Identifying the Nature of Bargaining between Workers and Large Firms William B. Hawkins∗ University of Rochester October 25, 2011

Abstract Can labor market policy improve welfare? Answering this question requires determining whether an observed allocation is constrained efficient. When firms with decreasing returns to labor employ workers in a frictional labor market, the equilibrium is constrained inefficient under the usual model of wage bargaining: firms hire too many workers. However, an econometrician who observes model-generated data on wages and on worker and job flows cannot tell whether they arise from this inefficient allocation under bargained wages or from the constrained efficient allocation in an alternative environment that differs in the average level of labor productivity. Data on firm profits, often not available in macroeconomic studies of the labor market, are required to distinguish the two models. JEL Codes: E24, J41, J64.

∗ Addresses: Harkness 232, Department of Economics, University of Rochester, Box 270156, Rochester, NY 146270156; [email protected]. An early version of results from this paper were originally included in a draft paper with the title ‘Privately Efficient Bargaining between Workers and Large Firms.’ Other results from that draft are now included in the paper ‘Bargaining with Commitment between Workers and Large Firms.’ I thank Mark Aguiar, Mark Bils, Leo Kaas, Philipp Kircher, Ryan Michaels, Giuseppe Moscarini, Moritz Ritter, and Ija Trapeznikova for very useful discussions and seminar participants at the Federal Reserve Bank of Richmond, Rochester, and Yale for helpful comments on the earlier draft. The fault for the remaining imperfections is, of course, my own.

1

Introduction

Perhaps the most important goal of structural modeling in economics is to be able to predict the effects of proposed policy changes and understand whether they will improve welfare. In the presence of externalities or trading frictions, the first welfare theorem may not hold, so that there can be room for intervention. A social planner, even one bound by the same technological and informational constraints as the agents in the economy, could instruct them to alter their behavior so as to generate a Pareto improvement. However, knowing whether this situation obtains requires an econometrician to be able to determine whether or not the data she observes were generated by an environment in which equilibrium is constrained inefficient. In this paper I study an important economic setting in which this general identification problem arises, and show that the usual data used to calibrate the standard models used to study this environment are not sufficient to solve the problem. My results thus serve as a warning for those using these and similar models in applied settings. The environment I focus on is one in which firms operate a decreasing returns to labor technology and hire workers in a frictional labor market. This setting is of great recent interest because of increased availability of microdata on the cross-sectional and time-series patterns of key labor market variables such as employment and wages across firms, provided by the U.S. Bureau of Labor Statistics and other statistical agencies. This has allowed study of models that account simultaneously for the joint dynamics of firm size and of employment and unemployment over the business cycle (Cooper, Haltiwanger, and Willis, 2007; Elsby and Michaels, 2011; Fujita and Nakajima, 2009; Hawkins, 2011b; Kaas and Kircher, 2011; Schaal, 2010). In addition, how firms interact with workers in a frictional labor market is important for the study of such disparate topics as the labor market effects of changes in trade patterns (Co¸sar, Guner, and Tybout, 2010; Helpman and Itskhoki, 2010) or the relationships between contracting and technological adoption (Acemoglu, Antr` as, and Helpman, 2007) or between product market and labor market regulation (Felbermayr and Prat, 2007; Delacroix and Samaniego, 2009; Ebell and Haefke, 2009; Maury and Tripier, 2011). The presence of frictions in labor markets often allows for a potential role for welfare-enhancing policies because of several kinds of externalities, and this also applies in the case I study here. When search is random, as in the tradition of Diamond (1981, 1982), Mortensen (1982), and Pissarides (1985), additional search by an unemployed worker generates a congestion externality which reduces the returns to search for other workers, along with a thick-market externality which increases the returns to search for the vacant jobs on the other side of the market. Only when the matching function exhibits constant returns jointly in the measure of searching workers and jobs and when the bargaining power of workers takes a particular value (Hosios, 1990) can the equilibrium be constrained efficient. All these considerations are already present in the simpler environment in which either each firm has only one job, or, equivalently for the aggregate labor market, operates a constant-returns technology, and they are also present in the model I study. However, these externalities are not the focus of the current paper; all my results apply to the case in which the 1

Hosios condition holds. The inefficiency I study here arises when the technology operated by firms exhibits decreasing returns to labor. Under the standard model of wage determination in this environment (Bertola and Caballero, 1994; Stole and Zwiebel, 1996a,b; Smith, 1999; Cahuc and Wasmer, 2001), firms continuously bargain over wages with each of their workers, treating each as marginal. An assumption of what Stole and Zwiebel (1996a, p. 377) term ‘extreme employment-at-will’ is made, so that firms are unable to commit to long-term contracts with their workers. Because bargained wages are closely related to the marginal product of labor at the firm, and because this marginal product decreases in the firm’s employment, hiring an additional worker drives down the wage paid not only to that worker, but to all workers previously hired. This gives an additional private incentive for hiring to the firm beyond what is efficient; accordingly, firms hire inefficiently many workers and welfare is reduced. This inefficiency is present even under the Hosios condition that would guarantee constrained efficiency of equilibrium in the case of a constant returns technology. Because hiring is excessive, policies that reduce firms’ incentives to hire can be welfare-enhancing.1 The inefficiency arises because of lack of commitment: under the alternative assumption that firms can commit to long-term contracts at the time of hiring, as in Hawkins (2011a), equilibrium is constrained efficient. Now consider two economies in each of which firms with decreasing returns to labor hire workers in a frictional labor market. The two economies differ in two respects. In economy A, wages are determined by continuous bargaining without commitment, following Stole and Zwiebel (1996a), while in economy B, at hiring, firms and workers sign long-term contracts which divide the joint surplus associated with their match, as in Hawkins (2011a). In addition, labor productivity of all jobs is higher in economy A than in economy B. Suppose that an econometrician observes data generated by one of these two economies. The main result of the paper, Theorem 1, is that (under natural assumptions), the econometrician will be unable to identify by which economy the data were generated, even if she observes all the information usually used to calibrate search and matching models of the aggregate labor market, such as employment and unemployment, vacancy posting, wages, and even the firm size distribution. Telling the difference between the two possible economies requires the econometrician to observe productivity directly, or equivalently to observe firm profits or the shares of labor and capital in its factor payments. The fact that such information is usually not used to calibrate standard macroeconomic models of the labor market suggests caution in using these models, so calibrated, to generate policy advice. The structure of the remainder of the paper is as follows. In Section 2, I introduce the economic environment, and in Section 3, I characterize equilibrium under the two alternative wage determination protocols. Section 4 is devoted to establishing the main result of the paper. After a discussion in Section 5, Section 6 concludes briefly. 1

For example, appropriately-rebated taxes on firm-level employment will work.

