Structural Econometrics: Equilibrium search models. Jean-Marc Robin

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1 Structural Econometrics: Equilibrium search models Jean-Marc Robin Plan 1. Facts on wage distributions 2. On-the-job search and wage posting { Burdett-Mortensen 3. Sequential auctions { Postel-Vinay Robin 1

2 Facts on wage distributions In the past ten years, after Abowd, Kramarz and Margolis's (Econometrica, 1999) initial push, many matched employer-employee datasets (MEED) have been constructed in Denmark, Italy, Sweden, Austria, etc, to estimate wage equations. MEED have the following basic structure: One panel of workers with individual index i 2 f1; :::; Ig and time index t 2 f1; :::; T g, One panel of rms with individual index j 2 f1; :::; Jg and same time index t 2 f1; :::; T g, A matching function j (i; t) 2 f1; :::; Jg that denes worker i's employer at time t. They usually are register data (employer payroll reports collected for tax purposes). Matching wage register data with rm data (accounting data like EBIT, book value, etc.), feasible by availability of rm ID, is not always done. Matching with other individual social security or health insurance data is realised in some countries (Denmark, Sweden). 2

3 Wage equations for MEED Basic model used by AKM is a standard error-component model with rm xed eects: where { w it is individual wage, ln w it = x it + j(i;t) + i + u it JX = x it + jd j it + i + u it j=1 { x it is a vector of time-varying individual characteristics (experience, tenure), and { d j it, j 2 f1; :::; Jg, are indicator variables: ( d j it = 1 if j (i; t) = j 0 otherwise AKM propose to estimate, person eects = ( 1 ; :::; N ) and rm eects = ( 1 ; :::; J ) by OLS. 3

4 Identication Requires VERY large datasets Only mobile workers, for whom d j it 6= dj i, contribute to identication, as the OLS estimator of is the within estimator: w it w i = (x it x i ) + JX j=1 j d j it d j i + u it u i One needs to observe mobile workers in every rm j for the identication of j. { E.g., if only one worker with no job mobility is observed in rm j, then j is not identied. { More generally, partition of the rm sample into disconnected \clusters" with no observed cross-cluster worker mobility creates multicolinearity and requires normalisation of some j 's. 4

5 OLS OLS estimator of consistent for xed J, large I, if rm-worker assignment is strictly exogenous: d j it t2f1;:::;t g j2f1;:::;jg [Acceptable if u it is transitory.]? (u it ) t2f1;:::;t g ; 8i 2 f1; :::; Ng : OLS estimator of : consistent for large T. JX b i = w i x i b b j d j i j=1 Conclusion: If too few individuals, too few time periods and too little mobility: Imprecise xed eect estimates; Spurious negative correlation between b i and b j(i;t) (across individuals). 5

6 Correlations of Components of Real Annual Wage Rates (Abowd, Kramarz, Lengerman, Roux, WP, 2003) France StD. ln w x β α = z γ + θ z γ θ ψ u Log real annual wage rate 0,9772 1,0000 Experience 0,4087 0,5377 1,0000 Person effect 0,5217 0,4569 0,0698 1,0000 Schooling 0,1522 0,1510-0,0469 0,2917 1,0000 Unobservable 0,4990 0,4316 0,0872 0,9565 0,0000 1,0000 Firm effect 0,4665 0,4287 0,1670-0,2225 0,0293-0,2415 1,0000 Residual 0,5545 0,5675 0,0000 0,0000 0,0000 0,0000 0,0000 1,0000 US StD. ln w x β α = z γ + θ z γ θ ψ u Log real annual wage rate 0,8941 1,0000 Experience 0,6965 0,2305 1,0000 Person effect 0,8434 0,4871-0,6085 1,0000 Schooling 0,2317 0,1733-0,1527 0,2748 1,0000 Unobservable 0,8110 0,4571-0,5893 0,9615 0,0000 1,0000 Firm effect 0,3586 0,4926 0,0635 0,0445 0,0824 0,0228 1,0000 Residual 0,3614 0,4042 0,0000 0,0000 0,0000 0,0000 0,0000 1,0000 A,K,L,R's estimate of spurious correlation around 0:10 for France and 0:02 for the US. 6

