Wage Adjustment in Efficient Long-Term Employment Relationships

Layer 1: Overview

This paper develops a tractable theoretical model of wage dynamics in long-term employment relationships, situated between two polar extremes in the existing literature: continual Nash renegotiation (Mortensen and Pissarides 1994) and wage adjustment only when participation constraints bind (MacLeod and Malcomson 1993). The central motivation is that neither polar extreme matches well-documented empirical facts about wage adjustment — wages are adjusted neither continuously nor as rarely as participation constraints alone would imply.

The model’s key ingredients are: (1) match-specific productivity that evolves as a geometric Brownian motion, generating persistent idiosyncratic shocks; (2) on-the-job search, whereby employed workers receive outside job offers at rate s*lambda; and (3) renegotiation costs modeled as breakdown probabilities (Delta_W for workers, Delta_F for firms) that apply whenever a party unilaterally initiates a renegotiation. These breakdown risks create a wedge between what each party can guarantee by threatening to renegotiate and the full Nash share, thereby generating inaction regions within which the wage remains unchanged. When either party’s surplus falls to the boundary of this inaction region, wage adjustment occurs by mutual consent at zero cost, keeping separations bilaterally efficient. The result is a “drunken walk” for wages: constant most of the time, adjusting minimally when productivity shocks or outside job offers drive the system to the boundary.

An analytical general solution for firm and worker surpluses is derived — a methodological innovation, since prior work with persistent idiosyncratic shocks has required numerical methods.

The model is calibrated at monthly frequency to: a 5% annual real interest rate; a 1% per month exogenous separation rate (from Farber 1999); a 6% steady-state unemployment rate; a 2.5% per month employer-to-employer (E-to-E) transition rate (from Fujita, Moscarini, and Postel-Vinay 2021); a standard deviation of annual log base wage changes among job stayers of 0.053; and an incidence of total compensation (base plus bonus) freezes of 17% (both from Grigsby et al. 2021). Worker bargaining power is set to beta=0.2, which delivers a wage pass-through elasticity of 0.22 (in range of Lamadon et al. 2022 and Kline et al. 2019), hiring costs of 1.4 months of wages (consistent with Oi 1962 and subsequent work), and a base pay share of compensation of 97% at the median (matching Grigsby et al. 2021). The breakdown probability calibrates to Delta=0.33 for both workers and firms.

Key quantitative findings:

First, the calibrated model generates a hump-shaped separation hazard peaking at just over 0.08 at around 3 to 5 months of tenure and declining thereafter, closely matching Farber (1999) — a nontargeted moment. Cumulative wage growth after 10 years of tenure is approximately 15%, lying between Topel’s (1991) estimate of over 25% and Altonji and Williams’ (2005) estimate of 11%.

Second, the model-implied distribution of annual base wage changes among job stayers features over 30% with zero change, substantially more wage increases than cuts, and limited downward flexibility — all key features documented in microdata (Altonji and Devereux 2000; Grigsby et al. 2021). The distribution of total compensation (base plus bonus) is far more symmetric and has lower incidence of freezes (targeted at 17%), consistent with Grigsby et al.’s finding that bonus pay drives most compensation flexibility. The sequential auctions special case (without renegotiation costs) greatly overstates pay freezes, underscoring that renegotiation costs are the mechanism generating empirically realistic intermediate wage adjustment.

Third, the model delivers a near-memorylessness property for hiring wages: because idiosyncratic shocks and outside job offers necessitate ex post wage adjustments that preserve bilateral efficiency, subsequent wages become independent of the initial hiring wage once the first adjustment occurs. Quantitatively, this largely negates Hall’s (2005) result that rigid hiring wages can generate substantial unemployment fluctuations: in the calibrated model with empirically realistic adjustment, the allocative effect of entry wage flexibility on labor market tightness is much smaller than in Hall’s special case.

