Wage Dispersion with Heterogeneous Wage Contracts

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Wage Dispersion with Heterogeneous Wage Contracts Cynthia L. Doniger 2015-023 Please cite this paper as: Cynthia L. Doniger (2015). “Wage Dispersion with Heterogeneous Wage Contracts,” Finance and Economics Discussion Series 2015-023. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.023. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Wage Dispersion with Heterogeneous Wage Contracts∗ Cynthia L. Doniger† Federal Reserve Board March 26, 2015 Abstract I study a labor market in which identical workers search on- and off-the-job and heterogeneous firms employ using either posted wages or wage contracts contingent on outside options. Firm level costs for contingent contracts generate a separating equilibrium in which less productive firms post wages. The model with heterogeneous contracts can achieve wage dispersion, labor share, employment transitions, and flow value of unemployment that are simultaneously consistent with empiricalobservationsevenwhenmostfirmspostwages. UsingGerman employee-level administrative data, I estimate roughly 70 percent of firms post wages and employ nearly 50 percent of workers under such contracts. ∗The views expressed in this paper solely reflect those of the author and not necessarily those of the Federal Reserve Board, the Federal Reserve System as a whole, or anyone else associated with the Federal Reserve System. †Email: Cynthia.L.Doniger@frb.gov. IwouldliketothankDmitriyStolyarov,JeffSmith, Matthew Shapiro, and Charlie Brown for advising and support. I also thank the insight and comments of the participants of the Second Joint Paris SaM Workshop and the Sixth Annual Joint Data Workshop, BI Oslo Group and LMDG (CAP) Group. Finally, I thank Aditya Aladangady, Isaac Sorkin, and the participants of the labor search reading group and informal macro and labor seminars at the University of Michigan. This study uses the weakly anonymous Sample of Integrated Labour Market Biographies (Year 2006). Data access was provided via on-site use at the Research Data Centre (FDZ) of the German FederalEmploymentAgency (BA)at theInstitute for Employment Research (IAB)in Ann Arbor, Michigan, USA.

1 Introduction Equilibrium search theories yield a fruitful theoretical paradigm for assessing the efficiency and design of labor market policy. A key component of search models is the mechanism used to divide the output of employment between an employer and employee and set the wage. However, different wage setting mechanisms yield different – sometimes radically – theoretical implications for market efficiency and the distribution of output between and across workers and firms, with some of these in contradiction with observables.1 I demonstrate here that a model featuring a mixture of wage setting mechanisms can simultaneously match a long list of policy-relevant observables. This suggests explicit modeling of wage contract heterogeneity as an important component of a policy-relevant, search-theoretic model of wage dispersion. In this paper, I demonstrate a mechanism by which two canonical wagesetting mechanisms – wage posting (WP) and sequential auction (SA) – exist side-by-side in the steady state of a random, on-the-job search equilibrium with homogeneous workers and heterogeneous firms. WP is characterized by a single and inflexible wage. In contrast, wages formed under the SA contract are contingent on a worker’s best-to-date outside option. A per-firm cost for employing under the SA contract induces firms to endogenously sort between WP and SA wage contracts such that the most productive firms select SA while all less productive firms choose WP. The cost may be thought of as reflecting the legal or human resources costs of offering a job in which wages are set under a contingent wage contract.2 1For example, Hornstein et al. (2011) give evidence that the pure wage posting model is incapableofgeneratingsignificantwagedispersionwithoutinconsistencywitheitherworkers search behavior or reasonable estimates of the outside options of the unemployed. One mightinterpretthisasanargumentagainstapplicationsofwagepostingmodelstostudying policies that affect unemployment, relying instead on models that generate observed wage dispersion from ex-ante differences in worker types or preferences. 2This simple cost structure enables me to isolate and exploit a single mechanism that drives contract choice: differentials in rent allocation between contract types. As a result, I amabletoconsideracontinuousdistributionoffirmtypesandoptimalwage-settingstrategies under both contract types. The environment is more general than that considered by Postel-VinayandRobin(2004)andHolzner(2011)andgeneralityisachievedbyabstracting from the micro-foundations of the cost of the SA contract. In contrast, Postel-Vinay and Robin (2004) and Holzner (2011) trade off a general labor market environment for micro- 1

Theintuitionunderlyingtheequilibriumisclearcutandreliesontwoobservations: 1) All else held equal, firms hiring under SA operate lighter expected wage bills, exclusive of the cost of SA, than they would if they hired under WP. To see this, note that SA firms offer starting wages that are contingent on historywhileWPfirmsofferthesamewagetoallworkers. ThismeansthatSA firms can share a smaller share of rents to workers with poorer labor market histories. Further, SA firms offer workers greater option value of employment than WP firms by promising to bid up wages when outside options arise. As a result, a portion of the SA compensation package rests on the promise of high wages at a future outside employer. These high wages are guarenteed, but not payed for, by the SA firm. 2) The more productive the firm, the bigger the wedge between wage bills since the more productive firm offers more lucrative future wages within the firm and at future employers. Thus, for a given cost, only sufficiently productive firms find that the difference in wage bills under the two contract types exceeds the cost of the ability to employ under SA. This mixed-contract equilibrium nests the antecedent pure-contract equilibria: under null costs, all choose SA, and the model becomes identical to Postel-Vinay and Robin (2002b); under sufficiently large costs, all choose WP, and the model becomes identical to Bontemps et al. (2000). Alimitedsetofempiricalstudieshavemeasuredincidenceofwage(re)negotiability. This evidence indicates that a significant fraction of employers hire under contracts that are nonnegotiable while others are willing to renegotiate: Brenzel et al. (2013) find 38 percent of German firms surveyed in the 2011 Germany Job Vacancy Survey negotiated the wage of the most recently hired employee; Barron et al. (2006) find 49 percent of U.S. firms surveyed in the 2001 Small Business Administration Survey reported willingness to renegotiate the wage of the most recently hired employee; Hall and Krueger (2012) find 31 percent of U.S. workers surveyed as part of the Princeton Data Improvement Initiative foundations of the cost of SA. Each provides micro-foundations that derive from costs to the worker of differential search effort under contract types. The general labor market considered here is at odds with such micro-foundation since a continuous distribution of productivitytypesandoptimalwagesettingstrategiesintheWPsectorimplythatworkers should have search incentive in both sectors, and search incentives are sometimes stronger in the WP sector. 2

in 2008 reported negotiating over pay at the time of hire.3 This paper characterizes the interaction and resulting wage when a WP firm and a SA firm compete over the same worker and, importantly, the impact of contract heterogeneity on the implied aggregate wage distribution and searchincentivesoftheunemployed. Anadvantagetoexplicitlymodelingwage contract heterogeneity is the ability to capture analytical and policy-relevant insight regarding firms that face information or negotiating frictions, as in the pure-WP model, at the same time as an improved overall fit to both microand macro- data. In particular, the mixed-contract model is able to produce substantial wage dispersion without contradicting data on unemployment and employment duration, the ratio of output to compensation of employees (labor share), or plausible ranges for the value of leisure. Further, good fit is achieved even when a large portion of firms hire under WP. The result is a model that is uniquely well suited to welfare analysis of labor market policy which takes into account both the impact of policy on firms’ decision making when firms are not fully flexible or not fully informed, as well as outcomes for workers. This result is significant since the pure-WP model struggles to achieve a reasonable labor share4 or flow value of non-employment when matching employment mobility and wage dispersion data. Meanwhile, the pure-SA mechanism fails to provide a theoretical incentive to search while unemployed. The former is an implication of the passive nature of competition in the WP mechanism, which is especially problematic in the right tail of the distribution.The latter is the opposite: fully contingent and back-loaded contracts extract the full value of search from the unemployed, resulting in a lack of search incentive 3Evidenceonthepropensityforrenegotiationasafunctionoffirmtypeis,unfortunately, limitedforseveralreasons. First,thesethreeexistingstudiesfocusonworkercharacteristics as determinants of contract type. Second, measures of firm attributes are limited and the relation between these attributes and the modeled “productivity” of a firm is not well established. However, these studies do find a notably larger propensity for negotiation for presently employed workers and workers who perform cognitive and/or supervisory tasks. Both are consistent with the notion that jobs higher up the job-ladder are the jobs that feature negotiable pay. Older studies – notably Topel and Ward (1992), which establishes that on-the-job pay gain features more prominently as a source of income growth later in careers,particularlywhencomparedtojob-to-jobmobility–alsoprovideevidenceconsistent with the story of this paper. 4Defining labor share as the fraction value added paid to labor. 3

and often implausible wage dispersion in the left tail.5 In the mixed-contract model developed in this paper, the WP sector puts upward pressure on entry wages in the SA sector, pushing these wages toward larger and more plausible values, and restores search incentive for the unemployed. Meanwhile, the SA sector provides a long right tail of wages even when the tail of the productivity distribution is relatively light, as well as additional moderated wage dispersion in the left tail. These features facilitate selection of a productivity distribution consistent with observed labor share and a positive, but moderated, value of search while unemployed. I estimate the parameters of the model using wages and labor market histories of West German workers in 2006 from the Sample of Integrated Labour Market Biographies provided by the Research Data Center of the German Federal Employment Agency at the Institute for Employment Research.6 I estimate that 73 percent of firms employ using the SA contract. Due to the job ladder this results in an estimated 47 percent of workers employed in a renegotiable contract. With the estimated parameters, the model implies a labor share of 54 percent and a ratio of the flow value of leisure to the average wage of 71 percent. For comparison, the EU-KLEMS database reports German labor share as 63.7 in 2006. My own estimates suggest a value of leisure well in excess of 30 percent of average wages.7 In contrast, constraining all firms to SA produces labor share in this range but a value of leisure nearly as large as the average wage. Meanwhile, constraining all to WP produces a labor share of only 23 percent and a large and negative value of leisure. 5Fully extracting rent from unemployed searchers can result in large, negative starting wages. Complementary solutions already present in the literature are to model workers as having some explicit bargaining power, as in Cahuc et al. (2006), and a high degree of time preference or risk aversion, as in Postel-Vinay and Robin (2002a). 62006 represents a relatively stable period postdating the Hartz reforms and predating the financial crisis. 7IfocusonWestGermanyinordertoabstractfromdifferencesinEastandWestGerman wagescalesthatarenotaddressedbythemodel. Allstatisticsotherthanlaborsharereflect thefocusonWestGermany. SurveyevidenceanalyzedinDoniger(2014)suggeststhatlabor share is not substantially different in the two regions. The 30 percent ratio of value of nonemployment to average wages reflects the ratio of the average unemployment benefits to average wages payed in the first week of January 2006. Due to selection on who becomes unemployed and nonpecuniary value of leisure, this 30 percent forms a very cautious lower bound. 4

Theremainderofthepaperproceedsasfollows. Section2laysoutthelabor market setting under consideration, including details of contracts that are available; describesworkerandfirmbehavior; anddemonstratesthat, forevery fixedper-firmcostfortherighttoSA,thereisaseparatingequilibriuminwhich low productivity firms employ under WP while higher productivity employ under SA. Section 3 provides comparative statics for a fixed distribution of firm types when the cost of the SA contract increases. As more firms join the WP sector, labor share increases but the distribution of wages becomes more compressed, setting up a trade-off whereby one can achieve substantial labor share even when many firms WP, since realistic degree of wage dispersion can be supplied by the remaining firms in the SA sector. Section 4 structurally estimates the model using social security register data for German workers. Section 5 reviews related literature and concludes. Proofs, derivations, and details of the estimation strategy are found in the For Online Publication: Appendix. 2 Model Iconsidertwowagesettingmechanisms: WPandSA.Aninnovationofthispaper is to characterize the interaction and resulting wage when a WP employer and a SA employer compete over the same worker. I also demonstrate that when the arrival rate of job offers is exogenous and the SA contract is costly to the firm, a separating equilibrium exists in which only low-productivity firms employ under the WP mechanism. Intuition of the separating equilibrium rests on noting that, all else equal, employing workers under SA results in a lighter steady-state wage bill than employing under WP. A cost for SA equivalent to the difference in wage bills for some threshold firm, thus, induces less productive firms to select WP and more productive firms to select SA. The remainder of this section is devoted to formalizing and proving this intuition. Sections 3 and 4 detail the impact on the implied equilibrium wage distribution and value of search. 5