7 Alternative wage equation We use French wage register data matched with rm accounting data. Regress log wages ln w it on employer's mean log productivity (measured by mean log value-added per worker): where ln w it = x it + i + y j = 1 T y {z j(i;t) + u it ; } = j(i;t) + j(i;t);t TX ln y jt : t=1 France, St.D. ln w it x it i y j(i;t) u it Log real annual labor cost (ln w it ) Quartic in age (x it ) Person eect ( i ) : Firm eect y j(i;t) Residual (u it ) Sample: all private sector employees, in initial year. 7

8 Conclusion The Law of One Price Does (thus) Not Hold in the Labour Market. Competitive models can generate interindustry wage dierentials (see for example Heckman and Honore (Ecta, 1990): worker sort themselves by industry according to particular characteristics). Dicult to extend this idea to interrm dierentials. Equilibrium search models oer a natural framework in which to analyse this multiform wage dispersion. { Labor market competition between employers is the fundamental determinant of wages. { Competition is limited by search frictions (information imperfection on job oer locations). This framework encompasses two extreme cases: { Competitive wages. When workers can freely force employers into competition, workers get paid their marginal productivity. { Monopsony wage. When the cost of nding alternative employers is innite, rms oer unemployed workers their reservation wages (Diamond, JET, 1971). 8

9 Equilibrium wage dispersion Diamond's (JPE, 1971) critique. Question: Can there exist wage dispersion at the equilibrium in a sequential search equilibrium model? When rm behavior is made endogenous, F () is shown to degenerate to a mass point at b. The search model thus collapses into a simple monopsony model of the labor market. Why? Suppose rms seek to maximize prot ow (p w)`(w) where `(w) is the labor force that they can expect if they oer a wage w. (Firms may dier in labor productivity p.) A rm oering less than in equilibrium attracts no worker and all rms oering more than attract the same number of workers. Hence, rms' best interest is to oer the smallest acceptable wage, that is. Now, if F is a mass point at, then = b R w F (w)dw = b: 9

10 Reservation wages heterogeneity Albrecht and Axell (JPE, 1984). Workers are heterogeneous by nature in this approach: They dier in their opportunity costs of employment, b, which are distributed according to some H. Heterogeneity in b is an exogenous source of heterogeneity in. This is enough to generate a non degenerate equilibrium wage distribution G, such that Supp (G) Supp(H). Low-wage rms make higher per capita prots, but they only attract low-b workers. High-wage rms make less prot per worker, but they make it up on volume since they are able to attract workers with higher values of b. Predicts negative duration-dependence of the exit rate of unemployment. 10

11 On-the-job search and wage posting Burdett and Mortensen (WP, 1989, and IER, 1998). Equilibrium wage dispersion can be generated among ex ante identical workers by allowing workers to search for a better job while employed. Firms play a wage posting game: Each rm randomly chooses a particular wage in some wage oer distribution F, and oers this wage to any worker employed or not that it meets. The unique non cooperative equilibrium of the game is characterized by a non degenerate distribution of wage oers F, which leads to a non degenerate earnings distribution G. The Burdett and Mortensen model predicts: 1. Equilibrium wage dispersion among ex ante identical workers; 2. \Larger rms pay higher wages"; 3. Workers with more tenure tend to be better paid and less mobile than \junior" workers; 4. Allows for worker and/or rm heterogeneity... 11

12 Framework L identical workers and N heterogenous rms. Mobility shocks: Matches exogenously destroyed at rate. Job oer arrival rate to unemployed = 0. Job oer arrival rate to employed workers = 1. Job values: 9 a job value index, y, such that job y 0 is preferred to job y i y 0 > y (wage, marginal productivity of labor, amenities, etc.). Sampling distribution of job values = F 6= = distribution of y across rms. f(y) = v(y) {z} (y) hiring eort of rm y Flow value of non labor time = b. 12

