Equilibrium Labor Turnover, Firm Growth
and Unemployment
Melvyn G. Coles
∗University of Essex
Dale T. Mortensen
†Northwestern University, Aarhus University, NBER and IZA
May 27, 2013
Abstract
This paper considers a dynamic economy in whichfirms cannot
precom-mit to future wages and workers do not observe persistentfirm productivity.
Workers search on the job and firms control hire rates at a cost. Wages
are determined as a separating signalling equilibrium with the property that
more productive firms always pay more so that workers transit from less to
more productive employers as they climb the job ladder. The speed of as-cent, however, depends on firm investment in hiring. There is firm turnover: new small start-up firms are created while some existingfirms die. Consis-tent with Gibrat’s law, firm growth rates are size independent but increase
with firm productivity which evolves stochastically. With endogenous
ag-gregate job creation rates and job-to-job transitions, the model provides a rich, coherent, non-steady state framework of equilibrium wage formation and worker flows. The existence of a steady state equilibrium for any finite number of firm productivity types is established. Furthermore, there is only ∗Melvyn Coles acknowledges research funding by the UK Economic and Social Research
Council (ESRC), award ref. ES/I037628/1.
†Dale Mortensen acknowledges research funding by a grant to Aarhus University from
one steady state whenfirm productivity is permanent and there are may tirm types. Finally, a unique non-steady state equilibrium exists in the case of one type and in the general case if the elasticity of the hire rate with respect to productivity is sufficiently small.
JEL Classification: D21, D49,E23, J42, J64
Keywords: Wage dispersion, signalling, labor turnover, unemployment.
Corresponding author: Dale T. Mortensend-mortensen@northwestern.edu
1
Introduction
This paper describes a tractable dynamic variant of the Burdett and Mortensen (1998) model [BM from now on] that can be used for both macro policy
applications and micro empirical analysis. The framework allows rich firm
size dynamics. Innovative start-up companies are born small but the more
productive grow quickly over time. Conversely large existing firms may
ex-perience adverse shocks and so enter periods of decline. Slow growing firms are characterized by low productivity, low wages, high quit rates, and worker recruitment rates insufficient to replace exiting employees. Firm size is an endogenous variable, one which is positively correlated with productivity and thus with observed wages and turnover rates. With endogenous aggregate job creation rates and job-to-job transitions (via on-the-job search), this pa-per identifies a new, coherent framework of equilibrium wage formation and labor force adjustment outside of steady state.
The recent equilibrium literature which considers on-the-job search with endogenous firm size, e.g. Robin (2011), Moscarini and Postel-Vinay (2010), has no investment margins. The resulting equilibrium dynamics are mechan-ical in the sense that workers simply quit from low to high productivityfirms atfixed (exogenous) rates. Here instead we introduce an investment margin:
firms choose how many new employees to hire at a cost. Equilibrium is more
interesting for, with incomplete markets, equilibrium price formation has a direct impact on unemployment, turnover and investment. Indeed the wage structure is that of an efficiency wage model, related to Weiss (1980), but set
in a dynamic, non-steady state equilibrium framework.1
1In BM the wage paid controls two margins, thefirm’s hire and quit rate. Here with
the introduction of a hiring margin, wages only target the employee quit rate, as in Weiss (1980). The directed search approach, as considered in Acemoglu and Shimer (1998), instead adopts the other polar case, that wages only target the hire rate.
The paper considers equilibrium wage formation with private informa-tion (workers do not observe firm productivity) and stochastic productivity
shocks. By assumption firms cannot precommit on future wages and so
each may cut its wage paid in the event of an adverse productivity shock. Firms have all the bargaining power but workers can, and do, quit for high wages. In equilibrium, the wage paid perfectly reveals thefirm’s type: higher
productivity firms pay higher wages. As this wage strategy signals current
productivity, employees use this information to predict future wages (given that firm productivity follows a positively autocorrelated Markov process). Should the firm cut its wage, employees infer thefirm has been hit by a
rel-atively unfavorable shock and so quit to firms that pay higher wages. Even
though this decision to quit is a dynamic forward looking problem, a ran-dom contact technology yields a tractable equilibrium wage structure, one which is not unlike afirst price auction with private independent values (e.g. McAfee and McMillan (1987)).