2

2

Environment

Time is discrete, t = 0, 1, . . . . There are two types of agents, workers and firms. All agents are risk-neutral and discount the future with discount factor β. The measure of workers in the economy is fixed at one; each worker can at any time be either employed by some firm or unemployed. There is a single good. Each period consists of four subperiods, in chronological order devoted to firm entry, firing, hiring, and production. At the beginning of the first subperiod, a continuum of measure ν > 0 of potential entrant firms is created. Each potential entrant firm can choose to pay an entry cost k > 0 in order to become active. During the second subperiod, incumbent firms with positive employment n can choose to fire any or all of their employees at no cost. In the third subperiod, each active firm chooses a measure v of vacancies to open, at cost γv units of the good. There is an aggregate matching function M (·) which determines the total measure of workers hired by firms as a function of u, the unemployment rate, and v¯, the total measure of vacancies posted by all active firms. I assume that M (u, v¯) = Zuψ v¯1−ψ is Cobb-Douglas, as suggested by Petrongolo and Pissarides (2001).2 Write θ = v¯/u for labor market tightness, that is, the vacancy-unemployment ratio, and define q(θ) = M (θ−1 , 1) and f (θ) = θq(θ) = M (1, θ). The measure of workers hired by a firm posting v vacancies is vM (u, v¯)/¯ v = vq(θ), and the probability with which an unemployed worker is hired is f (θ). In the fourth and final subperiod, production occurs and wages are paid. Each active firm operates a Cobb-Douglas production technology; if the firm employs n workers, then its output is y(n) = Apπnα units of the good. A and p both measure aggregate productivity, and π measures the firm’s idiosyncratic productivity. A is constant, while p ∈ {z1 , . . . , zmp } has expected value equal to one, takes finitely many possible values, and follows a first-order Markov process; the probability that the period-t + 1 shock takes the value p0 given that the period-t shock was p is denoted λ(p, p0 ). Similarly, π ∈ {ζ1 , . . . , ζmπ } has expected value one, takes finitely many possible values, and follows a first-order Markov process with transition function µ(π, π 0 ). The idiosyncratic productivity of a new entrant firm is drawn from some initial distribution Fπe after the entry cost is paid. I write pt = (p0 , p1 , . . . , pt ) for the history of aggregate shocks through period t, and πjt = (πj0 , πj1 , . . . , πjt ) for the history of idiosyncratic productivity shocks of a particular firm j, with the convention that if firm j is not active in period τ , then πjτ = 0. During this subperiod, the firm makes wage payments to its workers; I discuss wage determination in more detail in Section 3 below. A worker’s consumption is equal to her wage (if employed) or to an exogenous unemployment income b derived from home production (if unemployed). Finally, at the end of the fourth subperiod, any active firm may be destroyed; this occurs with probability δ, independently across firms and over time. There is no scrapping value. The workers previously employed by the firm begin the following period unemployed. 2

Throughout, I assume that parameters are such that M (u, v¯) ≤ min{u, v¯}.

3

3

Wage Determination and Equilibrium

I consider two models of wage determination. The first is due to Bertola and Caballero (1994), Stole and Zwiebel (1996a,b), and Smith (1999), and arises when firms are unable to commit to any form of long-term contracts. It is closely related to that studied in Section 3 of Hawkins (2011b). The second is due to Hawkins (2011a) and characterizes the opposite extreme of full commitment; in the environment of this paper it also decentralizes the constrained efficient allocation (at least when the bargaining power of workers satisfies the Hosios condition).

3.1

Lack of Commitment

I first characterize equilibrium when firms cannot commit to long-term employment contracts. I consider only equilibria in which the history of aggregate productivity shocks pt is a sufficient statistic for the current aggregate state of the economy (in particular, this implies that firm entry, hiring, and firing, together with aggregate wage payments to workers, are determined by pt alone). Firms and workers are subject to idiosyncratic shocks, both productivity and destruction shocks in the case of firms and the random process of hiring and firing in the case of workers, so that pt cannot be a sufficient statistic for the status of a particular firm or worker. However, it is sufficient to describe labor market aggregates such as the unemployment rate and the firm size distribution, and therefore to characterize the hiring, firing, and wage payments of an individual firm, which allows characterizing the equilibrium. Denote by JˆS (n; pt , π t ) the value of a firm which has n employees in period t after history pt before the labor market operates (that is, before the beginning of the second subperiod). Denote by J S (n; pt , π t ) the corresponding value after the labor market operates, but before production and wage payments occur (that is, at the end of the second subperiod). Then JˆS (n; pt , π t ) = max

0≤v 0≤f ≤n



−γv + J S (n − f + q(θ(pt ))v; pt , π t )



J S (n; pt , π t ) = Apt πt nα − nwS (n; pt , π t ) + β(1 − δ)Et JˆS (n; pt+1 , π t+1 ).

(1) (2)

The firm can fire any nonnegative number of workers up to the number of employees it has at the beginning of the period; it can post arbitrarily many vacancies at constant marginal cost. (Of course, it will not be optimal for any firm simultaneously to fire workers and post vacancies.) During the production phase of period t, the firm cannot adjust employment. ˆ S (pt ) the value of an unemployed worker in period t after history pt before the Denote by U labor market operates (that is, before the beginning of the second subperiod), and by U S (pt ) the corresponding value after the labor market has concluded (that is, after the second subperiod). Finally, denote by Vˆ S (n; pt , π t ) the value of a worker who enters period t after aggregate history pt employed at a continuing firm with n employees and idiosyncratic productivity shock history π t , and by V S (n; pt , π t ) the value of a worker employed at such a firm after the hiring phase of period

