Employment Fluctuations with Equilibrium Wage Stickiness ...

Employment Fluctuations with Equilibrium Wage Stickiness

By ROBERT E. HALL*

Following a recession, the aggregate labor market is slack–employment remainsbelow normal and recruiting efforts of employers, as measured by help-wantedadvertising and vacancies, are low. A model of matching friction explains thequalitative responses of the labor market to adverse shocks, but requires implau-sibly large shocks to account for the magnitude of observed fluctuations. Theincorporation of wage stickiness vastly increases the sensitivity of the model todriving forces. I develop a new model of the way that wage stickiness affectsunemployment. The stickiness arises in an economic equilibrium and satisfies thecondition that no worker-employer pair has an unexploited opportunity for mutualimprovement. Sticky wages neither interfere with the efficient formation of employ-ment matches nor cause inefficient job loss. Thus the model provides an answer tothe fundamental criticism previously directed at sticky-wage models of fluctuations.(JEL E24, E32, J64)

Modern economies experience substantialfluctuations in aggregate output and employ-ment. In recessions, employment falls and un-employment rises. In the years immediatelyafter a recession, the labor market is slack—unemployment remains high and the vacancyrate and other measures of employer recruitingeffort are abnormally low. Unemployment isdetermined by the rate at which workers losejobs and the rate at which the unemployed findjobs. I develop a model of fluctuations with amatching friction and sticky wages. The incor-poration of wage stickiness makes employmentrealistically sensitive to driving forces. Mycharacterization of wage stickiness is rather dif-ferent from earlier ideas of wage rigidity andmore closely integrated with the matching pro-cess. The model describes an economic equilib-

rium and overcomes the arbitrary disequilibriumcharacter of earlier sticky-wage models.

A line of research starting with Peter Dia-mond (1982), Dale Mortensen (1982), andChristopher Pissarides (1985)—nicely summa-rized in Pissarides (2000) and in Robert Shimer(2005)—provides an account of unemploymentas a productive use of time. I adopt many of theelements of their model—the DMP model—inthis paper. The DMP model views the labormarket in terms of an economic equilibriumwhere workers and employers interact purpose-fully. A friction in matching unemployed work-ers to recruiting employers accounts for theexistence of unemployment. Variations in theeconomic environment lead to fluctuations inunemployment. The DMP model portrays wagedetermination as a Nash bargain, where em-ployers receive a constant fraction of the matchsurplus. The payoff to recruiting activity—theemployers’ share of the surplus—is not verysensitive to driving forces. Hence the DMPmodel cannot explain the magnitude of move-ments in recruiting activity. In reality, the labormarket slackens substantially in recessions andworkers encounter difficulty in finding jobs,but the DMP model with Nash-bargain wagedetermination suggests stability in job-findingrates under plausible variations in the drivingforces.

* Hoover Institution and Department of Economics,Stanford University, Stanford, CA 94305 (email: [email protected]). This research is part of the research program onEconomic Fluctuations and Growth of the National Bureauof Economic Research. I am grateful to the editor and threereferees, to George Akerlof, Anthony Fai Chung, KennethJudd, Narayana Kocherlakota, John Muellbauer, GareyRamey, Felix Reichling, Robert Shimer, and Robert Solow,and to numerous seminar and conference participants forhelpful comments. Data and programs are available fromthe author’s Web site, http://www.stanford.edu/�rehall.

50

In a model with matching frictions, the bar-gaining set for wage determination is relativelywide, because the difficulty in locating matchescreates match capital the moment a tentativematch is made. The value of the match capitaldetermines the gap between the minimum wageacceptable to the worker and the maximumwage acceptable to the employer. From the per-spective of bilateral bargaining theory in gen-eral, any wage within the bargaining set couldbe an outcome of the bargain. The Nash bargainsets the wage at a weighted average of thelimiting wages, with a fixed weight over time.The alternative I offer permits variations overtime in the position of the wage within thebargaining set. When the wage is relativelyhigh—closer to the employer’s maximum—theemployer anticipates less of the surplus fromnew matches and puts correspondingly less ef-fort into recruiting workers. Jobs become hardto find, unemployment rises, and employmentfalls.

In the sticky-wage model I develop, whentemporary changes in the economic environ-ment shift the boundaries of the bargaining set,the wage remains constant, provided it remainsinside the bargaining set. The wage adjusts overtime in response to nonstationary changes in theenvironment. This mechanism guarantees thatwage rigidity never results in an allocation oflabor that is inefficient from the joint perspec-tive of worker and employer. Consequently, themodel provides a full answer to the condemna-tion of sticky-wage models in Robert Barro(1977) for invoking an inefficiency that intelli-gent actors could easily avoid. Unlike stickinessportrayed as an essentially arbitrary restrictionon the ability to set wages or prices—such asthe well-known model for prices of GuillermoCalvo (1983)—the stickiness considered herearises within an economic equilibrium. It satis-fies the criterion that no employer-worker pairforgoes bilateral opportunities for mutual im-provement. Peter Howitt (1986) made thispoint.

