Productivity and Wage Dispersion: Heterogeneity or Misallocation?∗ Jesper Bagger† Royal Holloway
Bent Jesper Christensen‡ Aarhus University
Dale T. Mortensen§ Northwestern University Aarhus University December 14, 2014
Abstract Labor productivity and wages vary and are positively correlated in a cross-section of workers. This may reflect ability heterogeneity, or resource misallocation coupled with rent sharing. In Danish data, misallocation is found to be the dominant source of log labor productivity variation and of the log labor productivity-log wage covariance. At the same time, worker ability differences account for the majority of the log wage variance. Ability adjusted worker flows are substantial, but more than half of the gross ability flows are ability churning. More productive firms have higher net ability flow rates. We estimate large marginal output and wage gains from labor reallocation. The finding that misallocation persists under these circumstances suggests that labor market frictions are important barriers to growth. Keywords: Productivity dispersion, Wage dispersion, Misallocation, Worker heterogeneity, Firm heterogeneity, Bargaining, Rent sharing, Matched employer-employee data, Worker flows, Ability flows, Returns to mobility JEL codes: C33, C78, J21, J24, J31
∗
Sadly, Dale T. Mortensen passed away on January 9, 2014. Dale worked until the end helping complete this paper. Earlier versions of this paper were circulated since 2009 under the title “Wage and Productivity Dispersion: The Roles of Rent Sharing, Labor Quality, Capital Intensity, and TFP.” We thank John Abowd, Paul Beaudry, Steve Bond, Melvyn Coles, Francois Fontaine, Manolis Galenianos, John Haltiwanger, John Kennan, Francis Kramarz, Peter Kuhn, Rasmus Lentz, Iourii Manovskii, David Rivers, Jean-Marc Robin, Juan Pablo Rud, Rob Shimer, Ija Trapeznikova and participants in numerous seminars and conferences for helpful and constructive comments. Henning Bunzel has been invaluable in the development of the data we use. The authors gratefully acknowledge financial support from the Danish Social Science Research Council, the Cycles, Adjustment, and Policy research unit, CAP, and the Aarhus Institute of Advanced Studies, AIAS, Aarhus University. † Department of Economics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom; E-mail:
[email protected] ‡ Aarhus Institute of Advanced Studies, Aarhus University, Høegh Guldbergs Gade 6B, DK-8000 Aarhus C, Denmark; E-mail:
[email protected] § Department of Economics, Northwestern University, 2001 Sheridan Road, Evanston, Illinois 60208, U.S.A.
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1
Introduction
Labor productivity, measured as average value added per worker, and wages vary substantially and are positively correlated in a cross-section of workers.1 We document that the intra-industry 90th to 10th percentile ratios for employment weighted labor productivity and wages in large Danish firms range between 2.1 and 2.6, depending on industry. The correlation between the two variables is between 0.17 and 0.34. Similar relations have been documented for other countries and data sources.2 Some argue that these robust empirical facts reflect unmeasured labor input heterogeneity as would be implied by the standard competitive model (see, e.g., Murphy and Topel, 1990). Treating unmeasured worker heterogeneity as ability heterogeneity, firms with higher average ability workers have higher labor productivity and pay higher wages, even if the competitive unit price of ability is common across firms. Others contend that dispersion reflects resource misallocation, resulting in marginal products of labor that vary across firms, and match rents that are shared between workers and firms (see, e.g., Krueger and Summers, 1988). Undoubtedly, both lines of explanation play a role, and the first contribution of this paper is to disentangle the relative importance of ability heterogeneity versus resource misallocation in shaping the joint distribution of labor productivity and wages.3 If production factors are misallocated there are potential output gains to factor reallocation.4 These gains are not realized and misallocation persists if factor mobility is insufficient, or not directed toward more productive firms. The second contribution of this paper is to quantify ability adjusted worker flows, document the amount of ability reallocation implemented by the labor market, and ascertain the extent to which ability adjusted labor reallocation is directed toward firms with higher marginal products of labor.5 Our interest in the joint distribution of labor productivity and wages is motivated by the fact that rent sharing ties the wage distribution to dispersion in marginal products of labor. Indeed, as wage dispersion stimulates job search (see, e.g., Christensen, Lentz, Mortensen, Neumann, and Werwatz, 2005), rent sharing is a potentially important vehicle for directing labor toward 1
Foster, Haltiwanger, and Syverson (2008) emphasize the distinction between productivity measures expressed in physical units, generally only available in the manufacturing industry, and value. In this paper we are concerned with labor productivity measured in value terms. 2 See, e.g., Baily, Hulten, and Campbell (1992), Foster, Haltiwanger, and Krizan (2001), Syverson (2004), Postel-Vinay and Robin (2006), and Bartelsman, Haltiwanger, and Scarpetta (2013). A related finding from the international trade literature is that firms that export tend to be more productive and pay significant wage premia, see, e.g., Bernard and Jensen (1995), Bernard, Eaton, Jensen, and Kortum (2003), Eaton, Kortum, and Kramarz (2004, 2011), and Pedersen (2009). 3 Krueger and Summers (1988), Murphy and Topel (1990), and Gibbons and Katz (1992) represent early attempts to quantify the role of unmeasured worker heterogeneity versus departures from the competitive model. These papers focus on inter-industry wage differentials and do not reach a consensus. Abowd, Kramarz, and Margolis (1999), Postel-Vinay and Robin (2002), Cahuc, Postel-Vinay, and Robin (2006), and Bagger and Lentz (2014) use matched employer-employee data to study intra-industry wage dispersion, but do not consider the impact of labor heterogeneity on the distribution of labor productivity. Iranzo, Schivardi, and Tosetti (2008), Fox and Smeets (2011), and Irarrazabal, Moxnes, and Ulltveit-Moe (2013) assess the role of ability heterogeneity on productivity dispersion, but do not consider the joint distribution of labor productivity and wages. 4 Restuccia and Rogerson (2008), Hsieh and Klenow (2009), and Bartelsman, Haltiwanger, and Scarpetta (2013) all find that gains from resource reallocation are substantial in the variety of settings that they study. 5 We focus on labor reallocation because labor flows and wages are observed in our data. Capital flows are not observed, nor are firm-specific prices of capital. This is not to say that an analysis of capital reallocation is not of interest, just that, given data availability, it is out of our reach.
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more productive employment. The third contribution of this paper is to measure output and wage gains associated with reallocation of labor to firms with higher ability adjusted marginal products of labor. These measurements are informative about the cost of labor market frictions, the monetary incentives offered by the labor market for workers to reallocate efficiently, and thus, ultimately, the magnitude of the labor market frictions that prevent this reallocation from taking place. The analysis is conducted within the context of a model of multi-worker firms producing output using capital and labor, facing firm-level TFP shocks. Workers differ in ability. Firms operate with multiple occupations. Ability is perfectly substitutable within occupations, and the occupation-specific labor inputs combine into an aggregate labor input. Labor market imperfections impede labor reallocation, both within-firm, between-occupations, and between-firms. We take no stance on the nature of these imperfections, and do not model labor reallocation explicitly. Hence, we allow, for example, internal (within-firm, between-occupations) and external (between-firms) labor markets to be organized differently and characterized by different types of imperfections, and we do not assume that workers are chasing more productive employers in the empirical analysis. A worker-firm match creates rent. The unit price of ability is determined by rent sharing, as in Stole and Zwiebel (1996) and Smith (1999), and varies across occupations, firms, and time. The joint distribution of labor productivity and wages reflects both ability heterogeneity and resource misallocation. Based on a Cobb-Douglas production function, we measure ability heterogeneity as cross-firm dispersion in the average ability of workers. Misallocation is a combination of occupational labor misallocation and aggregate factor misallocation. Occupational labor misallocation occurs when a firm distributes a given stock of ability inefficiently across occupations. Aggregate factor misallocation stem from dispersion of marginal products with respect to the aggregate labor input, and may thus reflect misallocation of both capital and aggregate ability adjusted labor input. In short, firms that employ higher ability workers, distribute the available ability efficiently across occupations, or combine capital and ability to achieve higher marginal products of aggregate labor will enjoy higher labor productivity and pay higher wages. The model is estimated on Danish matched employer-employee data with firm-level information on output and capital. Data with two-sided heterogeneity is essential. Unmeasured worker ability calls for worker effects, and labor market imperfections open up for firm heterogeneity. Data on capital allows controlling for intra-industry differences in capital intensity in the production function estimation. Output data allow us to anchor our analysis in the empirical joint distribution of labor productivity and wages.6 Our empirical analysis focuses on large firms in four industries: Manufacturing; Wholesale & Retail Trade; Transport, Storage & Communications; and Real Estate, Renting & Other Business Activities. Workers are assigned to one of four aggregate occupations: Managers, Salaried workers, Skilled workers, and Unskilled workers. To verify the robustness of our results, we present a full separate analysis of the four largest sub-industries within Manufacturing in online supplementary material. All empirical 6
Since the imperfections that cause factor misallocation in the first place also complicate the mapping from productivity to wages vis-a-vis the competitive benchmark, it is problematic to draw inference regarding input heterogeneity and factor misallocation from wage data alone.
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results are robust to this additional stratification. We estimate firm-level production functions while explicitly accounting for unobserved ability heterogeneity in a manner that is consistent with our theoretical framework. Worker ability is estimated as worker fixed effects from an individual level log wage regression, appropriately controlling for firm effects along the lines of Abowd, Kramarz, and Margolis (1999), although in our case, firm effects are endogenous time-varying objects. With measurement of individual ability, we construct occupation-specific ability adjusted labor inputs for each firm. These combine in the production function to a labor input aggregate. Firm effects from the log wage regression admit a structural interpretation as unit prices of ability derived from the proposed bargaining game. The estimated prices of ability, together with the estimated production function parameters, are used in the estimation of the occupation-specific sharing rules arising from the bargaining game. The estimated model offers novel decompositions of the intra-industry variances of log labor productivity and log wages, as well as their covariance. We find that misallocation is the dominant source of cross-sectional variance in log labor productivity. Within Manufacturing, ability heterogeneity accounts for 9% of the variance in log labor productivity, occupational labor misallocation accounts for 26%, and aggregate factor misallocation for 65%. However, heterogeneity in worker ability accounts for the lion’s share of within-occupation cross-sectional log wage variance. In Manufacturing, the within-occupation variation in log wages can be attributed to heterogeneity in worker ability, the firm level price of ability, and residuals in proportions 68%, 15%, and 17%. Finally, the cross-sectional log labor productivity-log wage covariance is primarily driven by misallocation. In Manufacturing, the covariance is decomposed into ability heterogeneity, occupational labor misallocation, and aggregate factor misallocation in proportions 8%, 59%, and 33%. Results in the other industries considered mirror those for Manufacturing. Our analysis of the ability flows implies annual turnover rates ranging from 26% to 56% across occupations and industries. That is, year on year, between 26% and 56% of the ability stock change employment. These rates pertain to total ability flows, combining internal (i.e., within-firm, between-occupations) and external (i.e., between-firm) ability flows. The implied external turnover rates range between 18% and 38%, roughly comparable to US figures as reported in Burgess, Lane, and Stevens (2000) who do not ability adjust the labor flows. However, more than half of our measured gross ability flows, in some cases up to 70%, are comprised of ability churning, i.e., ability flows over and above those needed to generate the realized ability reallocation.7 This is true for both total and external ability gross flows. Internal gross ability flow rates are substantially lower, involving relatively less churning and therefore more reallocation, compared to external gross flow rates. Regressing the net ability flow rate onto (lagged) marginal products, we find that ability adjusted labor flows are, on average, directed toward firms with higher marginal products of ability adjusted labor, although confidence intervals are 7
During a year, a firm hires some ability and separates from some ability. The total amount of ability involved in hiring and separation is the ability adjusted gross worker flow. Hiring and separation cause the firm to expand or contract (or remain at a constant size, as the case might be) in terms of the stock of ability it employs. The absolute value of this expansion/contraction is denoted ability reallocation, thus reflecting ability adjusted job flows. The difference between ability adjusted gross worker flows and ability reallocation is ability churning.
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wide in some of the occupation-industry combinations. Having estimated the ability adjusted marginal products of labor as well as the sharing rule, we quantify potential output and wage gains associated with reallocating a unit of ability from the 25th to the 75th percentile in the employment-weighted cross-firm, within-occupation distribution of marginal products of ability adjusted labor. We find large gains to output and wages at this margin. Marginal products more than double, and the wage gains often exceed 30%. Although large, these estimated wage gains are in line with results in other empirical studies of the return to labor mobility, most notably Bjelland, Fallick, Haltiwanger, and McEntarfer (2011). Taken together, our empirical analysis leads us to the following set of conclusions. First, resource misallocation is rife and the main source of log labor productivity dispersion and log labor productivity-log wage covariance. This result is not driven by lack of worker heterogeneity, as worker ability differences are the main source of log wage dispersion. Second, ability is relatively mobile, but up to 70% of the ability flows are merely churning. On average, the labor market does direct ability toward more productive employment, but the intensity of the reallocation process is evidently not sufficient to prevent substantial remaining misallocation. Third, our analysis predicts large marginal output and wage gains associated with labor reallocation. Therefore, labor market frictions are associated with significant output loss, and are not easily overcome. The rest of the paper is laid out as follows. The model is introduced in Section 2. Section 3 describes the data. Section 4 presents the estimation method and empirical results. Section 5 reports log labor productivity and log wage variance and covariance decompositions, and Section 6 analyzes ability adjusted labor flows and potential gains from labor reallocation in terms of output and individual wages. Section 7 concludes. A detailed data description and additional empirical results are provided in online supplementary material.