Fourth, the model provides a novel theory of recruitment and retention bonuses. Because persistent productivity shocks are best met with adjustments to the flow wage, while transitory outside offers are best met partly with lump-sum bonuses (flow wage increases are credibly capped by the firm’s inaction boundary), the model predicts non-base pay as an equilibrium outcome. Counterfactual experiments show that eliminating firms’ ability to pay retention bonuses reduces total match surplus at the date of new matches by approximately 15.1% and raises the employment-to-unemployment separation rate by approximately 9.5%; eliminating both retention and recruitment bonuses raises these figures to 16.0% and 10.3%, respectively.

The paper also extends the baseline model to accommodate positive inflation (nominal wages held fixed absent renegotiation), using a perturbation method due to Fleming (1971), generating a spike at zero nominal wage change that decays with inflation — consistent with the large empirical literature on nominal wage adjustment.

The implication for macroeconomics is that efficient long-term relationships with realistic sporadic wage adjustment cannot be the source of cyclical unemployment volatility, pointing toward either violations of bilateral efficiency (asymmetric information, wage-cut costs) or volatile labor demand as the necessary ingredient.

Layer 2: Deep Dive

What is the identification strategy and what are the main threats to it?

The paper is primarily theoretical and quantitative, not empirical, so it does not employ a conventional identification strategy. The model is calibrated to match a set of moments from existing microdata (Farber 1999; Fujita et al. 2021; Grigsby et al. 2021) and then evaluated on nontargeted moments such as the shape of the separation hazard by tenure. Threats to the model’s quantitative conclusions include: (a) the calibration sets beta=0.2 somewhat informally (targeted to four informal moments rather than formally estimated); (b) the baseline restricts mu=sigma^2/2 so that log match productivity is driftless, and Delta_W=Delta_F (symmetric breakdown risk) — the paper checks in the appendix that relaxing mu gives essentially unchanged main results; (c) the model abstracts from risk aversion, general human capital accumulation, and permanent firm heterogeneity, any of which could alter wage dynamics or calibrated parameter values; (d) the Grigsby et al. (2021) moments used for calibration pertain to a period of very low inflation, which the paper treats as approximately a zero-inflation environment.

What is the drunken walk and why is it called that?

The ‘drunken walk’ is the wage path that emerges from the model. The wage remains constant whenever both parties’ surpluses lie strictly within their respective inaction regions (neither party can credibly threaten to renegotiate). When idiosyncratic productivity hits the upper or lower boundary of the inaction set, the wage adjusts minimally upward (to restore the worker’s surplus to the threshold) or minimally downward (to restore the firm’s surplus to the threshold). The path therefore wanders irregularly, making small adjustments only when forced to by the boundaries, analogously to a drunken walk — a term echoing the dynamic contracting literature (Thomas and Worrall 1988), where the same path arises from insurance motives rather than renegotiation costs.

How does the paper characterize the surplus analytically and why is this novel?

The key innovation is that bilateral efficiency decouples the total match surplus (determined as an optimal stopping problem) from the division of that surplus between firm and worker. Total surplus S(x) is characterized analytically as a function of match productivity x alone, solving an ODE with boundary conditions (value-matching and smooth-pasting at the separation threshold). Given S(x), the firm surplus J(w,x) and worker surplus V(w,x) satisfy ordinary differential equations (not PDEs) for any fixed wage w, because wages change only at boundaries. This reduces the wage determination problem to one of iterating over constants rather than functions, allowing analytical general solutions (Propositions 2, 3, 4) that prior work with persistent idiosyncratic shocks could not obtain, requiring numerical methods instead (Yamaguchi 2010; Lise et al. 2016).

What are the two special cases studied and what do they reveal?