Although wage stickiness has no effect on theformation of a job match once worker and em-ployer meet and no effect on the continuation ofthe match, stickiness does have a profound in-fluence on the search process. If wages aretoward the upper end of the bargaining set, the

incentives that employers face to look for addi-tional workers are low. I start the paper withevidence about the remarkably strong procycli-cal movements of help-wanted advertising andvacancies. This evidence supports the mecha-nism proposed here.

I then turn to the model. I adopt the matchingfriction of the DMP model. But as Shimer(2005) and Marcelo Veracierto (2003) havestressed, the DMP model and others with thesame basic view of the labor market do not offera plausible explanation of observed fluctuationsin unemployment. The magnitude of changes indriving forces needed to account for the rise inunemployment and decline in recruiting effortduring slumps is much too large to fit the factsabout the U.S. economy. For this reason—andfollowing Shimer’s suggestion—I introducewage stickiness into the DMP setup. The result-ing model makes recruiting effort, job-findingrates, and unemployment remarkably sensitiveto changes in determinants. A small decline inproductivity results in a slump in the labormarket. With wage stickiness, these changesdepress employer returns to recruiting substan-tially. The immediate effect is a decline in re-cruiting efforts, a lower job-finding rate, and aslacker labor market with higher unemployment.

I focus on the points that sticky wages canarise in a full economic equilibrium and thatstickiness results in high volatility of employ-ment fluctuations. I do not venture into theterritory of explaining why the economy ap-pears to choose sticky wages from the widevariety of alternative equilibrium wage patterns.In addition, I do not try to demonstrate thataggregate or individual wages actually aresticky. The reason is simple: as the paper shows,the difference between the sticky wage and thecorresponding flexible, Nash-bargain wage issmall. The model proposes that the sticky wagevaries over time, but not by as much as does theNash wage. I do not believe that this type of wagemovement could be detected in aggregate data.

I. Variations in Recruiting Effort

The DMP model portrays recruiting effort interms of job vacancies. Prior to the beginning ofthe Job Openings and Labor Turnover Survey(JOLTS) in December 2000, no direct measures

51VOL. 95 NO. 1 HALL: EQUILIBRIUM WAGE STICKINESS

of vacancies had been available for the U.S. labormarket. Previously, authors have suggested—reasonably persuasively—that data on help-wanted advertising provided good evidenceabout variations in vacancies over time. Figure1 shows the Conference Board’s index of help-wanted advertising since 1951. Recruiting effortas measured by advertising is remarkably vola-tile. It is not uncommon for advertising to fallby 50 percent from peak to trough, as it didfrom 2000 to 2003.

Table 1 shows data from JOLTS on vacanciesby industry for the period of slackening of thelabor market since late 2000. The figures confirmthe high volatility of vacancies suggested by thedata on help-wanted advertising. The data showthat vacancies have declined in all industries. Al-though the forces that caused the downturn in theeconomy disproportionately affected a few indus-tries far more than others—notably computers,software, and telecommunications equipment—the softening of the labor market was economy-

wide. The new data strongly confirm the positionof Katharine Abraham and Lawrence Katz (1986)that recessions are times when the labor marketsof almost all industries slacken—not times whenworkers move from industries with slack markets

FIGURE 1. INDEX OF HELP-WANTED ADVERTISING

Source: The Conference Board, http://www.globalindicators.org

TABLE 1—CHANGE IN VACANCY RATES BY INDUSTRY IN

JOLTS, DECEMBER 2000 TO DECEMBER 2002

Industry

Ratio ofvacancy rates in12/02 and 12/00

Mining 0.36Construction 0.38Durables 0.45Nondurables 0.48Transportation and utilities 0.80Wholesale trade 0.52Retail trade 0.60Finance, insurance, and real estate 0.79Services 0.68Federal government 0.54State and local government 0.70

52 THE AMERICAN ECONOMIC REVIEW MARCH 2005

to others with tight markets. I conclude that arealistic model of the labor market needs to invokea market-wide force that has powerful effects onthe recruiting efforts of employers.

II. Model of the Labor Market

A. The Matching Process and RecruitingEffort

I adopt the standard view of the matchingfriction in the labor market. The flow of candi-date matches results from the application of aconstant-returns matching technology to vacan-cies, v, and unemployment, u (both are ex-pressed as ratios to the labor force). Let x be theratio of vacancies to unemployment and let �(x)be the per-period probability that a searchingworker will find a job. Let �(x) � �(x)/x be theper-period probability that an employer will filla vacancy. � is an increasing function and � isa decreasing function. Employers open vacan-cies and initiate the recruiting process wheneverit is profitable to do so.

The vacancy/unemployment ratio, x, servesas the indicator of labor-market conditions inthe model. In a tight market with a high ratio ofvacancies to unemployment, the unemployedfind it easy to locate new jobs, so the job-findingrate �(x) is high. Employers find it difficult tolocate new workers, so the job-filling rate �(x) islow. The matching model gives a precise mean-ing to the notion of tight and slack markets.