13 Steady-state Steady-state characterized by a triple (u; G; `) where u is unemployment rate, (1 u) G(y) is the number of employees in rms with job value less than y, and `(y) is the steady-state size of a rm oering y. u is the steady-state unemployment rate: Note that w ) F () = 1. 0 uf () = (1 u), u = + 0 : 13

14 Unemployment Rates From Stocks and Flows (French LFS and US CPS) (1) US (2) France From Stocks From Flows From flows From stocks 0,12 0,16 0,10 0,14 0,12 0,08 0,1 0,06 0,08 0,04 0,06 0,04 0, , From stocks: Fraction of unemployed in March of year t From flows: t/t+1 job destruction rate divided by t/t+1 job destruction rate plus t/t+1 re-employment rate Rates are computed by comparing the state in March of year t+1to the state in March of year t. 14

15 G is the steady-state distribution of job values across employees: 0 uf (y) = [ + 1 (1 F (y))] G(y) (1 u), G(y) = F (y) F (y) for 1 = 1 = is the average number of oers an employed worker can expect before the next layo. `(y) is the steady-state size of rm y: `(y) = = L N = L N L g(y) N (y) = F (y) F (y) 2 v(y): # workers in rms y # rms y f(y) (y) Note that `(y) needs not be increasing if v(y) is not monotonic. Equivalent ow equation: [ 0 u + 1 G(y) (1 u)] L {z } number of workers drawing an oer and willing to accept one of y f(y) {z} sampling proba = + 1 F (y) {z } quit rate `(y)(y)n {z } total employment in rms of type y 15

16 Wage posting equilibrium The Burdett-Mortensen model: Job value y = wage w. Uniform hiring eort: v(w) = 1. Firms dier in marginal labor productivity with p. A market equilibrium is a non cooperative, maybe dissymmetric, Nash equilibrium where rms seek to maximize prot ows (p) = max w (p w) `(w) subject to the steady-state employment condition: `(w) = L N F (w) 2 ; and workers apply the reservation strategy = b + ( 1 0 ) Z w F (w) dw F (w) : 16

17 Main equilibrium property There does not exist a mass point in the equilibrium wage oer distribution. If there existed a mass of rms oering the same wage w, then any one of them would be better o by oering slightly more. The additional wage cost would more than compensated by the additional labor force that this deviating rm would be able to attract. Technically: the function `(w) solving the (now exact) ow condition 0 u + 1 G(w ) (1 u)) Lf(w) = + 1 F (w) `(w)(w)n, + 1 G(w ) (1 u) L = + 1 F (w) `(w)n is discontinuous at any point of discontinuity of F or G. We write G(w ) = lim x!w G(x) instead of G(w) because only rms posting a wage strictly less than w loose their employees when they are contacted by a rm posting wage w. 17

18 Homogeneous equilibrium Even if rms are ex ante identical (degenerate distribution of labor productivities), they must oer dierent wages for an equilibrium to exist. The rms oering higher wages compensating higher wage costs by a bigger size. For the non degenerate equilibrium to exist, rms must be indierent in the wage they oer: where w = inf [Supp (F )]. As `(w) = L N 1+ 1 [1+ 1 F (w)] 2 = L N 8w 2 Supp (F ) ; (p w) `(w) = (p w) `(w) , the equal prot condition writes (1 + 1 ) 2 8w 2 Supp (F ) ; (p w) F (w) 2 (w) = (p w) r p w, F (w) = (1 + 1 ) p w : The optimal minimum wage oer is w =. The rm oering the lowest wage cannot raise its prot by oering a lower wage than. 18

19 The equilibrium wage oer density is increasing! f (w) = p : 2 1 (p w) (p w) The only reason that incites rms to oer a wage higher than the monopsony wage is the Bertrand competition induced by on-the-job search. Oering a higher wage is a way to reduce turnover. But the higher the wage oered the lower the competition pressure. Wage concentration hence also increases. 19

