Hysteresis via Endogenous Rigidity in Wages and Participation

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Hysteresis via Endogenous Rigidity in Wages and Participation Cynthia L. Doniger and David L´opez-Salido 2017-044 Please cite this paper as: Doniger, Cynthia L., and David L´opez-Salido (2017). “Hysteresis via Endogenous Rigidity in Wages and Participation,” Finance and Economics Discussion Series 2017-044. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.044r1. 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.

Hysteresis via Endogenous Rigidity in Wages and Participation Cynthia L. Doniger David Lo´pez-Salido Federal Reserve Board Federal Reserve Board First Draft: April 11, 2017 Current Draft: October 19, 2018 Abstract We document that the past three “jobless” recoveries also featured asymmetries in labor force participation and labor compensation, with each falling to new lows during each cycle. We model these asymmetries as resulting from a strategic complementarity infirms’wagesettingandworkers’jobsearchstrategies. Strategiccomplementarityresults in a continuum of possible equilibria with higher-wage equilibria welfare dominating lower-wage equilibria. Assuming that no economic agent deviates from an existing strategy unless deviation is a unilateral best response, the model exhibits (1) periods of endogenous rigidity in wages and participation, (2) persistent changes in wages, participation, and output in response to transitory movements in labor productivity, (3) sluggish recoveries including both a “jobless” phase, in which productivity recovers while unemployment remains elevated, and a “wageless” phase, in which employment recovers but wages remain depressed. Calibrating the model suggests that the U.S. unemployment rate may need to fall to around 3 percent before labor compensation recovers to pre-Financial Crisis levels. JEL Classification: D83, E24, J42 Keywords: KinkedLaborSupply,StrategicComplementarity,Hysteresis,RealRigidity, “Jobless” and “Wageless” Recovery, Monopsony This paper has previously been circulated as “Labor Demand Management in Search Equilibrium.” The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System. We thank Jim Albrecht, Susanto Basu, Jeffrey Campbell, Russell Cooper, St´ephane Dupraz, Jason Faberman,MarcusHagedorn,BenjaminJohannsen,MariannaKudlyak,EmiNakamura,ClaudioMichelacci, andSamSchulhofer-WohlforcommentsandCarterBrysonandSarahBakerforexcellentresearchassistance. WealsothankparticipantsinseminarsandconferencesattheSearchandMatchingWorkshopsinCambridge and Philadelphia, the ECB, Boston College, NASMES, and the Federal Reserve Board and Banks of New York and Chicago. Comments and suggestions can be directed to cynthia.l.doniger@frb.gov and david.lopez-salido@frb.gov.

1 Introduction It is well documented that the three most recent U.S. recessions were followed by “jobless” recoverees: i.e., measures of productivity recovered more rapidly than unemployment fell. We document that these three episodes also exhibit asymmetries in the behavior of labor force participation and labor share. In order to address these stylized facts we build a labor market model in which a strategic complementarity between participation and wage setting creates asymmetries in unemployment, participation, and compensation around productivity shocks. These asymmetries can bleed into medium run trends and generate hysteresis. A calibration of the model suggests that unemployment rates for prime-age workers below 3 percent would pertain when wages and participation recover to rates implied by pre-financial crisis trends.1 We model cyclical asymmetry and hysteresis as resulting endogenously from a strategic complementarity.2 Firms’ expectations regarding workers’ labor force participation inform their wage setting strategy, while workers’ expectations regarding wages inform their participation strategy. The strategic complementarity can be understood in a stylized two-player representation of the labor market. One worker and one firm can match and produce. The worker has an outside option with a value unknown to the firm but drawn from a known distribution and chooses a search strategy based on her beliefs about the wage to be offered by the firm. In particular, the worker may decide not to participate and drop out of the game before matching if her expectation for the wage offer is below her reservation wage. Meanwhile, thefirmpossessesalinearproductiontechnologyandchoosesandcommitsto a posted wage offer. In posting a take-it-or-leave-it offer, the firm exerts monopsony power. As such, the firm takes into account that a higher wage makes hiring more likely but lowers the profit from production. The firm’s choice depends on the distribution of the worker’s possible outside options and on the worker’s participation behavior, which in turn depends on the wage expected by the worker. As a result of the worker’s participation strategy, the labor supply curve that the firm expects to face is kinked: the marginal reduction in the probability of hiring from reducing the wage offer below the level expected by the worker discretely exceeds the marginal gain in the probability of hiring from increasing the wage above the expected level, since the worker with higher values of leisure may not be searching. Strategic complementarity results, since increasing the expected wage leads to higher participation, and higher participation lowers the marginal cost of high wage offers. Thus, many expected wage levels may be self-confirming. 1For reference, at the time of writing the unemployment rate for this population stood at 3.7 percent. This is the lowest rate in the past 49 years. 2For the canonical treatment of hysteresis in the labor market see Blanchard and Summers (1986, 1987). 1

We show the existence of an interval of wage and participation pairs which constitute a continuum of rational expectations equilibria. The lower boundary is pinned down as the minimum participation threshold such that marginal cost of hiring when approaching the kink in the labor supply curve (induced by the worker’s threshold) from below is equal to the marginal revenue. Meanwhile, the upper bound is pinned down by the same calculation taken from above the kink. These equilibria can be welfare ranked with higher-wage, higherparticipation equilibria dominating lower-wage, lower-participation equilibria.3 Finally, the highest wage-participation equilibrium coincides with the unconstrained monopsonistic solution studied by Diamond (1971). We consider the implications of multiplicity for asymmetry and persistence when the model incorporates productivity shocks under the plausible assumption that no player deviatesfromanexistingwageandparticipationpairunlessunilateraldeviationisabestresponse for at least one player. Real rigidity is then simply a corollary of the existence of a continuous set of equilibria. Further, we show that whenever a productivity shock does induce a unilateral deviation, one can trace out a sequence of simultaneous best responses–a Cournot tˆatonnement–that converges to a unique equilibrium in the set of equilibria consistent with the new productivity level.4 Thus, equilibria are learnable in the sense that rational agents could deduce the new equilibrium from knowledge of the shock and pre-shock equilibrium.5 The kink induced in the labor supply curve by the worker’s participation strategy results in upward wage revisions that fall farther from the constrained efficient wage level than do downward revisions. This disparity implies that transient productivity shocks lead to persistent decreases in output, wages, and participation. Further, in order to recover a particular pre-contraction wage and participation level, the subsequent expansion must overshoot the pre-contraction productivity level. Thus, contractions induce hysteresis. Inordertoprovidemicrofoundationsforthestylizedtwo-playerrepresentation, weembed 3A side result of this observation is that a social planner can potentially “prime the pump” and induce coordination on a welfare-dominant equilibrium, for example by an appropriately selected minimum wage. 4Specifically, our game shares in common many features of a supermodular game as detailed in Vives (1990, 2005) and Cooper (1994). In particular, the game contains enough structure that a Cournot tˆatonnement emanatingfromanypointthatliesonatleastoneplayer’sbestresponsefunctionwillconverge to an equilibrium. Further, we will show that the best response of the worker is independent of aggregate productivity shocks. Thus, after all shocks we achieve convergence. 5Therearealternativeapproachestotheexpectationalelementofgameswithstrategiccomplementarities. Onecanconstructbusinesscyclesdueto“animalspirits”inwhichplayersspontaneouslycoordinateexpectations on higher or lower output equilibria (Cass and Shell, 1983). Alternatively, one can refine the equilibria predictionvia“globalgames”andcanharnessthestrategiccomplementaritytoinduceamplification(Morris and Shin, 2001). Also alternatively, one may optimistically suppose that agents always coordinate on the highest equilibrium (Krugman, 1991). Since our aim is explicitly to generate asymmetry and hysteresis, we eschew these approaches in favor of the plausible supposition that agents do not deviate from an existing equilibrium unless unilateral deviation is the best response of some player. 2

the model in a labor market that features random search and we close our model with free entry into vacancy creation and a constant returns to scale matching function.6 This also providesamappingbetweenwage,participation,andproductivityleveltotheunemployment rate. We show that when some workers with high flow values of leisure search for work (but ultimately do not accept the equilibrium wage offer) the effect is congestion in the matching function.7 Further, changes in participation induced by shocks lead to changes in congestion, sincethesetofworkerswhosevalueofleisureexceedstheequilibriumwageswellswhenshocks induce negative wage revisions. Since wage revisions are persistent, the increased congestion persists beyond the duration of the disturbance and can result in “jobless” recoveries. As we have also already noted, recovery of the pre-contraction wage level requires that productivity over-shoots the pre-shock level. Recovery of wages and participation also imply resolution of the increase in congestion. Thus, unemployment must fall below pre-shock levels before pre-shock wages are recovered, resulting also in “wageless” recoveries. We calibrate the model to ascertain the degree to which the unemployment rate must undershoot pre-Financial Crisis levels if wages and participation are to recover to pre-shock levels. Our baseline calibration suggest that wage and participation should have been be expected to recover to their pre-shock trends when unemployment fell below 4.8 percent; however, acknowledging that this model–like others in the Diamond-Mortensen-Pissaredes class–fails to generate sufficient amplification and implementing a back-of-the-envelope adjustment suggests that unemployment may need to fall below 3 percent. Related Literature Our model brings together two branches of the modern Keynesian literature. The first follows Keynes’ suggestion that economic fortunes might be governed by “animal spirits.” This literature microfounds such economic fluctuations, at least in part, on strategic complementarity, multiplicity, and the potential for coordination on higher or lower output equilibria either spontaneously or under the direction of a social planner. Canonical examples include 6Thesuppositionofafrictionallabormarketendowsthefirmwithmarketpowerinwagesetting(monopsony) even for arbitrarily small market frictions (Diamond, 1971). Assuming massless agents and random search renders each worker (resp. firm) unable to affect the average participation threshold (resp. wage) throughunilateraldeviation. Thissupportsmodelingtheseaggregatesasexogenoustoeachdecisionmaker’s problemevenwhenexpectationsareinformedbyexistingaggregatevaluesasweshallconsider. Weclosethe model with the typical free-entry condition into vacancy posting. This provides a map from wages and participation levels to market tightness and unemployment rates. Free entry and the endogenously degenerate wage distribution also support the assumption that players have no continuation values. 7On the other hand, as we show in the two-player representation, when a higher proportion of workers with high flow value of leisure search the kink in the labor supply curve is less severe and recovery from contractions is more symmetric. 3