In contrast to the recent non-steady state search literature, equilibrium here is not constrained efficient. Constrained efficiency is useful as the equi-librium allocation can then be separately identified by solving the Planner’s problem. This is automatically a property of the directed search framework with risk neutral agents and full information. Menzio and Shi (2010) con-sider such a dynamic model of on-the-job search, but that framework cannot address firm size dynamics.
2
The Model
Time is continuous with ∈[0∞)and there are two types of agents: entre-preneurs and workers. Agents are risk neutral, equally productive, discount the future at rate 0and die at rate 0. New births replace the deaths so thatalso describes the inflow of new agents into the market where each new agent potentially starts life either as an entrepreneur or as a worker. Each entrepreneur is endowed with a single business idea which he/she sells to a set of anonymous shareholders. On the immediate transformation of that business idea into a new start-up company, the entrepreneur becomes the first employee of the new start-up firm. By assuming each entrepreneur only has one idea, becoming a worker is a permanent outcome (until death). In this version new start-up firms are created at an exogenous rate 0 ≤
and each starts life with a single employee.2
Every worker is either unemployed or employed, earns a wage if
em-ployed, and the flow value of home production, ≥ 0, if not. Firms are risk neutral and have constant returns to scale with marginal revenue product
of labour . Existing firms die at rate 0 and so the measure of firms
equals0 Φ()denotes the stationary distribution of productivities across existing firms with support denoted[ ] and . It is convenient, how-ever, to describe eachfirmby its corresponding rank=Φ()noting that productivity =() is given by the inverse function ofΦ()
At a start-up, the rank of a new firm is considered a random draw
from c.d.f. Γ0() Continuing firms with rank are subject to a technology
shock process characterized by a given arrival rate ≥ 0 and a new rank
0 randomly drawn from c.d.f. Γ
1(|) Throughout we require first order
stochastic dominance in Γ1(|) so that higher rank firms are more likely
to remain high rank into the future. If[0 ()]denotes the support ofΓ1(|)
we assume lim→0+() = 0 so that rank = 0 is an absorbing state [till firm death].
For any date let() denote the number of unemployed workers plus
those workers employed at firms with rank no greater than ∈[01] Hence (0) = is the number unemployed and (1) = 1 The state of any firm
is ( ()) where is its (integer) number of employees and() serves
as the information relevant aggregate state variable for the following reason.
When a continuingfirm hires a new employee, that new employee is
con-sidered a random draw from the set of all workers who strictly prefer the job to their current status. As job offers are random, each unemployed worker receives a job offer according to a Poisson process with parameter =()
which depends on the market state () Randomness implies all employed
workers also receive outside offers at this rate but, as job offers are hetero-geneous, not all job offers are preferred to the worker’s current job.3 Let
denote the expected lifetime payoffof an employee at a given firm. We use
( ) to describe the fraction of outside job offers which yield value to
2An alternative specification might assume it takes time to develop a business idea and
so describe turnover in the stock of entrepreneurs.
3It is often assumed in the literature that employed workers receive offers at a lower
rate than unemployed workers to match transition data. However, Christiansen et al. (2005) demonstate that one can explain the data with a model that allows for endogenous search effort. As the qualitative properties of that model are the same as assuming equal offer arrival rates, the simplification here is justified.
the worker no greater than in aggregate state () () and ( )
are endogenous objects which, in equilibrium, are determined by aggregating over the wage and recruitment strategies of all firms.
There is asymmetric information: is private information to the firm.