A standard specification for the matchingtechnology is

(1) ��x� � �x�.

The parameter � controls the efficiency ofmatching and the parameter � splits the varia-tion between changes in job-finding rates andchanges in job-filling rates. The underlyingmatching function gives an elasticity of � tovacancies and 1 � � to unemployment.

B. Separations

For simplicity, I assume a fixed hazard, �,that a job will end. In the U.S. labor market,separations that result in unemployment appear

to rise somewhat when unemployment rises, butseparations involving direct reemployment innew jobs decline. JOLTS measures the sum ofthe two flows; the sum declined moderatelyfrom December 2000 through the most recentlyreported data (see Hall, 2005). The situation isfurther complicated by the flows into unem-ployment of people who were previously out ofthe labor force and the flows of unemployedpeople back out of the labor force (see OlivierBlanchard and Diamond, 1990). My model inits present form does not claim to do justice tothese aspects of labor-market dynamics. It isstraightforward to extend the model to makeseparations endogenous. The key propertiesconsidered here would not be altered by thatextension. Higher separations in slack mar-kets would require higher vacancies to main-tain stochastic equilibrium in the market, andthis influence could flatten the Beveridge curveunrealistically (see Shimer, 2005; Hall, 2005).

In addition to ruling out endogenous move-ments of the separation rate, my assumptionalso rules out exogenous movements. That is, Ido not take spontaneous fluctuations in the sep-aration rate as a driving force in the model. Aspontaneous burst of separations raises both un-employment and vacancies, and shifts the Bev-eridge curve outward. The stability of theBeveridge curve argues against the importanceof such a driving force (see Abraham and Katz,1986).

C. Equilibrium with Matching Friction

The following is derived fairly directly fromPissarides (2000) and Shimer (2005). I use dis-crete time and a discrete random driving forceto facilitate computations. Initially, I consider astationary economy perturbed by a technologyshock drawn from a distribution that does notchange over time. I let � be the value a workerenjoys when searching (leisure value and unem-ployment compensation). The random state ofthe economy is s and the productivity of labor iszs. The economy transits from state s to state s�with probability �s,s�. The price of output isnormalized at one. It costs k in recruiting coststo hold a vacancy open for one period. Workersand firms are risk-neutral and discount the fu-ture at rate .


The model is conveniently specified in termsof Bellman value-transition equations. Let Us bethe value a worker associates with being unem-ployed and searching for a new job when thestate of the economy is s, and let Vs be the valuethe worker associates with being in a job afterreceiving that period’s wage payment, ws. Let Jsbe the value the employer associates with afilled job after making the wage payment. Iassume, as is standard in this literature, thatemployers expand recruiting effort to the pointof zero profit, so the value associated with anunfilled vacancy is zero.

The value transition equations are:

(2) Us � � �s�

�s,s� ��xs ��ws� Vs� �

� �1 � ��xs��Us�].

(3)

Vs � �s�

�s,s� ��1 � ��ws� Vs� � �Us� .

(4) Js � zs �1 � �� s�

�s,s� �Js� � ws� �.

(5) 0 � �k ��xs� �s�

�s,s� �Js� � ws� �.

Equation (5) captures a central aspect of themodel: Given the anticipated payoff from mak-ing a match, firms create vacancies up to thepoint where the payoff is canceled by the re-cruiting cost, k. As they create more vacancies,xs rises, recruiting success, �(xs) falls, and thepoint of zero net payoff is achieved. This pinsdown the key variable, xs, the state-contingentvacancy/unemployment ratio.

Conditional on the state-contingent wage, ws,equation (4) determines the value the employerderives from the match. Equation (5) then de-termines the amount of recruiting effort andtherefore the tightness of the labor market, xs,for each state. Finally, equations (2) and (3)determine the two state-contingent values forworkers. These do not enter the solution directlybut are needed to verify that the wage lieswithin the bargaining set.

Newly formed and continuing matches resultin the same wage-bargaining problem in thissimple setup. The worker’s reservation wage,ws, equates the unemployment value, Us, to theemployment value, Vs � w� s, so

(6) w� s � Us � Vs .

The employer’s reservation wage is the entireanticipated profit from the match,

(7) w� s � Js .

These values determine the boundaries of thebargaining set:

(8) Bs � �w� s , w� s .

Any wage in the bargaining set will result in theefficient formation or retention of a match, asboth worker and employer will benefit from thematch, in the sense of receiving a match value atleast as large as the non-match values repre-sented by the reservation wages.

D. Equilibrium Wages

Here I depart from the DMP model, whichviews wage determination as the outcome of aNash bargain. The symmetric Nash bargainwould be the average of the two reservationvalues:

(9)w� s w� s

2.