The costly renegotiation case (s=0, no on-the-job search) isolates adjustment driven purely by idiosyncratic productivity shocks and breakdown risk. In this case, the wage adjustment boundaries simplify to an upper bound from the worker’s threat and a lower bound from the firm’s threat; there is a fundamental asymmetry in that workers cannot credibly threaten a wage increase in the face of complete breakdown risk (Delta_W=1), since they receive no outside offers. The sequential auctions case (beta=0, Delta_F=1, on-the-job search only) recovers and extends Postel-Vinay and Robin (2002) to persistent productivity shocks with analytical solutions. In this case, wage adjustment is one-sided in a surprising direction: wage increases are triggered by reductions in match productivity, because lower productivity reduces the recruitment compensation that a worker could extract if an outside offer arrived, lowering her match value and necessitating a raise. This case greatly overstates pay freezes relative to data, confirming that renegotiation costs are essential to match empirical wage adjustment frequency.

What is the memorylessness property and what are its implications for Hall (2005)?

The memorylessness property states that, conditional on the occurrence of a wage adjustment, the subsequent path of wages is independent of the initial hiring wage. Once the wage is adjusted, the history is ‘forgotten.’ This arises because ex post wage adjustments are determined solely by contemporaneous productivity and the bilateral efficiency requirement, not by the history of wages up to that point. The implication for Hall (2005) is that the allocative effect of hiring wage rigidity on unemployment fluctuations — which rests on the hiring wage having an indefinite legacy (no adjustment ever needed in Hall’s special case of zero idiosyncratic shocks, zero on-the-job search, and full breakdown risk) — is largely negated once realistic wage adjustment is introduced. The decomposition in equation (27) shows that the entry wage effect on firm surplus and labor market tightness is much smaller in the baseline calibration than in Hall’s special case, and that general equilibrium effects (firms anticipating future wage adjustments in booms) further moderate volatility. This dovetails with the empirical literature initiated by Beaudry and DiNardo (1991) finding that economic conditions at the start of a job have little explanatory power for current wages once one controls for the history of conditions since job start.

What is the model’s theory of recruitment and retention bonuses and why does it matter?

Bonuses arise from the asymmetry between the type of shocks and the type of compensation instrument best suited to absorb them. When match productivity changes persistently, adjusting the flow wage is efficient; but when an outside offer arrives temporarily, the value delivered to retain a worker cannot always be committed credibly via flow wages — the firm can only raise the base wage up to the threshold at which the firm would immediately trigger another renegotiation to cut it back. Any remaining value above that threshold must be delivered as a lump-sum retention bonus. Analogously, when recruiting a worker from another firm, the new employer has an upper bound on the flow wage it can credibly offer; remaining value goes to a recruitment bonus. This provides an endogenous theory of non-base pay. The allocative stakes are large: eliminating retention bonuses reduces match surplus at new matches by 15.1% and raises the E-to-U separation rate by 9.5%; eliminating both retention and recruitment bonuses raises these figures to 16.0% and 10.3%. Even though bonuses are transitory and account for only a small share of overall compensation (the base pay share is 97% at the median in the calibration), they are allocatively important — the paper calls this an instance of the general principle that marginal variation can be allocatively consequential.

What heterogeneity is documented or analyzed?

The main model is deliberately parsimonious and abstracts from worker and firm heterogeneity. However, the paper notes that the model can accommodate permanent worker type differences in efficiency units: if x, b, and vacancy costs all scale with efficiency units, the log wage change distribution is identical across worker types while the initial wage scales proportionally. The paper also analyzes two sources of heterogeneity in wage outcomes that emerge endogenously: variation in wage change incidence with match tenure (separation hazard that is hump-shaped in tenure) and variation in base-wage versus total-compensation changes (base wages change less frequently and are more asymmetric than total compensation). The appendix contains an extended model allowing general drift mu, encompassing specific human capital accumulation, with results described as essentially unchanged.

What robustness checks are performed?