4

t is concluded. The values of the unemployed worker satisfy ˆ S (pt ) = f (θ(pt ))Et V S (n; pt , π t ) + (1 − f (θ(pt )))U S (pt ) U S

t

ˆS

U (p ) = b + βEt U (p

t+1

),

(3) (4)

where the expectation in (3) is taken with respect to the measure of vacancies posted by firms of different pre-existing employment n and idiosyncratic productivity histories π t , while that in (4) is taken with respect to the possible realizations of pt+1 conditional on pt . The value function for employed workers during the production phase of the period satisfies ˆ S (pt+1 ). V S (n; pt , π t ) = wS (n; pt , π t ) + β(1 − δ)Et Vˆ S (n; pt+1 , π t+1 ) + βδEt U

(5)

The first expectation in (5) is taken with respect to pt+1 and πt+1 , while the second is taken only with respect to pt+1 . I next describe the process of wage bargaining. I follow Bertola and Caballero (1994), Stole and Zwiebel (1996a,b), and Smith (1999) in assuming that firms bargain with each of their employees, treating each worker as marginal. The bargaining takes place while labor markets are still open during the hiring phase of period t, and determines the wage to be paid during the production phase of the same period.3 Given this timing assumption, a worker’s surplus from the employment relationship is ˆ S (pt ), W S (n; pt , π t ) = V S (n; pt , π t ) − U

(6)

since the appropriate outside option in bargaining is to return to the labor market and again search for a job. The firm’s surplus from the employment relationship is simply JˆnS (n; pt , π t ) = ∂ ˆS t t ∂n J (n; p , π ).

I assume that wages are bargained so that these marginal surpluses are related by

a constant ratio, that is, ηJnS (n; pt , π t ) = (1 − η)W S (n; pt , π t ).

(7)

This can be written as the solution to a Nash maximization problem in the usual way. The first implication of this bargaining assumption, together with the assumptions that hiring and firing proceed at constant marginal cost, is that the value of a worker employed at a firm does not change during the hiring and firing phase of period t. That is, Vˆ S (n; pt , π t ) = V S (˜ n(n; pt , π t ); pt , π t ).

(8)

Here the function n ˜ (n, pt , π t ) in (8) denotes the optimal choice of employment after hiring and firing of the firm that begins period t with n employees, as in (1). (8) is clear in the case of firms that neither hire nor fire. In a firm that hires a positive measure of workers, the firm’s optimal hiring decision in (1) ensures that the marginal value of an additional worker to the firm is γ/q(θ(pt )) 3

This timing assumption, while slightly artificial, is convenient in this discrete-time version of the model to generate a closed-form solution for wages. In a continuous-time version of the model, the outside option of the worker can be assumed to be simply the value of returning to unemployment contemporaneously.

5

independently of the initial value of employment n (provided that n is such that the firm wishes to hire at all). Intuitively, the firm’s hiring dynamics have a ‘bang-bang’ flavor: there is a target hiring level n∗ (pt , π t ), and if n < n∗ (pt , π t ), the firm immediately posts (n∗ (pt , π t ) − n)/q(θ(pt )) vacancies so as to hire n∗ (pt , π t ) − n workers. Therefore, the value of each additional incumbent worket to the firm in this range of employment is simply a saving of γ/q(θ(pt )) in foregone vacancy posting. According to the bargaining equation (7), this then ensures that the worker’s value at such a firm is given by ˆ S (pt ) + V S (n; pt , π t ) = U

η γ . 1 − η q(θ(pt ))

(9)

Firms with too many workers fire some. There is a target firing level n ¯ (pt , π t ) > n∗ (pt , π t ), so that any firm with employment n > n ¯ (pt , π t ) immediately fires n − n ¯ (pt , π t ) of these. Since the marginal firing cost is zero, the marginal value of a worker at a firm which fires a positive measure of workers is also zero. According to (7), the value of any worker who is retained is then just ˆ S (pt ), which is the same as the value of any worker who is fired. Using (6) and (8) in (5) then U establishes that W S (n; pt , π t ) = wS (n) − xS (pt ) + β(1 − δ)Et W S (˜ n(n; pt+1 , π t+1 ); pt+1 , π t+1 )

(10)

ˆ S (pt ) − βEt U ˆ S (pt+1 ) xS (pt ) = U

(11)

where

can be interpreted as the rental value of an unemployed worker during period t, taking into account the expected value of the jobs the worker may find, along with the unemployment income he receives in the event his search is unsuccessful. A final possibility is that a firm neither fires nor hires. This arises when the firm enters period t with an incumbent workforce n which lies between the hiring target n∗ (pt , π t ) and the firing target n ¯ (pt , π t ). This range of inaction arises because of the partial irreversibility of hiring: the marginal cost of adjusting employment downwards is zero since there is no firing cost, while hiring is costly because of the labor market friction. The fact that the marginal value of employment is constant across all firms that hire workers allows a simple expression for x(pt ) to be given. Substitute from (9) into the Bellman equation for an unemployed worker (3) and rearrange to observe that xS (pt ) = b +

ηγθ(pt ) . (1 − η)(1 − f (θ(pt )))

(12)

With this simplification, it is now straightforward to obtain a closed-form expression for wages. Using the wage bargaining equation (7) together with the Bellman equations for the firm (2) and for the worker (10) and simplifying using (12), a standard argument (for example, Lemma 1 of Hawkins, 2011b) establishes that there is a unique solution for wages which satisfies the boundary

6

condition that the wage bill for a small firm is finite. This solution is given by S

t

t

S

t

w (n; p , π ) = (1 − η)x (p ) + Apt πt n

− η1

Z

n

ν

1−η η

αν α−1 dν

0

αη ηγθ(pt ) + Apt πt nα−1 . = (1 − η)b + t 1 − f (θ(p )) αη + 1 − η

(13)

(13) writes wages as a sum of three terms, the worker’s unemployment income b, a term that increases in γ and in θ(pt ) and therefore measures the severity of labor market frictions, and a weighted average of all the inframarginal products of workers at employment levels from 0 to n. For the Cobb-Douglas case, this last weighted average can itself be written as a constant multiple of the marginal product of labor at the firm. (13) establishes that the wages paid by a firm do not depend directly on the history of aggregate or idiosyncratic productivity shocks in periods before t (although it is possible that the value of labor market tightness θ(pt ) may depend on the history of aggregate productivity). Nor do they depend per se on whether the firm hired, fired, or did not change its employment level this period. Because wages do not depend on the history of idiosyncratic productivity, neither does the value of the firm and likewise neither do the values of its employees. This allows me to write the values in (1) through (12) as depending only on current idiosyncratic productivity πt rather than on the history π t . It also allows me to use a consistent notation for incumbent and new entrant firms. The two state variables of the model are the firm size distribution and the unemployment rate. The evolution of the firm size distribution must follow from the firing and recruitment decisions of firms. Denote the measure of firms with employment at most n and idiosyncratic productivity π during the production phase of period t by G(n; pt , π); then G(n; pt , π) = (1 − δ)

mπ X j=1

Z µ(ζj , π) T (n;pt ,π)

dG(ν; pt−1 , π − ) + ne (pt )Fπe (π).