Heller (1986); Kiyotaki (1988); and Diamond (1982).8 Recently, Eeckhout and Lindenlaub (2015) have brought these techniques to bear on the issue of jobless recoveries; however, their paper is silent on the issue of wages. Our contribution also generates a degree of cyclical wage rigidity and thus relates to a second branch of the modern Keynesian literature, which seeks to harness real rigidities in order to amplify and propagate (nominal) disturbances. Mechanically, our firm’s problem resembles the kinked demand curve theory of real price rigidity. This theory originated in the Industrial Organization literature with Sweezy (1939) andHallandHitch(1939)andhasreceivedattentioninmacroeconomics, dueinlargepartto Kimball (1995), in the context of goods market rigidities. The innovations contained in this work are threefold. First, the empirically plausible size of real rigidities in the goods market provide insufficient amplification and propagation to match business cycle facts (Ball and Romer, 1990; Klenow and Willis, 2016). In contrast, plausible rigidities in the labor market are large. By switching the focus from product demand curves to labor supply curves, we open the possibility to harness the larger rigidity. Second, while the kinked demand curve is heuristically appealing, it is difficult to build a consistent equilibrium microfoundation. To these authors’ knowledge, no satisfactory equilibrium microfoundation has yet been published.9 Strategic labor force participation provides perhaps the first satisfactory equilibrium microfoundation of a kink. Third, labor market rigidities are often modeled as resulting from incentives for efficiency wage intended to deter “shirking” a` la Shapiro and Stiglitz (1984) and Yellen (1984); however, when the cost of job loss is increasing in the unemployment rate, efficiency wages may substantially procyclical. In contrast, our theory offers a microfoundation for wage rigidity that is based in an observable, participation, and robust to such criticism. Although it is not a focus of our paper, endogenous wage stickiness due to our coordination failure also speaks to the Shimer (2005) puzzle: the observation that the Diamond- Mortensen-Pissaredes (DMP) model of a frictional labor market fails to produce sufficiently volatile unemployment. As such, it relates to a larger literature that seeks to induce amplification through wage rigidity, for example Hall (2005), Gertler and Trigari (2009), and Kennan (2010). We differ from this literature in our focus of producing hysteresis as opposed to only amplification. A strict focus on amplification will typically yield rebounds of the unemployment rate following severe shocks that are steeper than the contractions that precede them. Consider a transient shock severe enough to yield a downward wage revision. Since this new wage is sticky, the value of a vacancy after the shock has dissipated must 8See Cooper and John (1988) and Solow (1998) for discussions. 9The often cited work, Woglom (1982), does not formally model the consumers’ problem. Dupraz (2017) provides the most complete microfouncation of kinked product demand of which these authors are aware. 4

Figure 1: Structural Breaks in Labor Force Participation and Labor Share 98 96 94 92 90 88 66 64 62 60 58 56 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 noitapicitraP ecroF robaL )45−52 neM( etaR Labor Share Note: Labor force participation (blue) and labor share (red) both trend downward. Grey shaded regions indicate NBER recession dates. Structural breaks following the 1990-1, 2001, and 2007-9 recessions coincide for the two series. These recessions were also followed by jobless recoveries. Source: U.S. Census Bureau; U.S. Bureau of Labor Statistics; authors’ calculations. be larger than it was initially. This implies an greater abundance of vacancies and a lower unemployment rate post-shock than pre-shock. Without cyclical variation in congestion, a rigid wage theory therefore implies “job-full” recoveries. Instead, the data show slow recoveries of unemployment and outward shifts in the Beveridge curve. Our model can generate both “jobless” recoveries, in which unemployment remains persistently high even after labor productivity has recovered, and “wageless” recoveries, in which unemployment falls to unprecedented lows before wages recover. 2 Cyclical Trends in Participation and Labor Share In this section, we present evidence of the presence of structural changes in labor force participation and compensation of employees over value added–labor share–at business cycle frequencies.10 In particular, we find that structural breaks in these series coincide following each of the past three contractions, which were all typified by jobless recoverees. This main 10U.S. Census Bureau, Current Population Survey, http://www.census.gov/programs-surveys/cps/datadetail.html and U.S. Bureau of Labor Statistics, Nonfarm Business Sector: Labor Share [PRS85006173], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/PRS85006173. 5

result is illustrated in Figure 1. To this end, we use Bai and Perron (1998, 2003) sequential tests that allow to find a number of breaks as well as estimating the timing and the confidence intervals of these breaks. Several statistics are useful in order to identify the break points: First, the sup F (k) test, is a F-type test of the null of no structural break (m = 0) versus the alternative T of a fixed (k) number of breaks (m = k). Second, two maximum tests of the null hypothesis of no structural break (m = 0) versus the alternative of an unknown number of breaks given some upper bound M (1 ≤ m ≤ M), i.e., UDmax test, an equal weighted version, and WDmax test, with weights that depend on the number of regressors and the significance level of the test. Finally, the sup F (k + 1|k) sequential test of the null hypothesis of k T breaks versus the alternative of k +1 breaks. Table 1 displays the results of applying these formal structural stability tests, allowing up to five breaks, to the labor force participation and the labor share. Both the UDmax and WDmax tests are highly significant, which implies that at least one break is present in both series. Next, all the sup F (k) tests are significant, with k running between 1 and 5 t breaks, so that multiple breaks are present in both variables during our sample period. In turn, the sup F (k +1|k) test is not significant for any k > 5 in the case of the labor force T participation and for any k > 4 in the case of the labor share. Thus, these tests identify fiver breaks for the latter variable and four for the former. Interestingly, as shown in Table 1, the sequential tests identify that last three of these breaks occur at the same times in both series: early-to-mid 90s, early-to-mid 2000s, and late 2000s. The left panel of Figure 1 shows that since the early 60s, the prime-age male labor force participation has exhibited a trend toward lower levels. As our statistical analysis has identified, trend exhibits an asymmetric cyclical responses to positive (expansions) and negative (recessions) shocks. Male nonparticipation in the labor force accelerates during contractions and is not reversed during expansions to the pre-recessions levels.11 As shown in the right panel of Figure 1, over the same horizon, labor’s share of income has been declining. The trend exhibits sharp reversals during economic downturns; but, after each recession, the labor share falls to a new low. Strikingly, in the past three jobless recoveries the pattern in the labor share and labor force participation series is synchronized. Thisevidenceisconsistentwithemploymentandcompensationshowingimportantasymmetries at business cycle frequencies and in the medium run. In particular, employment falls quickly around the onset of a contraction and recovers more slowly, with the “jobless” recov- 11In other words, while unemployment falls during expansions participation rates hardly recover. This is true even when accounting for demographic trends as well as disability status that are independent of cyclical variation. 6

Table 1: Bai-Perron (2003) tests of multiple structural changes. Labor Force Participation Labor Share UD 2105.8∗ WD 4621.0∗ UD 584.0∗ WD 584.0∗ max max max max Break date k F (k) F (k|k −1) Break date k F (k) F (k|k −1) T T T T 1971:1 1 672.8∗ – 1975:2 1 584.0∗ – 1981:3 2 751.0∗ 189.5∗ 1992:4 3 1627.3∗ 487.3∗ 1992:4 2 436.6∗ 80.9∗ 2001:2 4 1829.4∗ 62.7∗ 2001:2 3 374.3∗ 24.0∗ 2009:4 5 2105.8∗ 0.00 2009:4 4 305.9∗ 0.74 Sample period 1960:1 to 2018:1 for both series. * denotes significance at the 5 percent level. The critical values are taken from Bai and Perron (2003). The number of breaks corresponds to the sequential F-Statistics at 5 percent level size for the sequential test sup F (k+1|k). T eries following the 1991, 2001, and 2008 contractions being the most severe examples. Our evidence also points that participation in the labor market contributes to this phenomenon. In each contraction since 1975, participation falls and subsequently fails to recover to the pre-contractionlevel: compensationofemployeesovervalueaddedspikesduringcontractions but levels off below pre-contraction values during subsequent expansions. These “wageless” recoveries that started in early 90s are most stark after 2000. The rapid fall in employment and spike of compensation over value added are indicative of sizeable wage rigidities. Meanwhile, the persistent failure to recover pre-shock employment and rent sharing suggests hysteresis: that is, long-lasting scarring such that contractions bleed into a medium-run trend. 3 A Stylized Two-Player Game In this section, we consider a stylized single-shot, one-worker, one-firm labor market and show that there exist a continuum of rational expectations equilibria. Computing player’s best response functions requires positing a fixed expectation for the firm’s posted wage on the worker’s behalf and for the worker’s participation strategy on the firm’s behalf. We call the worker’s expectation of the wage w . We will see that we can summarize the worker’s 0 strategy as a participation threshold and we call the firm’s expectation of the threshold r . 0 Thus, w (resp. r ) is the belief that the worker (resp. firm) holds about the firm’s (resp. 0 0 worker’s) wage offer (resp. threshold). Worker and firm maximize income and profit. 7

3.1 Worker For the worker, the game proceeds in two stages. In the first stage, the worker draws a value of leisure, b, from a known distribution, H(b), with density h(b), defined on [b, ¯ b].12 In the second stage, the worker may costlessly seek to contact the firm.13 If she does not seek to contact the firm, we call this nonparticipation. If she does seek to contact the firm, she has a (1 − u) probability of making contact and being made a wage offer. If the worker accepts this wage offer we call her employed. If she rejects we call her voluntarily unemployed. Finally, in the case that there is no contact between worker and firm we call her involuntarily unemployed. We will see that the decision to participate will depend on her expectation of the wage offer, w . 0 As is typical for such problems, the worker’s strategy takes the form of threshold rules: a reservation wage and a threshold for labor force participation. Since search is costless, the reservation wage is equal to the value of leisure. Let VW(r,w ) be the expected payoff to a 0 worker of choosing threshold r when the expected wage choice of the firm is w .14 Observe 0 that there are two cases to characterize this value function: involuntary nonparticipation unemployment employment (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123)  (cid:122) (cid:125)(cid:124) (cid:123) (cid:90) r (cid:90) ¯b  (1−u)w 0 H(r)+u bh(b)db+ bh(b)db................... if r < w 0 VW(r,w 0 ) = b (cid:90) w0 r (cid:90) r (cid:90) ¯b  (1−u)w H(w )+u bh(b)db+ bh(b)db+ bh(b)db. if r ≥ w 0 0 0 (cid:124) (cid:123)(cid:122) (cid:125) b w0 r . employment (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) noninvoluntary voluntary participation unemployment unemployment Differentiating, itfollowsthat dVW(r,w0) > 0wheneverr < w . Inthiscase, theworkercan dr 0 improve her payoff by increasing her participation threshold. That is, a low threshold causes the worker not to seek work in some cases when there is a positive probability of receiving an acceptable employment offer. One can also see that dVW(r,w0) = 0 whenever r ≥ w : in this dr 0 12We do not rule out b=−∞ or¯b=∞. 13Under costless search, the reservation wage and threshold participation level coincide. We will see when we turn to the firm’s problem that this coincidence is essential for multiplicity. Costly search results in a uniqueequilibriumwithlessparticipationthantheminimalparticipationconsistentwiththesetofequilibria werecoverundercostlesssearch. Costlynon-participationresultsinauniqueequilibriumwithparticipation exceeding the maximal participation consistent with the set of equilibria we recover under costless search. Wenotethatevidenceonjobsearchbehaviordoesnotsupportsubstantialcost–seeMukoyamaetal.(2018). 14Note, the worker knows her b before she chooses to participate or not. VW(r,w ) considers the payoff 0 to participation threshold r from her perspective before she learns b. However, There are no values of b for which the ex-ante choice of r and the ex-post (after learning b) choice of r would differ. We write the problem in this way simply because it is analytically convenient to optimize from the ex-ante perspective. 8