2
The Model
We consider firms that employ multiple workers. At this point, we take the allocation of workers across firms as given, and abstract from worker heterogeneity and capital input. Let j index firms, and consider firm j with Nj identical workers. In the Stole and Zwiebel (1996) bargaining model, the firm bargains with each worker as if s/he were the marginal employee. The worker is assumed to have an outside option, which in the Cahuc, Marque, and Wasmer (2008) version of the model is taken to be the worker’s reservation wage. The gross profit flow of firm j is π(Nj , pj ) = pj f (Nj ) − φNj , where pj is firm j’s factor productivity, φ is the wage, and f (·) is the baseline production function, an increasing concave function of employment. The net value of the marginal worker to the firm is
∂π(Nj ,pj ) , ∂Nj
while the surplus value of employment to the worker is φ − b, where b
represents the worker’s reservation wage. We introduce asymmetric bargaining power by supposing that nature selects the worker to propose the deal with probability β and the employer with complementary probability 1 − 5
β at the beginning of each negotiation round. As information is complete by assumption, the proposer offers a wage that makes acceptance at least as attractive to the other party as continued negotiation. The agreed upon wage φ = φ(Nj , pj ) satisfies ∂φ 0 Nj = (1 − β) (φ − b) . β pj f (Nj ) − φ − ∂Nj
(1)
The worker receives a share β of the joint surplus. As Stole and Zwiebel (1996) show, the solution to (1), the wage bargaining outcome function, is Z
1
φ(Nj , pj ) = (1 − β)b + pj
z
1−β β
f 0 (zNj )dz.
(2)
0
In the constant marginal product case, the wage reduces to the average of the outside option b and the marginal product, (1 − β)b + βpj f 0 , as in the canonical search and matching model (Pissarides, 2000). In general, the wage is an increasing function of both workers’ share of the rent β, marginal product pj f 0 , and the outside option b.8
2.1
Heterogeneity
Cahuc, Marque, and Wasmer (2008) generalize the outcome of the Stole and Zwiebel bargaining problem to allow for any number H of different types of labor (indexed h = 1, 2, ..., H) that are imperfect substitutes in general, and other quasi-fixed factors, such as capital. In our empirical analysis, we treat labor types as aggregate occupations. In the rest of the paper we therefore refer to the H labor types as occupations. Hence, consider a general production function of the form Yj = pj f (Kj , Lj ), where pj now represents firm-level TFP, Kj is the capital stock, and Lj = (Lj1 , Lj2 , ..., LjH ) is firm j’s vector of occupation-specific labor inputs. Individuals are indexed by i. Let aih represent the ability of individual i in occupation h. For empirical tractability we restrict attention to the case where complementarities arise only between occupations; within each occupation, individual workers differ with respect to ability, but are perfect substitutes. Let J(i) = j if individual i is employed by firm j, and let H(i) = h if individual i is working in occupation h. Firm j’s occupation-h labor input is P thus Ljh = {i:J(i)=j∧H(i)=h} aih . We retain N to indicate the number of non-ability adjusted workers in the firm. Let ej = L
H X
Ljh
(3)
h=1
be the total amount of ability employed in firm j. We measure ability heterogeneity across firms using firm-level average ability
ej L Nj .
8 Of course, the bilateral bargaining outcome (2) determines the wage if and only if it is consistent with participation conditions. Mortensen (2010) shows that these do not bind in equilibrium.
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Interpreting φh as the unit price of ability for occupation-h labor, the profit flow reads π(Kj , Lj , pj ) = pj f (Kj , Lj ) −
H X
φh Ljh ,
h=1
and the generalization of (2) to the case of H different occupations is Z φh (Kj , Lj , pj ) = (1 − βh )bh + pj
1
z
1−βh βh
fh (Kj , Lj ◦ Ah (z))dz,
h = 1, ..., H.
(4)
0
Here, βh is the bargaining power, bh the reservation wage, and pj fh (·, ·) the marginal product of occupation-h labor. The vector βH β2 1−βh β1 1−βh Lj ◦ Ah (z) = Lj1 z 1−β1 βh , Lj2 z 1−β2 βh , ..., LjH z 1−βH
1−βh βh
,
h = 1, ..., H,
is the Hadamard product of the employment vector Lj and the vector Ah (z) with kth element βk
z 1−βk
1−βh βh
, for k = 1, ..., H.
Individual i’s wage is the price of ability agreed upon with the employer times the worker’s ability. That is, wi = φH(i) (KJ(i) , LJ(i) , pJ(i) )aiH(i) .
(5)
Wages vary across individuals because of differences in ability, occupation, and price of ability within occupation, φh . Differences in φh stem from variation in Kj , Lj and pj across firms. We focus exclusively on ability heterogeneity, treating capital as homogenous. Our data do not allow us to measure capital of heterogenous quality. This of course does not rule out capital misallocation.
2.2
A Tractable and Empirically Relevant Case
Our preferred empirical specification involves a Cobb-Douglas production function, Yj = pj f (Kj , Lj ) = pj KjαK L∗j
αL
,
(6)
where L∗j is the firm-level aggregate labor input, itself a log linear function of occupation-specific labor inputs, L∗j =
H Y
Lγjhh ,
(7)
h=1
reflecting both the total amount of ability employed in production, and its composition across occupations. In (6) and (7), αK and αL , respectively, are capital and labor aggregate elasticities P of output, and γh is a measure of the relative productivity of occupation h, with H h=1 γh = 1. With (6) and (7), the marginal product of occupation-h labor, denoted M P Lh , is M P Ljh = pj fh (Kj , Lj ) = γh αL
7
Yj , Ljh
(8)
such that the occupation- and firm-specific price of ability, see (4), takes the simple form φh (Kj , Lj , pj ) = (1 − βh )bh +
1+
βh 1−βh P βk αL H k=1 γk 1−βk
! M P Ljh .
(9)
The sharing rule (9) ties the unit price of ability, and thus individual wages, to the marginal product of labor. The strength of this link, and thus the magnitude of a given worker’s gain from reallocating to a more productive firm, depends on the model’s parameters.
2.3
Firm Dynamics
In this section, we relax the assumption that inputs and TFP are fixed. Instead, TFP is a stochastic process and labor and capital are quasi-fixed factors of production. Formally, we consider a discrete time formulation in which t = 1, 2, ... indexes periods and the TFP sequence {pjt } is a firm j specific first order Markov process. In the empirical analysis we shall take pjt to follow a linear AR(1) process. We assume that labor reallocation, i.e., hiring and separations, and investment in capital take place after current TFP is observed, and that investments in period t materialize within the same period. Wage bargaining takes place after labor reallocation and investment decisions. As a period in our empirical model is a year, this accommodates the possibility of a time-to-build of less than a year.9 Although one can generalize the formulation to any finite number of occupations, for expositional simplicity we sketch the basic model only for a single category. Given that hires, separations, and bargaining take place at the beginning of each period, labor input in period t is given by Ljt = Hjt + (1 − s)Ljt−1 ,
(10)
where Hjt represents new hires and s is the separation rate. Capital evolves according to the law of motion Kjt = Ijt + (1 − δ)Kj,t−1 ,
(11)
where Ijt represents gross additions to physical capital determined in period t and δ is a fixed depreciation rate. The hiring and investment decisions in period t are made contingent on Lj,t−1 , Kj,t−1 , and pjt . Of course, as in (9), the price of ability is the outcome of the wage bargaining in period t which is simultaneous with total labor and capital input. H
jt Suppose that costs of hiring and gross investment in period t take the forms cL ( Lj,t−1 )Lj,t−1
I
tj and cK ( Kj,t−1 )Kj,t−1 , where cL (·) and cK (·) are positive, increasing and convex functions. This
is implied if the cost functions are increasing, convex, and homogenous of degree one in stock 9 There is nothing salient about this particular timing assumption. We proceed this way only because our empirical analysis favors specifications where both labor and capital are allowed to be potentially endogenous in relation to contemporaneous TFP (see Section 4).
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and gross flow. The firm’s Bellman value solves V (Lj,t−1 , Kj,t−1 , pjt ) = max pjt f (Kjt , Ljt ) − φ(Kjt , Ljt , pjt )Ljt Hjt ,Ijt Ijt Hjt Lj,t−1 − cK Kj,t−1 + ΛE [V (Ljt , Kjt , pj,t+1 )|pjt ] (12) − cL Lj,t−1 Kj,t−1 subject to equations (10) and (11), where Λ ∈ (0, 1) is a discount factor. For purposes of the estimation that follows, we do not need to take a stand on the particulars of this dynamic formulation, other than the specification of what the employer knows when making the hiring and investment decisions. More complicated and possibly more realistic formulations, for example explicitly accounting for employed worker search, are consistent with the estimation procedure that follows, so long as the timing assumptions hold. The notation of the production function (6) and the price of ability equation (9) is easily amended to allow for dynamics, and will not be restated. It is however worthwhile to revisit the individual wage equation. Now let J(i, t) = j if individual i is employed in firm j in period t, and let H(i, t) = h if individual i works in occupation h in period t. The wage equation allowing for dynamics, here stated in logs, reads ln wit = ln φH(i,t) (KJ(i,t)t , LJ(i,t)t , pJ(i,t)t ) + ln aiH(i,t) .
(13)
Equation (13) is a two-way error component log wage regression. It is similar to the wage regression with firm and worker fixed effects studied by Abowd, Kramarz, and Margolis (1999). In (13), ln φH(i,t) (KJ(i,t)t , LJ(i,t)t , pJ(i,t)t ) represents the firm heterogeneity component and ln aiH(i,t) is the worker heterogeneity component. Our theoretical framework provides a structural foundation for the “firm effects” in strategic bilateral bargaining theory. These differ by occupation and are time varying, reflecting the firm’s capital stock, labor configuration, and TFP, thus warranting an interpretation as firm-, occupation-, and time-specific unit prices of ability. Our structural model of wage setting implies that a worker’s previous labor market search history does not affect the wage earned. This justifies the interpretation of the estimated “worker effect” in the wage equation as the worker’s individual ability.10
2.4
Misallocation
In the tractable and empirically relevant case of a Cobb-Douglas production function, see (6), extended to allow for TFP shocks, and a log linear labor aggregator, see (7), cross-firm dispersion in the marginal product of aggregate labor M P L∗jt = αL 10
Yjt L∗jt
(14)
As Postel-Vinay and Robin (2002) point out, this assumption is invalid in a job ladder model in which wages can be bid up when an employed worker finds an alternative employment opportunity. Then, as in their model, the worker fixed effect in the wage equation is likely to include a return to search and, consequently, is likely to be biased up as a measure of the worker’s productive abilities.
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and in the ratio of aggregate labor to total ability,
L∗jt e jt , L
measures resource misallocation. Indeed, γh Q γh , and the total = αL Yjt H h=1 M P Ljht
from (3), (7) and (8), aggregate labor input is L∗jt γh e jt = αL Yjt PH ability of a firm’s workforce is L
h=1 M P Ljht .
M P L∗jt
=
Hence,
H Y M P Ljht γh γh
h=1
and L∗jt 1 = e jt M P L∗jt L
H X h=1
γh M P Ljht
,
(15)
!−1 .
(16)
An efficient resource allocation equalizes marginal products across firms and occupations, i.e., production factors adjust to ensure M P Ljht = M P Lt . In this case, from (15) and (16), neither M P L∗jt nor
L∗jt e jt L
exhibits cross-firm dispersion. When resources are misallocated, M P Ljht
varies across firms and occupations, and, in general, both the marginal product of aggregate labor M P L∗jt and the ratio of aggregate labor to total ability
L∗jt e jt L
vary across firms.
With a constant returns to scale production function, M P L∗jt depends on TFP pjt and the ratio of capital to aggregate ability adjusted labor input M P L∗jt
Kjt L∗jt .
Cross-firm dispersion in
thus reflects a failure of allocating capital and aggregate labor efficiently in accordance
with the TFP shocks faced by firms. We label this phenomenon aggregate factor misallocation. Dispersion in
L∗jt e jt L
measures inefficiencies in firms’ allocation of labor across occupations and does
not depend on TFP pjt . We label this occupational labor misallocation. Effectively, occupational e jt generating a suboptimal labor misallocation results in a given stock of employed ability L level of aggregate labor input L∗jt . Total resource misallocation is the sum of occupational labor misallocation and aggregate factor misallocation. We return to these notions of misallocation in our empirical analysis.11
3
Data
The empirical analysis is carried out on Danish register-based matched employer-employee data. We rely on three different data sources. Employer data are secured from a firm level panel with accounting information from an annual survey conducted by Statistics Denmark. Employee data are drawn from the Integrated Database for Labor Market Research (IDA), an individual level annual panel of labor market histories. The third data element is the Firm-IDA integration (FIDA) that we use to link employers and employees in the last week of November of each year.
