemployment benefits to keep the reservation wage constant, a minimum wage increase raises the equilibrium wage distribution in the sense of first-order stochastic dominance. Proposition 3 proves that, comparing a high-commitment economy H (lower γH) and a low-commitment economy L (higher γL) that have identical pre-policy wage distributions (and therefore βH < βL), the high-commitment economy H exhibits strictly higher wages at every rank F after a small minimum wage increase. The mechanism is that a mass of workers at the minimum wage creates a dense region of outside options, making it worthwhile for firms to accept higher wages to reduce turnover — but only when committed wages are sticky enough to affect actual turnover.

What happens to the wage distribution spike at the minimum wage when renegotiation is frequent?

Under the baseline assumption that workers move when indifferent (no mass points), the equilibrium has no spike; the mass at the minimum wage spreads continuously upward. When this assumption is relaxed and workers may stay when indifferent (following Shimer 2006), an equilibrium with a mass point at the minimum wage exists. Equation (19)/(20) show the equilibrium mass point at the minimum wage is increasing in the renegotiation rate γ (higher γ → larger spike). This occurs because with frequent renegotiation, spillovers above the minimum wage are small, so the density just above the minimum is high, which in turn supports a large mass at the minimum. The paper parameterizes this with φ = 0.04 (ratio of mass at minimum wage to density just above) and illustrates with θ = 0.02 (long contracts) and θ = 0.5 (short contracts).

How does endogenizing the contract length change the predictions?

When firms choose the renegotiation frequency after observing the match type, two regimes emerge. In the first, the firm would not benefit from raising the wage above the continuous-renegotiation Nash-bargaining level: this happens when worker bargaining power is sufficiently high or productivity increments are large relative to profits. Firms then choose continuous renegotiation (γ = ∞) for that match type. In the second regime, lower turnover makes it profitable to commit to a higher wage via a longer contract; firms pick an interior γ satisfying the envelope condition. With all bargaining power to the firm (β = 0), the optimum is no renegotiation (infinite contract length). The equilibrium in the endogenous-contract model satisfies a differential equation that coincides with the wage-posting model differential equation in the interior region, providing a microfoundation for wage-posting results even when workers have some bargaining power. The model also provides a uniqueness justification for equilibria in Coles (2001) and Coles and Mortensen (2016).

How does this paper relate to Brügemann, Gautier, and Menzio (2015)?

Brügemann, Gautier, and Menzio (2015) identify a similar surplus-retention mechanism in a model where a single firm bargains successively with many workers: agreeing on a high wage with one worker is ‘cheap’ because the firm can recoup part of the cost through lower wages agreed with subsequent workers. Gottfries’ mechanism is the bilateral analogue: within a single match, a higher wage is cheap because it reduces wasteful turnover and extends the profitable match duration. Both models generate workers capturing a surplus share above their primitive bargaining power, but through distinct channels.

What assumptions are needed for uniqueness and what relaxing them implies?

Two key restrictions are imposed. First, Markov strategies are required and wage functions must be weakly increasing in match type F; without this, equilibria exist in which workers accept lower-productivity jobs for a higher current wage, creating decreasing wage functions. Second, workers must move with positive probability when indifferent between offers, which eliminates mass points on the support. Shimer (2006) showed that when indifferent workers never move, multiple equilibria with mass points exist. Relaxing the second restriction opens the door to a spike at the minimum wage in the minimum wage application. Alternative refinements — large-friction limits, the limiting case as λe → 0, or as β → 0 — all single out the same unique equilibrium.

What are the policy implications and their scope conditions?

The main policy implication is that the spillover effects of minimum wage increases depend critically on the degree of wage commitment in the labor market. In economies where wages are rarely renegotiated (higher θ), minimum wage increases spread substantially up the wage distribution; in economies with continuous renegotiation, only a spike at the minimum results with little or no spillover. This has direct implications for empirical studies of minimum wages: the observed pattern of spillovers is informative about the prevailing renegotiation regime. The scope conditions are: (i) partial equilibrium (firm entry and reservation wage are held fixed); (ii) all matches remain profitable at the minimum wage (wmin < x(0)); (iii) random rather than directed search. The paper does not provide an empirical test or identification strategy for the renegotiation frequency itself.