2.1 Setting I consider the steady state equilibrium of a search market in which firms and workers are brought together by a sequential process of random matching. MeasureN offirmsoperatetechnologiesthatproduceflowoutputpperworker. Technology is distributed continuously according to an exogenous distribution Γ(p) on support [p,p¯] with 0 < p and p¯ potentially infinite. Measure M of workers search both off- and on-the-job using uniform sampling, meaning the probability of sampling a firm of productivity p or less is Γ(p). Job offers arrive at exogenous Poisson arrival rates λ when unemployed and λ when 0 1 employed. Workers are exogenously separated from employment contracts at Poisson arrival rate δ and die and are replaced an equal mass of unemployed newborns at Poisson arrival rate µ. Workers receive flow b when unemployed. Each worker has linear utility and seeks to maximize the present discounted value of wages and unemployment benefits. Firms seek to maximize steady state current operating surplus: output less wages and costs paid for the right to SA, if any. Firms hire workers under one of two wage-setting mechanisms: WP or SA. If WP, the firm offers a nonnegotiable wage for as long as the worker wishes or until exogenous separation. In equilibrium, this wage offer is uniform for all workerswithinthesamefirm. IfSA,thefirmoffersawagechosentomatchthe value of each worker’s best-to-date outside offer whenever profitable. Wages are updated as outside offers evolve and the SA firm bids up the worker’s wage even in the event of a job-to-job transition. SA wages are thus described by the wage-setting mechanism and the productivity of both the incumbent firm and best-to-date outside option. If the firm chooses to set wages under SA it must pay a flow cost of c. The cost is independent of firm size. The micro-foundations of this cost are left for another paper; however, one possible story is that there are legal and/or administrative fees associated with posting a vacancy in which wages will be set by SA. These fees must be paid whether or not the vacancy is filled. Note that although firms differ in size in equilibrium, each offers an identical number of vacancies. Firm size is pinned down by the rate of vacancy filling and 6

the duration for which contracts persist. The cost structure used in this paper is highly stylized. Weaker stylization is possible with restrictions on the distribution of productivity or hazard parameters. For example, modeling costs as cost per worker may be desirable and is possible when firm size does not increase too rapidly in productivity. Micro-foundations of costs based on search incentive of employees are extremely interesting and have been explored in the literature; seePostel-VinayandRobin(2004)andHolzner(2011). Suchmicrofoundation poses a major drawback for the present project, however: with a continuum of firm types and optimally chosen WP wages, search incentive cannot be ranked between the WP and SA sectors.8 The remainder of this section proves the following proposition: Proposition 1. For each cost, c, there is a threshold, pˇ, such that a Nash equilibrium exists in which firms with productivity less than the threshold (p < pˇ) all strictly prefer WP while more productive firms (p > pˇ) all strictly prefer SA. Threshold productivity firms (p = pˇ) are indifferent. I begin by characterizing labor supply and wage schedules. I prove that for every threshold a cost exists such that no firm wishes to unilatrally deviate from the separating equilibrium. I then observe that the mapping pˇ (cid:55)→ c is continuous, that null costs are consistent with all firms selecting SA, and all firms select WP for sufficiently large costs. The claim then follows from the intermediate value theorem. Conditions for uniqueness are provided in the For Online Publication: Appendix. 2.2 Labor supply As stated, workers seek to maximize the value of their current employment contract. This yields the following lemma governing the flow of workers between firms: 8Note,thisdefiestheusualintuition. True,ifaSAfirmdeviatedtoWPitwouldprovide less search incentive to its current employees. However, one can always find an employee in the WP sector with more incentive to search than an employee in the SA sector when wages are optimally set: an easy example is the employee in the least productive WP firm compared to an employee with best to date outside option equal in productivity to her incumbent. The first has positive search incentive whenever the WP sector has nonzero mass while the second has zero search incentive. 7

Lemma 1. In the mixed contract model with proposed separating equilibrium, labor flows are constrained efficient. In other words, no worker ever rejects a job offer from a more productive employer than their current incumbent. Note that 1) flows within sectors remain efficient in the mixed equilibrium and 2) (more importantly) the flow between WP and SA sectors is efficient. The key ingredient is that in the proposed separating equilibrium, all SA firms are more productive than, and can outbid, all WP firms. Proof is provided in the For Online Publication: Appendix. Since workers accept any wage offer originating from a more productive firm, labor supply to a p-type firm can be pinned down by the method of mass balance. In steady state, the mass of workers flowing into firms of p-type or less must be equal to the mass flowing out: ¯ Uλ Γ(p) = [δ +µ+λ Γ(p)](M −U)L(p), 0 1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) in out ¯ where Γ(p) = 1−Γ(p) is the fraction of firms with productivity greater than p, U is the mass of unemployed workers, and L(p) is the mass of workers employed in a firm of productivity no greater than p. Evaluating the mass balance equation at the supremum of productivity types yields steady state unemployment rate: u = U/M = 1/(1+k ), where 0 k = λ /(δ + µ). Also, the fraction of workers working for a firm with tech- 0 0 ¯ nology p or less is L(p) = Γ(p)/(1 + k Γ(p)), where k = λ /(δ + µ) is the 1 1 1 expected number of job offers per employment spell. The supply of labor to a firm of type p can then be expressed as: expectedhiring expectedduration (cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123) λ U +λ (M −U)L(p) 1 1+k M −U 0 1 1 (cid:96)(p) = = . N δ +µ+λ Γ ¯ (p) [1+k Γ ¯ (p)]2 N 1 1 We selecting a wage schedule, firms consider the impact on both size, (cid:96)(p), and expected flow profit per worker, p−E[wages]. I now turn to the optimal wage schedule. 8

2.3 Wage choice Characterizing an equilibrium that features both kinds of wage contracts requires characterizing wages that arise when firms operating different wagesetting mechanisms compete for the same worker: I will call this the “transitional wage” and denote it w (q,p) where p is the productivity of the incum- PA bent and q is the productivity of the best-to-date outside option. Recalling that the SA firm’s optimal wage choice equates the value of employment in the SA firm at the optimal wage with the value of employment at the best-to-date outsideoptionatthatcompetitor’soptimalwagechoice. Thetransitionalwage can be characterized for an arbitrary SA firm of type p employing a worker with best-to-date outside option from an arbitrary WP firm of type q. Note that, since I am considering a separating equilibrium, if q is WP and p is SA then q < pˇ≤ p.9 Denote the values of employment in a WP and a SA contract are denoted as VP and VA. ThevalueofemploymentataSAfirmatsometreansitionalwage,w (q,p), PA consistent with best-to-date outside offer being a q-productivity WP competitor is: µVA(w (q,p),p) = w (q,p) PA PA +λ [Γ(pˇ)−Γ(q))][E[VP(w (x),x)|q < x < pˇ]−VA(w (q,p),p)] 1 PP PA (cid:124) (cid:123)(cid:122) (cid:125) on-the-jobwagegainduetoacrediblethreatfromaWPcompetitor +λ [Γ(p)−Γ(pˇ)][E[VA(x,x)|pˇ< x < p]−VA(w (q,p),p)] 1 PA (cid:124) (cid:123)(cid:122) (cid:125) on-the-jobwagegainduetoacrediblethreatfromaSAcompetitor +λ [Γ ¯ (p)][VA(p,p)−VA(w (q,p),p)] 1 PA (cid:124) (cid:123)(cid:122) (cid:125) job-to-jobtransitiontoaSAcompetitor +δ[VU −VA(w (q,p),p)], (2.1) PA (cid:124) (cid:123)(cid:122) (cid:125) unemploymentshock where w (x) and x are the optimal competing wage offers of x-type WP and PP 9The functional form of the full wage schedules under all combinations of contract types (WP, w (p); transitional, w (q,p); SA, w (q,p); and reservation entry wages from PP PA AA unemploymentintoeachcontracttypew (p)andw (·,p))arederivedintheForOnline UP UA Publication: Appendix. Thesearehelpfulforanalysisofequilibriumproperties,butarenot required to establish the existence of equilibrium. 9

SA firms with productivity less than p. The difference in the wage setting strategies of a WP competitor and a SA competitor is reflected in potential wages changes. If the incumbent SA firm meets a new WP competitor wages will rise only enough to just best the value of the competitor WP firm’s posted wage offer. On the other hand, if the incumbent SA firm meets a new SA competitor and is able to retain the worker, it must be the case that wages rise to just best the maximum value that the competitor SA firm is able to offer: the value of a wage equal to the productivity of the competitor SA firm at the competitor firm. This is reflected also in the option value of moving to a more productive firm. The transition would yield value just larger than the maximum the current incumbent is able to offer.10 Meanwhile, the value of employment in the employees’ best-to-date outside option, a q-productivity WP firm, is: µVP(w (q),q) = w (q) PP PP +λ [Γ(pˇ)−Γ(q)][E[VP(w (x),x)|q < x < pˇ]−VP(w (q),q)] 1 PP PP (cid:124) (cid:123)(cid:122) (cid:125) job-to-jobtransitiontoaWPcompetitor +λ [Γ ¯ (pˇ)][E[VA(w (q,x),x)|pˇ< x]−VP(w (q),q)] 1 PA PP (cid:124) (cid:123)(cid:122) (cid:125) job-to-jobtransitiontoaSAcompetitor +δ[VU −VP(w (q),q)]. (2.2) PP (cid:124) (cid:123)(cid:122) (cid:125) unemploymentshock Theoptimaltransitionalwagechoiceequatesthevalueofthetwocontracts: VA(w (q,p),p) = VA(w (q),q) for all q < pˇ≤ p and can thus be expressed PA PP as the posted wage at the best-to-date outside offer and the difference in the 10Note that the model is written as if the firm commits to a new flow wage yielding said value. It would be equivalent to write the firm as paying the flow value of leisure almost always and making lump sump payments equal to the innovations in the value of workers’ labor market history when outside offers arrive. This second contract is a poor fit for the world we observe, but illustrates why the firm cannot only temporarily raise the wage and then revert to a lower wage when the competition is “over.” 10

option values of the two employment contracts: w (q,p) = w (q)+ PA PP λ (cid:8) Γ ¯ (pˇ)VP(w (q),q)−[Γ(p)−Γ(pˇ)]E[VA(x,x)|pˇ< x < p]−Γ ¯ (p)VA(p,p) (cid:9) . 1 PP (cid:124) (cid:123)(cid:122) (cid:125) differenceinoptionvaluesintheSAcontractandbest-to-dateWPoutsideoption This expression is surprisingly simple. Notably, both the WP outside option and SA incumbent promise exactly the same schedule of option values when competing against passive WP competitors and in the event of an unemployment shock. Thus, the difference in option values depends exclusively on the difference between how the SA employer and WP best-to-date outside option would compete with SA competitors. Inspection of the option values reveals that the WP firm offers less option value than the SA firm due to its refusal or inability to renegotiate and bid up wages in Bertrand competition. Indeed, the option value of an encounter with a SA firm when employed at a WP firm is null, since the SA firm offers exactly the reservation wage for the transition. The result is that the SA firm is always able to employ workers at wages lower than the wages offered by the best-to-date outside option WP firms: w (q,p) < w (q) for all q < pˇ≤ p. (2.3) PA PP Further, a more productive SA firm offers even greater option value than a less productive one: it is able to promise a greater schedule of on-the-job raises and greater wages following job-to-job transition since it is willing to bid up wages to a larger value commensurate with its productivity. The result is that the more productive the SA firm, the larger the wage cut for any best-to-date outside option: dw (q,p) PA < 0 for all q < pˇ≤ p. (2.4) dp Conditions 2.3 and 2.4, which govern the transitional wage schedule, w , PA are all that is required to prove that a separating equilibrium is a Nash equilibrium under an appropriate cost of SA. Appropriate costs are identified in the following section. 11