(14)

 Here T (n; pt , π) = ν | n ≥ n ˜ (ν; pt , π) is the set of values of the workforce at the beginning of period t that lead an incumbent firm to hire and fire so as to result in employment at most n during the production phase of period t. ne (pt ) denotes the measure of entering firms during period t. The law of motion for the unemployment rate (measured during the production phase of period t) is u(pt ) = u(pt−1 )(1 − f (θ(pt−1 ))) + δ(1 − u(pt−1 )(1 − f (θ(pt−1 )))) mπ X mπ Z X f (n; pt , ζj )µ(ζi , ζj ) dG(n; pt−1 , ζi ). + (1 − δ)

(15)

i=1 j=1

Unemployed workers arise from three sources, those who were unemployed in the previous period and failed to find a job, those who were employed during the previous period but whose firm was exogenously destroyed, and those who were fired from the production sector at the beginning of the current period. (Here f (n; pt , π) denotes the optimal firing policy of a firm that enters period

7

t with n incumbent employees and receives the new idiosyncratic productivity shock π.) To close the model, market clearing and free entry conditions must be imposed. Market clearing takes the form of a consistency condition that the market tightness θ(pt ) should equal the ratio of vacancies to unemployment that is consistent with the hiring decisions of firms and with the laws of motion for the firm size distribution and the unemployment rate: θ(pt )u(pt ) = (1 − δ)

mπ X

Z µ(ζj , π)

v(n; pt , π) dG(ν; pt−1 , ζj ) + ne (pt ),

(16)

j=1

where v(n; pt , π) is the optimal vacancy-posting policy of the firm in (1). Finally, I assume that new potential entrant firms are born in sufficient measure each period that the value of entry net of the entry cost is driven to zero: mπ X

Fπe (ζj )JˆS (0; pt , ζj ) ≤ k

and

ne (pt ) ≥ 0

(17)

j=1

with complementary slackness. It is now straightforward to define equilibrium. Definition 1. An equilibrium under lack of commitment is a set of (aggregate productivity historydependent) sequences for unemployment u(pt ), market tightness θ(pt ), entry by new firms ne (pt ), and the firm size distribution G(n; pt , π), together with a set of (aggregate productivity historydependent and contemporaneous idiosyncratic productivity-dependent) sequences for target employment n ˜ (n; pt , π) and wages paid by firms w(n; pt , π), such that • wages satisfy (13); • firing and vacancy posting are chosen to maximize profits, so that f (n; pt , π) and v(n; pt , π) maximize the right side of (1) with f (n; pt , π) = min{0, n−˜ n(n; pt , π)} and q(θ(pt ))v(n; pt , π) = min{0, n ˜ (n; pt , π)}; • the free entry condition (17) holds with complementary slackness; • unemployment evolves according to (15); • individual firm sizes evolve consistently with target employment , and the firm size distribution evolves according to (14); and • θ(pt ) is consistent with the recruitment activities of firms and the number of unemployed workers, according to (16). I characterize the structure of equilibria under this assumption on wage dispersion more fully in Hawkins (2011b), and show that if aggregate shocks are small and not too frequent4 then an 4

The precise condition is that the transition matrix µ(·, ·) for the aggregate productivity shock should be strictly diagonally dominant.

8

equilibrium exists, in fact one of a very particular form in which labor market tightness depends only on current aggregate productivity pt , and not on other components of the aggregate history pt . The reader is referred to that paper for further details of the argument.

3.2

Commitment

In Hawkins (2011a), I introduced an alternative wage determination structure which I termed ‘bargaining with commitment.’ I assumed that firms and workers are able to sign a contract at the time of first meeting which determines the timing and structure of future payments and of the continuation of the employment contract. In this section I show how to introduce this wage determination concept in the environment of this paper, and define equilibrium when wages are so determined. Under full commitment, the timing of wage payments to an individual worker within a contract is indeterminate. It is simplest in terms of notation to assume that on meeting, firms pay workers a bonus h(n; pt , π t ) which could depend on the history of the aggregate economy as well as on the firm’s history of idiosyncratic productivity shocks. After the initial bonus, firms pay workers a value xE (pt ) which leaves them indifferent between continued employment at the firm and unemployment. The value of the bonus h(n; pt , π t ) will be determined by bargaining, to be described further below. With this notation, the value functions of a firm satisfy ( ˆE

t

t

J (n; p , π ) = max

0≤v 0≤f ≤n

E

t

t

t

Z

)

n−f +q(θ(pt ))v

−γv + J (n − f + q(θ(p ))v; p , π ) −

t

t

h(ν; p , π ) dν

(18)

n−f

J E (n; pt , π t ) = Apt πt nα − nxE (pt ) + β(1 − δ)Et JˆE (n; pt+1 , π t+1 ).

(19)

Here JˆE (n; pt , π t ) denotes the value of a firm before firing and hiring in period t, and J E (n; pt , π t ) denotes the value during the production phase of period t. (18) differs from the corresponding equation under lack of commitment, (1), in that the firm must pay the hiring bonus h(ν; pt , π t ) to each worker it hires. (19) differs from (2) in that the wage the firm pays to incumbent workers is given by xE (pt ). I can analogously write equations for the values of unemployed and employed workers. Denote ˆ E (pt ) the value of an unemployed worker in period t before the labor market operates, and by U by U E (pt ) the value of an unemployed worker during the production phase of period t. Denote by Vˆ E (n; pt , π t ) the value of a worker who begins period t employed at a firm with n employees and idiosyncratic productivity shock history π t , and by V E (n; pt , π t ) the value of a worker employed at such a firm during the production phase of period t. I assumed that the payment made by firms to continuing workers was such that these workers were indifferent between continued employment and unemployment. This implies, analogously to (11), that ˆ E (pt ) − βEt U ˆ E (pt+1 ) xE (pt ) = U