case the payoffs from increasing the participation threshold are nil. In the marginal case, the worker obtains the same value–the value of leisure–in voluntary unemployment as she does in nonparticipation. In the region of indifference (r ≥ w ) we posit a mixed strategy over the pure strategies 0 “voluntary unemployment” and “nonparticipate”: Assumption 1. When the worker expects to be indifferent between voluntary unemployment and nonparticipation, she randomises between the two states with probability i placed on voluntary unemployment. Under Assumption 1, the workers best response, r∗(w ), is to set a threshold: 0 r∗(w ) = w , (1) 0 0 such that she participates with probability one if b < r∗(w ) and with probability i < 1 if 0 b > r∗(w ).15 0 Lemma 1. This game exhibits positive spillovers and strategic complementarities for the worker. Proof. An increase in the posted wage–which we will see is the strategy of the firm–increases the payoff for the worker regardless of worker’s participation threshold strategy (positive spillovers). Also, an increase in the posted wage–which we will see is the strategy of the firm–increases the optimal participation threshold strategy of the worker (strategic complementarity).16 3.2 Firm The firm posts a wage offer, w, ex-ante and may meet the worker with probability (1−v). If the firm successfully hires the worker, it will earn rent p−w, where p is the output of the 15It is also possible that the worker adopts a more complex randomization strategy. In particular, one might posit that the weight placed on “voluntary unemployment” might depend on the realization of b. However, any randomization such that the probability of nonparticipation exceeds zero as the realized value of leisure approaches the expected wage level from above will produce a kink in the labor supply curve and thus will yield qualitatively similar results regarding rigidity and propagation. Further, if i → 1 as b (cid:38) w 0 thenwehavea“smoothed”kink,similartoKimball(1995),andwewillobtainresultsregardingprorogation. 16Formally, define positive spillovers for player j as the case in which an increase in the other player’s strategy increases the payoff to player j (Cooper and John (1988)). Now simply note that dVW(r,w) = dw (1−u)H(r) > 0 if r < w and dVW(r,w) = (1−u)H(w)+uwh(w) > 0 if r > w. Also, define strategic dw complementarities for player j as the case when an increase in the other player’s strategy increases the best response of player j (Cooper and John (1988)). This trivially follows from expression (1), that is dr∗(w) =1>0. dw 9

match. If the firm and the worker do not meet or if the worker rejects the wage offer then the payoff to the firm is zero. Let VF(w,r ) be the payoff to the firm of posting wage w when the firm’s expectation 0 for the worker’s threshold for participation is r . We can write the firm’s value function: 0   H(w) if w < r VF(w,r ) = (1−v) (p−w)× H(r0)+i[1−H(r0)] 0 0 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)  H(r0)+i[H(w)−H(r0)] if w ≥ r match revenue H(r0)+i[1−H(r0)] 0 probability from (cid:124) (cid:123)(cid:122) (cid:125) successful hiring expectedlaborsupplyschedule Thefirstterm, (1−v), andthesecondterm, (p−w), aretheprobabilityofmeetingtheworker and the payoff from successfully hiring. The third term contains the core of the problem, encoding the firm’s expectation of the labor supply schedule it faces. We can interpret the probability that a wage offer of w will be accepted as the labor supply curve faced by the firm. Notice that the participation threshold introduces a kink in the expected labor supply at the expected threshold. To the left of the threshold, where w < r the expected probability 0 that the worker participates for draws of the value of leisure in this region is one. Thus, posting a wage less than or equal to the expected labor force participation threshold results in hiring a worker with probability H(w) . To the right of the threshold the expected H(r0)+i(1−H(r0)) probability that the worker participates for draws of the value of leisure in this region is only i < 1. Thus posting a wage greater than the expected labor force participation threshold results in hiring a worker with probability H(r0)+i[H(w)−H(r0)]. H(r0)+i(1−H(r0)) The firm’s best response given a particular expectation for the worker’s strategy is to post a wage that satisfies: (cid:8) (cid:9) w∗(r ) = argmax VF(w,r ) . 0 0 w The problem takes the form of monopsony wage setting. The firm’s expectation over the worker’s participation threshold introduces a kink in the expected labor supply schedule. As a result the best response function–the optimal wage posting strategy of the firm–is defined piecewise on intervals of the expectation for the worker’s participation threshold. Our task is to determine these intervals and the optimal posted wage schedule within each of them. We distinguish three cases in what follows. Interior solution such that w < r 0 Suppose that the firm expects that the participation threshold is high enough that it is non-binding. In this case VF(w,r ) simplifies to (1−v)(p−w) H(w) and the firm’s 0 H(r0))+i(1−H(r0)) best response satisfies: 10

Figure 2: Optimal Wage Choice at Interior Solutions. Expected Threshold is High Expected Threshold is Low e ga y lp p n o itu e ga y lp p n o itu w u b w u b S ro b a L d e tc e p x E irts iD e p y T re k ro W S ro b a L d e tc e p x E irts iD e p y T re k ro W p Marginal Revenue p Marginal Revenue n w n o w d o r 0 k ra M d k ra M wC Cost w^ Cost Marginal r 0 Marginal H(r ) participation H(r ) participation 0 0 Note: The left panel illustrates an expected threshold such that the firm finds an unconstrained interior solution. The right panel illustrates an expected threshold such that the firm finds a constrained interior solution. The location of the kink in the expected labor supply curve depends on the firm’s expectation for the worker’s participation threshold, r . The angle of rotation and the associated jump 0 in the marginal cost curve depend on the probability that the worker searches when her value of leisure exceeds this threshold, i. Fore ease of illustration we take H(b) to be uniformly distributed on [b,¯b]. wC = argmax{H(w)(p−w)}, w noting that (1 − v) and (H(r ) + i[1 − H(r )]) are exogenous to the firm. The first order 0 0 condition can be written as follows:17 (cid:20) H(wC) (cid:21) p = wC 1+ (2) wCh(wC) (cid:124)(cid:123)(cid:122)(cid:125) marginal (cid:124) (cid:123)(cid:122) (cid:125) revenue marginal cost The left-hand side is the marginal revenue from an employee. The right-hand side is the marginal cost of hiring the worker in all the cases in which she will accept wages no less than wC. The wage choice, w , is illustrated in the left panel of Figure 2. The x-axis plots the C 17A sufficient condition such that wC is the unique maximizer is given by: 1 > d2H(b) 1 . This states p−b db2 2h(b) thatthedistributionH(b)isnotmoreconvexthanthehyperboladefinedbythefirmsiso-profitcurves: 1 . p−b This is trivially satisfied whenever the density is weakly decreasing – for example, the uniform and Pareto distributions as well as by the normal and logistic distributions when at least half of workers participate in the labor force. The remainder of this paper focuses on the case where the second order condition holds. 11

probability that a worker will accept a wage offer of w. From the firm’s perspective, this is the labor supply schedule. The y-axis plots wages. We plot both the underlying distribution of worker types and the labor supply schedule that the firm expects to face given the worker’s participation threshold in gray hashed and solid, respectively. Note that the expected labor supply schedule is rotated counterclockwise relative to the underlying distribution of worker types around the probability that the worker is type r or less, H(r ), creating a kink in 0 0 the labor supply curve. The angle of rotation depends on the probability, i, that the worker searches when her value of leisure exceeds this threshold, i. We also plot the marginal revenue and marginal cost faced by the firm. As in the typical monopsony problem, we find the optimal quantity of labor demanded, H(wC), at the intersection between marginal revenue and marginal cost. Since we have assumed that the firm is unconstrained by the worker’s expected participation choice the kink in the expected laborsupplycurvefallstotherightoftheintersectionofmarginalcostandmarginalrevenue. The wage and markdown are determined as in a normal monopsony diagram. Interior solution such that r < w 0 Suppose that the firm expects that the participation threshold is low enough that it is binding. In this case: wˆ = argmax{[H(r )+i[H(w)−H(r )]](p−w)}, 0 0 w and the first order condition is given by: (cid:20) (cid:21) H(r )+i[H(wˆ)−H(r )] 0 0 p = wˆ 1+ (3) ih(wˆ)wˆ (cid:124)(cid:123)(cid:122)(cid:125) marginal (cid:124) (cid:123)(cid:122) (cid:125) revenue marginal cost Again, the left-hand side is the marginal revenue from an employee. The right-hand side is the marginal cost of hiring the worker in all the cases in which she will accept wages no less than wˆ. The wage choice, wˆ, is illustrated in the right panel of Figure 2. Now we are considering that the firm’s labor demand is greater than H(r ) and thus that the kink induced by the 0 rotation of the expected labor supply schedule relative to the underlying distribution of worker types is binding. We again plot the marginal revenue and marginal cost faced by the firm. At the quantity of labor supplied at the expected participation threshold, H(r ), the 0 marginal cost jumps due to the kink in the expected labor supply curve. Again, we find the optimal quantity of labor demanded, H(r )+i[H(wˆ)−H(r )], at the intersection between 0 0 marginal revenue and marginal cost. 12