Key robustness exercises include: (1) The appendix provides the extended model with general mu (not restricted to mu=sigma^2/2), encompassing specific human capital accumulation; main results are stated to be essentially unchanged. (2) Recalibrated versions of the two special cases (s=0 for costly renegotiation; Delta_F=1 and beta=0 for sequential auctions) are examined separately to understand which mechanism drives empirical fit. (3) An alternative special case with Delta_W=Delta_F=1 and beta>0 is confirmed to generate a similarly counterfactual share of pay freezes (~75%), reinforcing that wage-adjustment-only-at-participation-constraints is empirically rejected. (4) The inflation extension in Section 3 uses an approximate analytical solution (Taylor expansion to first order in pi) following Fleming (1971) to show the model generates sensible nominal wage change distributions and a decaying zero-spike with inflation. (5) Proposition 2 result (ii) establishing the expected duration of wage spells provides an internal consistency check linking the allocative effects of wages to their duration.

How does this paper relate to and differ from closely related prior work?

MacLeod and Malcomson (1993) is the closest theoretical predecessor: it studies renegotiation by mutual consent with efficient long-term relationships and generates a drunken walk. This paper extends it by adding idiosyncratic productivity shocks and on-the-job search and making the model quantitative with analytically tractable solutions, moving beyond MacLeod-Malcomson’s polar case (Delta=1). Postel-Vinay and Turon (2010) study a similar environment to the sequential auctions special case but with i.i.d. productivity shocks, requiring numerical methods; this paper obtains analytical solutions even with persistent shocks. Postel-Vinay and Robin (2002) and Cahuc et al. (2006) are nested as special cases. Hall (2005) is nested and shown to be quantitatively non-generic: its result on hiring wages and unemployment fluctuations relies on special-case assumptions that are empirically rejected. Gertler and Trigari (2009) achieve large unemployment fluctuations via time-dependent staggered wage adjustment; this paper studies state-dependent adjustment and finds the opposite result. Grigsby et al. (2021) provide the key calibration moments on the incidence of pay changes; the paper replicates their finding that total compensation is more flexible than base pay and provides a theoretical interpretation. Balke and Lamadon (2022) study long-term contracts with directed search but without wage inaction, which is a central object here. Dupraz et al. (2022) model wage rigidities that generate inefficient separations; this paper instead maintains bilateral efficiency and generates wage rigidity endogenously.

What are the policy implications and their scope conditions?

The central policy-relevant conclusion is that, within a model of efficient long-term relationships with realistic sporadic wage adjustment, hiring wage flexibility (or rigidity) is much less consequential for unemployment fluctuations than Hall (2005) suggested. This implies that policies aimed at wage flexibility at the point of hiring are unlikely to substantially moderate unemployment fluctuations if the broader employment relationship is bilaterally efficient. The model instead points to wage-cut costs, asymmetric information, or impediments to matching outside offers as the necessary ingredients for hiring-wage stickiness to matter for unemployment. The allocative importance of non-base pay (retention and recruitment bonuses) suggests that regulations or institutional arrangements that restrict bonus pay could meaningfully retard match formation and raise separations, even when bonuses appear small as a share of total compensation. The scope conditions are bilateral efficiency, risk neutrality, and the absence of aggregate shocks (the paper focuses on idiosyncratic shocks in a stationary equilibrium, with only a perturbation analysis for aggregate shocks in the allocation-of-entry-wages section).

What does the user cost of labor framework reveal?

Section 1.6 extends the user cost of labor concept of Kudlyak (2014) — the shadow flow price of labor in long-term relationships — to this environment. The user cost in this model contains components absent from simple Diamond-Mortensen-Pissarides: turnover costs due to on-the-job search (proportional to the firm surplus of a new match, contributing sλ*J(w0,x0)), and the value of future productivity drift and variance (which act as a source of moderation of user cost). The key message is that idiosyncratic shocks and on-the-job search diminish the importance of the initial wage in the firm’s effective flow cost of labor, because future wage adjustments are anticipated. This provides a flow-based interpretation of the memorylessness property and complements the work of Doniger (2021) and Bils et al. (2023) on quality-adjusted labor costs.

How does inflation affect wage adjustment in the extended model?