3.1
Data Sources
Accounting data are available for the period 1995-2007. Industry coverage increases over time, starting in the initial year 1995 with Manufacturing and parts of Wholesale & Retail Trade.12 11
It is clear from (15) and (16) that aggregate factor misallocation and occupational labor misallocation are not independent. Indeed, if there is no aggregate factor misallocation, there can be no occupational misallocation, either. However, the converse is not true, implying that the distinction between occupational labor misallocation and aggregate factor misallocation has content. 12 We classify industries according to NACE rev. 1 (see EUROSTAT, 1996).
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Table 1: Sampling scheme for accounting data Nov. workforce
Fraction sampled
Years in/out
0.00 0.10 0.20 0.50 1.00
− 1/9 2/8 3/3 −
0-4 5-9 10-19 20-49 > 49
Note: In addition, firms that generated revenue exceeding DKK 100 mill. (Wholesale: DKK 200 mill.) in the preceding year are always sampled. Source: Statistics Denmark.
By 1999 most industries are included. The accounting data set has a rolling panel structure where firms are selected based on the size of their last week of November workforce in the previous year. All firms with at least 50 employees are surveyed, and some with less. The detailed sample selection rules applied are given in Table 1. If sampled, a firm is required to submit a standardized balance sheet to Statistics Denmark. Hence, we are able to compute value added, measures of the firm’s capital stock and various other relevant measurements, including annual workforce measured in full time equivalent (FTE) workers. Our online supplementary material contains details on these computations. We retrieve data on workers from IDA for the years 1995-2007 covered by our employer data. IDA is comprised of several files with distinct types of information: Person files, establishment files, and job files. The person files contain annual information on a wide range of socio-economic variables for the entire Danish population aged 15-70. The establishment files contain annual information on all establishments with at least one employee in the last week of November in each year. The job files provide information on all jobs that are active in the last week of November in each year. We use education and occupation information from the person files, industry code from the establishment files,13 and average hourly wage from the job files.
3.2
Analysis Data
Our empirical analysis uses both an individual level panel and a corresponding firm level panel, both extracted from the matched employer-employee panel constructed by merging employer and employee data using FIDA. Here, we provide a short description of the selection criteria. Details are relegated to the online supplementary material. 3.2.1
Individual Level Panel
First, we restrict attention to the following industries (NACE rev. 1 section and years in the sample in parentheses): Manufacturing (D, 1995-2007), Wholesale & Retail Trade (G, 1999-2007), Transport, Storage & Communications (I, 1999-2007), and Real Estate, Renting & Business Activities (K, 1999-2007). To verify the robustness of our results, we analyze the four largest sub-industries within Manufacturing in the online supplementary material. 13
Each firm has one or more establishments. We define the industry of a firm as that of its largest establishment.
11
Second, we discard observations on workers aged below 18 and observations with invalid or missing education information, and define four occupational categories according to ISCO88 grouping: Managers, Salaried workers, Skilled workers, and Unskilled workers. Workers with missing occupation data are assigned to an occupation based on length of education. The occupational categories are consistent with Christensen, Lentz, Mortensen, Neumann, and Werwatz (2005). Third, we discard observations with missing or non-positive wages and firm-years with nonmissing and non-positive value added, hours, or capital stock. The annual industry and occupation specific distributions of individual level wages and the (non-employment weighted) annual industry-specific distributions of hourly value added are trimmed at the top and bottom 1%. All nominal variables are detrended using the NACE rev. 1 section-specific implicit deflator in hourly value added. Fourth, we retain only firm-years where at least two workers are employed in each of the four occupations. This selection criterion excludes small firms, leading to a considerable loss in the number of observations in the individual level panel. The loss of information from the employer data is moderate since these are already large firms. Fifth, to ensure identification of the individual level log wage regression, we select the largest group of connected workers and firm-years in each occupation and industry.14 Viewing the matched employer-employee data as a bipartite graph, where workers and firm-years are the two sets of nodes, and employment relations are the edges, the largest group of connected workers and firm-years is the largest strongly connected component. As it turns out, the vast majority of workers and firm-years are connected in one large group in each occupation and industry. Table 2 reports the average number of workers and firms in an annual cross-section for the individual level panel. Manufacturing is the largest of the four industries in terms of both workers and firms, followed by Wholesale & Retail Trade. The smallest industry is Transport, Storage & Communications. 3.2.2
Firm Level Panel
The firm level panel is constructed from the individual level panel in the following way. First, we select only firm-years for which we have accounting data information. Second, we select only firm-years that are represented in all four occupations. Third, we select only firm-years that are part of a sequence of at least three consecutive firm-years, thus allowing the construction of lagged differences at the firm level. Table 3 summarizes the sizes of the final firm level analysis panels, reporting the average number of firms in the cross-section, and also the average number of Managers, Salaried workers, Skilled workers and Unskilled workers employed. The accounting data give us FTE labor input at the firm level, denoted F T Ejt . We distribute this across occupations in proportion to 14
The identification of this type of two-way error component regression is studied by Abowd, Creecy, and Kramarz (2002) in the case where the error components are a worker-specific permanent effect and a firm-specific permanent effect. In the absence of time-varying observable covariates, their identification result carries over to our context where the error components are a worker-occupation effect and a firm-year-occupation effect, with identification established within each occupation and industry separately.
12
Table 2: Individual Level Panel—Number of Workers and Firms
Years Managers Average # workers Average # firms Salaried workers Average # workers Average # firms Skilled workers Average # workers Average # firms Unskilled workers Average # workers Average # firms
Manufacturing
Wholesale & Retail Trade
Transport, Storage & Communications
Real Estate, Renting & Business Activities
1995-2007
1999-2007
1999-2007
1999-2007
28,195 1,229
10,370 518
5,956 106
26,729 473
37,396 1,309
23,251 857
16,834 163
21,755 493
156,393 1,418
66,174 925
47,514 186
25,105 516
31,999 1,216
19,629 703
12,833 159
22,434 398
Note: The reported averages are simple averages across annual cross-sections of employed workers.
their shares of employment.15 From the table, we are primarily dealing with firms with large workforces.16 Industry size in terms of number of firms is ordered as before, with Manufacturing largest and Transport, Storage & Communications smallest. In terms of size of firms, the ordering of industries within each occupation is roughly opposite, with most workers per firmyear in Transport, Storage & Communications, followed by Real Estate, Renting & Business Activities (except that firms here employ most Managers across all industries), and with firms of about equal size in Manufacturing and Wholesale & Retail Trade. While Managers is the largest occupation in Real Estate, Renting & Business Activities, it is the smallest in the other industries, and Skilled workers is instead largest.
3.3
Descriptive Statistics
Labor productivity is computed by dividing total value added Y by a headcount measure of employment, Njt = 12 × 166.33 × F T Ejt , where 166.33 hours is the monthly hours norm for jobs under the Danish Industry Confederation and F T Ejt is the reported workforce size (of firm j in period t) from the employer data. 15
Suppose firm j has Njht workers in occupation h in period t, observed in the individual level panel. We N then assign a share PH jht of F T Ejt to occupation h in period t in firm j (note that H = 4 in our empirical N k=1
jkt
implementation). 16 Comparison of the average number of FTE workers per year reported in Table 3 to the corresponding average number of FTE workers in each of the four industries (source: Statistics Denmark) reveals that the share of workers included in our analysis sample out of the industry total is about 0.55 for Manufacturing and about 0.25 for the other three industries.
13
Table 3: Firm Level Panel—Number of Workers and Firms
Years Average Average Average Average Average
# # # # #
Manufacturing
Wholesale & Retail Trade
Transport, Storage & Communications
Real Estate, Renting & Business activities
1995-2007
1999-2007
1999-2007
1999-2007
766 32 43 178 36
259 28 57 162 49
45 61 188 426 101
143 108 77 92 101
firms Managers (FTE) Salaried workers (FTE) Skilled workers (FTE) Unskilled workers (FTE)
Note: The reported averages are simple averages across annual cross-sections of firms. The firm level panel is constructed by aggregating individual level information from the individual level panel (see Table 2) to the firm level subject to selection criteria described in the text.
Table 4: Average 90/10 Percentile Ratios and Correlation Coefficients
Manufacturing
Wholesale & Retail Trade
Transport, Storage & Communications
Real Estate, Renting & Business Activities
Average 90th to 10th Percentile Ratios Labor productivity Individual wage
2.21 2.06
2.30 2.63
2.58 2.10
2.12 2.54
Average Correlation Coefficients Labor productivity and wage
0.18
0.17
0.22
0.34
Note: The reported average percentile ratios and average correlation coefficients are simple averages of percentile ratios and correlation coefficients in annual cross-sections of employed workers.
Table 4 quantifies cross-sectional dispersion in labor productivity,
Yjt Njt
(employment weighted),
and individual wages, wit , in terms of average 90th to 10th percentile ratios. For labor productivity, the ratios range between 2.12 and 2.58. For wages, the ratios range between 2.06 and 2.63. For Wholesale & Retail Trade and Real Estate, Renting & Business Activities, the 90th to 10th percentile ratios for wages exceeds those for labor productivity. Table 4 also reports average cross-sectional labor productivity-wage correlations. The correlation coefficients range between 0.17 and 0.34.17 In online supplementary material we document that the data patterns reported in Table 4 are preserved when we consider sub-industries within Manufacturing. A main part of our analysis involves decomposing the joint distribution of log labor productivity and log wages. Figure 1 plots estimates of the average cross-sectional densities of the log transformed variables. The distribution of log labor productivity is to the right of the distribution of log wages. 17
If we instead consider 90th to 10th percentile ratios for labor productivity (non-employment weighted) and firm-level average wages, the ratios for labor productivity always exceed those of wages, and the correlation between labor productivity and wages increases.
14
Figure 1: Log Labor Productivity and Log Wages—Employment Weighted Densities Wholesale & Retail Trade 2.0
2.0
Manufacturing
0.5
PDF 1.0
1.5
Log labor productivity Log wage
0.0
0.0
0.5
PDF 1.0
1.5
Log labor productivity Log wage
4.5
5.0 5.5 6.0 Log Danish Kroner
6.5
7.0
2.0
Transport, Storage & Communications
4.0
4.5
5.0 5.5 6.0 Log Danish Kroner
6.5
7.0
Real Estate, Renting & Business Activities 2.0
4.0
0.5
PDF 1.0
1.5
Log labor productivity Log wage
0.0
0.0
0.5
PDF 1.0
1.5
Log labor productivity Log wage
4.0
4.5
5.0 5.5 6.0 Log Danish Kroner
6.5
7.0
4.0
4.5
5.0 5.5 6.0 Log Danish Kroner
6.5
7.0
Note: The plotted densities are simple averages of densities in annual cross-sections of employed workers.
15
Figure 2: Regressions of Log Wages onto Log Labor Productivity—Employment Weighted
7.0
2.0 1.5 1.0 PDF 0.5
4.5 5.0 5.5 6.0 6.5 Log labor productivity
7.0
2.0 1.5 1.0 PDF 0.5
6.00 Log wage 5.50 5.75 5.25
7.0
Nonparametric (left axis) PDF (right axis)
0.0
2.0
5.00
4.5 5.0 5.5 6.0 6.5 Log labor productivity
0.5 4.0
0.0
5.00
5.25
1.0 PDF
Log wage 5.50 5.75
Nonparametric (left axis) PDF (right axis)
4.0
Real Estate, Renting & Business Activities
1.5
6.00
Transport, Storage & Communications
Nonparametric (left axis) PDF (right axis)
0.0
6.00 Log wage 5.50 5.75 5.25 5.00
4.5 5.0 5.5 6.0 6.5 Log labor productivity
0.5 4.0
0.0
5.00
5.25
1.0 PDF
Log wage 5.50 5.75
Nonparametric (left axis) PDF (right axis)
2.0
Wholesale & Retail Trade
1.5
6.00
Manufacturing
4.0
4.5 5.0 5.5 6.0 6.5 Log labor productivity
7.0
Note: The plotted regression lines and densities are simple averages of regression lines and densities in annual cross-sections of employed workers.
Figure 2 shows nonparametric regressions of individual log wages on log labor productivity. There is a clear positive relation between an employer’s log labor productivity and the log wages it pays its employees. The relation is close to linear where the data are dense. The relations documented in Figures 1 and 2 also appear within sub-industries of Manufacturing (see online supplementary material). In summary, labor productivity is dispersed, wages are dispersed, and high labor productivity firms pay high wages. These empirical regularities may result from labor heterogeneity, factor misallocation, or both. Our empirical analysis disentangles these mechanisms.
4
Empirical Analysis
Our model is an equation system involving a wage equation, see (13), a production function, see (6), and, for each occupation, an equation determining the price of ability, see (9). This system is taken to the data in three steps: First, the wage equation; second, the firm level
16
production function; and third, for each occupation, the ability price equation. Recall that we index individuals by i = 1, ..., I, firms by j = 1, ..., J, occupations by h = 1, ..., H, and time by t = 1, ..., T. To avoid clutter, it is suppressed in this notation that both individual and firm level panels are unbalanced.