Corner solutions Suppose that the constraint imposed by the expected participation threshold is binding and itinducesacornersolution(i.e., w = r ). Thisoccursifthemarginalrevenuestrictlyexceeds 0 the marginal cost when approaching the expected kink from below while the marginal cost strictly exceeds the marginal revenue when approaching from above:18 H(r ) H(r ) 0 0 p > r + and p < r + . 0 0 h(r ) ih(r ) 0 0 What remains is to find the lowest expected participation threshold such as the firm prefers to hire weakly more than H(r ). In other words, the lowest participation threshold, 0 rL, for which the corner solution is consistent must satisfy: H(rL) p = rL + . ih(rL) Figure 3 plots the range of participation thresholds for which the corner solution is optimal for the firm. The left panel plots the smallest threshold such that quantity of labor demanded coincides with the quantity supplied at the expected participation threshold: r = rL = wL. The right panel plots the largest threshold such that quantity of labor 0 demanded coincides with the quantity supplied at the expected participation threshold: r = rC = wC. The center panel plots a generic corner solution when rL < r < rH and the 0 0 strictly prefers to more than H(r −ε) and strictly less than H(r +ε) for arbitrarily small 0 0 ε. 18Formally, taking the derivative from the left, lim dVF(w,r0), and from the right, lim dVF(w,r0), w(cid:37)r0 dw w(cid:38)r0 dw respectively. Aparticularlyusefulformofrepresentingthefirm’sbestresponseisthatthewageisamarkdown from the marginal cost of labor, the markdown is given by the elasticity of the expected labor supply faced by the firm. That is, expression (2) can be written as p−wC = H(wC) = 1 , where η1 is the elasticity wC wCh(wC) η1 of the expected labor supply curve in the region below the firm’s expectation for the worker’s participation threshold. Notice that, η1 =η, where η = dH w denotes the elasticity of labor supply with respect to wages dw H would pertain if the worker participates for all values of b. Manipulating equation (3) one can show that, hiringinthissideofthemarket,themonopsonymarkdownisequalto p−wˆ = H(r0)+i[H(wˆ)−H(r0)] = 1,where wˆ ih(wˆ)wˆ ηi ηi is the elasticity of the expected labor supply curve above the kink. Note that ηi = η H(w)i . H(w)i+H(r0)(1−i) In other words, the markdown just above the kink market exceeds the markdown that the firm would choose if it were not constrained by the expected participation threshold. Notice also that, when i<1, the elasticity of the expected labor supply with respect to the wage at the expected threshold is discontinuous: ηi(r+) < η1(r−) and there is a jump in the marginal costs at the kink stemming from that discontinuity. 0 0 For a given r the size of the jump in the marginal cost depends on i, fraction of the time that the worker 0 drawsb≥w andsearchesforwork: whentheworkeralwayssearchesthenthemarginalcostissmooth; and 0 when all workers with b≥w non-participate, the marginal cost of hiring greater than H(r ) is infinite. 0 0 13

Figure 3: Optimal Wage Choice at Corner Solutions. Lower Corner Generic Corner Upper Corner e ga w p Marginal Revenue y lp p u S ro b a L d e tc e p x E n o itu b irts iD e p y T re k ro W e ga w p Marginal Revenue y lp p u S ro b a L d e tc e p x E n o itu b irts iD e p y T re k ro W e ga w p Marginal Revenue y lp p u S ro b a L d e tc e p x E n o itu b irts iD e p y T re k ro W wC w wL Cost Cost Cost Marginal Marginal Marginal H(rL) participation H(r 0 ) participation H(rC) participation Note: The left panel illustrates the smallest threshold such that the firm prefers to hire H(r ). The 0 right panel illustrates the largest threshold for which the firm weakly prefers to hire H(r ). The center 0 panel illustrates a generic corner solution. We can now write the firm’s wage best response (i.e., wage posting strategy) as follows:  wˆ if r < rL   0  w∗(w ) = r if r ∈ [rL,rC] (4) 0 0 0    wC if r > rC. 0 Lemma 2. For all expected participation thresholds, r , in the interval [rL,rC] the game 0 exhibits positive spillovers and strategic complementarities for the firm. Proof. Whenever the marginal searching worker would accept the posted wage offer an increase in participation increases the pay-off to the firm (positive spillovers). Moreover, whenever r ∈ [rL,rC] an increase in the participation threshold increases the optimal posted wage strategy of the firm (strategic complementarity).19 3.3 Rational Expectations Equilibria Definition 1. A rational expectation equilibrium of the two-player game is a pair– wage and participation threshold–such that wage and participation threshold are mutual best 19Formally, dVF(w,r) =(1−v)(p−w)h(r)(1−i)(i(1−H(w)]) >0wheneverw ≥r and dw∗(r) =1>0whenever dr [H(r)+i(1−H(r)]2 0 dr r ∈[rL,rC]. 14

responses. In other words, in any rational expectations equilibrium both worker’s and firm’s expectations must be self-confirming: r∗ = r and w∗ = w . 0 0 Proposition 1. For i < 1, a continuum of equilibria exists on the interval [wL,wC], with higher welfare for higher wage levels. Proof. Supposetheworkerexpectswagestobew . Herbestresponseistosetaparticipation 0 threshold such that r∗ = w . If this expectation is consistent with an equilibrium then it 0 must be the case that the firm’s best response is to confirm the expectation by setting a low wage such that w∗ = r∗ = w . This action will be consistent with the firm’s strategy as long 0 as w ∈ [wL,wC].20 Since we have already shown that both the firm and worker face positive 0 spillovers and strategic complementarities on this interval, the welfare ranking result is a straightforward application of Lemmas 1 and 2 and Cooper and John (1988) Proposition 5. Figure 4 illustrates the best response function of the worker (hashed) and the best response of the firm (solid). These are mutual best responses on the 45 degree line from (rL,wL) to (rC,wC). Note that as i approaches 1, wL approaches wC and the equilibrium set collapses to a unique equilibrium. This equilibrium is constrained efficient and coincides with the equilibrium found in Diamond (1971). Robustness Note that if there is non-zero probability that wage offer exceeds the worker’s expectation, then the worker’s threshold must be raised to exceed the support of the distribution of possible wage offers. At this point, one might be tempted to impose an equilibrium refinement with a “trembling hand” flavor, as in Selten (1975). If there is any chance that the firm’s hand “trembles” and it may make a mistake in some off-equilibrium path and post a high wage, such a consideration could be used to rule out all equilibria other than the constrained efficient. This consideration is a valid criticism of our two-player game. However, we have devised this simply to illustrate clearly the minimal assumptions that drive results in our model. In the full labor market that we present in Section 5, one can see that such a refinement is less plausible as it would require not just that a single firm’s hand “trembles” but that a discrete mass of firms all simultaneously mistakenly play an off-equilibrium wage. We suggest that such a deviation from equilibrium is too implausible to impose as a refinement. 20It is easy to construct an analogous example illustrating that (off equilibrium) expectations above wC or below wL are not confirmed. 15

Figure 4: Mutual Best Responses. expected wage 7b ) w0 r$( Worker's best response: wC Firm's best response:w$(r0) wL b b rL rC expected 7b participation threshold Note: The Firm’s and Worker’s strategies are mutual best responses for an interval of wage and participation thresholds on the on the 45 degree line from (rL,wL) to (rC,wC). Animal Spirits The stylized two player representation of the labor market can be used to generate endogenous business cycles. Consider that the worker and firm may experience a spontaneously and coordinated change expectations–for instance, due to a sunspot–such that the equilibrium shifts from a low wage to a high wage. From Proposition 1, we can infer that this shift will be attended by an increase in employment and an increase in output. Thus we can generate businesscycles motivatedpurely by the“animalspirits”of market participants. However, we find such cycles unsatisfying. In particular, such spontaneity has no dependence on history and thus can-not explain why contraction is quick relative to expansion. All these considerations lead us to prefer instead to investigate the potential there may be a degree of rigidity in wages and participation with respect to productivity shocks. Productivity shocks that are large enough to induce wage and participation revisions that may lead to long-lasting changes in expectations.21 21Further, when we embed the model in a two-sided frictional labor market in Section 5 we will see that without further assumptions cycles driven by sunspots may produce unemployment dynamics that are contrary to the data. Meanwhile, expectation shifts driven by realized productivity shocks are consistent with both the “jobless” and “wageless” phases of recovery observed in the data. 16

Priming the Pump A straightforward implication of Proposition 1 is that a policy-maker empowered with either the power to set a minimum wage or require participation could increase welfare by appropriately levering her instrument. While welfare-improving policies are interesting to consider, we focus our paper instead on the positive implications of the existence of these multiple equilibria. We turn now to demonstrating that the two-player game exhibits rigidity and hysteresis. 4 Productivity Shocks, Rigidity, and Hysteresis We have established the existence of a continuum of equilibria. We now consider the implications in the context of productivity shocks. In particular we compare equilibria before and after the arrival of shocks. Since the two-player game considered both before and after the shock is a single shot game, there is no need to distinguish between expected and unexpected shocks.22 4.1 Endogenous Rigidity Assumption 2. No player deviates from an existing wage and participation threshold pair unless unilateral deviation is the best response. Under this plausible assumption, the model predicts that there is a range of values for labor productivity over which both wages and participation are endogenously rigid. Proposition 2. Given an initial wage and participation threshold, {w ,r }, these values are 0 0 endogenously rigid for a range of labor productivity, [pL,pH], where H(w ) H(w ) pL(w ) = w + 0 pH(w ) = w + 0 . 0 0 0 0 h(w ) ih(w ) 0 0 Proof. The proof follows from considering the first-order conditions of the problem of a firm and worker. In this interval, no player has an incentive to unilaterally deviate. This is essentially a corollary to the existence of the interval of equilibria on [wL,wC] for any given productivity level. See Proposition 1. For i ∈ (0,1) this range of inactivity is depicted in Figure 5. The figure presents the firm’s monopsony problem constrained by the participation threshold as in Figures 2 and 3. 22WhenweconsiderafrictionallabormarketinSection5,wewilldiscusstheextenttowhichtheexpected distribution of shocks is orthogonal to player’s strategies. 17

Figure 5: Endogenous Rigidity. wages p H Marginal revenues consistent with inactivity p L w 0 H(w ) participation 0 Note: The firm’s monopsony problem constrained by the participation threshold as in Figures 2 and 3. The red shaded region highlights the range of productivities for which w is in the set of equilibria. 0 For productivity in [pL,pH], we see that the intersection between marginal cost and marginal revenue is at H(w ) and consistent with wage w and participation threshold r . 0 0 0 Endogenous Rigidity and the Barro (1977) Critique Barro (1977) suggests that any price rigidity that rules out a rent-producing trade is implausible. In addressing this criticising, it is useful to dissect rigidity in our model into ex-ante rigidity (before meeting the worker) and ex-post rigidity (after having met the worker). The firm in our model is free to choose any wage ex-ante (there are no menu costs). In this sense our rigidity is impervious to the Barro (1977) critique. Still, in equilibrium ex-ante posted wages are at times non-responsive to fundamentals due to coordination failure and, from Proportion 1, we know this has a welfare effect. On the other hand, the firm in our model experiences ex-post wage rigidity. We model wagesthiswayinordertocaptureaninformationalasymmetrybetweenworkerandfirm: the firm does not know and cannot learn the worker’s value of leisure. A typical critique of wage posting, in the vein of Barro (1977), is that in the sub-game in which a rent-producing match is ruled out, the informed party should reveal her type and bargaining should commence.23 23This criticism is valid for any model of monopolistic pricing: given the flexibility, the monopolist would 18