We only consider fully revealing equilibria. Specifically we consider Markov Perfect (Bayesian) equilibria where each firm posts a sequence of spot wages using an equilibrium wage strategy = ( ()) where () is strictly increasing in. Note this Markov structure assumes afirm cannot precommit on future wages.
Hiring is costly here not because there are vacancy posting costs, rather because employee time is required to screen new applicants and to assimilate
new hires into firm practice.4 Following Lucas (1967) and Merz and Yashiv
(2007), if a firm withemployees decides to recruit an additional worker at rate then the cost of hiring is() where= is the effort required per employee. () is continuously differentiable and strictly convex with Inada conditions (0) =0(0) = 0. Note that these recruitment costs exhibit
constant returns: recruitment costs double as employment and hires
double.
3
Markov Perfect (Bayesian) Equilibria.
3.1
Separating Equilibria with Firm Size Invariance.
For ease of exposition, we only consider Markov perfect (Bayesian) equilibria in which the set of equilibrium wage strategies=( ())for∈[01]
is
(R1) fully revealing; i.e. ( ) is continuous and strictly increasing in ∈[01]and,
(R2)firm size invariant; i.e. ()does not depend on
That price strategies are continuous and strictly increasing inis a standard property both in the BM literature and in the first price auction literature with independent private values (e.g. McAfee and McMillan (1987)). Firm size invariance (R2) occurs as there are constant returns to production and
4Recent empirical work suggests that "adjustment costs" of this form explain the data
better than the more traditional "recruiting cost" specification. For example, see Sala et al. (2012) and Christiano (2013).
in recruitment costs.5
In any equilibrium satisfying (1)(2) and for any announced wage
0 ∈ [(0 ) (1 )] Bayes rule implies the worker believes the firm’s rank is b(0 ) where b uniquely solves 0 = (b ) If the firm posts wage 0 (0 ) existence of an equilibrium requires belief b(0 ) = 0. If instead the firm posts wage 0 (1 )the out-of-equilibrium beliefs do not play any material role. For this case we specify belief b(0 ) = 1.
3.2
Characterization
Let()denote the value of being unemployed in aggregate state()Now
consider an employed worker infirm( )Given thefirm announces wage (R1) implies the worker’s belief of thefirm’s typeb( )is degenerateAs (R2) implies wages do not depend onfirm size, the expected discounted value of employment in this firm can be written as( )where =b( ).
We begin with a standard observation. If a firm pays wage = an
employee does not quit into unemployment: remaining employed at his/her current employer has a positive option value (the firm may increase its wage
tomorrow) while the worker can always quit tomorrow. Thus any firm with
≥ 1 must make strictly positive profit (as ) Strictly positive profit further implies ( )≥()for all ∈[01](otherwise all employees quit
into unemployment which yields zero profit). Thus along the equilibrium
path, it cannot be optimal for a worker to quit into unemployment and, as wages are fully revealing, ()is identified recursively by:
(+)( ) = ( ) +[()−( )] (1) + Z 1 0 [( )−( )]Γ1(|) +() Z max[0−( )0](0 ) + In words, theflow value of employment equalsflow wage income plus the cap-ital gains associated with the following possibilities: (a) a firm destruction shock and a consequent return to unemployment, (b) a firm specific produc-tivity shock with new realization ∼ Γ1(|), and (c) an outside offer with
5Wage strategies are notfirm size invariant in BM, Coles (2001), Moscarini and
value 0 ∼ () In a steady state the final term, denoted would be zero. Outside of steady state, the state variable evolves endogenously.
This latter term describes the expected capital gain through the dynamic
evolution of a term which is made precise once we have described the
equilibrium dynamics.
It is immediate that the value of employment () must be strictly in-creasing in : the wage paid() is strictly increasing inand there is first order stochastic dominance in Γ1(|) Thus along the equilibrium path, an
employee at firm which pays wage =( ) only quits to an outside offer from a higher productivityfirm0 which reveals its type by offering a strictly higher wage (0 ) .