20 Heterogenous equilibrium Now, suppose that is non degenerate and continuous. Each rm oers a wage w(p) so as to maximize prot ows (p w) `(w) given F and w. One rst shows that the wage oer function w(p) is continuous and increasing. For all p 1 > p 2, let w 1 = w(p 1 ) and w 2 = w(p 2 ), then, (p 1 w 1 ) `(w 1 ) (p {z } 1 w 2 ) `(w 2 ) > (p {z } 2 w 2 ) `(w 2 ) (p {z } 2 w 1 ) `(w 1 ) {z } =A 1 =A 2 =A 3 Hence w(p) is non decreasing. =A 4 ) A 1 A 4 A 2 A 3, `(w 1 ) `(w 2 ) ) w 1 w 2 : As there cannot exist mass points in F and as is continuous, then w(p) must be almost surely strictly increasing. 20

21 Equilibrium wage function The FOC of the prot maximization programme are: `(w) + (p w) `0(w) = 0, (p w) F (w) f(w) = 0 The equilibrium w(p) is increasing. Hence, F (w(p)) = thus be transformed into the following rst-order dierential equation: w 0 (p) (p w) (p) = 0: (p) (p) and f(w)w 0 (p) = (p): The FOC can One can solve it using the initial condition w p =. Indeed, nothing forces the least productive rm to post a wage higher than the smallest acceptable one by unemployed workers. 21

22 Alternatively, use the Envelope Theorem to show that 0 (p) = d dp [(p w(p)) `(w(p))] = `(w(p)) = L N Integrating 0 (p), it thus follows that (p) = [p = L N F (w(p)) 2 = L N (p) 2 : w(p)] L Z 1 + p 1 N (p) 2 = 0 (x)dx + (p) p Z! p (x) p w p 2dx + : p Hence, since w p =, w(p) = p (p) 2 Z p p (x) 2 dx + p! (1 + 1 ) 2 : 22

23 Estimation Bontemps, Robin, Van den Berg (IER, 2000). No point in estimating the homogeneous model as it predicts an increasing wage oer density f as well as an increasing earnings density g (the distribution of currently employed workers' earnings G as opposed to the wage oer distribution F ). Data: French Labor Force Survey. t0b t0f unemployed at time 0 Exits unemployment with wage w0 End of record t1b t1f New unemployment spell or new employment spell employed at time 0 at wage w1 Job ends End of record 23

24 Dependent variables: Employment state at the time the survey starts: x = 1 if employment, = 0 if unemployed. Unemployed at rst observation time: t 0b =elapsed, t 0f =residual unemployment duration d 0b = 1 or 0 whether unemployment duration left-censored d 0f = 1 or 0 whether unemployment duration right-censored w 0 = wage accepted by unemployed individuals Employed at rst observation time: t 1b =elapsed, t 1f =residual employment duration d 1b = 1 if job duration left-censored, otherwise = 0 d 1f = 1 if job duration right-censored, otherwise = 0 w 1 = wage of employees at time of rst interview v = 1 if job-to-unemployment transition,= 0 if j-t-j transition 24

25 Likelihood Worker unemployed at the time of the rst interview: 2 d 0b d 0f 0 exp [ 0 (t 0b + t 0f )] + {z } 0 {z } density of proba of being unemployed (t 0b ;t 0f ;d 0b ;d 0f ) Individual who is employed at the time of the rst interview: {z } proba of being employed g(w 1 ) {z } density of current wage f(w 0 ) 1 d 0f {z } density of accepted wage + 1 F (w 1 ) 1 d 1b exp + 1 F (w 1 ) (t 1b + t 1f ) {z } density of (t 1b ;t 1f ;d 1b ;d 1f ) h v 1 F (w 1 ) i 1 v 1 d 1f : {z } proba of exit destination : 25

26 Two-stage estimation procedure 1. First, estimate G and g using a non parametric estimator (a kernel estimator for example). For any 1, let F b (w; 1 ) = 1 G(w) b and f(w; b 1 ) = F b 0 (w; 1 + 1G(w) b 1 ): 2. Second, replace F and f by these expressions in the expression for the likelihood and maximize it with respect to 0 ; 1 and. 3. Use the FOC of the prot maximization programme to estimate a productivity value for each observed wage: (p w) F (w) f(w) = 0, p w = 1 + 1F (w) 2 1 f(w) ) bp i = w i b b 1G(w i ) : 2b 1 bg(w i ) 26