2.4 Cost and contract choice If the prescribed contract choice and wage schedule are consistent with Nash equilibrium, then there must not be any deviation that increases current operating surplus for any firm. Current operating surplus for WP firms is pinned down simply as rent per worker times labor supply: πP(p) = E[w|p,P](cid:96)(p) = [p−w (p)](cid:96)(p). Deriving current operating surplus for SA firms requires de- PP riving the mass of their employees earning each wage. Following the usual solution strategy, the mass flowing in and out of such wages must balance. Note that workers willing to accept wage w (q,p) (or w (q,p) if q is a PA AA SA firm) or less must have best-to-date outside option q or less. The mass flowing into such contracts will be Uλ Γ(q) and the mass flowing out must 0 be [δ +λ Γ ¯ (q)](M −U)L(q), which yields (cid:96)(w(q,p)|p) = (cid:96)(q).11 The current 1 operating surplus (exclusive of the cost of SA) for a firm of type p offering the SA contract is: πA(p) = E[w|p,A](cid:96)(p) = (p − w (p,p))(cid:96)(p) + (cid:82)pˇ (p − PA p (cid:82)p w (q,p))d(cid:96)(q)+ (p−w (q,p))d(cid:96)(q). PA pˇ AA Ifpˇisthethreshold, itmustbethecasethatthepˇ-typefirmsareindifferent between contract types and that all less productive firms prefer WP while all more productive firms prefer SA. For threshold productivity in the interior of the support of the productivity distribution, the threshold productivity firm’s willingness to pay for the right to SA is: c = πA(pˇ)−πP(pˇ) = {w (pˇ)−E[w|pˇ,A]}(cid:96)(pˇ). PP The strategies described constitute a Nash equilibrium if, when costs are equal to the threshold firm’s willingness to pay, firms maximize profit by offering the prescribed contract choice and wage schedule. The proof follows from noting that 1) the profit-maximizing WP wage of any firm is an increasing function of productivity, 2) for a given labor market history, SA firms pay a lower wage than WP firms, and 3) for any given labor market history, the SA wage is decreasing in the SA firm’s productivity. The intuition of the proof is summarized in figure 1. The center panel depicts wage schedules as a function 11Note, these results can also be found in Postel-Vinay and Robin (2002a, pg. 999-1001). 12

of the best-to-date outside offer for workers in a threshold productivity firm, p = pˇ, under WP and under SA. Note that the SA schedule is everywhere below the WP schedule. The region between these schedules, appropriately weighted by expected mass of employees with each labor market history, is the threshold firm’s willingness to pay and the cost of the SA contract. The left panel depicts possible wage schedules for a less productive firm: p < pˇ. The prescribed wage schedule, depicted as the solid line, is WP and results in fewer employees than employed in the threshold firm. Two deviations for this firm are also depicted. The firm could deviate to an alternate WP schedule, posting a higher wage and employing additional workers. This deviation yields fewer rents than the optimal WP schedule. The firm could also deviate to a SA schedule, employing these same additional workers at lower cost. The region between these schedules, appropriately weighted by the expected mass of employees with each labor market history, is an upper bound on the willingness to pay the right to SA of this firm with productivity less than the threshold firm. Although this p < pˇ firm could employ the workers more cheaply under SA, it cannot achieve wage cuts as large as the threshold p = pˇ firm because it cannot credibly bid up wages as high as the threshold firm if it were to encounter a SA competitor. Thus, the upper bound on willingness to pay for the less productive firm is strictly less than the cost of the SA contract. The right panel depicts possible wage schedules for a more productive firm: p > pˇ. The prescribed wage schedule, depicted as the solid line, is SA and results in more employees than employed in the threshold firm. A deviation to infinitesimally larger wages than offered by the threshold firm under the WP contractisdepicted. Anydeviationtoalowerpostedwageisdominatedbythis deviation. The region between these schedules, appropriately weighted by the expectedmassofemployeeswitheachlabormarkethistory,isalowerboundon the willingness to pay for the right to SA of this firm with productivity greater than the threshold firm. Note that if a deviation to a larger WP wage were optimal, the region of interest would be larger; thus, the illustrated is a lower bound on the willingness to pay. Since this firm can credibly bid up wages higher than the threshold firm, it can employ workers with equivalent labor 13

market histories at lower wages. Thus, the lower bound on the willingness to pay for the more productive firm is strictly larger than the cost of the SA contract. That a separating equilibrium exists for every choice of the cost of SA follows from the intermediate value theorem. For a fixed set of active firms, the mapping between cost, c, and threshold, pˇ, is unique when marginal wage schedule under WP, w (pˇ), increases more rapidly with respect to the change PP in threshold than the marginal wage schedule under SA, w (q,pˇ). This re- PA quires placing restrictions on how rapidly the value of search rises with threshold productivity. Formal conditions for this criteria are presented in the appendix. Under these conditions the productivity type of the threshold firm is an increasing function of the cost of SA. Formal proofs are in the For Online Publication: Appendix. 3 Comparative statics with respect to contract composition I compare models with mixed contracts of the type just developed when the cost of SA, c, varies. Throughout, I consider a fixed set of transition hazards, {λ ,λ ,δ,µ}, and distribution of productivity, Γ(p).12 The main takeaways 0 1 are 1) flow of labor between employers is constrained efficient in all models; 2) search is more valuable to workers with poor employment histories when more firms WP, which will have implications for the implied flow value of leisure; 3) the distribution of output is more favorable to labor when more firms employ under WP; and 4) wage distributions are more disperse when more firms SA. The third takeaway initially seems at odds with the stated need to generate a model that features WP and a large labor share. I reiterate, the problem is to develop a model that generates large labor share and substantial wage dispersion at the same time. With 3) and 4) established, I pursue the goal of matching observed wage dispersion in the section 4. The intuition that will 12As in both antecedents, both labor share and the value of leisure are larger when the right tail of the productivity distribution is lighter. Also, as in both antecedents, wage dispersion is smaller for productivity distributions with lighter right tails. 14

be established in this section and utilized in that section is that the larger the share of firms employing under SA in equilibrium, the lighter the tail of the productivity distribution required to match any given level of wage dispersion. 3.1 Efficiency In section 2, I established that, in equilibrium, flow of workers between firms is constrained efficient, meaning that an offer from a firm that is more productive than a worker’s current employer is never rejected. This result is somewhat surprisingsinceSAisatypicalframeforthecriticismthatWPisnotsub-game perfect. In such a criticism, deviating to SA enables a firm, in an otherwise WP equilibrium, to retain a worker who would otherwise exit to an alternate employer. Such a deviation would slow the flow of labor to more productive firms. This intuition stands as a critique of the lack of sub-game perfection of the pure-WP equilibrium. However, in the pure-SA equilibrium, constrained efficiency returns. No lower-productivity firm can profitably outbid a higherproductivityfirm,andworkersagainalwaysflowtowardmoreproductivefirms. In the present model featuring both contract types, constrained efficiency arises in the WP and SA sectors and flow between the sectors is efficient. The last occurs because all firms that opt to employ under SA are more productive than the most productive firm which employs under WP. The implication, somewhat unintuitively, is that the supply of labor to a firm of a given type is identicalinallthreemodels(pure-WP,pure-SA,andthemixedcontractmodel considered here) since the flow of labor in and out of firms is, in equilibrium, identical in all three models despite differences in wage-setting strategies. 3.2 Value of search RefertothevalueofemploymentinaWPfirm, equation2.2. Theoptionvalue of receiving an employment offer from a more productive WP firm is positive since posted wages are not contingent on workers’ labor market histories. The optionvalueofreceivinganemploymentofferfromaSAfirmbehavesquitedifferently. Pitting the fully flexible and informed wage setting policy of SA firms against the passive wage setting strategy of WP firm generates no increase in 15

the value of employment upon transition! Thus, the value of unemployment or employment in a WP contract depends only on the distribution of WP firms. Assumingthatw (x)isa differentiablefunction(thisis verifiedin theFor PP Online Publication: Appendix), the value of employment in a p-productivity WP firm is: w (p) λ (cid:90) pˇ Γ(pˇ)−Γ(x) δVU VP(w (p),p) = PP + 1 dwPP(x) + . PP µ+δ µ+δ µ+δ +λ [Γ(pˇ)−Γ(x)] dx µ+δ p 1 Similarly, the value of unemployment is: b λ (cid:90) pˇ Γ(pˇ)−Γ(x) VU = + 0 dwPP(x) µ µ µ+δ +λ [Γ(pˇ)−Γ(x)] dx p 1 The value in both states is larger when more firms operate the WP contract – VP(w (p))/dpˇ> 0 and VU/dpˇ> 0 – since, as the threshold rises, a greater PP share of firms yield rents and the new rent-yielding firms yield large rents. Thus, we have: Claim 1. For a given distribution of firms, as the cost of SA increases, the option value of unemployment rises and the flow value of unemployment consistent with equilibrium falls. Pure-SA has the least option value of unemployment with µVU = b, mixed contract value of unemployment increases in the threshold, and the value of unemployment is maximized under pure-WP. Conversely, one can rank equilibria in order of the flow value of unemployment consistent with a given set of active firms. The ranking falls in the reverse order. In other words, Claim 1 states that if two economies are each described by the same distribution of firms Γ(p), the same set of transition hazards, and different composition of wage contracts, it must be the case that workers in the economy with fewer SA contracts have a lower flow value of unemployment. As we will see, this is at odds with modeling the economy as containing too large a share of WP contracts: with many WP contracts and realistic wage dispersion, the flow value of unemployment must be extremely low. Similar logic and calculations yield: 16