4.1
The Wage Equation
To make the log wage equation (13) empirically implementable, we amend it with an additional w additive stochastic component εw it . We shall specify the statistical properties of εit below;
think of it as measurement errors in wages, shocks to individual ability, or other unmodeled heterogeneity. From (13), the log wage equation therefore reads ln wit = ln φJ(i,t)H(i,t)t + ln aiH(i,t) + εw it .
(17)
For estimation purposes, this is conveniently viewed as decomposing individual log wages into time and occupation specific firm effects ξjht ≡ ln φjht , occupation specific worker effects ζih ≡ ln aih , and the residual terms εw it unexplained by the former two. We impose the following identifying assumption on (17): Assumption A1 E[εw it |i, t, J(i, t)] = 0, for all i = 1, ..., I and t = 1, ..., T . Assumption A1 rules out that individuals change employer or occupation based on εw it . Under Assumption A1, we can estimate (17) by OLS.18 From the estimated ξjht ’s and ζih ’s we construct firm-year-occupation specific ability adjusted labor inputs Ljht . Indeed, since ζbih ≡ ln b aih , we construct estimates of Ljht as b jht = L
X
ωh exp(ζbih ),
(18)
{i:J(i,t)=j∧H(i,t)=h}
for j = 1, ...., J, h = 1, ..., H, and t = 1, ..., T , where ωh normalizes the grand mean of ability in each occupation to unity. Estimation errors on individual level ability are likely to be averaged out in this aggregation to firm level labor inputs. Estimates of ability prices are constructed in a similar manner: φbjht = ωh−1 exp(ξbjht ). 4.1.1
(19)
Results
Estimation of (17) is conducted on the full individual level panel. This includes workers employed by firms for which we do not observe accounting data. These firms are therefore not included in the firm level panel, and so are not used in the second and third step of the our estimation procedure. We use firms without accounting data in this first step estimation to 18
Identification of the (many) parameters in (17)) requires that εw it is orthogonal to the design matrix consisting of firm-year-occupation and worker-occupation indicators, and that the design matrix is of full rank. The former condition is met under Assumption A1. The latter condition is also met subject to a normalization of a worker or a firm-year effect within each occupation, since we are working with a set of connected workers and firm-years.
17
improve the precision of the estimated worker and firm effects. However, for purposes of comparison and interpretation, we report distributions of the estimated ability prices and ability adjusted labor inputs for the subsample of firms included in the firm level panel. Figure 3 depicts the average cross-section density of ability prices, see (19), across firms, by occupation and industry. The densities are non-degenerate. The average prices of ability rank as expected across occupations, with managerial ability commanding the highest average price, followed by Salaried workers. The same ranking applies with respect to price variation. Note that ability price dispersion suggests that misallocation is an empirically relevant phenomenon. Figure 4 exhibits the average cross-sectional density of ability adjusted labor input, see (18), across firms, by occupation and industry. Labor input also varies considerably in the cross-section. From the figure, firms in the Manufacturing industry employ relatively more Skilled labor and firms in Transport, Storage & Communications relatively less Management, compared to other industries. We return to the estimated wage equation when we decompose the cross-sectional distribution of wages. In what follows, we use the estimated ability prices to estimate the sharing rule coming out of our bargaining game, and the estimated ability adjusted labor inputs in the estimation of the production function.
4.2
The Production Function
The second step of the estimation procedure is concerned with the estimation of the firm level production function. Our empirical analysis favors a production function specification with elasticity of substitution of unity between occupations, i.e., a Cobb-Douglas labor aggregator. Although at times we refer to specifications involving the less restrictive CES aggregator, we therefore present the main estimation equations imposing the Cobb-Douglas labor aggregator. From (6) and (7), ln Yjt = αK ln Kjt + αL
H X
b jht + pjt , γh ln L
(20)
h=1
b jht is obtained from the estimated wage equation, see (18). where L Difficulties in estimating production function parameters are well known: Firm-level inputs Kjt and Ljt are likely to be chosen by the firm in response to realizations of TFP pjt . Dealing with this issue requires imposing some structure. We consider TFP processes on the following form: Assumption A2 The TFP process is pjt = pj + νjt where νjt = ηνjt−1 + εpjt , with E[εpjt ] = 0 for all j, t and E[εpjt εpjs ] = 0 for all j and s 6= t. This allows for both a time-invariant firm fixed effect pj and an AR(1) component that may be more or less persistent, depending on the value of η. The TFP innovations εpjt are idiosyncratic, and if η = 0 they are purely transitory. With respect to the timing of events in the model, we adhere to the following structure, consistent with the dynamic environment described in Section 2: The TFP innovation εpjt is realized at the beginning of period t. Following this, labor reallocates across firms and occupations, and
18
Figure 3: Occupation Specific Prices of Ability—Densities
400 600 Danish Kroner
800
0.012
Transport, Storage & Communications
200
400 600 Danish Kroner
200
400 600 Danish Kroner
800
Managers Salaried Skilled Unskilled
PDF 0.004 0.008 0.000
0
0
Real Estate, Renting & Business Activities
Managers Salaried Skilled Unskilled
PDF 0.004 0.008 0.000
0.000
200
0.012
0
Managers Salaried Skilled Unskilled
PDF 0.004 0.008
Managers Salaried Skilled Unskilled
PDF 0.004 0.008 0.000
Wholesale & Retail Trade 0.012
0.012
Manufacturing
800
0
200
400 600 Danish Kroner
800
Note: The plotted densities are simple averages of densities in annual cross-sections of firms included in the firm level panel.
19
Figure 4: Occupation Specific Ability Adjusted Labor Inputs—Densities
500
0.012
Transport, Storage & Communications
100 200 300 400 Ability adjusted labor input
100 200 300 400 Ability adjusted labor input
500
Managers Salaried Skilled Unskilled
PDF 0.004 0.008 0.000
0
0
Real Estate, Renting & Business Activities
Managers Salaried Skilled Unskilled
PDF 0.004 0.008 0.000
0.000
100 200 300 400 Ability adjusted labor input
0.012
0
Managers Salaried Skilled Unskilled
PDF 0.004 0.008
Managers Salaried Skilled Unskilled
PDF 0.004 0.008 0.000
Wholesale & Retail Trade 0.012
0.012
Manufacturing
500
0
100 200 300 400 Ability adjusted labor input
500
Note: The plotted densities are simple averages of densities in annual cross-sections of firms included in the firm level panel.
20
investments in capital are made. Given the set of workers available after labor reallocation, the firm negotiates wage contracts in accordance with the procedure outlined in Section 2. Finally, production takes place and payments are made. Hence, from an econometric point of view, labor and capital are both potentially endogenous, i.e., correlated with contemporaneous TFP innovations.19 Besides the timing assumption, we do not impose further restrictions on labor reallocation. In particular, we do not assume that firms attain the optimal labor input combination: Although they condition hiring and separation decisions on current TFP innovations, labor market frictions may imply that reallocation is imperfect. Assumption A2 allows quasi-differencing the production function to express the unobserved TFP innovations as functions of data and unknown parameters, εpjt (θ, pj ) = ln Yjt − η ln Yjt−1 − αK ln Kjt + ηαK ln Kjt−1 − αL
H X
b jht + ηαL γh ln L
h=1
H X
b jht−1 − (1 − η)pj , (21) γh ln L
h=1
where θ = (αK , αL , γ1 , ..., γH , η)0 . Using (21) and the assumed timing of events we construct orthogonality conditions to identify andhestimate the i production function parameters via GMM. Indeed, restricting pj = p, we p have E εjt (θ, p)Zjt = 0, where Zjt is a vector of instrumental variables in the information set at t − 1. With all pj , j = 1, . . . , J, left free instead, the standard procedure h i would be p to difference (21) once more and consider conditions of the form E ∆εjt (θ)Zjt = 0, where ∆εpjt (θ, pj ) = εpjt − εpjt−1 = ∆εpjt (θ) depends on data and θ, only, and Zjt in this case is a vector of instrumental variables in the information set at t − 2. The levels version nevertheless remains a relevant specification because AR(1) persistence (η > 0) with separate TFP innovations εpjt across firms allows capturing much the same empirical features in the data as separate fixed effects pj .20 In our empirical work, we collapse the system of orthogonality conditions by averaging across h i
firms and time (Roodman, 2009). Thus, our implementation of the conditions E εpjt (θ, p)Zjt = 0 reads
Tj J X X 1 εpjt (θ, p)Zjt = 0, M −J
(22)
j=1 t=2
with M =
PJ
j=1 Tj
and Tj the number of time periods available for firm j. We consider
instrumentation by lagged endogenous variables in levels. Both labor and capital are treated as endogenous, and when pj = p is imposed, Zjt contains functions of ln Kj,t−1 , ln Lj,t−1 , and ln Yj,t−1 . Notice that these instruments are valid even if capital and labor inputs are in fact predetermined. We use the minimum valid lag order only in order to maximize both the size 19
We did consider variations with capital predetermined, i.e., orthogonal to contemporaneous TFP innovations, and our results did not change much, but our empirical analysis nonetheless favored specifications with both capital and labor endogenous. 20 The approach of quasi-differencing to isolate exogenous variation follows the dynamic panel data literature, see, e.g., Holtz-Eakin, Newey, and Rosen (1988), Arellano and Bond (1991), and Blundell and Bond (1998). Olley and Pakes (1996), Levinsohn and Petrin (2003), Ackerberg, Caves, and Frazer (2006), and Gandhi, Navarro, and Rivers (2011) present alternative ways of dealing with endogenous inputs in production functions.
21
of the data set used and the power of the instruments. The exact elements in the vector of instrumental variables are specified when we present the empirical results below. 4.2.1
Results
For purposes of testing down from a more general specification in our empirical work, we consider P b ρ )1/ρ , with elasticity of a constant elasticity of substitution (CES) labor aggregator, ( H γh L h=1
jht
substitution between occupations of = 1/(1 − ρ). Thus, our most general production function specification is H X αL bρ ln Yjt = αK ln Kjt + ln γh L jht ρ
! + pjt ,
(23)
h=1
with (21) augmented accordingly. The log linear, or Cobb-Douglas, labor aggregator special case (see (7) and (20)) corresponds to the testable null of unit elasticity, = 1 (i.e., ρ = 0). The CES labor aggregator is consistent with the theoretical framework developed in Section 2, but does not admit a closed form solution for the price of ability, unless workers’ bargaining power is common across occupations. Least squares estimation We start with a brief presentation of estimates obtained from least squares estimation of the production function. This naive empirical strategy treats capital and labor inputs as orthogonal to the TFP innovation εpjt , and also imposes pjt = p + εpjt , i.e., η = 0. It is included here for comparison purposes. With CES labor aggregator, the production function (23) is nonlinear in parameters (in particular, ρ) and is estimated by Nonlinear Least Squares (NLS). With Cobb-Douglas labor aggregator, it is given as (20), amenable to estimation by Ordinary Least Squares (OLS). Estimates pertaining to both specifications are presented in Table 5. Here, we leave the returns to scale in capital and aggregate labor input unrestricted, i.e., αK + αL is not restricted to unity. In the bottom panel of Table 5 we report the p-values from F -tests of the restrictions = 1 (pCD ), αK + αL = 1 (pCRTS ), and the joint hypothesis = 1 and αK + αL = 1 (pCD∧CRTS ). The tests involving the restriction = 1 are only relevant for the specification with a CES labor aggregator. Empirically, we find that often approaches infinity, preventing us from computing the tests on in all industries except Wholesale & Retail Trade. In the latter, we fail to reject = 1 and αK + αL = 1. With or without = 1 imposed, we never reject αK + αL = 1 at a 1% significance level. The lowest p-value is 4%, in Real Estate, Renting & Business Activities, where point estimates nonetheless imply αK + αL = 0.94. Clearly, is not well identified in the NLS specification, and we focus attention on the columns imposing = 1 (labeled OLS in Table 5). Overall, the parameter estimates appear reasonable. Across industries, Wholesale & Retail Trade gets the lowest point estimate of αK , the capital input elasticity, at 0.08, with the highest, at 0.16, in Transport, Storage & Communications. The capital and overall labor input elasticity estimates αK and αL are relatively precisely estimated and sum to nearly unity in each industry, indicating that the data are consistent with the constant returns to scale restriction. In the OLS estimation, the lowest of the four occupation-specific relative productivities 22
Table 5: Production Function Parameters—NLS and OLS Estimation
Manufacturing NLS αK αL γ1 γ2 γ3 γ4
OLS
Wholesale & Retail Trade
Transport, Storage & Communications
Real Estate, Renting & Business Activities
NLS
NLS
NLS
OLS
OLS
OLS
0.102
0.115
0.066
0.082
0.152
0.155
0.124
0.113
(0.007)
(0.008)
(0.012)
(0.013)
(0.028)
(0.026)
(0.015)
(0.018)
0.910
0.885
0.928
0.907
0.886
0.875
0.820
0.827
(0.010)
(0.013)
(0.020)
(0.022)
(0.059)
(0.055)
(0.024)
(0.031)
0.309
0.178
0.368
0.208
0.511
0.170
0.408
0.374
(0.019)
(0.012)
(0.035)
(0.021)
(0.085)
(0.055)
(0.027)
(0.030)
0.346
0.239
0.300
0.291
0.202
0.290
0.260
0.274
(0.016)
(0.012)
(0.022)
(0.020)
(0.049)
(0.056)
(0.029)
(0.036)
0.177
0.453
0.154
0.341
0.193
0.428
0.197
0.197
(0.006)
(0.014)
(0.017)
(0.019)
(0.039)
(0.046)
(0.024)
(0.030)
0.168
0.131
0.179
0.159
0.093
0.111
0.134
0.156
(0.007)
(0.008)
(0.016)
(0.016)
(0.031)
(0.040)
(0.015)
(0.026)
∞
1.000
12.686
1.000
∞
1.000
∞
1.000
2,332
408 − 0.360 −
408
1,288 − 0.006 −
1,288
(13.592)
# observations pCD pCRTS pCD∧CRTS
9,962 − 0.074 −
9,962 0.966
2,332 0.390 0.682 0.674
0.496
0.438
0.040
Note: The constant term is not reported.