27 Results Estimations results for the frictional parameters 1= 1= 0 1= 1 All sample Food Intermediary goods Equipment Current consumption Construction Trade Transport Services

28 Earnings distribution min P 10 Q 1 Q 2 Q 3 P 90 P 90 P 10 Q 3 Q 1 All sample Food Intermediary goods Equipment Current consumption Construction Trade Transport Services

29 Properties of the estimated productivity distribution min P 10 Q 1 Q 2 Q 3 P 90 P 90 P 10 Q 3 Q 1 All sample Food Intermediary goods Equipment Current consumption Construction Trade Transport Services

30 Main empirical lesson of the BM model BM models ts turnover and wage distributions well. But estimated distribution of rm productivity implausible: exceedingly long right tail. Why? High-productivity rms have a lot of market power in the BM model. This tends to concentrate wages towards the upper part of the distribution. In order to generate the very long, thin tails of observed wage distributions, productivity distributions with much longer and thinner tails are thus necessary. Irreconcilable with actual productivity distributions. 30

31 Sequential Auctions Postel-Vinay and Robin (Econometrica, 2002). Match productivity: { Workers are perfect substitutes and dier in ability ". { Marginal productivity of ecient labor p is constant and rm-specic. { Marginal productivity of a match ("; p) is p". Matches break up exogenously at rate and workers meet employers at rate 1. Workers draw type p rms according to the same sampling distribution F, whatever their type. Wage contracts are negotiated between employers and employees under complete information about each other's type and can be renegotiated by mutual consent only. Lifetime value of unemployment for a worker of type " is V 0 ("). Lifetime value of current wage w for a worker " in rm p is V ("; w; p). 31

32 Wage contracts Employers have full market power. [Assumption relaxed in Cahuc, Postel-Vinay and Robin, 2005.] When an unemployed worker meets a potential employer, worker is paid a wage 0 ("; p) such that V ("; 0 ("; p) ; p) = V 0 (") : When an employed worker paid w in rm p receives an oer from a rm p 0 : { If p < p 0, moves to p 0 for wage ("; p; p 0 ) (possibly lower than w). { If p > p 0 and w < ("; p 0 ; p), then worker stays at rm p but wage rises to ("; p 0 ; p). { If ("; p 0 ; p) < w nothing happens. Wage value ("; p; p 0 ), p < p 0, solves the equation: V ("; ; p 0 ) = V ("; "p; p) : [Similar to a second-price auction.] 32

33 Solution All wages have the form ("; p; p 0 ), p < p 0, where: ("; p; p 0 ) = " (1; p; p 0 ) = " p 1 + Z p 0 p F (x) dx Within-rm wage distribution can be determined in closed-form.! : { In particular: E (wjp) = E (") p F (p) Z! p (1 ) 1 F (q) p min F (q) 2 dq ; where 1 = 1 = and = = ( + ). 33

34 Log-wage variance decomposition Use French wage register data (DADS). Not matched with rm accounting data (comparable to AKM). Decompose the log-wage variance as: Main results: Var (ln w) = Var (ln ("; q; p)) = 8 >< >: Var (ln ") (person eect) + Var E [ln (1; q; p) jp] (rm eect) +E Var [ln (1; q; p) jp] (eect of frictions): Person eect explains 40% of Var (ln w) for managers and quickly drops to 0 for unskilled categories. Contribution of market imperfections 50%. 34

35 Firm productivity Cahuc, Postel-Vinay and Robin (Econometrica, >2005). Match administrative data on wages with accounting rm data to obtain independent productivity estimates. Estimate production function with rm-level data: Q jt = p j ( 1 L 1t + 2 L 2t + :::) ; P j p j = 0; for unskilled (1) and skilled (2) workers. { 1 and 2 are mean ability " within each skill category; { p j is rm j's labor productivity; { A measure of p j is thus obtained, independent of wage data. Predict rm-level mean wages w kj using these estimates bp j and the theoretical E(w kj jbp j ). Then assess the t. 35

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