Claim 2. For a given distribution of active firms, as the cost of SA increases, the value of employment with best-to-date outside option a WP firm rises as does the transitional wage from such firms to the SA sector. Aswewillsee,anincreasingshareoffirmsoperatingWPcontractssignificantly impacts the support and shape of the wage distribution, shifting the support upward and increasing the skew of the distribution. 3.3 Distribution of output among factors The model divides output into three shares: current operating surplus, payment for the right to SA, and wages. These can be interpreted in comparison to statistics from national income and product accounts: capital input, intermediate service input, and compensation of employees. Interpretation of wages as compensation of employees is direct. Interpretation of aggregate payments for the right to SA as intermediate service input is consistent with conceiving of these as payment to a human resources or legal department or a head hunter that is responsible for negotiating employment contracts at least cost. Human resources does not produce final goods; rather it insures that final goods are produced by line workers who are employed with least cost. Under the structure of cost for the right to SA modeled in this paper, the human resources department or head hunter is equally effective regardless of the number of employees it is required to manage. Current operating surplus may be interpreted as capital input using the following logic. It is possible to micro-found the distribution of productivity on market-clearing conditions for capital input. Let there be some function f(K) = p, which transforms capital input K into labor productivity p with the usual conditions f(cid:48)(·) > 0 and f”(·) < 0. The capital market is in equilibrium when all firms are indifferent between all active productive technologies: Π = πi(p)−rf−1(p)forallchoicesofproductivityp ∈ [p,p¯]andcontracti ∈ {P,A}, where r is the rental rate of capital. Since πi(p) is continuous and increasing on the intervals [p,pˇ) and [pˇ,p¯] and πP(pˇ) = πA(pˇ), a function f(·) exists that is consistent with distribution of productivity Γ(p) (or visa versa). With free entry, Π = 0 and πi(p) = rf−1(p) for all productivities and both contract 17

types. Here I do not explicitly micro-found the capital market. However, notingthatsuchamarketcanbewritten, Iinterpretcurrentoperatingsurplus, πi(p), as rents payed to capital, rf−1(p), in the un-modeled capital market. I consider the distribution of output between capital, intermediate service, and labor for a fixed distribution of active productivity types, Γ(p), under different costs of SA. Throughout, I assume that the distribution of productivity, Γ(p), meets the condition for uniqueness of equilibria, and thus that threshold productivity, pˇ, is increasing in the cost of SA, c. Current operating surplus (capital input) Claim 3. As the cost of SA increases, the share of output paid to capital falls. This is straightforward to show. Consider a SA firm under an initial, low cost of SA. When costs rise the firm has two options: 1) continue to employ under SA but pay a larger share of output to intermediate service or 2) switch to WP, saving the cost of SA but paying a larger fraction of output to labor. Regardless of the firm’s decision, each firm that employed under SA under the initial low cost earns strictly less current operating surplus under high costs. Meanwhile, the current operating surplus of firms that employed under WP, even under low costs, are unaffected. In aggregate, firms earn less current operating surplus under high costs of SA. Interpreting current operating surplus as payments to a capital input, as described above, gives the result. Aggregate spending on cost of SA (intermediate service input) Claim 4. Aggregate spending on cost of SA (intermediate service input) is zero when all firms WP (pˇ= p¯), zero when all SA (pˇ= p), and maximized in some mixed-contract equilibrium. As the cost for the right to SA increases, some firms change wage-contracting strategies, so fewer firms pay the cost. The result arises immediately. The first follows since when all firms WP (pˇ = p¯), none pays any cost and aggregate payments are thus zero. The second follows since pˇ= p when c = 0, all firms pay a zero cost and the aggregate payment is zero. 18

Compensation of employees (labor input) Claim 5. As the cost of SA, c, increases, the share of output paid to labor rises. As the cost for the right to SA increases, more firms offer WP contracts that immediately yield rent to workers. These firms, obviously, operate larger wage bills when the cost of SA increases. On the other hand, firms that employ under SA come into Bertrand competition with other SA firms less often, leading to a drop in their wage bill, since wages are lower for workers whose best-to-date outside option no longer competes via SA. However, this drop is counteracted by a simultaneous increase in the wage schedule for hiring from WP firms. This increase arises since a larger share of WP firms implies a larger value of search in unemployment and in WP firms. The second effect dominates the first. The proof, which also shows that the wage bill rises within every firm, is provided in the For Online Publication: Appendix. 3.4 Distribution of wages Increasing the cost associated with SA, and therefore increasing the fraction of firms that select WP, decreases the spread and increases the skew of the distribution of wages. The impact is most starkly evident in the support of the distribution. Measuring wage dispersion as the spread between w and w¯ models can be ranked by increasing threshold: pure-SA has the most dispersion, mixed contract dispersion decreases in threshold, and pure-WP with the least dispersion. Every SA firm offers some wages smaller than the wage offered by the least productive WP firm: w (p,p) < w (p) for all p ≥ pˇ. Every SA firm PA PP also offers some wages larger than the most productive WP firm: w (p,p) < AA w (pˇ) for all p. Additionally, in all equilibria where some firms SA, both the PP largest and the smallest wages are paid by the most productive firm under the SA contract operated by this firm. Upper bounds, w¯, of implied wage distributions under equilibria can be ranked as follows: 19

• For pˇ< p¯ the upper bound of the implied wage distribution is equal to the upper bound of the productivity distribution: w¯ = p¯. When two p¯ productivity SA firms meet the resulting wage is p¯, and if such firms have positive probability, the interaction must also occur with some probability; thus the resulting wage must also have positive mass in the wage distribution.13 Lower bounds, w¯, of implied wage distributions under equilibria can also be ranked as follows: • As cost of SA, c, increases the lower bound of the wage distribution, w = w (p,p¯), increases. UA Remember that the value of unemployment is increasing in the fraction of firms that hire under WP. Since entry wages in SA firms are chosen to make workers indifferent between unemployment and accepting the wage offer, entry wages in each SA firm rise: dw (p)/dpˇ > 0. This is true, in particular, for UA the most productive SA firm, and this firm’s entry wage is the smallest wage in the economy: w = w (p¯). UA Expressions for the density of wages give a set of hazards {λ ,λ ,δ,µ} 0 1 and productivity distribution Γ(p) are provided in the For Online Publication: Appendix. Figure 2 plots wage distributions implied by a pure-SA (top), pure- WP (bottom), and mixed-contract models (middle three). The mixed contract models feature 25, 50, and 75 percent WP firms, respectively. These are decomposed into wages set under WP (or unemployment), wages set under SA withbest-to-dateoutsideoptionWP,andwagessetafterBertrandcompetition between two SA firms. As the fraction of WP firms climbs from zero to one, the distribution becomes more compressed. Compression is localized in the left tail, where 13Ifpˇ=p¯theupperboundoftheimpliedwagedistributionisstrictlylessthantheupper boundoftheproductivitydistribution: w (p¯)=w¯ <p¯. Thisfollowsfromobservingthatif PP allfirmsWPthenthelargestpostedwageisstrictlylessthanthelargestproductivity. This follows from the concavity of the posted wage schedule. See the appendix for the functional form of the posted wage schedule, w , and Bontemps et al. (2000) and Mortensen (2003) PP for further details. 20

increasing value of unemployment and employment in WP firms, due to increasing size of the WP sector, puts upward pressure on entry and transitional wages in SA firms. The result is increased skew of the distribution. 4 Empirical performance An econometrician or policy maker seeks a model that fits an observed distribution of wages and unemployment durations. The econometrician may also have auxiliary data on the relation between aggregate output and compensation of employees from national income and product accounts, and a notion of the generosity of social insurance with which the flow value of unemployment ought to be bounded. Therefore, from the econometrician’s standpoint, models ought to be compared on the plausibility of the implied distribution of productivity and implied flow value of unemployment when the observed empirical distributions of wages and (un)employment durations are matched. As demonstrated in the previous section, the mixed contract model presents a tradeoff between wage dispersion on the one hand and value of leisure and labor share on the other hand. The econometrician or policy maker can harness this tradeoff and use it to select a distribution of output that is both consistent with mirco data on workers (wage dispersion and transition rates) and aggregate and/or policy relevant moments (labor share and leisure value). SimulationstargetempiricalobservationsforGermanyin2006. Beforeproviding details of the data and method, one important choice should be highlighted. I target raw wage dispersion rather than residual wage dispersion. I acknowledged that individual characteristics account for some wage dispersion and that the exante identical workers who populate the model abstract from this. However, explicitly modeling worker types and, more importantly, transition between these is beyond the scope of the present work.14 Raw wage 14Considerthattheusualdecompositionbyindustryoroccupationmaskssignificanttransitionbetweenthesethatisunmodeled. Forexample: doesaworkerwhochangesoccupation changetypesandpopulateadifferentsegmentofthelabormarketorisoccupationalchange a feature described by the job ladder? An extreme interpretation of the results presented here would be to take the view that all sectoral or occupational changes are captured as “rungs” on the job ladder. I prefer to interpret my results as an illustration of the (large) 21

dispersion presents an upper bound on the inequality a model need achieve. That simulations presented here attain a good match while targeting raw inequality is indicative of capacity to exceed the level of residual inequality. The conclusion, contrary to some standing results in the literature, is that reasonable levels of inequality are well within the reach of a random search model even when respecting national accounting data and leisure value consistent with social insurance. Further, my results indicate that these are attainable even when a large majority of firms employ under the WP contract. Finally, as the amount of wage dispersion the model need supply decreases, the share of firms employing under the WP contract that can be consistent with the data increases. 4.1 Data Data come from the Sample of Integrated Labour Market Biographies (SIAB) from the Research Data Center (FDZ) of the German Federal Employment agency at the Institute for Employment Research (IAB). The SIAB is a 2 percent sample drawn from the population of individuals in Germany who are employed and subject to social security, marginal part-time employed, a benefit recipient to the German Social Code II, official registered as a jobseeker at the German Federal Employment Agency, or participate in programs of active labor market policies.15 I select 2006 as a relatively stable period that postdates the Hartz labor market reforms and predates the financial crisis. I calibrate the model to match labor market histories of West Germans. I select persons age 20 to 60 who were employed full time in the first 7 days of 2006.16 amount of dispersion that can be feasibly generated from a random search model with contractheterogeneity. Decompositionofdispersionintofrictionalandneoclassicalcomponents is the subject of a related active literature. 15See Dorner et al. (2010) for a full description of the data. 16I restrict the sample to West Germany to facilitate simulation. As will be noted, data arelimitedbycensoringatthevalueofthemaximummandatedsocialsecuritycontribution. ThisdiffersbyregionwiththestatesthatmakeuptheformerGermanDemocraticRepublic subjecttolowermaximumcontributions. Whilenationalaccountingdataareavailableonly for unified Germany, micro data on value added and wage bills indicate small differences in labor share across regions, with workers in West Germany capturing only minutely more of the surplus than in East Germany. See Doniger (2014) for further information. 22

From the cross section of workers employed in the first 7 days of 2006, I compute moments of the wage distribution. Wages contained in the SIAB data are daily wages computed as total pay during a registered employment spell divided by total days worked. Restricting the data to full-time workers minimizes the impact of differential hours on this measure of wages. Wages in the SIAB data are censored at the maximum mandated contribution to social security, 172 Euros in 2006. Mean and variance of this distribution are presented in Table 1. 8.56 percent of wages are censored. To account for this, I match moments from censored data to moments from analogously censored simulated data. From longitudinal data on the same sample, I compute the average duration of the initial employment spell and initial pay level, the fraction experiencing no job change, job-to-job transition, temporary job separation or persisting employment, and the fraction who experience nominal pay gain onthe-job. Job-to-job transitions are defined as transitions between employers with no greater than 7 days of nonemployment intervening and no registration of unemployment with the Employment Agency.17 Empirical moments are presented in Table 1. Censoring biases the duration of initial pay level and incidence of nominal pay gain. Again, I address this by matching moments of censored simulated data to censored empirical moments. In evaluating model performance, I exploit auxiliary estimates of labor share, share of firms that WP, and income replaced by social insurance. The first, labor share of 63.7% in 2006, comes from the EU KLEMS Growth and Productivity Accounts (O’Mahony and Timmer, 2009). The second, share of WP, firms is estimated as 62 percent by Brenzel et al. (2013) from a 2011 survey of German employers. To my knowledge, this is the only estimate for Germany outside of the present paper. The final, leisure value of 30%, represents a lower bound calculated from the cross section of SIAB data in January 2006, and is likely a severe underestimate. Shimer (2005) puts the figure at 41 percent. Subsequent work in the macro-search literature suggests 17Thisdefinitionissomewhatmorerestrictivethantypicallyappliedintheliterature. See Doniger (2014) for a discussion of alternate definitions of pay gain and job transition and checks of robustness to these. 23