within industry is always that of Unskilled workers, γ4 . In Real Estate, Renting & Business Activities, the γh s are ordered according to the standard education-skill hierarchy, with Managers highest, whereas in the other three industries, this ordering is reversed among the three other occupations besides the Unskilled. In the unrestricted NLS estimation, the standard ordering is obtained in all industries, except that the two highest occupations are switched in Manufacturing and the two lowest in Wholesale & Retail Trade. GMM estimation
We now turn attention to results of GMM estimation of the production
function parameters, relaxing the restrictions behind the NLS and OLS estimation to allow for potentially endogenous capital and labor choices on the part of firms, and allowing for η ∈ (−1, 1), i.e., potentially persistent firm-year effects in the TFP processes. We restrict pj = p for all j = 1, 2, ..., J. Hence, our empirical orthogonality conditions are on the form (22).21 Consider first the general case with a CES labor aggregator, and where the returns to scale in capital and aggregated labor input are unrestricted. That is, the production function is given P by (23). Recall that H h=1 γh = 1. Hence, in the CES case there are H + 3 parameters to be estimated in θ = (αK , αL , γ1 , ..., γH , η, ρ)0 . With four occupations in the data, we therefore estimate eight parameters in (θ, p). 21 We also compared to estimates treating pj as free, replacing εpjt by ∆εpjt in the orthogonality conditions (22) and modifying Zjt accordingly, lagging the endogenous variables once more (one observation per firm was lost). In these cases, we obtained much less stable results, inflated standard errors, and unrealistically low estimated returns to scale in production.
23
We employ an over-identified system of orthogonality conditions, treating both labor and capital as endogenous. We use lagged endogenous variables as instruments, but do not use lagb jt−1 , ln Kjt−1 , ln Yjt−1 , QH ln L b jht−1 , ging as a source of over-identification. Instead, we use 1, ln L h=1 b jt−1 ·ln Yjt−1 , and ln Kjt−1 ·ln Yjt−1 as instruments. With H = 4, this yields thirteen orthogln L onality conditions, and hence five over-identifying restrictions. We consequently use a two-step GMM estimator, where the first step is used for estimation of the optimal weight matrix, and report Hansen’s J-test of validity of the over-identifying restrictions (Hansen, 1982). Results appear in Table 6, in the columns labeled (1). Parameter estimates are sensible, with capital input elasticities αK ranging from 0.05 to 0.18, again with Wholesale & Retail Trade lowest and Transport, Storage & Communications highest. In Manufacturing, the capital input elasticity is estimated to 0.10. The GMM estimates of αK are slightly below the naive NLS estimates of Table 5, except that for Transport, Storage & Communications. Taken together, the unrestricted capital and aggregate labor input elasticities αK and αL indicate that the technology for each industry is not far from constant returns to scale. The GMM estimates of the relative productivities of the different occupations, γh , h = 1, ..., H, exhibit a clear pattern: Managerial labor is more productive than Salaried labor, Salaried labor is more productive than Skilled labor, etc. As in the NLS estimation, there are only two exceptions: Again, Managers and Salaried workers are switched in Manufacturing, and now the Unskilled are not the least productive in Real Estate, Renting & Business Activities. Point estimates of the elasticity of substitution between any two occupational labor inputs, = 1/(1 − ρ), are very imprecise and therefore vary considerably across industries. Finally, the AR(1) coefficient η from the firm level TFP processes is directly estimated in the GMM procedure, and the estimates imply strong persistence, but not unit roots, with η ranging from 0.65 in Transport, Storage & Communications to 0.81 in Real Estate, Renting & Business Activities. The lower panel of Table 6 reports p-values for Hansen’s J-test (pJ ), and Wald tests of the Cobb-Douglas labor aggregator restriction = 1 (pCD ), the constant returns to scale restriction αK +αL = 1 (pCRTS ), and the joint Cobb-Douglas and CRTS hypothesis, = 1 and αK +αL = 1 (pCD∧CRTS ). At a 5% level, we cannot reject any of the hypotheses (validity of over-identifying restrictions, CRTS or Cobb-Douglas labor aggregator) in any of the industries. This is in part because the elasticity of substitution in the CES labor aggregator is poorly identified in our data, a finding that is consistent with our naive least squares estimation in Table 5. Given the outcomes of the tests on the general model, we proceed to estimate a production function imposing constant returns to scale and a Cobb-Douglas labor aggregator on a justidentified system of orthogonality conditions. We still treat both labor and capital as endogenous. Since ρ is no longer estimated, there are now six parameters in (θ, p). The instruments b j1t−1 , ln L b j2t−1 , ln L b j3t−1 , ln Kjt−1 , and ln Yjt−1 .22 used are 1, ln L Results are reported in Table 6, in the columns labeled (2). In terms of the capital input elasticities αK , the restrictions imposed lead to a strengthening of the observed pattern, with the highest coefficient, again that in Transport, Storage & Communications, now even higher, 22
In our specification search we find that we get the most stable results with respect to both point estimates and estimated standard errors using exactly identified systems.
24
at 0.20, and the lowest, still in Wholesale & Retail Trade, considerably lower, now at 0.01. The estimate in Manufacturing is nearly unchanged, at 0.09. As in the least squares case, the relative productivities implied by the Cobb-Douglas labor aggregator are not as clearly ranked as in the CES case, except that Skilled labor gets higher coefficient than Unskilled, γ3 > γ4 , throughout. Indeed, it also gets higher coefficient than Salaried labor, γ3 > γ2 , i.e., a reversal among these two types, compared to the expected outcome. Manufacturing and Wholesale & Retail Trade exhibit the same pattern as in the OLS estimation, with the Unskilled least productive and the three higher occupations in reverse order, with Skilled labor most productive. In Transport, Storage & Communications, Skilled labor is similarly the most productive type. The high estimated relative productivity per ability unit of Skilled labor may be related to the fact that this category by the data definitions in the Danish case consists of relatively highly specialized blue collar workers, whereas Salaried labor consists of white color office workers in non-managerial positions, cf. Christensen, Lentz, Mortensen, Neumann, and Werwatz (2005). With Cobb-Douglas labor aggregator and constant returns to scale imposed, the estimated TFP process exhibits stronger persistence, with autocorrelation coefficients ranging between 0.74 and 0.82, but still no unit root. The strong persistence implies that some of the same empirical phenomena are picked up as if using permanent firm effects (i.e., pj free). Based on the tests carried out on the more general model and the fact that the parameter estimates appear reasonable, this production function, with constant returns to scale and CobbDouglas labor aggregator, estimated on a just-identified system of orthogonality conditions, constitutes our preferred specification. The remainder of the analysis in this paper is carried out using these estimates (i.e., columns labeled (2) in Table 6).
4.3
Ability Price Equations
The third step of our procedure is the estimation of the firm-level occupation-specific sharing rules—the ability price equations (9), extended to allow for dynamics. The bargaining game provides a deterministic link between the unit price of ability, φjht = φh (Kjt , Ljt , pjt ), and the Y
jt marginal product of ability adjusted occupation-h labor, M P Ljht = γh (1 − αK ) Ljht . To break
this link, we amend each ability price equation with an error-term εφjht , which produces φbjht = (1 − βh )bh +
βh 1−βh
1 + (1 − α bK )
!
PH
βk bk 1−β k=1 γ k
\ M P Ljht + εφjht ,
(24)
for h = 1, 2.., H. In (24), we have imposed constant returns to scale. The notation emphasizes that our measurements of the price of ability φjht , the marginal product of labor M P Ljht , the capital elasticity αK , and the relative occupational productivities γh are estimates obtained from the wage equation and the production function. Hence, the error-term εφjht is in part intended to capture the effect of estimation errors, but can also be interpreted as a tremble to the bargaining process, or simply a reduced form residual containing variation in the price of ability not accounted for by our model. In either case, the following assumption on (εφj1t , ..., εφjHt ) is of interest:
25
Table 6: Production Function Parameters—GMM Estimation
Manufacturing (1) αK αL γ1 γ2 γ3 γ4
(2)
# observations pJ pCD pCRTS pCD∧CRTS
(1)
(1)
(2)
(2)
Real Estate, Renting & Business Activities (1)
(2)
0.096
0.089
0.053
0.006
0.175
0.198
0.120
0.054
(0.013)
(0.015)
(0.030)
(0.039)
(0.067)
(0.077)
(0.034)
(0.066)
0.924
0.911
0.949
0.994
0.870
0.802
0.848
0.946
(0.022)
(0.015)
(0.056)
(0.039)
(0.181)
(0.077)
(0.067)
(0.066)
0.275
0.144
0.334
0.066
0.424
0.016
0.440
0.417
(0.069)
(0.052)
(0.188)
(0.167)
(0.706)
(0.605)
(0.094)
(0.605)
0.426
0.354
0.324
0.412
0.286
0.014
0.168
0.130
(0.077)
(0.054)
(0.076)
(0.122)
(0.304)
(1.001)
(0.200)
(1.453)
0.165
0.433
0.255
0.486
0.249
0.822
0.155
0.281
(0.075)
(0.036)
(0.200)
(0.089)
(0.313)
(0.394)
(0.105)
(0.484)
0.134
0.068
0.087
0.035
0.040
0.148
0.237
0.172
(0.047)
(0.040)
(0.065)
(0.086)
(0.183)
(0.512)
(0.088)
(0.382)
54.778
1.000
1.886
1.000
10.886
1.000
∞
1.000
(1375.2)
η
Wholesale & Retail Trade
Transport, Storage & Communications
(3.452)
(170.07)
0.725
0.753
0.740
0.771
0.649
0.744
0.812
0.823
(0.015)
(0.012)
(0.022)
(0.025)
(0.053)
(0.152)
(0.037)
(0.066)
8,521 0.060 0.969 0.121 0.212
8,521
1,907 0.168 0.797 0.941 0.901
1,907
335 0.176 0.954 0.714 0.772
335
1,058 0.055 1.000 0.516 0.804
1,058
Note: Constant term not reported. Columns labeled (1) contain estimates obtained under CES labor aggregator and unrestricted returns to scale in capital and aggregated labor. Columns labeled (2) contain estimates obtained with a Cobb-Douglas labor aggregator and constant returns to scale in capital and aggregated labor. The reported standard errors are not corrected for the first step estimation of individual abilities.
26
h i \ \ Assumption A3 E εφjht |M P Lj1t , ..., M P LjHt = 0, for all j = 1, ..., J, h = 1, ..., H, and t = 1, ..., T. To proceed, consider the reduced form, linear in levels system, \ φbjht = ψ0h + ψ1h M P Ljht + εφjht ,
h = 1, ..., H.
(25)
Under Assumption A3, (25) can be considered as a seemingly unrelated regressions system (Zellner, 1962). Our structural model implies specific structure on the reduced form parameters (ψ01 , . . . , ψ1H ) in terms of the structural parameters b1 , ..., bH and β1 , . . . , βH . Thus, upon estimation of the 2H reduced form parameters, estimates of the structural parameters may be recovered as βbh =
ψb1h ψb1h + (1 − (1 − α bK )
PH
bk ψ1k ) k=1 γ
and
bbh =
b
ψb0h , 1 − βbh
(26)
for h = 1, ..., H, with estimates of (b αK , γ b1 , . . . , γ bH ) from the second step. To assess the uncertainty about the structural parameter estimates, we might in principle calculate standard errors using the delta-method, thus accounting for error stemming from εφjht in the estimation of the reduced form, and transformation from reduced form to structural parameters. Such an assessment would require a further fitted regressor adjustment, along the \ lines of Pagan (1984), since M P Ljht in (25) contains estimates from the first and second step of our analysis. However, because these estimates depend non-linearly on coefficients from the individual level wage equation, and because the estimates from the second step of (b αK , γ b1 , . . . , γ bH ) 23 b in (26) depend on Ljt , too, the exact form of the adjustment would be complicated. To circumvent these complications we consider instead reverse regression, PL \ M P Ljht = ϑ0h + ϑ1h φbjht + εM jht ,
h = 1, ..., H,
following, e.g., Friedman and Schwartz (1982). Reverse regression estimates of ψ0h and ψ1h are b then given by ψbr = − ϑ0h and ψbr = 1 . The true values of the reduced form parameters 0h
b1h ϑ
1h
b1h ϑ
belong to the intervals within the Frisch bounds given by ψ0h ∈
ψ0h −
−1 ψ1h
and ψ1h ∈
E [M P Ljht ] V ar(εφjht ) V ar(M P Ljht )
, ψ0h + ψ1h
E [M P Ljht ] V ar(εφjht ) V ar(M P Ljht ) + V ar(εφjht )
V ar(εφjht ) V ar(ujht ) −1 ψ1h − ψ1h , ψ1h + ψ1h V ar(M P Ljht ) \ V ar(M P Ljht )
! ,
! ,
23 Further, any errors-in-variable problem and resulting violation of Assumption A3 would induce attenuation bias toward zero in ψb1h , and thus potentially βbh .