However, in the limiting case of our model as i falls to nil, this suspect sub-game never occurs, since the worker only searches if she will accept posted wage and all matches are consummated. Although ex-post wage rigidity never binds and ex-ante wages are flexible, wages are endogenously rigid on a vast region of values for labor productivity: for all realizations of productivity above p , wages remain unchanged and for shocks that return p L in this region, wages are endogenously rigid. In other words, this limiting case eliminates the sub-game that is the source of criticism but leads to the largest possible range of the fundamental for which wages are rigid in our model. This limiting case, however, provides results that are too strong for our purposes: subjected to shocks, an economy in which the worker always chooses non-participation when her value of leisure exceeds the expected wage will eventually converge to zero participation, zero employment, and zero output. We take an intermediate value of i, allowing i ∈ (0,1), in order to admit the possibility of macroeconomic recovery following contractions. When we have i > 0 we are partially subject to Barro’s criticism in the sense that ex-post some rent producing matches are ruled out.24 We turn now to analyzing equilibrium revisions for intermediate values of i. 4.2 Hysteresis Assumption 2 imposes that the equilibrium wage-labor force participation pair changes only when productivity evolves in such a way that the existing pair is no longer in the set of possible equilibria. At such a point the pre-existing wage and participation pair constitute a disequilibrium. Thus we require a protocol for equilibrium revelation given the observation of some disequilibrium. We consider Cournot tˆatonnement as a plausible equilibrium revelation protocol. Definition 2. A Cournot tˆatonnement is a sequence indexed by k = 1,2,... such that {w ,r } ∈ [b, ¯ b]2, and {w ,r } = {w∗(r ),r∗(w )} is the simultaneous best-response of 0 0 k k k−1 k−1 the firm and worker at iteration k to the k−1 value of the wage and participation threshold, respectively. TheCournot tˆatonnement revealsthenewequilibriumifthereexistssomek forwhichw and k always prefer to price discriminate and price discrimination is more efficient than monopoly pricing. 24Of course, if negotiation is allowed in this sub-game, there will be cases when a mutually beneficial deal canbestruck. Further,ifsuchnegotiationispermittedthiswillbeknownbyworkersand,asaresult,beliefs consistent with inactivity will be difficult to support (at least at the threshold for participation predicted in the baseline model). One solution to this sub-game that preserves our results is to assume that workers can trigger bargaining if they wish. If they do their bargaining power is zero. The result is that workers with b < r = w will never trigger bargaining. Workers with b ∈ (w ,p] may, and if they do they will receive 0 0 0 wages equal to their value of leisure: so, they won’t care if they do or don’t bargain. 19

r are mutual best responses. Note, while the Cournot tˆatonnement is iterative it is indexed k by layers of rationality–e.g., how many best-responses to best-responses each player must compute before convergence. We are agnostic as to the relation between number of iterations required to archive convergence and the calendar time elapsed and consider instead serial equilibria in the following results regarding comparative statics. As already noted, shocks that result in p ∈ (pL(w ),pH(w )) trigger no player to unilaterally deviate and the Cournot 0 0 tˆatonnement converges in the first iteration. Lemma 3. Given any shock to labor productivity, p, such that p > b, the ensuing Cournot taˆtonnement converges to an equilibrium. Further, positive (resp. negative) shocks lead to weakly positive (resp. negative) wage and threshold innovations. Proof. 1) The optimal posted wage conditional on a fixed expected participation threshold is weakly positively dependent on productivity, dw∗(r0,p) ≥ 0. This enables the initial deviation dp from an existing equilibrium. 2) The optimal threshold conditional on a fixed expected wage is independent of productivity, dr∗(w0,p) = 0. Thus, following any shock, even iterations fall dp on the worker’s best response function while odd iterations fall on the firm’s. 3) Posted wages exceed the threshold everywhere to the left of equilibria, wˆ(r ) > r ; posted wages 0 0 fall short of the threshold everywhere to the right of equilibria, wC(r ) < r ; and r∗(w ) is 0 0 k positive monotone. Thus, each iteration raises r∗(w ) if w < wL and lowers r∗(w ) if k−1 k−1 k−1 w > wC. 4) Finally, w∗(r ) is positive monotone for r > rC. Thus, whenever w > wC k−1 k k 0 we have monotone convergence from above to wC and whenever w < wL we have either 0 monotone convergence from below or there exists an iteration in which w∗(r ) > wC after k which we have monotone convergence from above. 5) Whenever the second order condition, 1 > d2H(b) 1 , holds we have monotone convergence from below to w∗ ∈ [wL,wC] where p−b db2 2h(b) wC is only attainable if r = b. In other words, efficiency is only attainable if the pre-shock 0 market was in collapse. Note that this second order condition is identical to that required for the unconstrained monopsony’s problem to have a unique solution. The left panel of Figure 6 illustrates the firm and worker’s best response functions before and after a negative productivity shock large enough to render p− < pL. The blue dotted line represents the workers best response function both before and after the shock. Note that the workers best response function is independent of productivity. The red lines represent the firm’s best response function before (solid) and after (dashed) the productivity shock. The negative productivity shock weakly decreases the wage that the firm is willing to pay for all participation thresholds. In particular, the illustrated shock is large enough to render w > w∗(r |p−). Thus, a Cournot tˆatonnement commences from {w ,r }. We label this as 0 0 0 0 k . In the first iteration of the Cournot tˆatonnement the worker best responds to w with 0 0 20

Figure 6: Equilibrium Revelation Following Large Productivity Shocks. Negative Shock Positive Shock Note: The left panel illustrates a negative productivity shock and the right panel a positive productivity shock. Hashed lines indicate post-shock schedules. Iterations of Cournot tˆatonnement induced by the productivity shock are indicated as the points labeled k , k , and k . As before, for ease 0 1 2 of illustration we take H(b) to be uniformly distributed on [b,¯b]. r = r∗(w |p−) and the firm best responds to r with w∗(r |p−) < w . We denote this set of 0 0 0 0 0 actions as k . Note that r and w∗(r |p−) are not mutual best response. In a second iteration 1 0 0 the worker’s best response function now dictates a deviation from k and the worker best 1 responds to w∗(r |p−) and sets her participation threshold at r∗(w∗(r |p−)) < r . Meanwhile 0 0 0 the firm again best responds to r with w∗(r |p−). We denote this set of actions as k . One 0 0 2 more iteration reveals that the actions specified by k are an equilibrium given p− and the 2 Cournot tˆatonnement converges in three iterations. TherightpanelofFigure6illustratesthefirmandworker’sbestresponsefunctionsbefore and after a positive productivity shock large enough to render p+ > pH. The blue dotted line again represents the workers best response function both before and after the shock. Again, the workers best response function is independent of productivity. The red lines represent the firm’s best response function before (solid) and after (dashed) the productivity shock. The positive productivity shock weakly increases the wage that the firm is willing to pay for all participation thresholds. In particular, the illustrated shock is large enough to render w < w∗(r |p+). Thus, a Cournot tˆatonnement commences from {w ,r }. We again label 0 0 0 0 this as k . As with the negative productivity shock, in the first iteration of the Cournot 0 tˆatonnement the worker best responds to w with r = r∗(w |p+). Meanwhile the firm best 0 0 0 21

Figure 7: Persistent Response to Transitory Productivity Shock. Pre-Shock Shock Recovery of MPL wages wages wages p p 0 2 p 1 w 0 w w 1 2 H(w ) part: H(w ) part: H(w ) part: 0 1 2 Note: At time 0 the economy is at steady state at wage level w . At time 1 a negative productivity 0 shock hits rendering p <p <p and inducting a negative revision in the wage to w <w as firm’s 1 L 0 1 0 restrict hiring. The shock is temporary and at time 2 the pre-shock productivity level returns: p =p . 2 0 Despite recovery in the fundamental, however, wages remain depressed: w >w =w . 0 2 1 responds to r with w∗(r |p+) > w . We denote this set of actions as k . Again, note that r 0 0 0 1 0 and w∗(r |p+) are not mutual best response. In a second iteration the worker’s best response 0 function now dictates a deviation from k and the worker best responds to w∗(r |p+) and sets 1 0 her participation threshold at r∗(w∗(r |p+)) > r . Meanwhile the firm again best responds 0 0 to r with w∗(r |p+). We denote this set of actions as k . One more iteration reveals that 0 0 2 the actions specified by k are an equilibrium given p+ and the Cournot tˆatonnement once 2 again converges in three iterations. Proposition 3. A transient productivity shock leads to persistent changes in wages, participation, and output. Proof. We construct a proof by example and illustrate in Figure 7. The left panel depicts an equilibrium at time 0 in which productivity is within some inaction range. Wages and participation are at levels w and H(w ), respectively. At time 1, a shock arrives such that 0 0 productivity at time 1 is outside and below the time 0 inaction rage. Lemma 3 guarantees that wages and participation fall to w < w and H(w ) < H(w ) respectively. A new 1 0 1 0 inaction range is established following the shock. Figure 7 illustrates that it is possible that the two inaction ranges overlap and, indeed, that p may fall inside the new inaction 0 range. At time 2 productivity recovers to the pre-shock level: p = p . However, when 2 0 22