substantially higher values; for example, Hall and Milgrom (2008). 4.2 Estimation method In estimation, I model productivity as distributing according to a Pareto distribution, Γ(p) = 1−(p/p)α, with unknown shape, α, and scale, p. Parameters of the Pareto distribution also have convenient interpretation. The shape parameter, α, is inversely related to the density in the tail of the distribution. Meanwhile, with free entry the scale parameter pins down the reservation wage of the unemployed for employment in the WP sector: w (p) = p. The UP functional form assumption is not innocuous, but selection of the Pareto distribution is supported on both theoretical and empricial grounds.18 Further, up to a minor adjustment to account for censoring parameters {λ ,λ ,δ,µ,s } 0 1 f can be identified without any knowledge of the distribution of firm types, and estimates of these five parameters obtained without modeling productivity do not differ meaningfully from estimates obtained when all seven parameters are estimated jointly. Parameters to be estimated are the hazards of job offer arrival, separation, andexit; theshareoffirmsemployingunderSA;andtheshapeandscaleofthe productivity distribution: {λ ,λ ,δ,µ,s ,p,α}. I estimate these via the Simu- 0 1 f lated Generalized Method of Moments, minimizing the sum of squared percent differences between empirical moments regarding job mobility, pay changes and mean and variance of wages (summarized in table 1), and analogues in simulated labor market histories for 10,000 workers. Specifics and further data description are in the For Online Publication: Appendix. Broader analysis, including analysis of robustness to coding of job mobility and pay change and discussion of the (de)merits of using linked employer-employee data, can be found in Doniger (2014). An advantage of the simulation methods employed is the ability to be agnostic about the relation between modeled productivity 18On theoretical grounds, the Pareto distribution is consistent with a balanced growth path for growth driven by technology adoption Perla and Tonetti (2014); Kortum (1997); Jones (2005) and Eaton and Kortum (1999). Empirically, Gabaix (2009) give evidence that firm size distributes Pareto, and ? give evidence productivity across industries and time distributes Pareto. 24

and measured firm-level characteristics. All that is required for identification is that there exists some common ranking of firms such that when presented with the opportunity to move from a firm of type q to a firm of type p, all workers do so whenever q < p. 4.3 Results Table 2 reports parameter estimates. I estimate that 73 percent of firms employ under the WP contract. Together with estimated hazard of on-the-job offer arrival, λ , of 0.2129, and separation and exit hazards, δ and µ, of 0.0993 1 and 0.0098, this implies that 47 percent of workers are employed with a nonnegotiable wage contract. This estimate is broadly in line with the estimate of Brenzel et al. (2013) using survey data from 2011.19 With these estimated parameter values the model produces labor share equal to 54 percent and a ratio of the flow value of nonemployment to the average wage of 71 percent, significantly outperforming the nested pure contract equilibria. Recalibrating the model such that hazard parameters are unchanged but all firms are constrained to SA produces labor share in this range but a value of leisure that is nearly as large as the average wage. Meanwhile, repeating the same exercise but constraining all firms to WP produces a labor share of only 23 and a large and negative value of leisure. Recall, that the EU-KLEMS database reports German labor share as 63.7 in 2006. My own estimates suggest a value of leisure well in excess of 30 percent of average wages, while Shimer (2005) suggests a value closer to 40 percent and Hall and Milgrom (2008) suggest a value in excess of 70 percent (both for the United States).20 For robustness, I repeat this exercise an additional time constrain- 19Note that the model implies a strict ranking: the share of firms with WP is always less than the share of workers with WP (except in the pure-WP equilibrium). Survey data summarized in the introduction appears to contradict. Note, however: 1) surveys are not conducted contemporaneously and the two U..S. surveys are not designed for comparison. 2) As Hall and Krueger (2012) note, even if a worker is employed in a negotiable contract it is not necessary that actual negotiations have occurred and thus no negotiation may be reported. Unfortunately, no survey asks a prospective question about negotiating to employees. 20IfocusonWestGermanyinordertoabstractfromdifferencesinEastandWestGerman wagescalesthatarenotaddressedbythemodel. Allstatisticsotherthanlaborsharereflect 25

ing the share of WP firms to 62 percent as estimated by Brenzel et al. (2013). Constraining the model to this share of WP firms requires a productivity distribution with a lighter tail and results in a slightly higher labor share, 58 percent, and a larger ratio of the flow value of leisure to average wages, 84 percent. These relations persist broadly over the range of shares of WP firms. Table 3 records the scale and shape parameters required to achieve wage dispersion comparable to that observed for a range of possible shares of WP contracts while constraining hazards to the estimated values. The table also presents the implied cost of the SA contract, labor share, and ratio of the flow value of leisure to average wages. Costs are expressed in Euro valued terms, as percent of aggregate, and as percent of output in the threshold firm. That costs are increasing and take an increasingly large share of the threshold firm’s output whenthethresholdrisesisstraightforward. Themagnitudeofcostsissensitive to the choice of the size of the mass of workers, M, as compared to the size of the mass of firms, N. In the simulation the ratio is calibrated as 1. A larger ratiowouldproducelargerfirmsizesandthuslargeraggregatecosts. However, a larger ratio also entails larger output. In simulations the fraction of output that must be paid to service the cost of SA never exceeds eight percent. As the portion employing under wage posting approaches 100%, the required distribution of productivity has an increasingly heavy tail (the shape parameter falls substantially). Implausibility of such a heavy tail is captured by implausibly low labor share, also documented in Table 3. In contrast, the tail remains quite light for compositions of contracts up to seventy percent WP. The difficulty is not the presence of WP, rather it is the presence of WP in the tail if the distribution. Table 3 also records the ratio of the value of leisure to the average wage. When nearly all firms select WP, the best-fit simulation is unable to produce substantial value of leisure. The difficulty arises from the supposition that thefocusonWestGermany. SurveyevidenceanalyzedinDoniger(2014)suggeststhatlabor share is not substantially different in the two regions. 30 percent replacement rate reflects the ratio of the average unemployment benefits to average wages paid in the first week of January 2006. Due to selection on who becomes unemployed and nonpecuniary value of leisure, this 30 percent forms a very cautious lower bound. 26

starting wages may be drawn from the full empirical distribution, resulting in an implausibly large option value of search for the unemployed. Inclusion of a SAsectormitigatesthisproblem: largewagesariseonlyafteron-the-jobsearch and competition for a worker by two firms employing under SA. The result is that the simulated distribution of posted wages is more compressed, resulting in a more reasonably sized option value of search while unemployed. The ratio of the flow value of leisure to average wages is problematic for both extremes, largely WP and largely SA. In the largely WP case, the problem is again the presence of WP in the tail: the possibility of receiving such wage offers out of unemployment coupled with short unemployment spells contradict a large flow value of unemployment. In the other extreme, when many firms employ under SA the problem stems from the full rent extraction assumed. With flow value of leisure very near to or higher than the average wage, the question arises: why are the unemployed searching for work at all? Inclusion of the WP sector supplies a search incentive, and for intermediate mixtures the value of leisure is a (large) proportion of the average wage.21 Insum, thecalibrationdemonstratesthatthemixedcontractmodelcanattain labor share at or near the observed value and reasonable value of the flow value of unemployment when matching data on employment transitions, incidence of wage increases, and on raw wage dispersion. Further, this is possible for homogeneous workers with null bargaining power and linear preferences, suggesting that this model, as well as more nuanced versions, are capable of matching any level of residual wage dispersion, in particular whatever portion for which search frictions are accountable. 5 Related literature 5.1 Nested models and empirical performance The on-the-job search model laid out in Burdett and Mortensen (1998) propels random search as a contender for equilibrium analysis of inequality and 21An alternate and complementary solution would be to model workers with preferences that reduce the option value of search: increased discounting and/or risk aversion. 27

policy in labor markets. The model provides a solution to the Diamond (1971) paradox / ? critique that does not require ex-ante heterogeneity to generate disperse wages and search incentive. This feature makes on-the-job search a particularly fruitful backdrop for welfare analysis. WP models describe a labor market in which informational or negotiation failures prevent firms from extracting full labor rents. A policy-relevant insight from such models is that since firms’ hands are bound in negotiations they overreact to policy, shifting wages and/or hiring more radically than if firms were fully informed or fully flexible. For example, when uniform unemployment insurance becomes more generous, the entire distribution of wage offers shifts toward higher values. As a result, changes in worker incentives affect not only what job offers are acceptable but also the schedule of offers that are available. Unemployment responds both to the set of firms that remain active in the market and to the new schedule of wage offers made by those firms. The baseline random on-the-job search model with pure WP is difficult to rationalize against data into two main features: 1) the magnitude of the value of unemployment and search behavior of the unemployed when generating significant wage dispersion and 2) the relation between the distribution of output and the distribution of wages. A conclusion one might draw from these difficultiesisthatsearchfrictionsimplycannotexplainarelevantshareofwage dispersionandthat, instead, dispersionmustderivefromex-anteheterogeneity in worker’s types or preference. Such a conclusion casts doubt about the relevance of policy insights dependent on information asymmetries modeled. The results of this paper suggest instead that the introduction of (realistic) heterogeneity in the contracting mechanism can generate a model that can produce wage dispersion as large as observed raw wage dispersion without any ex-ante heterogeneity among workers. In other words, information frictions captured in this (type of) search model can capture an enormous amount of (policy relevant) dispersion in the price of labor, restoring inquisition into the policy implications of WP contracts as a relevant and important line of research. TheSAmodelofPostel-VinayandRobin(2002b)providesasolutiontothe Diamond (1971) paradox / ? critique with a very different flavor. The poten- 28

tial to generate competition between employers inspires workers’ search on the job. Anticipation of future competition factors into workers’ expected payoff of employment at any given firm. As a result, the model generates significant heterogeneity in wages within firms, with workers low on the within-firm wage ladder being compensated largely by the value of future competition. Within firm wage heterogeneity leads to significant wage dispersion in aggregate, with the distribution extending from wages equal to the marginal product of the most productive firm to wages that are actually less than the flow value of unemployment. Although, for a fixed distribution of productivity, output dominates wages more severely than in the WP model (Postel-Vinay and Robin, 2002b), greater dispersion inherent in the model facilitates estimation of a productivity distribution, with a lighter tail making the model consistent with labor share (Postel-Vinay and Robin, 2002a). Meanwhile, this dispersion does not contradict unemployment-to-employment transition data because wages in the right tail are only attainable via job-to-job transition. However, in the baseline SA model, firms extract full rents from the unemployed, leaving them with no reason to search.Full rent extraction detaches unemployment from the mechanism of the model and leaves it unable to address questions of extensive and intensive inequality jointly. The problem of providing a search incentive to the unemployed can be mitigated by introducingworker’sbargainingpowerasinCahucetal.(2006), andonecouldconsider endogenizing the matching function. Still, these solutions forgo the possibility that the set of opportunities faced by workers may shift in response to policy in a way more profound than left censoring. The results here demonstrate that the benefits of the SA model, in terms of abilitytogeneratedispersion,canbeharnessedevenwhenthemajorityoffirms employ under WP, suggesting the mixed-contract model as a viable context for welfare analysis in the context of information frictions being binding on a large fraction of employment contracts. 29

5.2 Optimal contracting with on-the-job search Optimal contracting in the context of on-the-job search is an active literature important in its own right, independent of questions of goodness of fit.22 A key insight, as noted earlier, is that different contracts provide differential search and transition incentives to employees. Matching outside offers rewards and incents search on-the-job. On the other hand, failure to match leads to potentially preventable turnover. The model presented here is most closely related to Postel-Vinay and Robin (2004) and Holzner (2011). Each pursues the question of optimal contract type in a similar environment as considered here. An important distinction, however, is that each explicitly models workers’ search incentive and the implication of firms’ contract choices for search intensity and turnover. The intuition pursued in these papers is that SA rewards workers for on-the-job search and thus might provides greater incentive for search effort. This argument is clearly true at the level of an individual firm – the complication is to find a setting in which differential incentive for search effort is global, with search always being more attractive in the SA sector than the WP sector. As noted in footnote 1, to gain a handle on the problem, each paper simplifies some elements of the setting. Postel-Vinay and Robin (2004) require all WP firms to post identical wages, thus eliminating search incentive entirely in the WP sector. Holzner (2011) restricts the set of firm types to simply a high type and a low type. Without such extreme assumptions, search effort muddies the water and a tradeoff between lower average wages and higher costs micro-founded by search effort and turnover is not clear. As noted in both papers, the problem quickly becomes intractable. By abstracting from search effort, I obtain a clear-cut prediction here. The question of what the optimal wage setting mechanism is remains. Unfortunately, as the product search literature reveals, answers are largely model driven. Choice of search mechanism, competing wage-setting mechanisms, and which actions/parameters are endogenized yield different results. 22Someprominentandsomenewercontributions: Stevens(2004),Postel-VinayandRobin (2004), Burdett and Coles (2010), Holzner (2011), and Lentz (2014) 30