9

(20)

ˆ E (pt ) and is the rental value of an unemployed worker. It also implies that Vˆ E (n; pt , π t ) = U ˆ E (pt ) for all n, pt , and π t . Finally, analogously to (3) and (4), the values U ˆ E (pt ) V E (n; pt , π t ) = U and U E (pt ) satisfy h i ˆ E (pt ) = f (θ(pt ))Et h(n; pt , π t ) + U ˆ E (pt ) + (1 − f (θ(pt ))U E (pt ) U

(21)

ˆ E (pt+1 ). U E (pt ) = b + βEt U

(22)

The expectation in (21), analogously to that in (3), is taken with respect to the measure of vacancies posted by firms of different pre-existing employment n and idiosyncratic productivity histories π t , and that in (22), analogously to that in (4), is taken with respect to the possible realizations of pt+1 conditional on pt . I can immediately rearrange (21) and (22) to deduce that xE (pt ) = b +

f (θ(pt )) Eh(n; pt , π t ). 1 − f (θ(pt ))

(23)

To complete the discussion of wage determination, I assume that the hiring bonus paid to a worker newly hired by a firm with n incumbent employees satisfies the following surplus-sharing equation.5 η JˆnE (n; pt , π t ) = h(n; pt , π t ).

(24)

To see the implications of (24) for the value of the hiring bonus, note that as in the no commitment case, the firm’s hiring dynamics take a ‘bang-bang’ form: the firm hires until JˆnE (n; pt , π t ) = JnE (n; pt , π t ) = γ/q(θ(pt ))+h(n; pt , π t ), or, equivalently using (24), until JˆE (n; pt , π t ) =

γ (1−η)q(θ(pt )) .

Because of the bang-bang hiring dynamics, the value function of the firm is linear for lower employment levels: having an additional incumbent worker saves the firm the vacancy-posting cost together with the hiring bonus. It follows that h(n; pt , π t ) =

η γ 1 − η q(θ(pt ))

(25)

at all firms that hire a positive measure of workers. Thus (23) can be simplified as xE (pt ) = b +

ηγθ(pt ) . (1 − η)(1 − f (θ(pt ))

(26)

This equation takes the same form as (12). I can also simply the expression for the firm’s optimal 5 (24) does not appear to be comparable to (7) since the bonus on the right side is not multiplied by 1 − η. In fact, the equations both indicate that the surplus associated with the match should be split between worker and firm in ratio η : (1 − η); the difference in form arises from the fact that JˆE (n; pt , π t ) denotes the value of a firm that bonus h(n; pt , π t ) in the past when it hired its nth worker. Rewriting (24) as i h has already paid the hiring E t t t t η Jˆn (n; p , π ) − h(n; p , π ) = (1 − η)h(n; pt , π t ) makes the relation between the two equations more transparent.

10

choice of hiring and firing in (18) using (25) to yield that JˆE (n; pt , π t ) = max

0≤v 0≤f ≤n

  γv + J E (n − f + q(θ(pt ))v; pt , π t ) . − 1−η

(27)

It is also apparent that the history of the firm’s idiosyncratic productivity π t−1 is not payoff-relevant, so that the value functions of firms and workers can be written in terms only of current employment n and current idiosyncratic productivity πt , as well as of the aggregate history pt . I abuse notation and drop the dependence of the value functions on π t−1 while preserving the notation JˆE (n; pt , πt ), and J E (n; pt , π). To complete the definition of an equilibrium, I need to assume that the two state variables of the model, namely the firm size distribution and the unemployment rate, follow the appropriate laws of motion. As in the case of no commitment, denote by G(n; pt , π) the measure of firms with idiosyncratic productivity π in history pt with employment no greater than n, and by u(pt ) the unemployment rate during the production phase of period t. Then (14) and (15) still describe the evolution of the two state variables, provided that the optimal hiring and firing policies of the firm are reinterpreted as those optimal in (27) rather than in (1). Likewise, market clearing still requires that (16) hold. The free entry condition takes a form closely analogous to (17), namely that mπ X

Fπe (ζj )JˆE (0; pt , ζj ) ≤ k

and

ne (pt ) ≥ 0

(28)

j=1

with complementary slackness. I can then define equilibrium analogously to Definition 1. Definition 2. An equilibrium under commitment is a set of (aggregate productivity historydependent) sequences for unemployment u(pt ), market tightness θ(pt ), entry by new firms ne (pt ), and the distribution of firms by employment and idiosyncratic productivity G(n; pt , π), together with a set of (aggregate productivity history-dependent and contemporaneous idiosyncratic productivitydependent) sequences for target employment n ˜ (n; pt , π), hiring bonuses h(n; pt , π), and wages for incumbent workers xE (pt ), such that • hiring bonuses satisfy (25) and incumbent wages satisfy (26); • firing and vacancy posting are chosen to maximize profits, so that f (n; pt , π) and v(n; pt , π) maximize the right side of (27) where f (n; pt , π) = min{0, n−˜ n(n; pt , π)} and q(θ(pt ))v(n; pt , π) = min{0, n ˜ (n; pt , π)}; • the free entry condition (28) holds with complementary slackness; • unemployment evolves according to (15); • individual firm sizes evolve consistently with target employment and the firm size distribution evolves according to (14); and 11

• θ(pt ) is consistent with the recruitment activities of firms and the number of unemployed workers, according to (16). The existence of an equilibrium under commitment follows as an immediate corollary from the proof of existence of an equilibrium under lack of commitment, together with the main nonidentification result Theorem 1 below. I therefore defer the proof of existence for now.