In the extension (Section 3), the nominal wage is held fixed absent renegotiation, so the real wage drifts downward at the inflation rate pi. This creates an additional source of value to the firm (and loss to the worker), valued at -piwJ_w. Because J_w<0 (higher wages reduce firm surplus), inflation raises firm value and consequently shifts the adjustment boundaries inward: for a given productivity, firms are less likely to demand nominal wage cuts and workers are more likely to demand nominal wage increases. The zero-change spike in the distribution of nominal wage changes decays as inflation rises, a well-established empirical feature. The analytical solution uses a first-order Taylor expansion in pi (following Fleming 1971), which the authors note may also be extendable to approximate solutions for aggregate shocks.

Key Concepts

Drunken walk (wage dynamics): The equilibrium wage path in the model: wages remain constant for extended periods and adjust minimally — only enough to prevent a unilateral renegotiation — when idiosyncratic productivity shocks or outside job offers drive firm or worker surplus to the boundary of their respective inaction sets. The name reflects the irregular, boundary-regulated wandering of wages over time.

Renegotiation costs (breakdown risk): The cost of unilaterally initiating a wage renegotiation, modeled as a probability Delta_W (Delta_F) that the match breaks down if the worker (firm) forces a renegotiation. These costs generate inaction regions in which neither party can credibly threaten a unilateral renegotiation, so the wage remains unchanged. They are the key parameter governing the frequency of equilibrium wage adjustment, nesting both continual bargaining (Delta=0) and adjustment only at participation constraints (Delta=1) as polar cases.

Inaction set: For any current wage w, the set of match productivities x within which neither the firm nor the worker can credibly issue a unilateral threat to renegotiate. The wage remains constant when productivity lies in the interior of both parties’ inaction sets. The boundaries of these sets are the thresholds x_W(w) and x_F(w) at which wage adjustments are triggered by mutual consent.

Memorylessness (of hiring wages): The property that, once a wage adjustment occurs, the subsequent path of wages is independent of the initial hiring wage. This arises because ex post adjustments are determined solely by contemporaneous productivity and the bilateral efficiency requirement. As a result, the legacy of any hiring wage is truncated to the duration of the first wage spell, negating the allocative importance of hiring wage rigidity for unemployment fluctuations in Hall’s (2005) sense.

Recruitment and retention bonuses: Lump-sum payments made by the current or prospective employer when an employed worker receives an outside job offer, in situations where the value to be delivered to retain or recruit the worker exceeds what can credibly be committed via increases to the flow base wage (which face a ceiling imposed by the firm’s inaction boundary). The model predicts these bonuses as an equilibrium outcome of bilateral efficiency, arising from the asymmetry between persistent productivity shocks (best absorbed by flow wage changes) and transitory outside offers (partially absorbed by lump-sum bonuses).

Bilateral efficiency (in long-term employment relationships): The property that firm and worker jointly maximize total match surplus, so that separations occur if and only if total surplus is exhausted, and wages are set to preserve this condition. In this paper, bilateral efficiency is preserved on the equilibrium path because costless mutual-consent wage adjustments preempt costly unilateral renegotiations. The term is used specifically for bilateral efficiency of individual relationships (not equilibrium efficiency of aggregate allocations).

User cost of labor: The shadow flow price of labor in a long-term employment relationship, extending Kudlyak (2014) and the Jorgenson (1963) capital user cost concept to this environment. It equals flow output at a new match and consists of the flow wage plus flow-equivalent discounting and separation costs, minus the capital gains from anticipated future wage adjustments induced by productivity drift, variance, and on-the-job search. Idiosyncratic shocks and on-the-job search reduce the importance of the initial wage in this user cost, providing a flow-based expression of the memorylessness property.

Wage pass-through elasticity: The elasticity of the equilibrium wage with respect to a change in match-specific productivity — the log change in wages induced by a one log-point rise in match productivity. In the calibrated model this equals 0.22, reflecting that efficient renegotiation shares only part of idiosyncratic productivity gains with the worker (bounded by the worker’s bargaining power beta=0.2 and the renegotiation cost structure). This is the model’s analogue to empirical rent-sharing elasticities in Lamadon et al. (2022) and Kline et al. (2019).