Instead, I characterize wage determination interms of a Nash (1953) demand game or auction(see also Kalyan Chatterjee and William Sam-uelson, 1983; Roger Myerson and Mark Satter-thwaite, 1983). In the auction, worker andemployer know one another’s reservation val-ues. The worker proposes a wage, wL, and thefirm, without knowing the worker’s proposal,makes its own proposal, wH. If wL � wH, thematch is made or continues, and the wage isagreed to be w � wL � (1 � )wH with 0 1. The auction has the property that any win the bargaining set [w� s,w� s] is a Nash equilib-rium. Believing that the worker is bidding wL,


the firm will bid wL as well, provided that wL �w� s. Similarly, believing that the firm is biddingwH, the worker will bid wH as well, providedw� s � wH. Thus any w � wL � wH � Bs is aNash equilibrium.

I use the demand-game auction as a metaphorfor the unstructured bargaining that normallyoccurs in the labor market. I do not mean tosuggest that the bargaining takes the particularstructured form of the demand game. I am alsoaware that an auction setup leaves certain issuesunsettled (see Abhinay Muthoo, 1999). In par-ticular, the Nash equilibrium arises because theplayers believe that no bargain will occur unlessthey bid in a way that will achieve the bargain.They believe that they cannot make a deal laterif the auction fails to make one. Muthoo (Ch. 8)studies the case where the two players in abargaining problem make initial simultaneousbids but can revise the bids at a cost if the firstround fails to reach a bargain. The later roundsfollow the game studied by Ariel Rubinstein(1982) which converges to the Nash bargain.Because the Nash bargain leaves the volatilityof employment unexplained, I rule out thissetup. It is a topic for further work to develop aricher model of wage bargaining that retains theindeterminacy that is central to the view of thispaper.

Nash proposed the celebrated equilibrium se-lection rule—the Nash bargain—adopted in theDMP model. I explore different equilibrium se-lection rules to pin down the wage within thebargaining set. I begin by considering rules thattake the form of assigning a different wage ineach state of the economy—the wage is ws, afunction of the state, s. Thus I exclude variables,often called sunspot variables, which might playa role even though they have no direct role inthe substance of the bargaining problem. I alsoexclude the history of the economy. My exclu-sions are similar in spirit to the ones that defineMarkov-perfect equilibrium in dynamic games,though this setup lacks the state variable thatusually is a central element of a dynamic game.

A wage rule ws is an equilibrium in the econ-omy if it results in a solution to equations (2)through (5) with ws � Bs. There is a rich spaceof equilibria, including the symmetric Nashbargain.

Because wages are frequently regarded as

less flexible than a full spot market might imply,my next step is to consider the class of constantwages, where ws � w for all states s. For thispurpose, define J̃s as the solution to the linearsystem,

(10) J̃s � zs �1 � �� s�

�s,s�J̃s� .

J̃s is the value an employer would attach to anew hire who never receives any wage—it is thepresent value of the revenue generated by aworker hired when the economy is in state s.Then

PROPOSITION: A constant wage w is an equi-librium of the model if

(11) � � w � mins

�1 � �1 � ��J̃s

that is, the wage lies between the flow value ofbeing unemployed, �, and the annuity value,[1 � (1 � �)]J̃s of the lowest-expected-profitstate.

PROOF:First, I verify that the constant wage does not

fall short of the reservation wage ws in any state.For this purpose, let Ys � w � (Us � Vs), theexcess of the constant wage over the reservationwage. Subtracting equation (3) from equation(2) yields

(12)

Ys � w � � �1 � ��xs � � �� s�

�s,s� Ys� .

In matrix notation, this equation is

(13) �I � A�Y � b.

Because the characteristic values of A all havemodulus less than one, the equation has thesolution

(14) Y � �I A A2 ...�b.


Because w � �, b � 0, so all elements of Aare positive. Hence Y � 0 and so w � w� s.

To show that w � Js for all s, let

(15) Zs � J̃s � Js

� �1 � �� s�

�s,s� �Zs� w�.

The solution is

(16) Zs ��1 � ��

1 � �1 � ��w.

Thus

(17) Js � J̃s ��1 � ��

1 � �1 � ��w.

By hypothesis, w � [1 � (1 � �)]J̃s. Hence

(18)

w � �1 � �1 � ��Js �1 � ��

1 � �1 � ��w� .

or w � Js, as required.A constant wage rule may be interpreted as a

wage norm or social consensus. A related con-cept is a focal point. Much of the discussion ofwage norms considers resistance to wage reduc-tions—George Akerlof et al. (1996) discuss thistype of a wage norm and Truman Bewley(1999) provides evidence about the operation ofa modern labor market constrained by socialforces. Those authors focus on the avoidance ofdownward wage adjustments, but many of theirideas point toward the absence of immediateupward wage adjustments as well. My specifi-cation is limited in a way not previously con-sidered in the literature on wage rigidity—I donot permit the norm to lie outside the bargainingset. The earlier work implied inefficient out-comes, especially the loss of a job under con-ditions where both worker and employer couldhave been better off with a wage adjustment.The wage norm I consider interferes neitherwith the formation of efficient matches once theparties are in touch with one another nor with

the preservation of jobs with positive surplus.Inefficient separations cannot occur. As a result,the model provides a full answer to the indict-ment of sticky wage models in Barro (1977) forinvoking unexplained inefficiencies in eco-nomic arrangements.