4

Identification

I can now state the main result of the paper, which is that identification of the nature of bargaining in the model is not possible using the standard data used in macroeconomic models of the labor market. Theorem 1 (Observational equivalence). Let u(pt ), θ(pt ), ne (pt ), and G(n; pt , π) be sequences of unemployment, market tightness, entry, and the firm size distribution (conditional on the history of aggregate productivity) and let n ˜ (n; pt , π) be a sequence of target employment (conditional on current employment and idiosyncratic productivity and on the history of aggregate productivity). Define wS (pt ) by (13), h(n; pt , π) by (25), and xE (pt ) by (26). Then (u(pt ), θ(pt ), ne (pt ), n ˜ (n; pt , π), w(n; pt , π), G(n; pt , π)) is an equilibrium under lack of commitment of an economy with permanent aggregate productivity A, entry cost k, and worker bargaining power η if and only if (u(pt ), θ(pt ), ne (pt ), n ˜ (n; pt , π), h(n; pt , π), xE (n; pt , π), G(n; pt , π)) is an equilibrium under commitment of an economy with permanent aggregate productivity A/(αη+1−η), entry cost is k/(1 − η), and worker bargaining power η. The proof of Theorem 1 is straightforward. If JˆS (·) and J S (·) are the equilibrium value functions of the firm under lack of commitment, then they solve (1) and (2). Substitute for wages from (13) into (2) to see that J S (·) solves 1−η ηγθ(pt ) Apt πt nα − n(1 − η)b − n + β(1 − δ)Et JˆS (n; pt+1 , πt+1 ) αη + 1 − η 1 − f (θ(pt ))   A α E t = (1 − η) pt πt n − x (p ) + β(1 − δ)Et JˆS (n; pt+1 , πt+1 ). (29) αη + 1 − η

J S (n; pt , πt ) =

It is then apparent that if JˆS (·) and J S (·) solve (1) and (29), then

1 ˆS 1−η J (·)

and

1 S 1−η J (·)

solve (19)

n ˜ (n; pt , π)

and (27). Because the target employment policy was optimal in (1), it must therefore S ˆ also be optimal in (27). Similarly, if J (·) solves the free entry condition (17) given the entry policy ne (pt ) and the entry cost k, then ne (pt )

and entry cost

k 1−η .

1 ˆS 1−η J (·)

solves the free entry condition (28) given entry policy

The remainder of the conditions in Definition 2 follow immediately from

the corresponding properties for Definition 1. The converse argument is similar. An immediate corollary of Theorem 1 is the existence of equilibrium under commitment. Since by Proposition 1 of Hawkins (2011b) an equilibrium under lack of commitment exists, at least if aggregate shocks are sufficiently small and the transition matrix λ(p, p0 ) is strictly diagonally 12

dominant, Theorem 1 ensures the existence of an equilibrium under commitment under similar conditions. A particularly important case of Theorem 1 arises when the bargaining power of workers, η, is equal to the unemployment elasticity of the matching function, ψ. This case ensures in oneworker-per-firm models of the labor market that constrained efficiency obtains: it causes firms to internalize the congestion externality their vacancy-posting imposes on other firms by ensuring that they only receive less than the full value of the matches so created. In the environment of this paper, matching is not one-to-one, but constrained efficiency still obtains under the Hosios condition when firms can commit to long-term contracts. The intuition for this is straightforward. The ability of firms to commit ensures that the incentive for over-hiring present in the absence of commitment, and particularly emphasized by Stole and Zwiebel (1996a,b) and Smith (1999), is not present. The Hosios condition then ensures that firms choose vacancy posting and entry in accordance with the constrained efficient prescription. Because of the constant returns hiring technology, the marginal value of an additional hire is the same to all firms that hire a positive measure of workers. This ensures that the problem studied in Hawkins (2011a) (that the relative intensity of vacancy posting by firms of different sizes might not be efficient) different firms post vacancies with inefficient relative intensities) in the case where hiring is time-consuming does not arise.6 Of course, in the absence of commitment, equilibria are not constrained efficient even under the Hosios condition, due to the incentive of firms to hire excessively so as to reduce the wages of incumbent workers. Thus under the Hosios condition Theorem 1 shows that a constrained efficient equilibrium in one economy can be observationally equivalent to a constrained inefficient equilibrium in another.

5

Discussion

Theorem 1 shows that establishing the nature of bargaining in an economy of the class considered in this paper is not straightforward. In this section I consider this result in more detail, and ask how the nature of bargaining can in principle be distinguished. Observations on firm dynamics are clearly not sufficient for distinguishing employment: the optimal employment policy of a firm is identical in either equilibrium in the statement of the Theorem. What about wages? The rental equivalent value of an unemployed worker under commitment, xE (pt ) characterized by (26), is the same as that under lack of commitment, xS (pt ) characterized by (12). It follows that the average wage paid to all employed workers is the same in the two economies. In fact the expected present discounted value of wage payments to workers hired in period t after aggregate productivity shock history pt by firms with idiosyncratic productivity π is 6 A formal proof of the efficiency of equilibrium under commitment under the Hosios condition follows immediately from writing the optimization problem of a social planner who maximizes the present discounted value of output, taking as given the period-0 firm size distribution and unemployment rates. The marginal value to the planner of an additional firm with n employees and idiosyncratic productivity π in aggregate productivity shock history pt is then given by JˆE (n; pt , π) at the beginning of period t and J E (n; pt , π) after the labor market has closed in period t; ˆ E (pt ). The first-order conditions similarly, the marginal value to the planner of an unemployed worker is given by U for the planner’s problem then coincide with (19), (21), (22), (27), and (28).

13

the same in the two models. To see this, observe that ˆ S (pt ) + V S (n; pt , π) = U

η γ ˆ E (pt ) + h(n; pt , π t ) =U 1 − η q(θ(pt ))