5.3 Broader literature The question of which heterogeneous pricing mechanisms (or combinations thereof) are optimal is an active one in the product search literature.23 The literature regarding search with heterogeneous wage setting, however, is small. Ellingsen and Ros´en (2003) consider a market in which unemployed workers have heterogeneous outside option and search for a job and firms may choose to bargain or post a nonnegotiable wage offer. With wage posting only, this setting imposes a tradeoff on firms – low wages extract more rent but may be rejected and thus lead to lower odds of matching. Meanwhile, bargaining allows firms the flexibility to hire all profitable applicants at a cost of higher expected wages. Ellingsen and Ros´en (2003) characterize the situation - market tightness and degree of worker heterogeneity – which induces all firms to bargain. Michelacci and Suarez (2006) come at the problem from the perspective of directed search. Again, wage setting affects the odds of matching with a high-type worker; however, in Michelacci and Suarez (2006), the mechanism is different. Bargaining enables firms to reward skills that cannot be contracted upon. Posted commitment to bargaining thus attracts higher-skilled applicants. Michelacci and Suarez (2006) give conditions under which some, but not all, firms bargain. These authors observe that more flexible contracts imply more inequality. The work presented here captures these insights: flexible contracts allocate a greater share of surplus to firms and induce greater dispersion in outcomes for workers. These results suggest a line of inquiry into trends in both labor share and inequality experienced in developed countries in the last several decades. The present work goes further. In particular, the equilibrium described hereinteractswithworkers’on-the-jobsearchinaninterestingway. Asworkers climb the job ladder, they are more likely to be employed in a SA firm. An implication is that as workers gain (continuous) employment experience, onthe-job pay gain becomes an increasingly important source of wage growth as compared to job-to-job transition. This result is consistent with empirical 23A nonexhaustive list of examples: Bester (1993), Arnold and Lippman (1998), and Camera and Delacroix (2004). 31

evidence (Topel and Ward, 1992) but suggests a new interpretation of patterns in job mobility and wages over the life cycle. 6 Conclusion This paper offers a model of on-the-job search in which firms select between hiring under a WP and a SA contract. The mechanism described, a fixed perfirm cost for the right to hire under SA, is quite elegant. This cost structure enables me to characterize the equilibrium under any continuous distribution of productivity and for optimal WP strategies. The equilibrium described is separating, with some intermediate firms being indifferent between contract types, all less productive firms selecting WP, and all more productive firms selecting SA. Compared with its nearest neighbors, Postel-Vinay and Robin (2004) and Holzner (2011), the model presented here accepts general productivity distributions and allows for a WP sector in which wages are optimally set by firms and, more importantly, in which wages are disperse, relaxing the restrictions on wage-setting strategy imposed by Postel-Vinay and Robin (2004) and on the domain of firm types imposed by Holzner (2011). Further, my model nests the continuous productivity case of the pure-WP with on-the-job search (Bontemps et al., 2000) and the pure-SA model (Postel-Vinay and Robin, 2002b) as limiting cases. These results are achieved by accomplished by abstracting from the micro-foundations of the cost structure and instead directly exploiting a single, separation-driving mechanism: the differential in wage bills for like-firms employing under different wage contracts. The cost structure and the mechanism that it isolates could be extended to include more complex contract types. Particularly interesting candidates are tenure-contingent contracts. In principle, as long as flows between sectors of thelabormarketareconstrainedefficient, itshouldbepossibletofindasimilar separatingequilibriumevenwithtenure-contingentcontracts. Theequilibrium plausibly features three sectors, with the least productive firms populating the WP sector, moderately productive firms offering tenure-contingent contracts, and the most productive firms offering fully contingent SA contracts. I leave 32

full exposition of such an equilibrium to future work. In intermediate cases of the model, when both contract types coexist, one canmatchemploymenttransitionsandwagedispersion, whichcanbeobserved by a policy maker or econometrician, while at the same time the implied productivity dispersion does not contradict observed labor share or plausible values of the flow value of unemployment. This is novel for a model that features WP. Further, since the value of search when unemployed depends only on the distribution of firms in the WP sector, and wages in this sector need not be as disperse as the aggregate wage distribution, a large flow value of leisure is no longer at odds with moderate unemployment duration. Also, in intermediate cases featuring both contract types, distributions of wages are more positively skewed when more firms WP, mitigating a shortcoming of the pure-SA model. The main alternatives – increasing workers’ impatience or risk aversion or introducing explicit bargaining power for workers – are substitutable (Postel-Vinay and Robin, 2002a; Cahuc et al., 2006). When a mixture of contracts is considered, the model can be consistent with data for patient workers with linear utility and no bargaining power. I demonstrate that the model can match labor market histories and cross-sectional wages of Germans without being at odds with data on labor share or the share of income replaced by social insurance. These features make the new model an excellent candidate for welfare analysis of labor market policy and social insurance. The model captures features of wage posting: in particular that policy impacts both the set of acceptable jobs and the characteristics of offered jobs. Particular policy changes of interest include manipulation of need-based social insurance such as food stamps, labor history contingent social insurance such as unemployment insurance, and minimum wages. Full analysis of such policy changes is reserved for future work. 33

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Figure 1: Prescribed strategies and best deviations Prescribed WP Threshold Prescribed SA p<pˇ p=pˇ p>pˇ 110 110 110 best wage deviation schedule deviation with WP under WP to WP 100 100 100 ↓ ↓ ↓ ↑ ↑ prescribed 90 90 90 prescribed wage wage schedule schedule ↓ 80 80 80 ↑ 70 best 70 70 deviation ↑ with SA wage 60 60 schedule 60 under SA contract type of contract type of contract type of best−to−date best−to−date best−to−date 50 50 50 outside offer: outside offer: outside offer: ← WP SA → ← WP SA → ← WP SA → 90 100 110 90 100 110 90 100 110 Productivity of best−to−date outside offer Note: Center panel: wage schedules as a function of the best-to-date outside offer for workers in a threshold productivity firm, p = pˇ. The region between these schedules, appropriately weighted by expected mass of employees with each labor market history, is the willingness to pay of the threshold firm and the cost of the SA contract. Left panel: possible wage schedules for a less productive firm: p < pˇ. The firm could deviate to an alternate WP schedule, posting a higher wage and employing additional workers. This deviation yields fewer rents than the optimal WP schedule. The firm could also deviate to a SA schedule, employing these same additional workers each at lower cost. The region between these schedules, appropriately weighted by the expected mass of employees with each labor market history, is an upper bound on the willingness to pay the right to SA. Right panel: possible wage schedules for a more productive firm: p > pˇ. A deviation to infinitesimally larger wages than offered by the threshold firm under the WP contract is depicted. The region between these schedules, appropriately weighted by the expected mass of employees with each labor market history, is a lower bound on the willingness to pay for the right to SA. 37

Figure 2: Wage distributions 0.04 0.03 0.02 0.01 0 0 100 200 300 AS eruP Decomposition by wage setting mechanism 0.04 0.04 0.04 0.02 0.02 0.02 0 0 0 0 100 200 300 0 100 200 300 0 100 200 300 WP v. WP Unemp v. SA SA v. SA 0.04 0.03 0.02 0.01 0 0 100 200 300 PW 4/1 ,AS 4/3 Decomposition by wage setting mechanism 0.04 0.04 0.04 0.02 0.02 0.02 0 0 0 0 100 200 300 0 100 200 300 0 100 200 300 Unemp v. SA WP v. SA Unemp v. SA 0.04 0.03 0.02 0.01 0 0 100 200 300 PW 2/1 ,AS 2/1 Decomposition by wage setting mechanism 0.04 0.04 0.04 0.02 0.02 0.02 0 0 0 0 100 200 300 0 100 200 300 0 100 200 300 Unemp v. SA WP v. SA Unemp v. SA 0.04 0.03 0.02 0.01 0 0 100 200 300 PW 4/3 ,AS 4/1 Decomposition by wage setting mechanism 0.04 0.04 0.04 0.02 0.02 0.02 0 0 0 0 100 200 300 0 100 200 300 0 100 200 300 Unemp v. SA WP v. SA Unemp v. SA 0.04 0.03 0.02 0.01 0 0 100 200 300 PW eruP Decomposition by wage setting mechanism 0.04 0.04 0.04 0.02 0.02 0.02 0 0 0 0 100 200 300 0 100 200 300 0 100 200 300 WP v. WP WP v. SA SA v. SA Note: Distributions of wages for a fixed distribution of output for different proportions of posting and countering firms. The distribution is decomposed based upon whether the current employer or best-to-date outside option employees under SA. The wage distribution is more positively skewed when the WP sector is larger. The wage distribution is more disperse when the SA sector is larger. 38

Table 1: Empirical Moments Wage distribution (2006 Euros) Mean 98.04 Variance 39.70 Duration (in years) of ... Employment .9235 Nonemploymenta .0348 Initial pay .9063 Fraction of workers with ... Job staying .8421 Job-to-job change .0581 Job change w/ unemp. .0532 Job losing .0467 On-the-job raise .0352 Wage distribution (2006 Euros) Mean 98.04 Standard deviation 39.70 Observations 268,721 aDuration of unemployment is coded to nil if no unemployment spell is experienced. Table 2: Estimated Parameters Share of WP firms : s 0.7267 (0.0666) f Unemployed offer arrival : λ 1.7832 (0.4719) 0 Employed offer arrival : λ 0.2129 (0.0015) 1 Job losing : δ 0.0993 (0.0060) Quit to nonparticipation : µ 0.0098 (0.0052) Reservation productivity : R 95.07 (0.7019) Shape of productivity distribution : α 2.6101 (0.1429) k = λ /(δ+µ) 16.3446 0 0 k = λ /(δ+µ) 1.9514 1 1 Share of WP workers : s = s /(1+k (1−s ) 0.4739 w f 1 f Note: Standard errors in parentheses. 39

rednu segaw naem ot erusiel fo eulav wofl fo oitar dna erahs robal elbisaef :noitarbilaC :3 elbaT .stcartnoc fo snoitisopmoc gnireffid noitcuA laitneuqeS fo tsoC eulav wofl fo oitaR fo % fo % smriF fo tnecreP tnemyolpme-non fo tuptuo etagergga 6002 ni gnizilitU gaw egareva ot erahS robaL mrfi ˇp ni tuptuo soruE epahS elacS gnitsoP egaW 699.0 755.0 0 0 0 42.3 69.001 0 599.0 365.0 17.5 71.1 53.2 42.3 59.001 01 299.0 275.0 16.01 23.2 02.5 42.3 97.001 02 189.0 085.0 59.41 84.3 98.8 42.3 74.001 03 169.0 585.0 46.81 16.4 57.31 02.3 39.99 04 329.0 785.0 97.12 56.5 55.02 21.3 90.99 05 168.0 875.0 5.42 55.6 47.03 59.2 97.79 06 257.0 355.0 10.72 2.7 80.84 76.2 58.59 07 935.0 974.0 9.92 24.7 28.88 71.2 09.29 08 690.0 363.0 51.92 61.6 40.391 46.1 48.97 09 557.0- 592.0 001 0 ∞ 92.1 78.94 001 .ynamreG tseW fo noitalupop )elamef dna elam( deloop eht fo stnemom tegrat snoitalumiS :etoN PW eht sA .tnecrep 04 yletamixorppa ot pu stcartnoc PW fo serahs rof egral sniamer retemarap epahS :etoN eht troper snmuloc retneC .etar gnisaercni na ta sllaf retemarap epahs eht ,tniop siht dnoyeb sesaercni erahs dlohserht eht ni tuptuo fo erahs a sa dna tuptuo etagergga fo erahs a sa ,sorue ni tcartnoc AS eht fo tsoc deilpmi llams morf sesaercni smrfi PW fo erahs eht sa sesir erahs robaL .tuptuo fo tnecrep 5.7 deecxe reven stsoC .mrfi ytivitcudorp elbats ylevitaler a htiw tnetsisnoc ,epyt tcartnoc hcae fo tnecrep 05 dnuora dezimixam si dna serahs retemarap epahs eht sa etar gnisaercni na ta sllaf erahs robal ,tniop siht dnoyeB .noiger siht revo noitubirtsid .noitarbilac PW-erup eht ni eulav evitagen a ta pu gnidniw ,sknirhs 40