The idea that the wage is constrained to lie inthe bargaining set of the employment relation-ship but may be insensitive to current conditionsapart from that constraint has an extensive his-tory in the literature on employment theory (seeJames Malcomson, 1999, for many citations).The new feature of my model is the effect ofwage stickiness on the pre-match recruiting ef-forts of employers and thus the implications ofstickiness for unemployment. Because the vari-ations in unemployment and vacancies respondto expectations formed when workers are hired,the essential stickiness in the model is in thoseexpectations. If only post-employment wageswere sticky, and wages paid in the first period ofemployment fluctuated to offset anticipatedlater wages, the model would deliver muchsmaller fluctuations in labor-market conditions.In Hall (2005) I formulate a related model inwhich the expected present value of wages overthe life of a job is the sticky variable.

E. Wage Rules in a NonstationaryEnvironment

A realistic environment for wage determina-tion is nonstationary. The stochastic upwardtrend in productivity rules out a constant wagerule: eventually the bottom of the bargaining setwill rise above any constant wage and that wagecan no longer be an equilibrium. To extend theidea of a wage norm to a nonstationary envi-ronment, suppose that productivity evolves asthe product of two components:

(19) zt � ztPzst

M.

The component ztP is a slow-moving trend

known to the public. The component zstM is a

mean-reverting process similar to the singlecomponent studied earlier—it depends on a dis-crete state st as before. The analog of the con-stant wage rule in this economy is

(20) wt � wztP.


The Bellman equations for the nonstationaryeconomy can be written in terms of values in theform zt

PUst, zt

PVst, zt

PJst, and zt

Pwst. For example,

the equation for the job value is

(21) ztPJs � zt

Pzst

M ̃t �1 � ��

� �st � 1

�st,st � 1zt � 1P �Jst � 1 � wst � 1�.

I rewrite as

(22) Js � zst

M �zt � 1P

ztP � ̃t �1 � ��

� �st � 1

�st,st � 1�Jst � 1 � wst � 1�.

I let � (zt�1P /zt

P)̃t, the inflation and growthadjusted discount, which I assume to be con-stant. Then equation (22) and its counterpartsfor U and V are the same as equations (2)through (4). I assume that � and k also share theupward trend of zt

P and reinterpret these param-eters as the constant detrended values.

The effects of the mean-reverting componentof productivity, zst

M, on unemployment in thenonstationary economy are the same as the ef-fects of the single productivity shift, zs, in theearlier stationary economy. In particular, thesticky wage rule wst

� w has exactly the sameallocational consequences in the nonstationaryeconomy as in the earlier stationary economy.

So far in the paper I have not specified theunits for measuring the variables involving eco-nomic values. The stationary model wouldmake little sense unless the units had stablepurchasing power, but in the nonstationarymodel, the drift component zt

P could be nomi-nal, in which case part of its drift would arisefrom drift in the overall price level. In thiscase, there is a connection between the modelof this paper and the idea of a Phillips curve.The Phillips curve describes short-run de-viations around a nominal path that is inter-preted as reflecting inertia in wage and pricedetermination.

Milton Friedman (1968) and Edmund Phelps(1967) launched a rich literature on nominal

inertia. They pointed out that the wage determi-nation process adapts to persistent inflation andproductivity growth. A key implication is thatthe unemployment effects of wage movementswould be insulated from these longer-termtrend-like movements. Friedman put the pointthe other way around: an attempt to keep un-employment low would result, in the longer run,in ever-increasing inflation. Experience in manycountries in the ensuing three decades generallyconfirmed this proposition. The wage pro-cess summarized in equation (20) captures theFriedman-Phelps hypothesis.

The huge literature on wage determination inthe Phillips-curve and related frameworks hasdistinguished backward-looking or adaptive be-havior from forward-looking behavior. My ap-proach sidesteps this issue by associating thecomponent of wages that represents shifts of thePhillips curve with the trend variable zt

P and thecomponent that represents movements along thePhillips curve with the random variable zst

M.

III. Parameters

To estimate the elasticity of the matchingfunction, �(x), I use the aggregate data fromJOLTS shown in Table 2. I calculate x as theratio of vacancies to unemployment and thejob-filling rate as the job-finding rate divided byx, and estimate the elasticity as the change in thelog of the job-finding rate divided by the changein the log of the vacancy/unemployment ratio, x.The resulting estimate is 0.765.

I assume that productivity takes on five dis-crete values zs uniformly spaced in the interval[1 � �, 1 � �]. I assume that the transitionprobabilities are zero except as follows: �1,2 ��4,5 � 2(1 � � ), �2,3 � �3,4 � 3(1 � � ), withthe upper triangle of the transition matrix sym-metrical to the lower triangle and the diagonalelements equal to one minus the sums of thenondiagonal elements. The resulting serial cor-relation of z is �.