using respectively (9) and (25). Although the present values of wages are the same both with and without commitment, I did assume for algebraic convenience, that there is a difference in the timing of payments. In the equilibrium under commitment, all wages above the value of an unemployed worker are paid at the beginning of the employment relationship as a signing bonus, while without commitment, wages are paid as a flow during the employment relationship. However, under full commitment, the expected value of the hiring bonus is the same as that of the wage stream paid by the firm in the equilibrium under lack of commitment (less the value of the component of wages in that equilibrium that gives the unemployment value xE (pt ) = xS (pt ) each period). Because there is full commitment, it would be equivalent for workers to be paid the same wage stream as would be observed under incomplete markets. Thus wage payments do not distinguish between the economies with and without commitment either. In summary, an econometrician who observes data generated either by an economy with bargaining with commitment or bargaining without commitment will find it difficult to distinguish the two possibilities. Even if the econometrician has data on wages, the value of an unemployed worker, and firm entry, exit, vacancy posting, and firing, this still does not suffice to distinguish the nature of bargaining. When the Hosios condition obtains, the difficulty of identifying the nature of bargaining from data on wages, employment, unemployment, hiring, and firing becomes particularly striking. Because in this case the equilibrium with commitment is constrained efficient, while the equilibrium under lack of commitment is not, it is clear that the econometrician cannot recommend whether labor-market policies, for example firing taxes and subsidies, will be welfare-enhancing. If firms cannot commit to long-term contracts, their incentive to hire excessively many workers can potentially be corrected by appropriate taxes and subsidies. If the firm can commit to long-term contracts, the equilibrium will be constrained efficient. Because the data usually used to calibrate macroeconomic models of the labor market in the class considered in this paper (for example, Elsby and Michaels, 2011; Hawkins, 2011b) are consistent with both constrained efficient and constrained inefficient equilibria. Since the two economies described in the statement of Theorem 1 are different, it must be possible to tell them apart somehow! The key observation for this is that the two economies differ in the share of output that is spent on entry costs. In a model in which capital was modeled more fully, this corresponds to a difference in the shares of capital and labor in aggregate income. Intuitively, under lack of commitment, firms hire excessively so as to drive down wages for inframarginal workers. In equilibrium, although wages fall because firms’ employment level is high, the total wage bill is still a greater share of total output in the economy with lack of commitment than in

14

the economy with commitment. To gain intuition, it is helpful to consider the simple special case of the model in which there are neither aggregate nor idiosyncratic productivity shocks. Thus p ≡ 1 and π ≡ 1. To characterize equilibrium in this case, note that on entry, firms immediately hire to reach their target employment level, then never hire or fire thereafter. Thus the steady-state equilibrium is characterized by two numbers, n, the size of all active firms during the production phase of any period, and θ, the market tightness. The first interesting comparison between the two allocations under the two bargaining protocols is a partial equilibrium comparison. Taking labor market tightness as given, how do the sizes of active firms, n, compare across the two bargaining arrangements? In this comparison, wages are allowed to adjust either according to (13) or according to (25) and (26), given the value of θ. This is the subject of the following Proposition. Proposition 1. Let θ > 0 be given such that f (θ), q(θ) ∈ (0, 1). Then the firm size under bargaining without commitment, nS , is greater than the firm size under bargaining with commitment, nE , with nS = (αη + 1 − η)α−1 . nE

(30)

The proof of Proposition 1 is in the Appendix. Note that because αη+1−η = 1−η(1−α) ∈ (0, 1) and α − 1 < 0, the right side of (30) exceeds unity. The Proposition is a statement of the classic over-hiring result of Stole and Zwiebel (1996a,b) and Smith (1999): the inability of firms to commit to long-term wage contracts gives them an incentive to hire excessive workers so as to drive down wages. This incentive is absent under commitment. A second comparison between the two bargaining protocols allows for the market tightness to adjust according to the free entry condition. Proposition 2. In the steady-state equilibrium with no productivity shocks, the firm size in the equilibrium under lack of commitment, nS , is greater than the firm size in the equilibrium under commitment, nE , with nS = nE



αη + 1 − η 1−η

1

α

.

(31)

The market tightness under lack of commitment, θS , is less than that under commitment, θE . Again, because

αη+1−η 1−η

> 1, the right side of (31) exceeds unity. Proposition 2 shows that the

over-hiring result survives in equilibrium: when the number of firms is allowed to adjust so that entrants make zero profits in expectation, then market tightness falls as fewer firms enter. This leads to a further increase in the relative firm size under lack of commitment, as wages also fall with market tightness, and then fall further with increased firm size. However, the statements of Proposition 1 and Proposition 2 do not directly relate to the main result of the paper, namely Theorem 1. In the two Propositions, parameters of the model such as the level of productivity A and the entry cost k are held constant, and the comparison is between 15

different equilibria under different bargaining protocols. The result of Theorem 1, on the other hand, states that the equilibrium allocation under lack of commitment in an economy in which the level of technology is A and the entry cost is k closely resembles that arising under commitment in an economy with technology A/(αη + 1 − η) and entry cost k/(1 − η). Arguing as in the proof of Proposition 2, it can be verified that in the simple case without productivity shocks, the firm size in both economies is  n=

αη + 1 − η 1−η

1  α

(1 − β(1 − δ))k A(1 − α)

1

α

(32)

and market tightness θ in both economies is given implicitly as the unique solution to    1 1−α 1 ηθ (1 − η)1−α α γ 1 − β(1 − δ) A α α(1 − α) α = −b + + 1−α 1−α . 1 − η 1 − f (θ) q(θ) αη + 1 − η (1 − β(1 − δ)) α k α

(33)

Because of (12) and (26), it follows also that total wage payments to workers are also the same in the two economies. That is, the two economies are indistinguishable in terms of the size of firms, firms’ vacancy-posting behavior, and wages. The difference between the two economies shows up in the share of firms’ output spent on payments to workers. In the case of the economy with commitment, (32) implies that the share of firms’ output spent on payments to workers is α.7 In the economy without commitment, however, the same expression implies the the share of firms’ output spent on payments to workers is 1 − 1−η > α.8 It follows that if the econometrician can estimate the returns to labor α in (1 − α) αη+1−η

firms’ production function directly (perhaps using microdata and directly estimating the production function itself), then by observing whether the labor share matches this value, she can verify whether bargaining proceeds with or without commitment. Of course, the identification strategy just outlined only works if the econometrician has independent information on α. If the econometrician simply observes the labor share, this cannot be used to determine α without knowing on whether firms can commit to long-term contracts when bargaining. In summary, if the econometrician knows how bargaining works, she can use data on profits or the labor share to estimate α. If she knows α (and η), she can use this data to determine how bargaining works. But she cannot identify both α and the nature of bargaining from this single data point alone. In summary, distinguishing the economy where bargaining occurs under commitment with that in which bargaining occurs without commitment, as in Stole and Zwiebel (1996a,b) or Smith (1999) requires an econometrician to observe not just wages, firm sizes, entry, growth, and vacancy posting, but also directly to observe the labor share. This is equivalent to requiring that profits be observable, something not usually true in macroeconomic models of the labor market. In addition, the econometrician must have separate information on the returns to labor parameter α. 7

To see this, used to pay the k entry cost 1−η . 8 To see this,

A k rewrite (32) as (1 − α) αη+1−η nα = (1 − β(1 − δ)) 1−η .This establishes that share 1 − α of output is expression on the right side of this equation, the stream of which payments has EPDV equal to the The remaining α share goes to workers. 1−η rewrite (32) as (1 − α) αη+1−η Anα = (1 − β(1 − δ))k.