Figure 8: scarring Effects of Contractions Pre-Shock Shock Recovery of MPL wages wages wages p p 0 2 p 1 w 0 w 2 w 1 H(w ) part: H(w ) part: H(w ) part: 0 1 2 Note: At time 0 the economy is at steady state at wage level w . At time 1 a negative productivity 0 shock hits rendering p <p <p and inducting a negative revision in the wage to w <w as firm’s 1 L 0 1 0 restrict hiring. The shock is temporary and at time 2 the pre-shock productivity level returns: p =p . 2 0 Despite recovery in the fundamental, however, wages remain depressed: w >w >w since firms are 0 2 1 constrained by the lower participation threshold induced by the shock. recovering from the shock, the firm is constrained by the shock-level participation threshold andassociatedinactionrange. DuetoAssumption2,wagesandparticipationdo not rebound when productivity rebounds. Proposition 4. Transient productivity shocks may lead to scarring. That is wages, participation, and output pruning below pre-shock levels. Proof. The proof follows from noting that during the contraction the firm’s optimization problem coincides with an unconstrained monopsony, equation 2, while during expansion the firm’s optimization is constrained and follows, equation 3. Further wˆ(r ) < wC whenever 0 r ∈ (b,wC) whenever the second order condition 1 > d2H(b) 1 holds. This condition 0 p−b db2 2h(b) coincides with the condition required for a unique wC. Note that Lemma 3 guarantees convergence to an equilibrium following each shock and that the conditions noted are those requiredforconvergencetow∗ ∈ [wL,wC). Further,fromProposition1wecanestablishthat, since wages and participation are larger in the pre-shocks equilibrium than the post-shocks, equilibrium output must also be larger. Figure 8 illustrates, the scarring effects on wage and participation. The left panel depicts an equilibrium at time 0 in which productivity is within some inaction range. Wages and participation are at levels w and H(w ) respectively. At time 1, a shock arrives such that 0 0 23

productivity at 1 is outside and below the time 0 inaction range. As a result wages and participation fall to w < w and H(w ) < H(w ) respectively. A new inaction range is 1 0 1 0 established following the shock. As discussed in Figure 7, it is possible that the two inaction ranges overlap and, indeed, that p may fall inside the new inaction range. Figure 8 displays 0 that it is possible that p may also fall outside and above the new inaction range. At time 0 2 productivity recovers to the pre-shock level: p = p . However, when recovering from 2 0 the shock, the firm is constrained by the shock-level participation threshold and associated inaction range. As a result, wages and participation do not fully rebound when productivity rebounds. Further, from Proposition 1 we can see that output and welfare are larger in the pre-shocks equilibrium than the post-shocks equilibrium. 5 A Two-Sided Frictional Labor Market We embed this stylized game in a frictional labor market that is endowed with appropriate features to justify the assumptions of our two-player game. In the two-player game we assumed that the probability of a worker (firm) meeting a firm (worker) is exogenous to that player’s strategy. Here, this assumption is micro-founded by the assumption that every agent is atomistic. Thus, the equilibrium job finding and filling hazards are exogenous to each worker’s and each firm’s strategy. In the two-player game we also assumed that the firm holds monopsony power and posts wages ex-ante. Here, as Diamond (1971) shows, sequential random matching endows the firm with this monopsony power and the monopsony wage level prevails even in the limit as search friction fades. Finally, in the two-player game we assumed no continuation payoffs. This is justified in the two sided game by the observation that Diamond (1971) guarantees a degenerate wage distribution, and thus no option value of search for the worker and free entry drives the value of a vacancy to zero for the firm. In the Appendix, we present the Bellman equations associated with each worker’s and each firm’s decision problem. These yield participation and wage posting strategies that are identical to the two-player game up to the scaling of the fraction of unemployed who will not accept the going wage offer i. The best response of workers is a participation threshold strategy as in equation (1), and the best response of firms is a wage posting strategy as in equation (4), where i is replaced by ι = i/uˆ and uˆ is the equilibrium unemployment rate of worker who will accept the equilibrium wage offer. WeclosethemodelbypositingastandardmatchingfunctionasintheDMPmodel. Firms post vacancies at flow cost, c and workers engage in search at zero cost. Both discount the future at rate ρ. As in the baseline DMP, the flow of new matches is determined by the matching function, denoted as m(U,V), where U is the mass of unemployed workers and 24

V is the mass of vacancies. Imposing Inada conditions and constant returns to scale, the job-finding rate of unemployed workers, f(θ) ≡ m = m(1,θ), is increasing and concave in the U market tightness defined as the ratio of vacancies to the unemployed, θ = V . Analogously, U the rate at which vacancies meet unemployed workers, q(θ) ≡ m = f(θ), is a positive and V θ decreasing function of market tightness.25 Note that since we have assumed random matching, the matching rate of a worker with a high flow value of leisure and a low flow value of leisure are the same whenever they participate in the labor market. Thus, the mass of workers who search for work but will reject wage offers of w∗, i(1 − H(w∗)), are just as likely to meet a firm as the uˆH(w∗) mass of workers who are unemployed and will accept wage offer w∗. Since the i(1−H(w∗)) mass of workers reject the wage offer whenever they meet a firm the vacancy filling rate is g(θ) = q(θ) , where Λ(w∗) ≡ H(w∗)+ι(1−H(w∗)) is a measure of the severity of the congestion Λ(w∗) H(w∗) imposed by workers with leisure value above the equilibrium wage level searching for work while at the same time rejecting all offers. Λ = 1 occurs when workers with flow value of leisure above the expected wage offer never search and is the case when congestion is minimized. For ι > 0 we have Λ > 1 and congestion drives a wedge between the rate at which vacancies meet employees and the rate at which jobs fill. For ι < uˆ we have a kink in the labor supply curve as in the two-player representation. Given the wage, free entry into vacancy creation pins down the labor market tightness as the value of a vacancy is driven to zero. Thus equilibrium tightness satisfies: cΛ(w∗) p−w∗ = . (5) q(θ) ρ+δ Conditional on a participation threshold, the job creation condition is downward sloping.26 Since the wage schedule is flat, this guarantees a unique equilibrium tightness for every productivity and wage pair. From equation ?? we can see that there are two sources of amplification in our model. The first is the proximity of p to w∗, as highlighted by Hagedorn and Manovskii (2008). The second is the elasticity of Λ with respect to w∗, which is new. The second sources captures the change in congestion following a wage revision, which is also a source of propagation. 25It is important to note here that our model differs in an important way from the classic model of upward sloping labor demand: Diamond (1982). In that model, multiplicity derives from a thick market externality generated by increasing returns to scale in the matching function. In our model, multiplicity derives from a pair of externalities, a thick market externality derived from workers’ participation decision and a pecuniary externality derived from firms’ wage posting decision. We follow the main-stream DMP literatureandimposeconstantreturnstoscaleonourmatchingfunction. Weappealtotheempiricalresults summarized in Pissarides and Petrongolo (2001) to justify this assumption. 26To see this note that dΛ <0 and dΛ >0. dw dθ 25

In Section 6 we show how these forces can combine to create a “jobless” and a “wageless” state following a transitory contraction. In Section 7 we exploit this intuition to identify key parameters of the model. Definition 3. A symmetric rational expectations equilibrium of the frictional labor market is a triple – wage, participation threshold, labor market tightness – such that: 1. the wage and participation threshold are mutual best responses. 2. labor market tightness satisfies the free entry condition, equation (5). Notethatallfirms(resp. workers)facethesameobjectivefunctionandwehaverestricted our attention to equilibria that are symmetric in the sense that every firm (resp. worker) plays the same strategy as every other firm (resp. worker). Thus our equilibria, rigidity, and convergence results from the two-player game naturally extend to the two sided frictional labor market. Finally, we pin down the unemployment rate. In steady state, the flow into and out of unemployment for the subset of workers who will accept offer w∗ must be equal. Since job destruction is an exogenous shock that arrives with Poisson arrival δ, we have uˆ = δ . δ+f(θ(w∗)) Also note that a mass equal to i[1−H(w∗)] are perpetually unemployed. Finally, the total mass of labor force participants is H(w∗) + i[1 − H(w∗)]. Thus we have the steady state unemployed rate u = uˆH(w∗)+i[1−H(w∗)]. H(w∗)+i[1−H(w∗)] The slope of the job creation condition, equation (5), depends on congestion. For a given wage, more congestion implies a steeper job creation condition and therefore a looser labor market. This follows from the decrease in the expected value of a vacancy when there is an increase in the probability of meeting a worker who will reject the wage offer. Since congestion lowers the value of a vacancy, fewer are produced. 6 “Jobless” and “Wageless” Recoveries TheeffectofcongestiononmarkettightnessisillustratedinFigure9. Intheleftpanelweplot a market in which the fraction of workers who search even when the expected wage, i, falls shortofthevalueofleisureishigherthanintherightpanel. Consequently, foranywagelevel congestion is higher in the left panel than the right, this is reflected in the steeper slope of the job creation conditions in the left panel than in the right. Comparing θ+ (resp. θ−) across panels we can see that higher congestion implies lower labor market tightness conditional on the wage level. Higher i increases the steady state unemployment rate. This can be decomposed into two effects. The direct effect is that the mass of perpetually unemployed, 26

Figure 9: Wage Effect (WE) and Congestion Effect (CE). Congestion Effect Dominates Wage Effect Dominates wages wages p p J C (w ! J C (w+ ) J C (w ! ) JC (w+ ) ) w+ w+ w! w! WE WE CE CE 3! 3+ tightness 3+ 3! tightness Note: Job creation is a downward sloping curve pinned down by the free entry condition, equation (5), while the wage is a horizontal line pinned down by the posted wage. Thus we are guaranteed a unique labor market tightness for every wage and productivity pair. An increase in congestion rotates the job creation condition clockwise. The left panel illustrates equilibrium labor market tightness at a high wage (solid) and low wage (hashed) equilibrium and when the congestion effect is large relative to the wage rigidity effect. The right panel illustrates the converse. i[1−H(w∗)], increases. The indirect effect is that the mass of unemployed who are willing to work at the equilibrium wage, uˆH(w∗), increases since congestion reduces the value of a vacancy and loosens the labor market. Decreasing the wage from w+ to w− also increases congestion since the mass of searching workers who will not accept the wage offer rises from i[1 − H(w+)] to i[1 − H(w−)]. This mechanically increases the fraction of workers who are voluntarily unemployed. However, the effect on the unemployment rate of workers who will accept the wage offer w− and on the overall unemployment rate is ambiguous. If the increase in congestion triggered by moving from the high wage to the low wage equilibrium is large enough, then unemployment will be higher in the low-output equilibrium. This is illustrated in the left panel: the congestion effect (labeled CE) is larger than the wage effect (labeled WE). If the increase in congestion ismild, unemploymentamongworkerswhowillacceptw0 falls andthedropmaybesufficient to offset the rise in unemployment among the perpetually unemployed, with the total effect being a drop in overall unemployment. This is illustrated in the right panel: the congestion 27