27
r and for h = 1, . . . , H.24 The lower and upper bounds on ψ0h are consistently estimated by ψb0h d , with r and d indicating reverse respectively direct regression, and the bounds on ψ ψb0h 1h by d r ψb and ψb . We report bounds on structural parameters by backing them out at the reduced 1h
1h
form bounds using the relations (26), with interval midpoints serving as point estimates. 4.3.1
Results
Estimates of the structural parameters of the ability price functions are reported in Table 7. Generally, the reported ranges for the bargaining powers βh are compatible with those reported in other relevant studies (e.g., Cahuc, Postel-Vinay, and Robin, 2006). By the point estimates, the received patterns are very clear. Within each industry, the outside options (per unit of ability adjusted labor supply) bh are ordered according to the usual education-skill hierarchy, except that they are slightly better for Skilled than for Salaried workers. This is consistent with our results on the relative productivities in the production function estimation above, and the findings by Christensen, Lentz, Mortensen, Neumann, and Werwatz (2005) that Skilled workers face lower search costs than Salaried workers, possibly because these relatively highly specialized blue collar workers participate in well connected occupational networks. In fact, from Table 7, in the two largest industries, Manufacturing and Wholesale & Retail Trade, the outside options of Salaried workers are hardly any better than those of the Unskilled, and the bargaining power β2 of Salaried workers is indeed the lowest across occupations—point estimates at 0.22 and 0.13, respectively. The two other industries have fewer but larger firms (see Table 3), and here, Managers and Salaried workers have higher bargaining power than the other groups, possibly in excess of 0.5 for both occupations within Transport, Storage & Communications, based on the bounds. Comparing across industries, workers in Manufacturing have bargaining powers between those of workers in the same occupation in other industries, the exception again being Skilled labor, with β3 considerably higher in Manufacturing than in other industries. In terms of outside options, those in Wholesale & Retail Trade dominate those in Manufacturing, and also those in Transport, Storage & Communications, except that here, Managers find the best outside options across industries. The last industry, Real Estate, Renting & Business Activities, offers very favorable outside options, the best of all for Skilled and Salaried workers, and the secondbest for Managers. Again, this is the pattern by point estimates. The bounds do have bite, though: Throughout, they rule out that workers have all the bargaining power (βh = 1), and in most cases, they also rule out zero bargaining power or outside option. They show that Managers in Transport, Storage & Communications have higher bargaining power than those in Real Estate, Renting & Business Activities, and higher than blue collar (Skilled and Unskilled) workers in their own industry—intervals do not overlap. Less formal comparisons suggest that these blue collar workers by the Frisch bounds cannot be facing as favorable outside options as those indicated by the point estimate for Managers in 24
\ This form of the bounds is for error-ridden measurements M P Ljht = M P Ljht + ujht where M P Ljht is the true variable, the measurement error ujht is uncorrelated with M P Ljht and εφjht , and all variables are treated as uncorrelated across equations h for simplicity. The consistency result extends to more general forms of the bounds than those exhibited, e.g., for correlation across equations, Leamer (1987), although more complicated cases require modified estimators, e.g., if ujht and εφjht are correlated, Krasker and Pratt (1986).
28
Table 7: Ability Price Function Estimates
Manufacturing Managers Bargaining power β1 Outside option b1 Salaried workers Bargaining power β2 Outside option b2 Skilled workers Bargaining power β3 Outside option b3 Unskilled workers Bargaining power β4 Outside option b4
Wholesale & Retail Trade
Transport, Storage & Communications
Real Estate, Renting & Business Activities
0.308
0.373
0.608
0.093
[0.019,0.596]
[0.047,0.699]
[0.356,0.861]
[0.007,0.180]
158.4
171.2
214.3
208.6
[0.0,316.8]
[0.0,342.4]
[0.0,428.7]
[115.2,302.0]
0.219
0.131
0.485
0.241
[0.002,0.435]
[0.001,0.261]
[0.063,0.906]
[0.015,0.466]
116.0
128.6
115.1
168.7
[0.0,232.0]
[10.8,246.4]
[0.0,230.2]
[86.9,250.6]
0.363
0.139
0.051
0.071
[0.006,0.720]
[0.006,0.272]
[0.003,0.100]
[0.012,0.130]
120.7
132.3
128.2
173.5
[51.2,190.3]
[88.9,175.7]
[95.6,160.8]
[143.6,203.5]
0.287
0.365
.141
.070
[0.015,0.560]
[0.117,0.613]
[0.000,0.281]
[0.010,0.133]
115.7
127.2
77.6
107.7
[53.5,177.8]
[67.1,187.2]
[0.0,155.2]
[32.4,183.1]
Note: The reported estimates are conditional on the production function parameter estimates reported in columns labeled (2) in Table 6. Numbers presented in square brackets are bounds obtained using reverse regression techniques.
the same industry, at 214. The same is true in Real Estate, Renting & Business Activities. On the other hand, the bounds contain all the point estimates of outside options when looking across industries within each occupation, Skilled workers in Real Estate, Renting & Business Activities being the only exception. In this sense, market opportunities are more similar within occupation than within industry, especially for higher occupations. The results from estimation of the wage equation, the production function, and the ability price equations are used in assessing the relative importance of heterogeneity and resource misallocation, measure ability flows, and evaluate potential output and wage gains from labor reallocation.
5
Heterogeneity or Misallocation?
Labor productivity and wages exhibit pronounced intra-industry cross-section dispersion and positive covariance, see Figures 1 and 2, and Table 4. These observations may reflect either ability heterogeneity or factor misallocation, i.e., dispersion in marginal products across firms. The estimated model allows us to conduct a novel set of measurements shedding light on this distinction. Throughout, we consider variance and covariance in a cross-section of workers.
29
5.1
Log Labor Productivity Variance
e jt = PH Ljht is total ability in a firm, such that Lejt is the average ability of firm Recall that L h=1 Njt QH γh ∗ j’s period-t workforce, Ljt = h=1 Ljht is aggregate ability adjusted labor input, and M P L∗jt = Y
Y
25 Since ln jt = (1 − αK ) Ljt ∗ is the marginal product of aggregate ability adjusted labor input. Njt jt
e L
ln Njt + ln jt
L∗jt e jt L
Y
∗ + ln Ljt ∗ and M P Ljt ∝ jt
Yjt L∗jt ,
Y
Y
Y
jt ∗ such that Cov(ln Njtjt , ln Ljt ∗ ) = Cov(ln N , ln M P Ljt ), jt jt
Y
it is natural to decompose the cross-section t variance of log labor productivity ln Njtjt as ! e jt Yjt Yjt L |t = Cov ln , ln |t V ar ln Njt Njt Njt | {z } Ability heterogeneity
! L∗jt Yjt Yjt ∗ |t + Cov ln , ln + Cov ln , ln M P Ljt |t , (27) e jt Njt Njt L | {z } {z } | Aggregate factor misallocation
Occupational labor misallocation
|
{z
Misallocation
}
where, to avoid cluttering the notation unnecessarily, it is implicitly understood that j = J(i, t) Y
as we consider variation in ln Njtjt in a cross-section of employed workers. The first term in (27) is variation in observed log labor productivity accounted for by ability heterogeneity. Firms that employ workers who are on average more able will have higher observed labor productivity. The second term in (27) captures occupational labor misallocation. e jt more efficiently across occupations have a higher Firms that distribute a total stock of ability L ratio of aggregate labor input to total ability,
L∗jt e jt , L
and higher labor productivity. The third
term in (27) captures aggregate factor misallocation. Firms with higher marginal products of aggregate labor input, M P L∗jt , have higher labor productivity. The total effect of misallocation is the sum of occupational labor misallocation and aggregate factor misallocation. The variance-covariance decompositions are conducted on observed data. We can therefore not hold other influences, such as TFP and capital misallocation, fixed when evaluating the role of, say, ability heterogeneity. Hence, the decomposition (27), as well as other decompositions to follow, should be interpreted in an accounting sense. Another implication of this is that variance shares may come out negative. For example, ceteris paribus, a firm with a more able workforce has higher observed labor productivity. However, if the firm happens to be hit by a negative TFP shock, or has a low capital-labor ratio, we might observe a negative covariance between labor productivity and average ability of the workforce due to the confounding effect of TFP or capital misallocation.26 Table 8 reports the average cross-section variance of log labor productivity, and the average percentage shares of ability heterogeneity, occupational labor misallocation, and aggregate factor misallocation according to (27). The decomposition is quantitatively similar in the two largest industries, Manufacturing and Wholesale & Retail Trade. Here, ability heterogeneity accounts for 9% and 11%, respectively, of the variance in cross-section log average labor productivity. The 25
See (3), (7), and (14), imposing constant returns to scale in capital and aggregate labor input. Moreover, in the long run, variation derives fundamentally from TFP shocks. Hence, our decompositions captures short- and medium-term variation, where production factors can be considered quasi-fixed. 26
30
Table 8: Employment Weighted Log Labor Productivity Variance Decomposition
h i Y E V ar ln NJ(i,t),t |t J(i,t),t
Ability heterogeneity Occupational labor misallocation Aggregate factor misallocation
Manufacturing
Wholesale & Retail Trade
Transport, Storage & Communications
Real Estate, Renting & Business Activities
0.100
0.128
0.256
0.162
9% 26% 65%
11% 24% 65%
3% −22% 119%
−8% 65% 43%
Note: The decomposition is given by (27). J(i, t) = j if worker i is employed by firm j in period t. Results are based on the production function estimates reported in columns labeled (2) in Table 6.
remaining 91% and 89% of log labor productivity variance stem from misallocation. Aggregate factor misallocation generates more than twice the amount of variance in log labor productivity, compared to occupational labor misallocation. The log labor productivity variance decompositions for Transport, Storage & Communications and Real Estate, Renting & Business Activities bear qualitative similarities to the decompositions for Manufacturing and Wholesale & Retail Trade, albeit with some quantitative differences. Ability heterogeneity is still (by far) the smallest variance component, contributing 3% in Transport, Storage & Communications, and −8% in Real Estate, Renting & Business Activities. The negative covariance may arise due to the confounding effect of, say, TFP shocks. Misallocation accounts for 97% and 108% in Transport, Storage & Communications and Real Estate, Renting & Business Activities. Aggregate factor misallocation is the main contributor, at 119%, in Transport, Storage & Communications, whereas occupational labor misallocation accounts for the largest share, 65%, of the variance in log labor productivity in Real Estate, Renting & Business Activities. Still, aggregate factor misallocation contributes 43% of the variance in log labor productivity in this industry. Summing up, misallocation emerges as the primary reason for intra-industry dispersion in observed log labor productivity. Ability heterogeneity accounts for no more than 11% of intra-industry dispersion in log labor productivity across the industries considered.27 Our decompositions show that aggregate factor misallocation dwarfs occupational labor misallocation in all but one industry.
5.2
Log Wage Variance
Cross-sectional log wage variance contains both within- and between-occupation variation. By iterating expectations, V ar(ln wit |t) = E [V ar (ln wit |h, t) |t] + V ar (E [ln wit |h, t] |t). | {z } | {z } Within-occupation
27
(28)
Between-occupation
The limited impact of ability heterogeneity on labor productivity is consistent with existing empirical results, e.g., Jorgenson, Gollop, and Fraumeni (1987), Irarrazabal, Moxnes, and Ulltveit-Moe (2013), and Fox and Smeets (2011), and with observations in Bartelsman, Haltiwanger, and Scarpetta (2013, footnote 5).
31
Table 9: Log Wage Variance Decomposition
Manufacturing
Wholesale & Retail Trade
Transport, Storage & Communications
Real Estate, Renting & Business Activities
0.099
0.164
0.108
0.253
75% 25%
72% 28%
78% 22%
72% 28%
E [E [V ar (ln wit |h, t) |t]]
0.074
0.118
0.085
0.182
Worker ability Firm-level price of ability Residual
68% 15% 17%
84% 6% 10%
80% 8% 12%
77% 15% 8%
E [V ar (ln wit |t)] Within occupation Between occupation
Note: The decompositions are given by (28) and (29).