16

The preceding discussion suggests that the intra-firm bargaining process in the absence of commitment has a similar effect to a factor-specific tax or subsidy. More specifically, imagine a tax on capital or a subsidy on labor that is then rebated to the firm lump-sum (so that there is no net cost to the firm). This policy distorts the effective relative factor prices faced by the firm, and leads it to utilize capital and labor in different proportions than would an undistorted firm. Using standard data on wages and firm entry and dynamics, a similar result to Theorem 1 applies in this case; an econometrician would find it difficult to distinguish distorted and undistorted versions of such an economy. The presentation of the main results of the current paper made some stark assumptions in order to make the main result as striking as possible. To conclude this section, I discuss briefly the extent to which the main assumptions can be relaxed while preserving the result. First, if the production function is not Cobb-Douglas, the result of Theorem 1 remains true, although the relationship between the production functions required in the two economies will not in general be as simple as in the Cobb-Douglas case in which the two production functions differ only in the level of productivity.9 Second, if the vacancy-posting technology does not exhibit constant returns, so that the hiring dynamics of firms are more sluggish than those arising here, then Hawkins (2011a) shows that the growth pattern of firms may differ across the two economies, as the interaction of the wage bargaining protocol with the decreasing marginal product of labor affects the relative incentives of small and large firms for hiring. In principle, this could be used to identify the nature of bargaining.

6

Conclusion

This paper establishes a cautionary result. Under the most commonly-used bargaining protocol for negotiations between workers and large firms, the equilibrium is not constrained efficient: firms hire excessively many workers so as to drive down the wage for inframarginal employees. However, identifying whether bargaining has this property can be difficult. Data on profits, or on the factor shares of labor and capital, are required to distinguish an economy where bargaining takes place without commitment from one in which firms and workers sign long-term contracts at the time of hiring. Since under the Hosios condition equilibrium in the latter economy is constrained efficient, this result shows that it is challenging to identify whether there is room for welfare-enhancing labor market policy. Because applied studies especially in trade and in the study of labor market regulations often assume that bargaining between firms and workers proceeds under lack of commitment, the welfare conclusions of such papers are subject to a caveat: they are true only to the extent that the privately-inefficient model of intra-firm bargaining they assume is an accurate description of reality. This paper shows that whether that is true cannot be identified from aggregate data on the labor market alone. 9

The two production functions in general need to be related so that y S (n; p, π) R n 1 −1 E 0 ν η (y ) (ν; p, π) dν. 0

1 1− η

n

17

=

y E (n; p, π) −

A

Omitted Proofs

Proof of Proposition 1. (29) implies that 1−η nα − n(1 − η)xS (θ) A αη+1−η Anα − nwS (n) = , 1 − β(1 − δ) 1 − β(1 − δ)

J S (n) =

(34)

ηγθ where xS (θ) = b + (1−η)(1−f (θ)) according to (12) and the second equality in (34) follows using (13). The first-order condition for optimal vacancy posting in (1) can be written 1−η A αη+1−η αnα−1 − (1 − η)xS (θ)

1 − β(1 − δ)

=

γ ,. q(θ)

(35)

(35) can be rewritten as 1 γηθ γ(1 − β(1 − δ)) Aαnα−1 = b + + . αη + 1 − η 1 − f (θ) (1 − η)q(θ)

(36)

A similar analysis applies in the case of equilibrium under commitment. (19) implies that J E (n) =

Anα − nxE (θ) , 1 − β(1 − δ)

(37)

ηγθ where xE (θ) = b + (1−η)(1−f (θ)) according to (26). The first-order condition for optimal vacancy posting in (27) can be written

Aαnα−1 − xE (θ) γ = . 1 − β(1 − δ) (1 − η)q(θ)

(38)

(38) can be rewritten in the form Aαnα−1 = b +

γ(1 − β(1 − δ)) γηθ + . 1 − f (θ) (1 − η)q(θ)

(39)

In partial equilibrium, the value of θ does not change according to whether firms can commit to long-term contracts, so the right sides of (36) and (39) coincide. Because αη+1−η = 1−η(1−α) < 1 and because α − 1 < 0, the statement of the Proposition follows immediately from comparing the left sides of these two equations. Proof of Proposition 2. The free-entry condition under bargaining without commitment, (17), when combined with (1), takes the form γn k = J S (n) − . (40) q(θ) Equations (34), (35), and (40) can be combined to establish that in the equilibrium under lack of commitment, firm size is S

n =



αη + 1 − η 1−η

1  α

18

(1 − β(1 − δ))k A(1 − α)

1

α

(41)

and market tightness θS satisfies    1 1−α 1 A α α(1 − α) α γ ηθS (1 − η)1−α α 1 − β(1 − δ) = −b + + 1−α 1−α . 1 − η 1 − f (θS ) q(θS ) αη + 1 − η (1 − β(1 − δ)) α k α

(42)

Note that the left side of (42) is strictly increasing in θ and under the Cobb-Douglas functional form assumed for the matching function, converges to 0 as θ → 0+ and diverges to +∞ as θ increases to the point where f (θ) = 1, that is, where M (1, θ) = 1. Thus there is a unique solution for θ. Similarly, the free-entry condition (28), combined with (27), takes the form k = J E (n) −

γn . (1 − η)q(θ)

(43)

Equations (37), (38), and (43) can then be combined to establish that in the steady-state equilibrium under commitment, all firms have size E

n =



(1 − β(1 − δ))k A(1 − α)

1

α

(44)

and market tightness θE satisfies   1−α 1 γ ηθE 1 − β(1 − δ) A α α(1 − α) α + = −b + 1−α 1−α . 1 − η 1 − f (θE ) q(θE ) (1 − β(1 − δ)) α k α

(45)

The first part of the Proposition follows from dividing (41) by (44). The second part follows since the left side of (42), and equivalently, the left side of (45), is strictly increasing in θ, while  1 1−α α < 1. The right-hand inequality follows because by Taylor’s theorem, for some 0 < (1−η) αη+1−η ξ ∈ (0, η), (1 − η)1−α = 1 − η(1 − α) − α(1 − α)ξ 2 < αη + 1 − η.

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