What are the limits and caveats?

The model treats the renegotiation frequency as an exogenous parameter (except in Section 6). The calibration does not structurally identify the renegotiation frequency from data; it instead illustrates sensitivity. The analysis of minimum wages is partial equilibrium — firm entry and reservation wages are held fixed — and the paper notes that general equilibrium effects (entry, reservation wages) are ambiguous in sign and difficult to identify empirically. The model has no on-the-job search effort endogeneity or worker heterogeneity (workers are homogeneous ex ante). The wage-posting and counteroffers models studied in the literature require strong commitment assumptions that this model relaxes but does not fully endogenize in a dynamic contracting sense.

Key Concepts

Renegotiation (frequency parameter γ): The Poisson rate at which a contracted wage expires and a new wage is bargained. In the paper’s own sense, γ indexes the degree of wage commitment: γ = 0 means the contracted wage lasts the entire match (perfect commitment, no renegotiation); γ → ∞ means the wage is continuously reset (no commitment). The frequency γ governs how much the contracted wage — versus future renegotiated wages — matters for the worker’s turnover decision, and hence how much of the match surplus the worker captures.

Bilateral inefficiency of transitions: The paper defines a job-to-job transition as bilaterally inefficient when the value to the worker at the new job is less than the total surplus of the existing match. Since the firm loses its profits when the worker quits, the pair jointly would prefer the worker to stay — yet the worker moves whenever her individual value is higher elsewhere. The gap between individual and joint incentives is the source of bilateral inefficiency; it is what makes turnover-reduction through higher wages mutually beneficial and gives workers extra bargaining power beyond β.

Match type (F) and wage expectation: In the model, F is a match quality drawn from the uniform distribution on [0,1] upon meeting. F determines both the productivity x(F) and the wage expectation — the anticipated outcome of future renegotiations. Critically, the wage expectation is the payoff-relevant state variable that differs across types and thereby distinguishes matches, restoring equilibrium uniqueness. Higher F is associated with higher wage expectations, lower turnover, and greater match surplus.

Commitment parameter (θ): Defined for the homogeneous-productivity case as the expected fraction of the expected discounted match duration for which the currently agreed wage remains in force. θ = 1 corresponds to no renegotiation; θ → 0 to continuous renegotiation. A one-unit wage increase reduces turnover by θ in equilibrium, so θ directly scales the turnover-retention channel and the extra surplus share flowing to workers beyond their primitive bargaining power β.

Minimum wage spillover: The paper uses ‘spillover’ to mean the upward shift in wages paid by firms above the minimum wage that results from a minimum wage increase. Mechanically, a minimum wage creates a mass of workers at the floor; if turnover responds to wages (i.e., commitment is high), firms above the minimum prefer to raise wages to avoid losing workers to the mass point competitors, spreading the effect. The paper proves (Proposition 3) that spillovers are strictly larger in higher-commitment (lower γ) economies.

Markov-perfect equilibrium (MPE) of the bargaining game: The equilibrium concept applied to the alternating-offers bargaining game. In an MPE, offer and acceptance rules depend only on the current match type F, not on prior bargaining history. This restriction, combined with the renegotiation structure, is what allows the paper to derive a unique differential equation for the wage function w(F) and a unique initial condition, yielding the unique equilibrium wage distribution.

Turnover-retention channel: The mechanism by which a higher contracted wage reduces the worker’s quit probability and thereby increases the joint match surplus. Because marginal quits are bilaterally inefficient, a small wage increase generates a surplus gain proportional to the density of arriving outside offers times the expected fraction of the match covered by the contracted wage times firm profits — exactly the extra term that elevates the worker’s effective surplus share above β. This channel is the paper’s central contribution to understanding why workers capture more than their bargaining power suggests.