Using the log wage equation (17), the within-occupation component is further decomposed as E [V ar (ln wit |h, t) |t] = E [Cov (ln wit , ln aih |h, t) |t] | {z } Worker ability
+ E [Cov (ln wit , ln ϕjht |h, t) |t] + E [Cov (ln wit , ln εw it |h, t) |t], (29) {z } {z } | | Price of ability
Residual
where j = J(i, t) and h = H(i, t). Decomposition (29) reflects that more able workers command higher wages, some firms pay higher prices for ability, and some residual dispersion may remain. Table 9 shows the decompositions (28) and (29) for the four industries we consider, averaged over the available annual cross-sections. From the upper panel in Table 9, within-occupation variation accounts for between 72% and 78% of the variance in log wages, clearly dominating between-occupation variation. From the lower panel in Table 9, the largest contributor to within-occupation log wage variance is ability heterogeneity across workers, accounting for between 68% and 84%. Dispersion in the prices of ability across firms accounts for only 6% to 15% of the within-occupation cross-sectional variance in log wages. Residual variation accounts for the remaining 8% to 17%. While ability heterogeneity contributes only little to labor productivity dispersion (Section 5.1), worker ability co-varies strongly with wages at the individual level. The contrasting variance decompositions for log labor productivity and log wages reinforce the importance of employing both firm level output data and matched employer-employee wage data for a comprehensive understanding of the joint distribution of productivities and wages. Moreover, it highlights the fact that the dominant role of misallocation in generating variance in observed labor productivity does not simply reflect that worker heterogeneity is absent in our data.
32
5.3
Log Labor Productivity-Log Wage Covariance
The final step in our analysis of the joint distribution of labor productivity and wages relates to the covariance between the two variables in a cross-section of workers. We consider the following covariance decomposition, again utilizing that M P Ljt ∝ Cov(ln wit , ln M P L∗jt ):
Yjt L∗jt
Y
such that Cov(ln wit , ln Ljt ∗ ) = jt
! e jt Yjt L Cov ln wit , ln |t = Cov ln wit , ln |t Njt Njt | {z }
Ability heterogeneity
! L∗jt Cov ln wit , ln |t e jt L {z } |
+
! + Cov |
Occupational labor misallocation
|
ln wit , ln M P L∗jt |t {z
, (30) }
Aggregate factor misallocation
{z
Misallocation
}
where, to avoid clutter, we have again used the shorthand j = J(i, t). The first term on the right-hand side of (30) is the portion of covariance between log wages e L
and log labor productivity that can be attributed to firm-level ability heterogeneity, ln Njt . jt Firms that employ more able workers have higher labor productivity, and high ability workers command higher wages. The second term represents the effect of occupational misallocation on the covariance between log wages and log labor productivity. Firms that allocate ability efficiently across occupations have higher labor productivity, and high labor productivity firms pay higher wages. The third term captures the effect of aggregate factor misallocation. Firms with higher marginal product of aggregate ability adjusted labor input, M P L∗jt , have higher labor productivity, and high labor productivity firms pay higher wages. The two last terms combine to the total effect of misallocation. Table 10 tabulates the log labor productivity-log wage covariances, and the percentage shares of the three components in (30). For Manufacturing, we see that most of the covariance, 59%, arises through occupational labor misallocation, with aggregate factor misallocation accounting for 33% and ability heterogeneity for 8% of the log labor productivity-log wage covariance. Occupational labor misallocation is also the most important covariance contributor in Wholesale & Retail Trade. Here, ability heterogeneity accounts for 20% and aggregate factor misallocation only 8% of the log labor productivity-log wage covariance. For the first of the two smaller industries, Transport, Storage & Communications, we similarly find that ability heterogeneity does not contribute much, only 6%, to the log-labor productivity-log wage covariance. From Table 8, in this industry there is negative correlation between log labor productivity and our measure of efficiency in occupational labor allocation, L∗jt e jt . L
Since more productive firms pay higher wages in the data, this manifests itself as a neg-
ative contribution of occupational labor misallocation to the log labor productivity-log wage covariance. Consequently, aggregate factor misallocation comes out as the major contributor to the observed log labor productivity-log wage covariance. In Real Estate, Renting and Business Activities, ability heterogeneity contributes negatively to the covariance (due to the negative correlation between averge ability
e jt L Njt
and labor productivity reported in Table 8). The positive 33
Table 10: Log Labor Productivity-Log Wage Covariance Decomposition
Manufacturing
Wholesale & Retail Trade
Transport, Storage & Communications
Real Estate, Renting & Business Activities
0.020
0.044
0.045
0.096
8% 59% 33%
20% 72% 8%
6% −26% 120%
−33% 78% 55%
h i Y E Cov ln NJ(i,t),t , ln wit |t J(i,t),t
Ability heterogeneity Occupational labor misallocation Aggregate factor misallocation
Note: The decomposition is given by (30). J(i, t) = j if worker i is employed by firm j in period t.
log labor productivity-log wage covariance is thus entirely driven by misallocation, with occupational labor misallocation accounting for about 50% more than aggregate factor misallocation. With some qualitative differences across industries, our decomposition of the joint distribution of log labor productivity and log wages can be summed up as follows. Misallocation is the principal source of labor productivity dispersion and the log labor productivity-log wage covariance, with ability heterogeneity being of modest importance. However, worker ability differences is the primary source of wage variation in the cross-section. Firm heterogeneity, arising due to resource misallocation, also contributes, but substantially less.
6
Ability Flows and Gains from Ability Reallocation
Resource misallocation is rife. This suggests that there are potentially large output gains from reallocating resources, and that this reallocation is not taking place. Reallocation entails adjusting the ratio of capital to aggregate ability adjusted labor, ability adjusted labor input to total ability,
L∗jt e jt ; L
Kjt L∗jt ,
and the ratio of aggregate
see Section 2.4 for details. The former adjust-
ment may be achieved by capital and labor reallocation, the latter only by labor reallocation. In this section, we focus on labor reallocation, as our data are particularly rich in this dimension. First, we document the magnitude and direction of ability adjusted labor flows. Second, we use the estimated model to measure potential gains from efficiency enhancing reallocation, in terms of both output and individual wages.
6.1
Ability Flows
Our model accommodates both between-firm and within-firm, between-occupation ability flows. Between-firm flows are denoted external flows.28 Within-firm, between-occupation flows are denoted internal flows. The overall flows of ability to and from a given occupation in a given firm are denoted total flows. We use superscripts IN , EX, and T O to distinguish between internal, external, and total flows. We extend the labor flow accounting framework of Burgess, Lane, and Stevens (2000) to 28
External flows may be within- or between-occupation.
34
O,H ability adjusted worker flows. Thus, let LTjht be the total amount of ability hired (internally or O,S externally) into occupation h by firm j during period t. Let LTjht be the total amount of ability
separated (internally or externally) from occupation h in firm j. Gross ability flow for firm j, occupation h between t − 1 and t is the sum of hired and separated ability between t − 1 and t, T O = LT O,H + LT O,S . The corresponding net ability flow for firm j, occupation h between LGFjht jht jht T O = LT O,H − LT O,S . Finally, ability reallocation is LRT O = |LN F T O |. t and t − 1 is LN Fjht jht jht jht jht
All the above concepts can be defined separately for internal (superscript IN ) and external (superscript EX) hires and separations.29 We use the average of the ability adjusted labor inputs in time t and time t−1 as denominator when computing the flow rates, see, e.g., Davis and Haltiwanger (1990). Hence, the gross ability T O = LGF T O /L TO TO flow rate is LGF Rjht jht , the net ability flow rate is LN F Rjht = LN Fjht /Ljht , jht T O = LRT O /L and the reallocation flow rate is LRRjht jht , where Ljht = (Ljht +Ljh,t−1 )/2. Internal jht
and external rates are defined analogously. With these concepts in hand we may write TO TO TO LGF Rjht = LRRjht + LCF Rjht ,
(31)
T O is the total ability churning flow rate, i.e., ability flows in excess of what is where LCF Rjht T O . Analogous decompositions can needed to attain the observed ability reallocation rate, LRRjht
be obtained for internal and external ability flows. Table 11 reports the occupation-specific average annual gross ability flow rates, split into average net reallocation rates and average ability churning rates according to (31). We consider both total (column labeled T O), internal (IN ) and external (EX) ability flows. The total annual gross ability flow rate ranges between 0.52 and 1.11 across occupations and industries. If hires and separations balance, these gross flow rates translate into annual worker turnover rates ranging between 26% and 56%. The annual turnover rates for internal ability flows range between 7% and 20% (gross flows between 0.13 and 0.40) and for external flows between 18% and 38% (gross flows between 0.36 and 0.76). That is, internal gross ability flows—the ability flows between occupations within a firm—account for up to a third of total flows. Existing empirical evidence on labor flows considers only external flows and do not ability adjust the flows. Still, the external flow rates in Table 11 are comparable to the non-ability adjusted labor flows for the US reported in Burgess, Lane, and Stevens (2000), and the relatively high internal flow rates corroborate evidence presented in Kramarz, Postel-Vinay, and Robin (2014) for France. In Table 11, with only few exceptions, Manufacturing has the lowest total gross ability flows and Real Estate, Renting & Business Activities the highest. The Unskilled labor market exhibits larger flows than any of the other occupations in all of the four industries. For Manufacturing and Wholesale & Retail Trade, there is only little difference between flow rates in the three higher occupations. In Transport, Storage & Communications and Real Estate, Renting & Business 29 Internal and external hires, separations, and gross and net ability flows add up to their corresponding total O,H O,S TO IN EX TO measures: LTjht = LIN,H + LEX,H , LTjht = LIN,S + LEX,S jht jht jht jht , LGFjht = LGFjht + LGFjht , and LN Fjht = IN EX TO LN Fjht + LN Fjht . The sum of internal and external reallocation weakly exceeds total reallocation: LRjht ≤ IN EX LRjht + LRjht .
35
Activities, the gross ability flow rates rank according to occupation, Managers exhibiting the lowest rates. Ability reallocation rates range between 0.16 and 0.47. The ranking of occupations and industries is similar to the rankings found for the total gross flows, with the exception that the ability reallocation rate among Managers in Real Estate, Renting & Business Activities is the lowest across the four industries. This implies that the total ability churning rates range between 0.34 to 0.64 across occupations and industries. The ranking of the churning rates mirrors the ranking based on total gross ability flows. Putting these numbers together, ability churning accounts for 55% to 69% of the total gross flows.30 That is, more than half, and up to almost 70% of the observed annual ability flows are in excess of the flows needed to generate the observed annual ability reallocation. Ability churning is an empirically relevant phenomenon for both internal and external flows, but is less pronounced for internal flows. Internally, ability churning comprises 27% to 46% of gross ability flows, with the shares for external flows being 52% to 69%. 6.1.1
Ability Flows and Employer Productivity
We now turn to the question of whether ability reallocation is directed toward more productive T O is the total net ability flow rate for firm j, occupation h during firms. Recall that LN F Rjht T O > 0. Let Rank M P L be the period t. A firm is an ability-adjusted net job creator if LN F Rjht jht
percentile rank of firm j in the cross-section t distribution of marginal products across firms, M P L = 0.25 if firm j is at the first quartile in the within occupation h. For example, Rankjht
distribution of marginal products across firms in cross-section t. T O on Rank M P L .31 For each industry, Figure 5 plots nonparametric regressions of LN F Rjht jh,t−1
we plot four regressions, corresponding to each of our four occupations. By the fitted regressions, T O and Rank M P L in all we observe an increasing, close to linear, relationship between LN F Rjht jh,t−1
occupations and in all industries. The least productive firms are shedding ability weighted labor, while the most productive firms, on average, are ability adjusted net job creators. For most industries and occupations, firms turn into ability adjusted net job creators once their (lagged) marginal product of labor is in the top quartile. Taking into account the 95% confidence bands around the regression lines in Figure 5, we see that for the two smaller industries, Transport, Storage & Communications and Real Estate, Renting & Business Activities, the confidence T O = 0.00 for most Rank M P L values. intervals are wide, and include LN F Rjht jh,t−1
The evidence presented in Figure 5 thus suggests that ability flows are, on average, directed toward more productive firms. Nonetheless, misallocation persists (see Table 8), suggesting that labor reallocation from less to more productive firms is not sufficiently strong to equalize marginal products across firms. This is consistent with our finding, reported in Table 11, that more than half of the observed gross ability flows do not contribute to ability reallocation. The evidence in Table 11 and Figure 5 is also corroborated by findings in Jolivet, Postel-Vinay, and Robin (2006), Bagger and Lentz (2014), and Bagger, Fontaine, Postel-Vinay, and Robin (2014). 30
Measured as the ratio of mean rates. Computing the mean of the ratio of churning to gross flows yield similar results. 31 We use the lagged marginal product rank as ability flows directly impact contemporaneous marginal products of labor.