effect is smaller than the wage effect.27 Referring to Proposition 1, we note that output under w+ and w− can be ranked with output under w+ exceeding that under w−.28 We see that unemployment and output move in opposite directions only when the congestion effect is severe enough. Thus, without knowledge of the magnitude of the congestion effect the relation between fluctuations in unemploymentandoutputisambiguous, andwefailtoconsistentlygeneratesatisfyingbusiness cycle regularities, such as counter-cyclical unemployment, simply from animal spirits.29 This seemingly undesirable ambiguity actually works to our benefit and enables us to generate instances of both “jobless” and “wageless” phases of recovery from productivity contractions–that is, a period during which unemployment remains elevated even after productivity has recovered and a phase during which unemployment falls below pre-contraction levels while wages remain depressed. Joblessness operates through the increase in congestion induced by the fall in the participation threshold during a severe contraction. Wagelessness operates through an increase in the wage effect induced by persistence of the lower severe-contraction wage level beyond the duration of the disturbance. When the increase in congestion is not particularly severe, wagelessness may dominate throughout a recovery. However, whenever the drop in the participation threshold is severe enough to trigger a jobless recovery, joblessness will occur during the early part of the recovery and will be followed by a period during which unemployment falls to unprecedented lows while wages continue to lag: a ”wageless” phase. Figure 10 illustrates. The first two panels illustrate a steady state at time 0. The firm’s wage setting problem is illustrated in the left panel and the job creation condition in the right. At this pre-shock steady state, wages, participation, and labor market tightness are w , H , and θ , respectively. The economy is then hit by a shock that lowers productivity to 0 0 0 p−, a value below the pre-shock inaction range. This is illustrated in the second set of panels. As a result of the shock, wages and participation fall to w− and H−. Labor market tightness also falls. The drop in labor market tightness is the result of three forces: the drop in productivity, an increase in congestion due to the drop in the participation threshold, and a drop in wages. The first two clearly push tightness down. That these dominate the third is a result of firms’ optimization over wages. Note that a fall in labor market tightness implies an increase in unemployment even among searchers who will still accept the new, lower, wage: 27To see that this case exists for every parameterizations of H(b), m(U,V), p, and c for which there are equilibria, consider the limit as i(cid:38)0 or the limit as w− (cid:37)w+ for fixed i. 28Whenever both are candidate equilibrium: w+ ∈[wL,wC] and w− ∈[wL,wC]. 29Asnotedpreviously,themodelcouldbeusedtocreateeconomicfluctuationsviacoordinatedchangesin expectations(i.e.,animalspirits). Forsuchfluctuationstomatchabasicbusinesscyclesfact–countercyclical unemployment–we would need to impose that congestion is large enough. 28

Figure 10: “Jobless” and “Wageless” Recoveries. :A lenaP........ kcohs-erP....... wage wage p p 0 0 w w 0 0 H(w ) part: 3 tightness 0 0 :B lenaP........ kcohS.......... wage wage p 0 p! p! w w! w!0 H(w!) part: 3! 3 tightness 0 :C lenaP........ LPM fo yrevoceR... wage wage p p 0 0 w w! w!0 H(w!) part: 3J 3 tightness 0 :D lenaP........ egaW fo yrevoceR.. wage wage p+ p+ p 0 w0 w 0 H(w ) part: 3 3W tight: 0 0 Note: (Panel A) The economy begins at steady state at wage level w . (Panel B) A negative 0 productivity shock hits rendering p− <p and inducting a negative revision in the wage to 0 w− <w and in labor market tightness to θ− <θ . The shock induces a clockwise rotation 0 0 in the job creation condition as the decrease in the wage offer increases congestion. (Panel C) Productivity recovers to p but wages remain depressed at w− <w . The congestion effect 0 0 dominates the wage effect and market tightness also remains depressed at θJ <θ . (Panel D) 0 Productivity overshoots, p+ >p and wages recover the initial level, w . Congestion 0 0 dissipates when wages recover, isolating the wage effect, and we have θW >θ . 0 29

match efficiency has fallen and the Beveridge curve has shifted outward. Unemployment increases further due to an increase in the share of workers who search but will not accept employment, i[1−H(w )], increases to i[1−H(w−)]. 0 The third set of panels illustrates wages, participation, and labor market tightness if the economy is subsequently hit by a productivity shock that returns productivity to the preshock level. From Proposition 3 and Proportion 4 we see that wages and participation may not fully recover despite the recovery in productivity. As a result, labor market tightness rises. Now tightness is a result of two of the three forces: the increased in congestion due to the lower participation threshold and the lower wage level. As was illustrated in Figure 9, it is possible that the congestion effect dominates the wage effect. In this case, both the mass of unemployed who will accept and who will not accept the equilibrium wage offer remainelevated. Inparticular, congestionfromthesearchingworkerswhowillnotacceptthe equilibrium wage offer increases the unemployment rate of those who will: match efficiency remains low and the Beveridge curve shifts outward. We have a jobless recovery. The final set of panels illustrates the productivity shock required to return the economy to the pre-shock wage level following the contraction. When wages and participation recover to the pre-shock level so too does the magnitude of the congestion effect and the mass of workers who search but will not accept wage offer w . However, due to the asymmetry 0 induced by the inaction range, the productivity level required to return the pre-shock wage level exceeds the pre-shock productivity level. Thus, labor market tightness in the period in which wages and participation recover must exceed the tightness experienced just before the impact of the first shock, since all that remains from the contraction is the wage effect. Thus, we have that at least the end phase of each recovery from any contraction severe enough to yield a downward revision in the wage must be wageless. Thus, every jobless recovery must be followed by a wageless recovery. 7 How Much Room is Left to Run? The purpose of this section is to use our model to discuss its implications for the unemployment rate during the latest recovery phases. Our model predicts that unemployment must fall below pre-shock levels before wages recover to pre-shock levels. Quantitatively, how far below? Here, we describe how to identify key parameters of our model given 1) a shock in which participation falls followed by 2) a jobless recovery. We then simulate the unemployment rate predicted to arise when wages and participation recover. 30

7.1 Identification In general, since our model exhibits multiplicity, identification is challenging. A continuum of equilibria are consistent with a given set of fundamentals; thus, observation of a given equilibrium or sequence of equilibria may be consistent with an interval of parameter values. However, identification can be achieved in certain situations by making three observations. 1. Givenasevereenoughnegativeproductivityshock, theproposedequilibriumrevelation mechanism–Cournot tˆatonnement–predicts a unique equilibrium from the feasible set. Importantly, such a shock generates an empirically observable flag: a decline in labor force participation. 2. The difference between market tightness before a severe shock and after productivity has recovered is in part a function of the change in the participation threshold, r∗, and the probability that a worker is voluntarily unemployed, i. Thus, we can use information about the change in participation and the change in tightness to help identify the participation threshold and the probability of voluntary unemployment. That is we compare the labor market tightness and associated unemployment rate impliedbyEquation5priortotheshockandatthemomentwhenproductivityrecovers to the pre-shock level: (cid:18) cΛ(w )(ρ+δ) (cid:19) (cid:18) cΛ(w−)(ρ+δ) (cid:19) q−1 0 = θ(p,w ) q−1 = θ(p,w−), p−w 0 p−w− 0 recalling that Λ(w) ≡ H(w)+ι(1−H(w)) measures the congestion in the labor market at H(w) wage level w. 3. What remains is to identify the firm’s surplus, p − w∗(r∗). Note that there are two sources of amplification in our model–the wedge between productivity and wages and thechangeincongestionassociatedwithchangingtheequilibriumselection. Wejointly use the information about the change participation and difference in tightness between the pre-shock and shock and between shock and recovery of productivity to attribute amplification to the two sources. This provides the remaining variation needed to pin down the firm’s surplus in the initial equilibrium. Thus, data consisting of observations of unemployment and participation rate before, during, and after a negative productivity shock severe enough to trigger a downward revision in participation, along with severity of the shock itself are sufficient to identify the key parameters of our model so long as the distribution of workers value of leisure–H(b)–can be 31

described by two or fewer parameters. In the following, we take a local linear approximation to H(b) which is equivalent to modeling it as uniform. 7.2 Data We consider the two most recent contractions–the dot-com bust and the financial crisis– both of which feature jobless recoveries.30 We measure productivity as deviations from GDP trend as identified by the filter proposed by Hamilton (2017).31 We then identify six dates (three for each recession). The first four such identified dates–1999;Q1, 2003;Q3, 2005;Q1, and 2011;Q1–feature nearly identical deviations of GDP above trend. At the two remaining dates–2002;Q2 and 2009;Q1–GDP makes its maximum deviation below trend in the respective business cycle episodes. The difference in detrended GDP between 1999;Q1 and 2002;Q2 is 3.8% and between 2005;Q1 and 2009;Q1 is 9.2%. These magnitudes may seem large for relative to those derived using an HP filter, but are consistent with the series reported in Hamilton (2017). Next we construct the unemployment and labor force participation rates for workers with 1-30 years of potential experience who are not in school, ill or disabled using the Current Population Survey basic monthly files and compute quarterly averages.32 Observing that unemployment peaks roughly five quarters after the respective negative productivity shocks we construct our vector of target moments as the unemployment and labor force participation rates attained on average in the 4-7 quarters after each of the aforementioned dates selected using the TFP series. These are summarized in Table 2. 7.3 Calibration We calibrate the model by matching moments. A priori, we adopt a constant returns to scale Cobb-Douglas matching function–m(U,V) = AUηV(1−η)–and parameterizations for the matching function, discount rate, separation rate, and vacancy cost from the literature. Specifically we follow Barnichon and Figura (2015) and take η = 0.67; and follow Shimer (2005) and take δ, ρ, and c to be 0.1, 0.012, and 0.231 respectively. The remaining parameters–match efficiency, A, the linear approximation of H, and the location of p within the interval [pL,pH] prior to the shocks–are calibrated to minimize the 30Secular trends toward increased female labor force participation dissipated after 2001. Following this date, maleandfemaleparticipationmovetogether. Sincetherearenogenderspecificproductivitymeasures we restrict our attention to the recessions that have occurred since this synchronization. 31Bureau of Economic Analysis, Gross Domestic Product (GDP), https://www.bea.gov/data/gdp. 32U.S. Census Bureau, Current Population Survey, http://www.census.gov/programs-surveys/cps/datadetail.html. 32