ONLINE APPENDIX A Proofs and Derivations A.1 Further details of the proof of Proposition 1 A.1.1 Constrained efficient labor flows The largest possible WP wage is, at most, equal to the productivity of the threshold firm. In the proposed separating equilibrium, the least productive SA firm has larger productivity than the most productive WP firm. So, each SA firm can profitably pay a wage at least as large as the largest possible WP wage and the SA contracting mechanism permits the firm to offer such a wage. Therefore, every SA firm must hire employees of every WP firm, if it meets them, in equilibrium, and the flow between sectors is efficient.24 Within the WP sector, firms continue to behave as if the whole economy posted wages, and so flows are efficient. The presence of SA firms does not alter the solution to the WP firms’ maximization problem since 1) profitable wage choices are only competitive against other WP firms and 2) there is no mass of WP firms at the threshold productivity, since Γ(p) is continuous. The proof due to Burdett and Mortensen (1998, pg.268) applies: more productive firms can employ workers of less productive firms at trivially greater wages and at greater profits. Finally, within the SA sector, more productive firms are still able to outbid less productive firms and so the flows are efficient. In total, flows of employed workers in the separating equilibrium are efficient. A.1.2 Separating Nash equilibrium for {c,pˇ} The proposed separating equilibrium is a Nash equilibrium of the labor market ifeachfirm prefers theprescribedwagecontractand wage schedule conditional on all other firms playing the assigned contract and wage schedule and labor flowing toward more productive firms. To prove this, I must show that current operating surplus from the proposed strategies exceed current operating surplus from each firm’s best deviation. 24We will see that, in equilibrium, the SA firm is actually able to hire these workers for less than their best-to-date posted wage. The reason is that the SA firm compensates its workers partially through the option value of contingent pay. 41

ONLINE APPENDIX Suppose WP is prescribed: A firm for which WP is prescribed must have p < pˇ. Forthep-productivityfirm, currentoperatingsurplusfromplayingoptimal wage under the prescribed wage contract, WP, and the best deviation to SA can be written as πP(p) =[p−w (p)](cid:96)(p) PP and (cid:90) p` πBD(p) =[p−w (p,p)](cid:96)(p)+ [p−w (q,p)]d(cid:96)(q)−c(pˇ) PA PA p where p` is the productivity of the most productive firm that offers a posted wagelessthanp(e.g., themostproductivefirmthatthep-typefirmcanoutbid by switching to SA). Simplifying, πBD(p) =[p−w (pˇ)+w (p,pˇ)−w (p,p)](cid:96)(p) PP PA PA (cid:124) (cid:123)(cid:122) (cid:125) <0,since dwPA(p,p) <0 dp (cid:90) p` + [p−w (pˇ)+w (q,pˇ)−w (q,p)]d(cid:96)(q) PP PA PA p (cid:124) (cid:123)(cid:122) (cid:125) <0,since dwPA(q,p)<0 dp (cid:90) pˇ − [w (pˇ)−w (q,pˇ)]d(cid:96)(q) PP PA p (cid:124) (cid:123)(cid:122) (cid:125) ≥0 <[p−w (pˇ)](cid:96)(p`) PP ≤πP(p). The last line follows from noting w (p) was the unique profit-maximizing PP posted wage choice for the p-type firm. In other words, the WP firm could increase its labor supply by deviating to SA. However, the firm could also increase its labor supply by the same amount by deviating to a larger posted wage. Willingness to pay for the right to SA is then strictly less than the difference between the wage bill under the deviation 42

ONLINE APPENDIX to SA and the deviation to a higher posted wage, which in turn is strictly less than the cost of SA. Figure 1 depicts wages schedules under WP and SA in the threshold firm (left) and for a less productive firm (right). The cost of SA and bound on the willingness to pay for the right to SA are represented by the shaded regions. The cost or willingness to pay are calculated as the mass in these regions weighted by the supply of labor to the firm with each possible best-to-date outside option. Suppose SA is prescribed: A firm for which SA is prescribed must have pˇ≤ p. For the p-productivity firm, current operating surplus from playing the prescribed SA wage schedule and deviating to the best posted wage are (cid:90) pˇ (cid:90) p πA(p) =[p−w (p,p)](cid:96)(p)+ [p−w (q,p)]d(cid:96)(q)+ [p−w (q,p)]d(cid:96)(q)−c(pˇ) PA PA AA p pˇ and πBD(p) =[p−w`](cid:96)(w`). Note that w` ≥ w (pˇ) since p ≥ pˇ. Simplifying, PP (cid:90) pˇ (cid:90) w` πBD(p) =[p−w`](cid:96)(p)+ [p−w`]d(cid:96)(q)+ [p−w`]d(cid:96)(q) p pˇ (cid:90) pˇ (cid:90) w` <[p−w (pˇ)](cid:96)(p)+ [p−w (pˇ)]d(cid:96)(q)+ [p−w (q,p)]d(cid:96)(q) PP PP AA (cid:124) (cid:123)(cid:122) (cid:125) p (cid:124) (cid:123)(cid:122) (cid:125) pˇ (cid:124) (cid:123)(cid:122) (cid:125) ≤w` ≤w` <w` <[p−w (p,p)−w (pˇ)+w (p,pˇ)](cid:96)(p) PA PP PA (cid:124) (cid:123)(cid:122) (cid:125) >0,since dwPA(p,p) <0 dp (cid:90) pˇ + [p−w (q,p)−w (pˇ)+w (q,pˇ)]d(cid:96)(q) PA PP PA p (cid:124) (cid:123)(cid:122) (cid:125) >0,since dwPA(q,p)<0 dp (cid:90) p + [p−w (q,p)]d(cid:96)(q) AA pˇ <πA(p). The best deviation to WP involves a reduction in the SA firm’s labor supply. I can find a bound on the minimum willingness to pay for the right to 43

ONLINE APPENDIX SA by considering only the labor supply that would arise under the smallest possible best deviation the SA firm might select: w (pˇ). Willingness to pay PP fortherighttoSAisthenlargerthanthedifferencebetweenthewagebillunder thedeviationtoWPandthewagebillfortheseemployeesundertheprescribed SA contract, which in turn is strictly greater than the cost of SA. Figure 1 depicts wages schedules under WP and SA for the threshold firm (left) and for a more productive firm (right). The cost of SA and bound on the willingness to pay for the right to SA are represented by the shaded regions. The cost or willingness to pay are calculated as the mass in these regions weighted by the supply of labor to the firm with each possible best-to-date outside option. Since no firm wishes to unilaterally deviate, the pair {c,pˇ} form a Nash equilibrium. A.1.3 Existence of pˇ for any c. Since pˇ was chosen arbitrarily, c is defined for any possible threshold in the support of Γ. First, consider Γ(p) with finite support [p,p¯]. To show that for every cost, c, there exists a threshold, pˇ, I must first extend the definition of c to include the boundaries of the support of Γ(p). • c(p) = (−∞,0] (when SA is subsidized or free, all firms select SA). • c(p¯) ⊃ [(cid:96)(p¯)p¯,∞) (if the cost of SA exceeds the output of the most productive firm, (cid:96)(p¯)p¯, then no firm selects SA). So, c is upper hemicontinuous on support [p,p¯] and continuous on support (p,p¯). The intermediate value theorem implies that there exists at least one threshold for every cost. The result can be generalized to Γ with infinite upper support by considering the limit as p¯ → ∞: for every p¯ there exists a c = (cid:96)(p¯)p¯ such that all firms SA. 44

ONLINE APPENDIX A.1.4 Uniqueness of equilibria Equilibrium is unique if [(w (pˇ)−p)k ]−1 ≥ dΓ(pˇ) for all pˇ. This condition PP 1 requires that the distribution of productivity be “thin enough” everywhere in the tail that the shift dwPA(q,pˇ) due to indirect upward pressure on schedules dpˇ w (q,p) from the now larger WP sector is dominated by the direct downward PA pressure on the schedule in the marginal firm due to the now larger productivity of the marginal firm. Proof, which stems from differentiating the marginal wage schedule, is available upon request. This guarantees that dc is increasing for all pˇin the interior of the support dpˇ of Γ(p) and the mapping from pˇto c is one-to-one. A.2 Wage schedules SA vs. SA If the best-to-date outside offer originates from a SA firm the problem is analogoustothatconsideredbyPostel-VinayandRobin(2002b),sinceallfirmswith productivity greater or equal to the best-to-date outside option also SA. The least productive firm that is able make a credible threat to hire the worker is a SA firm, and it is the firm that is able to offer a value equal to VA(w (q,p),p) AA with a wage offer of q. Postel-Vinay and Robin (2002b) observe that VA(q,q) = (q +δVU)/(δ + µ). Equating VA(q,q) and VA(w (q,p),p) identifies the reservation wage AA for accepting a job at the p-productivity SA employer when the best-to-date outside option is a q-productivity SA employer: (cid:90) p ¯ w (q,p) =q −k Γ(x)dx for pˇ< q ≤ p, (A.1) AA 1 q where k = λ /(µ+δ) is the expected discounted number of job offers per em- 1 1 ployment spell.25 This pins down the portion of the optimal wage schedule for a p-productivity SA firm and the best-to-date outside option a q-productivity SA firm: when pˇ≤ q ≤ p. 25Note that k →κ when µ→0. 45

ONLINE APPENDIX WP vs. WP In the equilibrium, if the p-productivity firm selects WP then it must be the case that all less productive firms also select WP. The problem facing the firm is thus very similar to the problem considered by Bontemps et al. (2000). The optimal WP wage offer, w (p), maximizes the expected profit from the WP PP contract: πP(p) = [p−w (p)] (cid:96)(p) . (A.2) PP (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) rentperworker laborsupply Bontemps et al. (2000) show that when all firms WP and make optimal wage choices, constrained efficiency of labor flows arrives and labor supply is pinned down by firms’ productivity type. The intuition is that more productive firms prefer to post higher wages, and continuity of Γ(p) yields the required oneto-one mapping between w (p) and p.26 To see that the result extends to PP the separating equilibrium, I must verify that the mapping remains one-toone in the presence of (more productive) SA firms. All SA firms can outbid the highest profitable posted wage of the most productive WP firm and the measure of pˇWP firms is zero since Γ(p) is continuous. WP firms’ maximization problem can be solved by applying the envelope theorem and solving the implied differential equation: (cid:90) p w (p) = p−[1+κ Γ ¯ (p)]2 [1+κ Γ ¯ (x)]−2dx for p < pˇ, (A.3) PP 1 1 wUP where w is the reservation wage of a worker to accept a job at a WP firm UP from unemployment. SA vs. WP Returning to optimal wages for workers transitioning between contract types, nearly all the elements of equation ?? are now pinned down. What remains is to solve for the value of employment in a WP firm. Integration of equation 26See Burdett and Mortensen (1998) and Bontemps et al. (2000) for details and proofs of these results. 46