As positive profit implies( )≥() all job offers are acceptable to
the unemployed. The value of being unemployed is thus given by
(+)() =+() Z [0−()](0 ) + (2)
We now consider optimal firm behavior.
3.3
Firm Strategies.
LetΠ( )denote the expected discounted lifetime profit offirm( )
using an optimal wage and recruitment strategy. Suppose thefirm posts wage As each employee believes thefirm’s type is=b( )this posted wage induces employee quit rate ( b )(determined below). Hence firm( )
chooses wage and per employee recruitment effort = to solve the
Bellman equation: (+)Π( ) = max ≥0 * [()−]−() +[Π( + 1 )−Π( )] +[+(b )] [Π( −1 )−Π( )] +R01[Π( )−Π( )]Γ1(|) +Π +
In words the flow value of the firm equals its flow profit less recruiting costs plus the capital gains associated with (i) a successful hire (→+ 1)(ii) the loss of an employee through a separation (→−1)and (iii) afirm specific productivity shock with new draw ∼ Γ1(|) The last term captures the
The constant returns structure implies Π( ) = ( ) solves this Bellman equationwhere( )the value of each employee infirm( ) is given by: (+++)( ) = max ≥0 ¿ ()−−(b( ) )( ) +( )−() +R01( )Γc1(|) + À (3) The following transversality condition is also necessary for a solution to this dynamic programming problem:
lim →∞0 £ −( )|0 0 ¤ = 0 (4)
The Bellman equation (3) and the definition of ∗() implies the firm’s
optimal recruitment effort strategy ( ) = ∗(( )) Optimal
recruit-ment effort per worker is thus independent of firm size and so, consistent
with Gibrat’s law, hiring inflow =is proportional to firm size.
The Bellman equation (3) implies the optimal wage strategy minimizes the sum of the wage bill and turnover costs; i.e.
( ) = arg min
[+(b( ) ))( )] (5)
This wage structure is not unlike afirst price auction with a stochastic num-ber of bidders (McAfee and McMillan (1987)). In that literature, the buyer’s optimal bid depends on the likely best outside price received by the seller. Here this information is encapsulated by the worker’s equilibrium quit func-tion
( b ) =()[1−(( b ))] (6) where () describes the (Poisson) probability that the worker receives an outside offer and1−() is the probability the outside offer yields expected value 0 ( b ) and the worker quits.
3.4
Aggregation
Determining () and () which by (6) are the parameters of each firm’s decision problem (5), requires aggregating over the equilibrium wage and re-cruitment strategies of all firms. Recall that () describes the fraction of workers who are either unemployed or employed atfirms with rank no greater
than Consider then a firm which hires at equilibrium rate = As job offers are random, then each job offer by thisfirm is only accepted with
probability () (as an equilibrium offer is only accepted by workers
em-ployed at some firm0 ) Thus an expected hiring rate of =( ) at firm ( ) requires the firm makes job offers at rate () Aggre-gating across all firms implies the aggregate flow of job offers is
Z 1 0
( )()
()
where ()describes the measure of workers employed at type firms. As there is a unit measure of workers and job offers are random, the rate any given worker receives a job offer must therefore be
() =
Z 1 0
( )()
() (7)
To determine the composition of those job offers, defineb( )as the
frac-tion of job offers made by firms with type no greater than in aggregate
state The same aggregation argument implies
()[1−b( )] =
Z 1
( )()
() (8)
where the right hand side is the aggregated flow of job offers by firms with
type greater than .
3.5
The Equilibrium Distribution of Wages
By (8) and the definition ofb() the equilibrium quit rate function is
(b ) = Z 1 ( )() () (9)
noting that all firm types boffer wage ( ) (b ) and the worker quits Equation (5) now implies the equilibrium wage strategy solves:
( ) = arg min ∙ +( ) Z 1 () ( )() () ¸ (10)
Consider first the lowest productivity firm = 0 and recall that positive profit implies (0 ) ≥ () A contradiction argument now establishes
equilibrium implies (0 ) = ().6 The intuition is that the lowest
productivity firm cuts its wage until (0 )≥ () binds. It cannot cut
its wage further as its employees would then quit into unemployment. This equilibrium condition now yields lemma 1.