Table 2: Calibrated Simulations. Pre-shock Shock Recover p dot-com 1999:Q1 2002:Q2 2003:Q3 data model data model data model Labor Force Participation Rate 82.7 83.1 81.2 80.9 81.0 80.9 Unemployment Rate 4.1 4.1 6.3 4.8 5.5 4.2 Pre-shock Shock Recover p Financial Crisis 2005:Q1 2009:Q2 2011:Q3 data model data model data model Labor Force Participation Rate 81.2 82.1 80.2 79.6 79.7 79.4 Unemployment Rate 4.9 4.9 10.0 6.0 8.5 5.0 Sample: Workers with 1-30 years of potential experience who are not in school, ill, or disabled. Source: U.S. Census Bureau; U.S. Bureau of Economic Analysis; authors’ calculations. Table 3: Parameters. ¯ ι A pos b b dot-com 0.74 1.13 0.85 0.75 0.88 Financial Crisis 0.69 0.79 0.87 0.51 0.76 Fitted values minimizing squared deviations from the three observed participation rates, the initial unemployment rate, amplification ( shockunemployment ), and persistence pre−shockunemployment ( shockunemployment ) underweighting unemploymentatrecoveryofproductivity the loss associated with missing the amplification moment by a factor of 100. sum of squared errors between model simulated moments and empirical moments. Specifi- ˜ cally, the linear approximation to H is captured by two parameters: b such that H(b) = 0 ¯ ˜ ¯ ˜ and b such that H(b) = 1 where H is the uniform distribution that linearly approximates H at the pre-shock wage and participation. We express the location of p within [pL,pH] as a weighted average of pL and pH and estimate as pos the weight placed on pH. We note several countervailing forces: 1. On the one hand, our model contains an amplification mechanism in the form of rigid wages. Thus, if only amplification were targeted, a calibration in which the probability of voluntary unemployment were very low and the initial productivity level were well above the lower bound of the range of inactivity could easily match the data. However, such a calibration predicts little persistence of the unemployment rate or perhaps an 33

even more rapid rise than fall of the unemployment rate. The intuition of such a calibration is illustrated in the right panel of Figure 9. 2. On the other hand, our model contains a propagation mechanism in the form of congestion on the part of the voluntarily unemployed. Congestion is largest when many workers with high reservations search; however, the higher is these workers’ probability of participation then the smaller is the interval of equilibria and thus the smaller is the wage rigidity. Thus, targeting propagation will tend to imply calibrations which obtain less amplification due to wage rigidity. Weaddressthesecountervailingforcesinourselectionoflossfunction. Wetargetsquared deviations from the three observed participation rates, the initial unemployment rate, amplification ( shockunemployment ), and persistence ( shockunemployment ). In adpre−shockunemployment unemploymentatrecoveryofproductivity dition, we acknowledge the difficulty in obtaining amplification in this class of models and thus underweight the loss associated with missing the amplification moment by a factor of 100. Table 2 presents the participation and unemployment rates predicted by calibrated models which target participation and unemployment data moments from the dot-com and from financial crises. Table 3 reports parameter values obtained for each episode. 7.4 Results Some items jump out immediately. First, the fit contains limited amplification. This, as we have noted, is unsurprising as this is a variant of the DMP model. Additionally, it is unsurprising since we have selected an objective function that underweights this amplification. Adjusting the model to admit amendments that enhance volatility by reducing the firm’s surplus produce similar results. The intuition is that these adjustments bind the wage setting problem as well as the vacancy creation problem. In other words, small changes in labor force participation in response to the observed productivity shocks rule out a very small firm surplus in the wage setting problem. Thus, in the end, these model adjustments lead to largely similar projections for the undershooting in unemployment commensurate with wage recovery.33 Second, and more interestingly, independently calibrated models call for similar values of share voluntarily unemployed, ι, but wildly different values of the matching efficiency, A. This suggests a structural break in the value of the match efficiency around the time of the financial crisis. This is a view that is consistent with the observations of Diamond and S¸ahin 33AnalternativeistoadmitdiscountrateshocksasinHall(2017). Suchshocksenterthevacancycreation problem but not the wage setting problem and thus could increase the amplification in our model. 34

(2015); S¸ahin et al. (2014) and Barnichon and Figura (2015) that match efficiency fell during the financial crisis. Third, calibrations imply wage levels at the onset of recessions between 60 and 75 percent of productivity prior to the onset of crisis. This is, unsurprisingly, significantly lower than the estimated implied by Hagedorn and Manovskii (2008) and follows both from the presence of endogenous wage rigidity in our model and from the relative weight placed on matching persistence. Still our estimate is significantly larger than that implied by Shimer (2005). Indeed, our estimate is most closely in line with Hall and Milgrom (2008) and our relative poor performance with respect to amplification indicates that our wage setting mechanism, so calibrated, is relatively more flexible. Again, this is not surprising as our main objective is to generate persistence and not amplification. Forth, we match persistence, measured as ( shockunemployment ), quite well. unemploymentatrecoveryofproductivity In the data, persistence stands at 1.15 and 1.18 for the dot-com and financial crisis respectively. In the calibrated model we have 1.15 and 1.19 respectively.34 This is the goal of our calibration. 7.5 Counterfactual Unemployment Rates We turn now to projecting the level of productivity and associated unemployment rate that would need to arise in order to return wages and participation thresholds to their pre-shock levels. Forthedot-comcrisisourmodelsuggeststhatovershootingpre-shockproductivityby 2 percent is required to regain the pre-shock wage level. For the financial crisis the equivalent figure is 3 percent. These imply unemployment rates of 4.0 and 4.8 percent compared to the pre-shock rates of 4.1 and 4.9. While these undershoots seem modest they should be taken in the context of a model which predicts very low volatility in the unemployment rate. A back of the envelope calculation which divides these deviations by the shortfall in volatility suggests unemployment rates of 3.0 and 2.8 might have needed to pertain after each of these recessions respectively before wages and participation thresholds recover to pre-shock levels. 8 Conclusion We have considered the possibility that when workers are faced with a labor market in which the anticipated wage falls short of their outside option they may choose to not search. This may induce a kink in the labor supply curve faced by firms. From this consideration, we have 34While respecting a sum of mean squared errors for the block of participation moments of 2.38E-5 and 1.27E-4, respectively. 35

shown that a strategic complementarity between workers’ job search and firms’ wage posting exists and leads to a continuum of welfare-ranked equilibria. We show that a correlate of the existence of a continuum of equilibria is intervals of endogenous rigidity with respect to wages and labor force participation. We consider this framework in the context of productivity shocks and posit that neither workers nor firms deviate from an existing wage and participation threshold pair unless unilateral deviation is a best response. We then show that, in the event of deviations, players can forecast opponents’ best response and learn the new equilibrium. Further, the kink in the labor supply curve induces asymmetry such that wages and participation may exhibit persistent responses to temporary shocks and that upward revisions in wages and participation fall farther from efficiency than do downward revisions. These are features of hysteresis. We embed these intuitions in a model of a frictional labor market and show that such a market can exhibit both “jobless” and “wageless” phases of recovery from productivity contractions–that is, a period during which unemployment remains elevated even after productivity has recovered and a phase during which unemployment falls below pre-contraction levels while wages remain depressed. Joblessness stems from a congesting effect of search on the part of workers whose outside option exceeds the equilibrium wage offer. Wagelessness stems from asymmetric responses to shocks. In order to recover a wage level, the rebound must overshoot the pre-shock productivity level and during such a rebound unemployment consequently undershoots the pre-shock unemployment rate. A calibration suggests that the required undershooting in unemployment could be substantial. 36

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A Value Functions in the Two-Sided Labor Market We can write out the asset value equations faced by a generic worker: ρI(b,w ) = b 0 ρU(b,w ) = b+f(θ)max{0,[W(b,w )−U(b,w )]} 0 0 0 ρW(b,w ) = w +δ[U(b,w )−W(b,w )] 0 0 0 0 Thefirst,I(b,w ),capturestheassetequationforaninactiveworkerwithflowvalueofleisure 0 b. This worker simply consumes b.35 The second, U(b,w ), captures the asset equation for 0 a searching workers. While searching the worker consumes flow value b and at hazard f(θ) receivejobofferswhichyieldoptionvaluemax{0,[W(b,w )−U(b,w )]}. Thethird, W(b,w ) 0 0 0 captures the asset equation for employed workers, these consume flow value w and at hazard 0 δ are separated to unemployment. Webeginbysolvingforthegenericworker’sreservationwageconditionalonparticipating. This sets the value of unemployment equal to the value of employment: W(b,w ) = U(b,w ). 0 0 Thus we have a reservation wage equal to this generic workers realization of the flow value of leisure: b. In the event that w exceeds b this worker anticipates higher value from 0 the employed state than the unemployed state: given the opportunity she accepts all wage offers.36 If w falls short of b then the worker anticipates lower value from the employed state 0 than the unemployed state: given the opportunity, she rejects all wage offers. Now we turn to considering the threshold participation decision. We can observe that if b < w then the worker strictly prefers unemployment to inactivity and employment to 0 unemployment: I(b,w ) < U(b,w ) < W(b,w ). Conversely, if b > w then the worker 0 0 0 0 is indifferent between inactivity and unemployment and strictly prefers either of these to employment: I(b,w ) = U(b,w ) > W(b,w ). Thus we have that whenever b < w the 0 0 0 0 worker always prefers to participate and accepts employment whenever matched. Meanwhile if b > w the worker is indifferent between participation and non-participation and strictly 0 prefers not to accept employment whenever matched. As in the two player game we break the worker’s indifference with Assumption 1. 35Notethatinsteadystatethisisindependentofthewagelevel. Whenaddingshockswenotethataslong as there is no barrier to reentering the unemployment pool following a shock the independence is preserved. 36As we have seen in the two player game and will see again here, information frictions prevent the firm from observing b and thus from setting w =b for each worker and extracting the full rent. 40

We can also write out the asset value equations faced by a generic firm: (cid:104) ρV(w,r ) = −c+q(θ) I uH(w) 0 {w≤r0}uH(r0)+i(1−H(r0)) (cid:105) +(1−I )uH(r0)+i[H(w)−H(r0)] [J(w,r )−V(w,r )] {w≤r0} uH(r0)+i(1−H(r0)) 0 0 ρJ(w,r ) = p−w+δ[V(w,r )−J(w,r )] 0 0 0 The first asset equation, V(w,r ), captures the value of a open vacancy. The vacancy costs 0 the firm a flow of c and with hazard q(θ) the firm meets and makes the posted wage offer w to a worker. The bracketed term follows the logic of the firm’s problem in the two player 0 game: higher wage offers are accepted by a larger fraction of the workers that the firm might meet with the return to increasing the wage offer being discontinuous at the expected participation threshold. The final term is the option value of forming a match. The second asset equation, J(w,r ) captures the value of a filled job. This is the flow rent (p−w) and 0 the hazard of separation times the option value of separation. Wepositfreeentryintovacancycreation. Thisdrivesthevalueofavacancytozero. Thus the firm’s objective function in the two-sided game is isomorphic to the objective function of the firm in the two-player game. 41