ONLINE APPENDIX 2.2 by parts gives w (q) 1 (cid:90) pˇ k [Γ(pˇ)−Γ(x)] δVU VP(w (q)) = PP + 1 dwPP(x) + . PP µ+δ µ+δ 1+k [Γ(pˇ)−Γ(x)] dx µ+δ q 1 Plugging in VP and VC appropriately and manipulating gives the result: (cid:90) pˇ w (q,p) = w (q)−k Γ ¯ (pˇ) (cid:2) pˇ−w (q)− k1[Γ(pˇ)−Γ(x)] dwPP(x)(cid:3) PA PP 1 PP 1+k1[Γ(pˇ)−Γ(x)] dx q (cid:90) p ¯ −k Γ(x)dx for q ≤ pˇ≤ p. 1 pˇ (A.4) Hiring out of unemployment Finally, the reservation wages for entering employment at a WP firm or a SA firm of type-p are pinned down as the wages equate the value of unemployment to the value of employment in the relevant firm. The value of unemployment (when all firms follow the proposed strategies) can be written as: (cid:110) (cid:111) µVU = b+λ Γ(pˇ)[E[VP(w (x),x)|q < x < pˇ]−VU]+[Γ ¯ (pˇ)][E[VA(w (x),x)|pˇ< x]−VU] . 0 PP UA (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) hiredbyaWPfirm hiredbyaSAfirm (A.5) Since the SA firm selects the wage offer to yield no rents, the third line is equal to zero. Equating this to the value of employment in the least productive WP firm gives the reservation wage for employment in a WP firm, w (ε), as: UP (cid:90) pˇ [Γ(pˇ)−Γ(x)] w = b+(k −k ) dwPP(x). (A.6) UP 0 1 1+k [Γ(pˇ)−Γ(x)] dx wUP 1 Meanwhile, when entering employment in a SA firm, reservation wages depend on the SA firm’s productivity, p. However, since the value of unemployment equals the value of employment in the least productive WP firm, VU = VP(p), and the SA firm selects the wage to match the value of the best-to-date outside option, the schedule of reservation wages is equal to the wage required to hire from the least productive firm: w (p) = w (p,p). UC PC 47

ONLINE APPENDIX Thus, all wages are pinned down as a function of worker type and labor market history. All equations have closed form solutions for the case when λ = λ , since in this case w = b. 0 1 UP A.3 Proof of Claim 5 The claim is proved if every firm’s wage bill weakly rises. Consider a small increase in the costs of SA. Firms are of three types: always WP, switch from SA to WP, always SA. Wage bills for always WP firms are clearly unaffected by the change in threshold productivity induced by the change in cost (note that I am considering an identical set of active firms). Switching firms strictly increases their wage bill, as before the cost increase they paid for the right to SA,butafterthecostincreasetheyprefertopaylargerWPwagebillstoevade the higher cost of SA. The third category of firms requires heavier lifting. First, note that the wages of workers with best-to-date outside option a WP are set under schedule w (q,p) in each p-productivity SA firm and that PA the mass of such workers in each firm is (cid:96)(pˇ). Also, note that d2wPA(q,p) < 0.27 dqdpˇ So the change in the wage bill associated with these workers when pˇrises is at least dw (pˇ,p) dw (pˇ) PA ¯ PP = [1+k Γ(pˇ)] +k dΓ(pˇ)[pˇ−w (pˇ)] > 0. 1 1 PP dpˇ dpˇ Meanwhile, each p-productivity firm can lower the wage paid to employees whose best-to-date outside option was a SA firm before the cost increase and is now a WP firm. This reduces the p-productivity firms wage bill by d(cid:96)(pˇ)[pˇ− w (pˇ)]. PP The increase in the schedule of wages for workers with best-to-date outside option a WP firm dominates, which is made explicit by noting that dwPP(pˇ) = dpˇ 2k1dΓ(pˇ)[pˇ−w (pˇ)] and d(cid:96)(pˇ) = 2k1dΓ(pˇ)(cid:96)(pˇ). 1+k1Γ(pˇ) PP 1+k1Γ¯(pˇ) Since the wage bill for every productivity firm weakly rises, the total wage bill rises. 27Proof, which follows from performing the derivatives, is available on request. 48

ONLINE APPENDIX A.4 Wage distributions The distribution of wages can be derived by aggregating across firms within sectors and then by aggregating across sectors. Distribution of wages within firms Within WP firms the distribution of wages is a mass at the posted wage. For SA firms, however, wages are disperse. Following the discussion in section 2.4 and Postel-Vinay and Robin (2002a, pg. 999-1001), the distribution of wages within SA firms can be expressed as a function of the distribution of employees best-to-date outside options:   1 , if p < pˇ G(w|p) = (cid:96)(q(w,p)) = w≥wPP(p) (cid:96)(p)  (cid:104) 1+κ1[1−Γ(p)] (cid:105)2 , if pˇ≤ p 1+κ1[1−Γ(q(w,p))] Aggregate distribution of wages Within the WP sector, the wage distribution can be expressed as: G(w|WP) = L(q(w )) PP where q(w ) is the productivity of the firm that optimally posts wage w . PP PP Meanwhile, within the SA sector, the wage distribution can be expressed as: (cid:90) w G(w|SA) = G(w|p)dL(p)+L(p)G(w|p). w(p,p¯) Summing the two gives the aggregate wage distribution. 49

ONLINE APPENDIX B Estimation strategy B.1 Identifying Assumption Ideliberatelyavoiduseofproxiesforestablishmentqualityinestimationofparameters since these are, at best, measured with significant noise. For further discussion of the merits of using linked employer-employee data in this context see Doniger (2014). The estimation strategy employed relies only on transition and duration data, which is recorded with high fidelity in the German register. Identification requires one assumption: Assumption 1. There exists some ranking of establishments such that if p(cid:48) > p then a worker given the opportunity to work for a firm with ranking p or p(cid:48) always chooses to work for the firm with ranking p(cid:48). This assumption provides a more hands-off approach to the issue of measuring firm quality; however, it is still reasonably restrictive. Identification requires that some ranking of firm quality exists, is agreed upon by all observed workers, and that workers climb the job-ladder described by this ranking at every available opportunity. These assumptions preclude explicit wage-tenure contracts and heterogeneous preferences. B.2 Moments Estimation targets means of the following variables: d average duration of the initial employment spell ei d average duration of the initial pay level pi jcen fraction with first employment spell censored i jtj fraction with job-to-job transition ends first employment spell i jn fraction with nonemployment ends first employment spell i r fraction with at least one positive wage change in the initial employment spell i E[w] mean wages V[w] wage variance 50

ONLINE APPENDIX Constructing these variables requires a definition of what constitutes a jobto-job transition and what constitutes a pay-gain. For the primary analysis, I define a job-to-job transition as any change of employer with 7 days or fewer of intervening nonemployment and no overlapping registration as unemployed with the Employment Agency. An on-the-job pay gain is defined as a mid-year pay change that results ina nominal increase inaverage daily pay. SeeDoniger (2014) for a discussion of the construction of these moments and robustness to alternative measures. B.3 Estimation procedure I estimate the model by minimizing the sum of squared errors between simulated and empirical moments. Simulations contain a cross-section of 10,000 simulated employment histories. Workers’ simulated initial firm is drawn from the theoretical steady state distribution of workers across firms given some parameter choice for λ and δ: 1 ¯ L(p) = Γ(p)/(1+κ Γ(p)). 1 Workers’best-to-dateoutsideoptionisdrawnfromthetheoreticalsteadystate distribution of outside offers given incumbent employer type: (cid:96)(q) (cid:18) 1+k [1−Γ(q)] (cid:19)2 1 G(q|p) = = . (cid:96)(p) 1+k [1−Γ(p)] 1 One year (365 days) of employment history is then simulated for each worker. Each day, each worker may receive a new job opportunity, or may become separated from her employer. These events occur with Poisson probabilities λ and δ to be estimated.28 1 If the worker receives a new job opportunity, it may come from a more productive firm than the incumbent – with [1−Γ(p)] probability – or a more productive firm than the best-to-date outside option – with [1 − Γ(q)] prob- 28At daily frequency and plausible parameter values it is extremely unlikely to simulate more than one shock per day. Daily is, however, only an approximation to the continuous time process in which the model is formulated. 51

ONLINE APPENDIX ability. Note again that the functional form of Γ is irrelevant so long as it is a differentiable c.d.f. In the first case, I record a job-to-job transition. If the second case but not the first holds – if the new draw quality is between q and p – the worker may experience an on-the-job raise; however, the raise occurs only if incumbent employer quality is above some threshold, pˇ. Note that cardinal value of the threshold, pˇ, is not substantive. s ∈ [0,1] is the share of F employers with quality greater than this threshold. s = δs /(δ+λ [1−s ]) w F 1 F is the parameter of interest, which is to be estimated. In estimation, I simulate moments and match these to empirical moments. Simulation allows me to integrate out the impact of firm quality on transition and pay change, matching aggregate incidence of these as opposed to employee-level incidence. From the year of simulated data, I compute the fraction that experience no employment change, job-to-job transition, job-tononemployment transition followed by re-employment, job-to-censored-nonemployment transition, and on-the-job pay gain in the initial employer. Note that the first four of these sum to one by construction. I also record simulated initial employment and initial pay level durations. Finally, I simulate the mean and variance of initial wages, which are the analogues to the empirical moments recorded in table 1. To take into account censoring in the data, I censor simulations at the maximum social security contribution. This affects the mean and variance of initial wages and the incidence and duration of on-the-job pay gain.29 I estimate an optimally weighted, over-identified model, minimizing the sum of squared errors between simulated moments and empirical moments: θ ˆ = argmin {m(θ)(cid:48)Wm(θ))}, (B.1) θ where m(θ) = 1 (cid:80) [S(θ)−d(x )] is the vector of moment conditions. S(θ) is n i i the vector of simulated moments conditional on θ and d(x ) is the vector of i data for each observation. W is the optimal weight matrix. Estimation is conducted in two steps. First, I minimize the sum of squared 29SimulationsareconductedinMATLAB.Allcodeisavailableuponrequest. Contactby email: Cynthia.L.Doniger@frb.gov. 52

ONLINE APPENDIX errors with W as the identity matrix and estimate the optimal weight matrix as: (cid:34) (cid:35)−1 1 (cid:88) W ˆ = Ω ˆ−1 = (S(θ)−d(x ))(cid:48)(S(θ)−d(x ) . (B.2) i i n i I then repeat the minimization using the estimated weighting matrix. Standard errors are computed by means of numerical derivatives. The variance covariance matrix is: NV(θ ˆ ) = (D(cid:48)ΩD)−1(D(cid:48)WΩWD)(D(cid:48)ΩD)−1 (B.3) where D = dm(·)| . Numerical derivatives are computed from a 1 percent dθ θ=θ0 deviation around the point estimate of the parameter value. Since the optimal weight matrix is such that W = Ω−1, equation B.4 simplifies to: NV(θ ˆ ) = (D(cid:48)ΩD)−1. (B.4) Empirical moments and covariances are computed in STATA. Simulation and estimationareconductedinMATLAB.Optimalvaluesarefoundusingfminsearch. 53