Lemma 1. In equilibrium the lowest wage offer is(0 ) =
Proof. Follows directly by putting = 0 and using equations (1), (2) and the equilibrium requirement (0 )=()7
As firms may grow arbitrarily large, ensuring the value of a firm is
bounded and that distribution of employment () is well defined requires sufficient discounting; a restriction on primatives is needed in general. The following result provides a sufficient condition which implies∗()≤+.
Lemma 2. In an equilibrium, () is increasing in and is bounded by the unique solution to
= −+ max≥h−()i
+ (11)
if ≤+(+)
Proof of Lemma 2 is in the online appendix.
For simplicity from now on assume () is differentiable with 0 0 for all ∈ [01] and (0) = 0. As is differentiable, (10) implies the necessary first order condition for optimality is:
1−( )(b ) 0(b) (b) b = 0 (12) where b( ) solves = ( b ) As b = £1 ¤ (12) implies (13)
below is a necessary condition for equilibrium The proof of Proposition 1
establishes it is also a sufficient condition for equilibrium wages ().
Proposition 1. Equilibrium implies () is the solution to the initial value problem: = ( )( )0() () for all ∈[01] (13) 6if (0 )
() firm = 0deviates with wage 0 (0 ) As its employees
believe b= 0, (10) implies anouncing wage0 (0 )is cost decreasing, which is the required contradiction
7Note this result requires = 0 is an absorbing state. If instead = 0 were not
absorbing, there would be a positive probability thefirm pays higher wages in the future. This option value would then imply(0 ) .
with (0 ) =
Proof of Proposition 1 is in the online appendix.
This wage outcome with random search has the structure of afirst price
auction with independent private values: firm does not “bid” 0 ( ) as it yields a too high quit rate, while 0 ( ) is not optimal as the corresponding fall in the quit rate is not worthwhile. As firms make strictly positive profit, Proposition 1 implies () is indeed continuous and strictly increasing in . The firm’s optimal wage = ( ) thus fully reveals its type.
4
Existence of Equilibria.
Characterizing non-steady state equilibria is potentially complex as the state variable () evolves endogenously over time and is infinitely dimensional. Moscarini and Postel-Vinay (2010) consider a related framework but assume precommitment on future wages and impose a solution concept of "stationary equilibrium". As a general principle, however, allowing precommitment on future wages is inconsistent with a stationary framework. To be precise, note that at the start the world, = 0 the outside option of each employed
worker is to be unemployed. Equilibrium contract posting at = 0 thus
implies all firms∈ [01] post contracts with starting value0( 0) =
(0) where 0 describes the initial firm size distribution. Coles (2001)
considers a related framework and describes how those equilibrium values ( )evolve over time. That evolving distribution of value paths, however,
is not stationary. Precommitment on future wages cannot be consistent with a stationary payoff structure for if = ( ) then the equilibrium condition ( 0) =
(0) at = 0 with 0 an arbitrary initial value,
necessarily yields the Diamond paradox (all firms pay =)
In contrast assuming (i) no precommitment on future wages, (ii) a hiring marginand (iii) asymmetric informationyields a tractable Markov perfect (Bayesian) equilibrium framework. In this section we fully characterize such equilibria for the (limiting) case that firms are equally productive () = and then provide results for the heterogenous firm general case.
4.1
The Homogenous Firm Case.