The model operates at a monthly frequency. Icalibrate as follows: According to JOLTS, theaverage value of the vacancy/unemployment ra-tio, x, during the period from December 2000 toDecember 2002 was 0.539. I solve the modelwith Nash wage bargaining—equations (2)through (5) and (9)—for the recruiting cost k


and all of the endogenous variables except x3,which I set to 0.539. The resulting estimate of kis 0.986, measured in units of output per workerproduced in the median state (z3 � 1 in mynormalization). Then I set the fixed wage to theNash-bargain wage for the median state andsolve the fixed-wage model—equations (2)through (5). I set the serial correlation of pro-ductivity equal to the historical serial correla-tion of the U.S. unemployment rate. I set thedispersion parameter � so that the fixed-wagemodel matches the observed standard deviationof unemployment. Table 3 shows the results ofthese calculations.

The solved values of the variables in themedian state for the fixed-wage model areshown in Table 4. The worker’s career is worthabout 230 units of monthly productivity. In themedian state, the worker assigns almost exactlythe same value to unemployment and to

employment—the worker’s reservation wage isclose to zero. This implies that the wages to beearned in the future are sufficiently high that theworker—if pushed to the wall—would be will-ing to work for the first month for free. Theemployer values the relationship at a little be-low two units of monthly productivity. Thewage of 0.96 units is 96 percent of the totalvalue created from work. The remaining 4 per-cent compensates the employer for the initialcost of recruiting.

TABLE 2—CALCULATIONS FROM JOLTS DATA

December 2000 December 2002

New hires 4.070 million 3.187 millionUnemployed 5.264 million 8.209 millionVacancies 4.036 million 2.558 millionJob-finding rate, � 0.773 per month 0.388 per monthJob-filling rate, � 1.008 per month 1.246 per monthUnemployment rate, u 3.6 percent 5.7 percentVacancy rate, v 2.8 percent 1.8 percentx 0.767 vacancies per

unemployed worker0.312 vacancies per

unemployed worker�, elasticity of job finding with

respect to x0.765

�, efficiency of matching 0.947

TABLE 3—PARAMETERS

Parameter Interpretation Value Source

� Separation rate 0.034 JOLTS� Flow value while searching (leisure

or unemployment compensation)0.4 Corresponds to a flow value while

searching that is about 40 percent of theflow wage

k Flow cost of a vacancy 0.986 Matches vacancy/unemployment ratio inmedian state to average, 2000–2002

Discount factor 0.995 Corresponds to 5-percent annual rate� Serial correlation of mean-reverting

component of productivity0.9899 Serial correlation of U.S. unemployment,

1948–2003� Dispersion parameter for mean-

reverting component ofproductivity

0.00565 Matches standard deviation ofunemployment to U.S. level of 1.54percent

TABLE 4—VALUES OF ENDOGENOUS VARIABLES IN THE

MEDIAN STATE

Variable Interpretation Value

U Value while searching 229.34V Value while working 229.28J Value of worker to the firm 1.8698w Wage 0.96572


IV. Properties of the Model

In the model, the unemployment rate is astate variable. Unemployment is not a functionof the current state, as are all of the othervariables, but depends on the history of theeconomy. But, because the job-finding rate is sohigh, unemployment is a fast-moving state vari-able, and it departs only slightly from the value

(23) u*s ��

� ��xs �.

See Hall (2005) for a further discussion of thispoint and a comparison between the actual un-employment rate and the rate inferred from thisformula. For the moment, I will treat the unem-ployment rate as a jump variable along with allthe other variables, which are true jump vari-ables. In a later section, I will show the fulldynamic response of unemployment.

Figure 2 shows the basics of the model.

When productivity is high, toward the right ofthe figure, unemployment is low, vacancies arehigh, and the job-finding rate is high. The labormarket is tight—it resembles conditions in theU.S. labor market in 2000. The higher produc-tivity level, with the wage held fixed, results inhigher profit per worker. Employers put moreresources into recruiting because they receive ahigher fraction of the surplus. Consequently, thejob-finding rate is higher and the unemploymentrate is lower.

The curves in Figure 2 display properties thatare central to the view of the labor marketembodied in the model. In the following discus-sion, I will use figures associated with the me-dian state; the figures for the other states aresimilar. If productivity falls, unemploymentrises substantially. The rise occurs because jobsbecome hard to find.

The high sensitivity of labor-market condi-tions to productivity when the wage is fixedarises for the following reason: the value that anemployer achieves from a success in recruiting

FIGURE 2. JOB FINDING, VACANCY, AND UNEMPLOYMENT RATES, FIXED WAGE


is J. Recruiting cost exhausts this value in equi-librium. The response of recruiting effort—andtherefore of conditions in the labor market—depends on the change in J induced by a changein productivity. J is the present value to the firmof the profit margin generated by a worker in thecourse of the job and, with exogenous separa-tion, does not depend on any other variables inthe model. In the fixed-wage model, when pro-ductivity rises from state 3 to state 4, J riseswith a slope of 21 units of J for each unit of z.The result is a large increase in recruiting effort.By contrast, with a symmetric Nash wage bar-gain, as in the DMP model, almost all of thisincreased profit goes into wages, because ahigher z raises both w and w� , so the slope is only1.4 units of J per unit of z. The productivitychange has little effect on the employer’s jobvalue and thus little effect on recruiting effort.