36
37
0.567 0.215 0.352 0.621 0.521 0.162 0.358 0.688 0.923 0.386 0.537 0.582 8,521
Salaried workers Gross ability flow Ability reallocation Ability churning Churning/Gross flow
Skilled workers Gross ability flow Ability reallocation Ability churning Churning/Gross flow
Unskilled workers Gross ability flow Ability reallocation Ability churning Churning/Gross flow
Number of firm-years
8,521
0.395 0.235 0.160 0.405
0.130 0.070 0.060 0.461
0.213 0.120 0.093 0.438
0.177 0.111 0.066 0.372
IN
8,521
0.528 0.228 0.300 0.569
0.390 0.122 0.269 0.688
0.355 0.144 0.211 0.594
0.398 0.177 0.221 0.555
EX
1,907
0.882 0.359 0.523 0.593
0.660 0.209 0.451 0.683
0.603 0.223 0.380 0.631
0.626 0.281 0.345 0.551
TO
1,907
0.285 0.194 0.092 0.321
0.183 0.104 0.079 0.432
0.189 0.112 0.076 0.405
0.192 0.131 0.062 0.321
IN
Wholesale & Retail Trade
1,907
0.597 0.228 0.369 0.618
0.477 0.147 0.331 0.693
0.414 0.156 0.258 0.623
0.434 0.209 0.225 0.519
EX
335
1.026 0.435 0.591 0.576
0.813 0.276 0.538 0.661
0.686 0.260 0.426 0.621
0.665 0.288 0.378 0.567
TO
335
0.345 0.240 0.105 0.304
0.263 0.153 0.110 0.417
0.210 0.131 0.078 0.374
0.189 0.126 0.063 0.333
IN
335
0.681 0.253 0.428 0.629
0.551 0.176 0.375 0.681
0.476 0.165 0.311 0.653
0.476 0.210 0.266 0.559
EX
1,058
1.112 0.473 0.639 0.575
0.904 0.322 0.582 0.644
0.719 0.265 0.454 0.631
0.705 0.229 0.476 0.675
TO
1,058
0.357 0.262 0.095 0.267
0.284 0.183 0.101 0.355
0.235 0.144 0.091 0.387
0.186 0.105 0.080 0.432
IN
1,058
0.755 0.278 0.477 0.632
0.620 0.195 0.425 0.685
0.483 0.175 0.308 0.638
0.519 0.175 0.344 0.662
EX
Real Estate, Renting & Business Activities
Note: Omitting superscripts T O, IN and EX for Total, Internal and External. Gross ability flow is the average gross ability flow rate E[LGF Rjht |h], Ability reallocation is the average ability reallocation rate E[LN RRjht |h], and Ability churning is the average ability churning rate E[LCF Rjht |h]. LGF Rjht , LN RRjht , and LCF Rjht are defined in the text. Churning/Gross flow is E[LCF Rjht |h]/E[LGF Rjht |h].
0.575 0.237 0.338 0.588
Managers Gross ability flow Ability reallocation Ability churning Churning/Gross flow
TO
Manufacturing
Transport, Storage & Communications
Table 11: Annual Ability Flow Rates
Figure 5: Net Ability Flow Rates and Employer MPL Rank
Managers Salaried workers Skilled workers Unskilled workers
0.00
0.25 0.50 0.75 Lagged MPL rank
1.00
Transport, Storage & Communications Annual net ability flow rate −0.3 −0.1 0.1 0.3
Annual net ability flow rate −0.3 −0.1 0.1 0.3
Wholesale & Retail Trade
Managers Salaried workers Skilled workers Unskilled workers
0.00
0.25 0.50 0.75 Lagged MPL rank
Managers Salaried workers Skilled workers Unskilled workers
0.00
0.25 0.50 0.75 Lagged MPL rank
1.00
Real Estate, Renting & Business Activities Annual net ability flow rate −0.3 −0.1 0.1 0.3
Annual net ability flow rate −0.3 −0.1 0.1 0.3
Manufacturing
1.00
Managers Salaried workers Skilled workers Unskilled workers
0.00
0.25 0.50 0.75 Lagged MPL rank
1.00
Note: The shaded areas are 95% confidence intervals.
These papers stress that job-to-job transitions on average reallocate workers to higher rungs on the job ladder, but that not all job-to-job transitions are up the ladder. In other words, job-to-job transitions not directed toward more productive firms are empirically relevant.
6.2
Marginal Output and Wage Gains from Labor Reallocation
The final step in our analysis uses the estimated sharing rules to quantify how the labor market (X)
incentivizes individual workers to seek out productive employers. Let M P Lh
be the Xth
percentile in the employment weighted distribution of marginal products of ability adjusted occupation h labor.32 On the margin, the return to output from reallocating a worker from a firm with marginal product at the Qth percentile to a firm at the P th percentile is (P )
RhM P L (Q, P ) = ln M P Lh
(Q)
− ln M P Lh .
(32)
By the individual wage equation (17), the return to individual wages from reallocating from firm j to firm j 0 is ln φj 0 ht − ln φjht . The sharing rules (24) imply that ln φj 0 ht − ln φjht depends 32
Empirically we use the average cross-sectional percentiles.
38
on marginal products of ability adjusted labor and error terms εφjht . We fix εφjht at its zero (Q)
(P )
expected value. The wage return from reallocating from a M P Lh -firm to a M P Lh -firm is then " Rhw (Q, P ) = ln (1 − βh )bh +
1+
βh 1−βh PH βk αL k=1 γk 1−β k
!
# (P )
M P Lh
" − ln (1 − βh )bh +
1+
βh 1−βh P βk αL H k=1 γk 1−βk
!
# (Q) M P Lh
. (33)
In Table 12 we report the reallocation gains in terms of labor productivity, RhM P L (Q, P ), and wages, Rhw (Q, P ), for inter-quartile mobility in the distribution of marginal products. That is, we take (P, Q) ∈ {(25, 75), (25, 50), (50, 75)}. Consider first the returns to output, (32). Across occupations and industries, the log 75 to 25 percentile ratios range from 0.45 (Unskilled workers in Real Estate, Renting & Business Activities) to 1.69 (Skilled workers, Transport, Storage & Communications). Hence, on the margin, moving a worker from a firm at the 25th percentile to one at the 75th would in general more than double the worker’s productivity. Looking within industries, both the highest and lowest returns are found in the lower occupations (Skilled and Unskilled workers).33 Comparing returns across industries within occupation, we find no discernible pattern. With a few exceptions, the distribution of marginal products is skewed to the left, resulting in higher gains for transitions at the right tail, (Q, P ) = (50, 75), than at the left, (Q, P ) = (25, 50). The estimated marginal productivity gains from labor reallocation are sizable, a manifestation of our finding that misallocation is pervasive. Other recent studies have also found large gains from resource reallocation. For example, Hsieh and Klenow (2009) study misallocation of labor and capital in China and India and find that the two countries would enjoy TFP increases of 30% to 50% and 40% to 60%, respectively, if inputs were reallocated to reduce resource misallocation to the US benchmark. Bartelsman, Haltiwanger, and Scarpetta (2013) find relatively large negative impacts on consumption from resource misallocation in a number of European countries, relative to the US benchmark. What are the wage gains for workers from moving to firms with higher marginal products? The second column for each industry in Table 12 reports the individual wage returns according to (33). These range between 0.02 (Unskilled workers, Real Estate, Renting & Business Activities) and 0.37 (Managers, Manufacturing). Ignoring the lowest return of 0.02, which is somewhat of an outlier, on the margin, a transition from a firm at the 25th percentile in the employmentweighted distribution of marginal products to one at the 75th leads to at least a 9% wage gain for that worker, and typically more, up to 45%, depending on the occupation and industry considered. Across occupations, Managers tend to have the highest wage gains, the exceptional industry being Real Estate, Renting & Business Activities, where Salaried workers have higher gains from reallocation. For Manufacturing and Wholesale & Retail Trade, Skilled workers have the lowest 33 The exceptional industry is Transport, Storage & Communications where salaried workers have the lowest returns.
39
returns, for Transport, Storage & Communications, Salaried workers have the lowest returns, while, as already mentioned, in Real Estate, Renting & Business Activities, Unskilled workers face very low wage gains from reallocation. Looking across industries within occupation, we see that the wage returns are largest in Manufacturing, and lowest in Real Estate, Renting & Business Activities, except for Salaried workers where the lowest returns are found Transport, Storage & Communications. On the margin, wage returns are substantial.34 Of course, comparison with existing estimates of the return to labor mobility should be made with caution. We estimate the marginal return to reallocation of a worker from a firm at the 25th percentile to one at the 75th percentile in the employment-weighted distribution of ability adjusted labor productivity. The existing empirical studies of the wage return to job-to-job transitions often reports the average realized wage return across all observed job-to-job transitions, see e.g., Topel and Ward (1992) and Keith and McWilliams (1999).35 Bjelland, Fallick, Haltiwanger, and McEntarfer (2011) stand out in this literature in that they report summary statistics on the distribution of earnings changes following a job-to-job transition. Using US data, they find that workers making job-to-job transitions experience a median earnings change between 5% and 9%, but also that the 75th percentile is 37%. Hence, the distribution of earnings changes associated with job-to-job transitions includes most of the marginal returns reported in Table 12. In this light, our estimates of the marginal wage gains are not out of line with findings in the empirical literature. We can put the magnitude of the wage returns in Table 12 further in perspective by comparing them to estimates of the average returns to education. The caveat regarding comparison between the return to a specific marginal reallocation and average returns of course applies here, as well. Nonetheless, Badescu, D’Hombres, and Villalba (2011) report a wage gap in Denmark of 28% between workers with primary and tertiary education, with 7% between primary and secondary, and 21% between secondary and tertiary. Hence, at the lower end, the individual wage gain from reallocation, as reported in Table 12, is of similar magnitude to the average wage return from attaining secondary education (in Denmark). At the higher end, the wage gain from reallocation exceeds that from attaining tertiary education. In summary, on the margin, the gains to labor reallocation are substantial in terms of output, indicating that the labor market frictions preventing this reallocation from taking place are associated with a significant output loss. Private wage gains from reallocation are also high. The labor market does provide strong monetary incentives for workers to seek out employment in more productive firms. The fact that misallocation persists under these circumstances suggests that labor market frictions are pervasive, and that the barriers to mobility that they represent are not easily overcome. 34
It is possible that the Stole and Zwiebel (1996) bargaining protocol exaggerates the link between labor productivity and wages, thus overstating the potential wage gains relative to true incentives. Shimer (2005) points out that Nash bargaining overemphasizes the labor productivity-wage link. Still, the magnitude of the estimated wage gains suggests that considerable incentives to reallocate are in place. 35 One might speculate that average returns exceed marginal returns, but the relation between the two is clouded by job-to-job transitions that are not directed toward more productive employers, an empirically important phenomenon, see Section 6.1.1.
40
41
0.629 0.343 0.285
0.542 0.213 0.329
1.424 0.636 0.789
Salaried workers Transition: (Q, P ) = (25, 75) Transition: (Q, P ) = (25, 50) Transition: (Q, P ) = (50, 75)
Skilled workers Transition: (Q, P ) = (25, 75) Transition: (Q, P ) = (25, 50) Transition: (Q, P ) = (50, 75)
Unskilled workers Transition: (Q, P ) = (25, 75) Transition: (Q, P ) = (25, 50) Transition: (Q, P ) = (50, 75) 0.350 0.111 0.238
0.249 0.089 0.160
0.342 0.175 0.168
0.371 0.144 0.227
Rhw
1.049 0.495 0.553
0.701 0.196 0.505
0.903 0.352 0.551
0.916 0.320 0.596
RhM P L
0.205 0.076 0.130
0.121 0.027 0.094
0.286 0.091 0.195
0.370 0.107 0.263
Rhw
1.328 0.178 1.150
1.688 0.256 1.432
0.615 0.229 0.386
0.727 0.315 0.412
RhM P L
0.302 0.024 0.277
0.222 0.016 0.206
0.093 0.029 0.064
0.361 0.140 0.221
Rhw
Transport, Storage & Communications
0.448 0.147 0.301
1.293 0.565 0.728
0.986 0.521 0.464
0.730 0.235 0.496
RhM P L
0.016 0.004 0.011
0.098 0.029 0.069
0.224 0.097 0.126
0.087 0.022 0.065
Rhw
Real Estate, Renting & Business Activities
Note: RhM P L = RhM P L (Q, P ) is given by (32). Rhw = Rhw (Q, P ) is given by (33). Computations are based on the production function estimates reported in columns labeled (2) in Table 6, and in Table 7.
0.839 0.374 0.466
Managers Transition: (Q, P ) = (25, 75) Transition: (Q, P ) = (25, 50) Transition: (Q, P ) = (50, 75)
RhM P L
Manufacturing
Wholesale & Retail Trade
Table 12: Output and Wage Gains from Labor Reallocation
7
Conclusion
Labor productivity and wages exhibit cross-sectional dispersion and are positively correlated. We develop a model of wage bargaining with heterogeneous firms and workers of different ability. The joint distribution of labor productivity and wages may reflect ability heterogeneity, or resource misallocation—dispersion of marginal products of ability adjusted labor across firms— and rent sharing, depending on the model parameters. The model is estimated on Danish matched employer-employee data for four major industries. Resource misallocation is the dominant source of cross-sectional variance in log labor productivity and the log labor productivity-log wage covariance. At the same time, workers are heterogenous, with ability differences the most important determinant of cross-sectional log wage variance. The presence of resource misallocation suggests that there are output gains from labor reallocation. These gains are not realized because ability flows are either too low, or not sufficiently directed toward the most productive employers. We document substantial gross ability flows in our data, but also that more than half of these flows are comprised of churning, i.e., ability flows in excess of those needed to generate the observed ability reallocation. Firms with higher ability adjusted marginal products of labor are more likely to be ability adjusted net job creators. However, the pull by these more productive firms is evidently not strong enough to prevent substantial remaining misallocation. On the margin, the estimated model predicts large potential gains associated with labor reallocation, both in terms of output and wages. Indeed, the labor market offers strong monetary incentives for workers to seek out employment in more productive firms. The fact that misallocation persists under these circumstances suggests that labor market frictions are pervasive and therefore represent substantial obstacles to growth.
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