The sensitivity of recruiting effort to produc-tivity depends on the distribution of rents be-tween workers and employers. If everyemployer makes take-it-or-leave-it offers to its

workers and captures all the rent, workers areindifferent between unemployment and em-ployment and their wage is �. Employers havelarge incentives to recruit workers at all times,but the elasticity of the value is unity and theresponse of recruiting effort to price changes isnot very elastic. Thus the high amplification ofprice or productivity shocks that occurs in themodel depends on the assumption that the typ-ical worker shares a significant fraction of thejoint surplus from the employment relationship.

Figure 3 shows the factors relating to thewage across the productivity states. The hori-zontal line in the middle is the actual fixedwage. The curves at the top and bottom are theupper and lower limits of the bargaining set forwages in each period, based on the expectationthat the fixed wage will be paid in all subse-quent periods. The actual wage lies at the mid-dle of the bargaining set for the medianproductivity state.

The line just above the actual wage is thehighest possible wage in that state, as defined in

FIGURE 3. WAGE ELEMENTS


equation (10). The horizontal line beneath theactual wage is the flow value of unemploymentcompensation and leisure. The actual wage liesinside these bounds as well.

Notice that the actual wage lies close to itshighest permissible value and far above its low-est value, the leisure value �. The model wouldbe strained if the actual wage were well belowthe maximal level. In that case, the labor marketwould be exceptionally tight because the em-ployer’s payoff from a hire would be high.Employers could reasonably be expected to dealwith the tight market by recruiting techniquesthat lie outside the model, such as advertisingwages above the wage norm. The simple modelprovides a reasonable account of the marketwithout those techniques, in the DMP tradition.Of course, a richer model would consider manypossible recruiting techniques as part of a moredetailed characterization of recruiting. Thatmodel would also consider a more active rolefor workers in the job-matching process.

A. Comparison to the Same Model with NashWage Bargain

A model in the DMP family can be created byreplacing the wage determination process de-veloped above with a symmetric Nash wagebargain, as in equation (9). Figure 4 displays itsproperties in the same format as Figure 2. Thefigure reveals the finding about the DMP modelstressed by Shimer (2003) and Veracierto(2002)—the small shifts in productivity thatsuffice to explain movements in the labor mar-ket of typical magnitude in the fixed-wagemodel cause almost no visible movements withthe Nash-bargain wage.

B. Dynamic Response

I derive the model’s dynamic response to aproductivity shock by comparing the expectedunemployment rates of two economies. The firststarts with the level of unemployment associated

FIGURE 4. JOB FINDING, VACANCY, AND UNEMPLOYMENT RATES, NASH-BARGAIN WAGE


with the median state but is in the state belowthe median state. The second starts with thesame level of unemployment and is in the me-dian state. The difference is the response overtime to the shock, which is the transition be-tween the median and lower states that occurredat time zero.

Figure 5 shows the dynamic response to anegative productivity shock in the sticky-wagemodel, along with the univariate moving-average representation of actual U.S. unem-ployment. Unemployment in the model matchesthe actual response reasonably closely. Bothrise quickly in the early months following ashock and then decline gradually over a periodof several years.

Figure 6 shows the response of the job-finding rate, �, to the same impulse. As soon asproductivity drops, the labor market slackens

and the job-finding rate falls by 11 percentagepoints from its normal level of about 60 percentper month. With a constant inflow to unemploy-ment and a diminished outflow, unemploymentbuilds rapidly to a maximum effect of about onepercentage point. Because the productivityshock is mean-reverting, the expected job-finding rate rises continuously after the shock.At six months, improved job finding and higherunemployment combine to equate the outflowfrom unemployment to the exogenous inflow,and unemployment reaches its maximum. Fromthat point forward, further improvements in jobfinding bring the unemployment rate back downto its unconditional mean. The vacancy rate (notshown) moves in the same way as the job-finding rate.

The persistence of slack conditions after anegative shock comes essentially entirely from

FIGURE 5. RESPONSE OF UNEMPLOYMENT TO NEGATIVE IMPULSE, MODEL AND ACTUAL


the persistence of low productivity and hardly atall from the time required for workers to findnew jobs. Michael Pries (2004) considers acomplementary explanation of the persistenceof unemployment—the workers who find newjobs after an adverse shock leave those jobsrelatively soon and experience multiple spellsof unemployment before finding stable jobs.

C. Comparison to Shimer’s Results

Shimer (2005) compares the response of thevacancy/unemployment ratio (x in my notation)to productivity shocks (z in my notation). InShimer’s version of the DMP model with Nashwage bargaining, the elasticity of that responseis 1.7 (his Table 3—1.7 is the ratio of thestandard deviation of the log vacancy/unem-ployment ratio to the standard deviation of pro-ductivity). In my version of the same model, theresponse is 1.8, the slope of the log unemploy-

ment/vacancy ratio with respect to log produc-tivity in Figure 4. The similarity of the twofigures demonstrates that our basic calibrationsare similar.

Shimer concludes that the response of theDMP model with Nash bargaining is well belowthe value needed to understand the volatility ofunemployment and vacancies. In filtered quar-terly data for the United States for the years1951 through 2003, he shows that the regressioncoefficient of the log vacancy/unemploymentratio on log productivity is 7.2 (the correlationcoefficient of 0.391 multiplied by the ratio ofthe standard deviations of the two variables).The response in my fixed-wage model is muchstronger—the elasticity is 94 (the slope fromFigure 2). Part of the difference arises fromnoise in Shimer’s measure of productivity. Pre-sumably another explanation is that not allwages are literally fixed. I conclude that thefixed-wage model is easily capable of explaining

FIGURE 6. RESPONSE OF JOB-FINDING RATE TO NEGATIVE IMPULSE IN THE MODEL


the observed high volatility of labor-market al-locations, even if it does not apply everywherein the market.

V. Other Equilibrium Wage Rules

I commented earlier on the richness of the setof equilibrium wage rules that make the wagedepend only on the productivity state, s, and onthe trend, zt

P. I have focused on the subset ofconstant wages. One example of a nonconstantwage is the partially smoothed wage,

(24) wsP � �w �1 � ��ws

N.

Here wsN is the state-contingent Nash-bargain

wage and 0 � � � 1 indexes the amount ofsmoothing. If w is an equilibrium wage, then ws

P

is also an equilibrium wage. With partialsmoothing, the effects of productivity shocks onemployment will be smaller than in the case ofa constant wage. The volatility of employmentwill be controlled by the smoothing parameter �.

Interesting examples of equilibrium wage rulesarise outside the class of purely productivity-state-dependent wages. One possibility is anadaptive wage,

(25) wtA � �1 � ��wt � 1

A �wsN.

The wage becomes a new state variable of themodel. It must be constrained to remain in thecurrent bargaining set, though this is unlikely tohave any practical effect. In Hall (2003) I ratio-nalize an adaptive wage in terms of the aggre-gation of individual wage decisions, eachperturbed by a match-specific random compo-nent. If the random component results in a bar-gaining set for the match that does not containthe aggregate wage norm, the wage is reset tothe nearest boundary of the bargaining set. Theaverage wage in one period becomes the normin the next period. The match-level adjustmentof the wage to keep it inside the bargaining setwas studied earlier by Jonathan Thomas andTim Worrall (1988). The reason for the rule intheir model is to keep wage volatility as low aspossible but to retain efficient matches.

With an adaptive wage, employment is lesspersistent than productivity. The model with anadaptive wage delivers stationary unemploy-

ment even with nonstationary productivity.Thus the adaptive mechanism provides an alter-native way to characterize the role of nonsta-tionary elements of the environment. But theadaptive mechanism is essentially arbitrary andcould take many other forms.

VI. Concluding Remarks

Strong evidence supports the following viewof fluctuations in employment and unemploy-ment: When the labor market is tight andunemployment is low, employers devote sub-stantial resources to recruiting workers. Job-finding rates for the unemployed are high. Bycontrast, when the market is slack and unem-ployment is high, employers recruit less aggres-sively and job-finding rates are low. Data onhelp-wanted advertising, vacancies, and unem-ployment confirm these relations. Further,transitions from strong markets with low unem-ployment and high vacancies to weak marketswith high unemployment and low vacanciesseem to occur without large measurable changesin driving forces. Rather, small shocks stimulatelarge responses of unemployment.

I have offered a model of fluctuations in thelabor market that mimics all of these properties.In the model, the labor market becomes slackwhen recent events have lowered the benefit tothe employer from hiring. These events, such asa small decline in productivity or a small rise ininput prices, substantially reduce the payoff tohiring if the wage is sticky. Stickiness is plau-sible, because it occurs only within the rangewhere the wage does not block efficient bar-gains from being struck and maintained. Theoutcome of the bargain between worker andemployer is fundamentally indeterminate andwage stickiness is an equilibrium selectionmechanism. The stickiness can be interpreted interms of a wage norm that provides the equilib-rium selection function.

The wage-stickiness model developed in thispaper, based on a wage norm as an equilibriumselection mechanism, achieves a strict standard ofpredictive power in one respect—that the wagenever falls outside the bargaining set—but ispermissive with respect to wage-determinationmechanisms that keep the wage inside the bar-gaining set. Application of the model in practice


needs to be guided by evidence about actualwage determination, because theory is unrestric-tive apart from the role of the bargaining set.

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