Given equally productive firms, let = () denote the value of an
em-ployee in aggregate stateAs optimal recruitment effort∗()is the same
for all firms, (7) implies equilibrium=() given by:
= Z 1 0 ∗( )() () =−∗() ln (14)
Equilibrium thus depends on and the unemployment rate but is
otherwise independent of Putting= 0 in (3) and using (14) now yields
˙ = (++−∗() ln)− µ −+ max ≥0 [−()] ¶ (15) As workersflow out of unemployment at rate +, successful entrepreneurs enter at rate 0 and the unemployedfind jobs at rate,
·
= (+)(1−)−0+∗() ln (16)
= (+−0)(1−)−(0−∗() ln)
Equatons (30) and (29) describe a pair of autonomous differential equations for ( ) The surprising result is the information relevant market state re-duces from the entire distribution to a singleton in the limiting case, the unemployment rate =(0). (30) reveals the underlying economic
struc-ture of the model: each firm’s recruitment effort is a best response to the recruitment strategies∗ of the otherfirms. In particular increased
competi-tion for a firm’s employees, via higher recruitment rates ∗ of outside firms,
makes employee retention more costly. Given equilibrium wages determined according to Proposition 1, (15) and (16) determine equilibrium recruitment rate∗(
)where =∗()is the best response of eachfirm given all others
choose ∗
Coles and Mortensen (2012) establish the system of differential equations composed of (15) and (16) has a single steady state which is a saddle point.
Figure 1 depicts its phase diagram. Because on a solution trajectory
above the stable saddle path must eventually exceed, while any one below
the steady state ultimately yields zero (which contradicts optimal firm
Phase Diagram
The equilibrium unemployment dynamics are stable but adjustment is
slow. Periods of high unemployment imply lower job offer rates which, by
reducing competition for a firm’s employees, increases the value of an
em-ployee As firms respond by increasing recruitment effort∗this sustains
a gradual recovery to the steady state.
Proposition 1 implies the equilibrium wage is: ( ) =+∗()ln µ () ¶
Inother words, wages are disperse even in the limiting one firm type case.
Wages are tied down by = at = 0 but are otherwise forward looking,
noting that =() depends on how unemployment rates are expected to
evolve in the future. Note that all wages generally increase with the value
of a worker, , but decrease with the measure of unemployment along the
adjustment path at each decile of the wage distribution. Hence, the model offers the possibility of rich wage dynamics in response to aggregate shocks.
As firm growth rates are independent of firm size, firm size evolution is consistent with Gibrat’s law. Coles and Mortensen (2012) provides a com-plete discussion of the micro-implications of this framework.
4.2
Heterogeneous Firms.
Consider now a finite number of firm productivity states (1 ) where
1 and where −−1 = ∆ 0 for all = 2 Corresponding to
this grid is a partition (−1 ] ⊆ [01] with 0 = 0 −1 and = 1
such that firms with rank∈(−1 ]have productivity() =. Assume
Γ1() satisfies Γ1(|)−Γ1(−1|) = for all ∈ (−1 ] so that the
transition probability from productivity state to state is independent of thefirm’s rank in state . = denotes the transition rate from state
to andΓ0 is the probability that a start-up has type no greater than 8
Let = () denote the number of unemployed workers plus those
employees in firms of type or less, so 0 = and = 1 We now
show that the relevant market state () reduces to the finite state vector
G=(0 1 2 −1). Let(G) denote the value of an employee in an
firm in aggregate state G and v=(1 2 )the vector of employee
val-ues. Lemma 3 now solves the equilibrium conditions for wage =(vG)
which is the wage paid by firm = and its corresponding quit rate
= (vG) This yields an autonomous differential equation system for
(v·G· )
Lemma 3. For= 1 equilibrium implies: · = (+++()) (17) −−() + max ≥0 {−(}) + X 6= ( −) · = 0Γ0+ (+−0) (1−)−(0+()) (18) + X =+1 £ Σ≤ ¤ (−−1)− X =1 £ Σ ¤ (−−1) 8Note
1= 0for all 1as state= 1is absorbing for sufficently smallA simple
alternative is to specify 1 = and invoke a tie breaking assumption that employees
quit when indifferent to doing so. Reaching state1 = would then be equivalent to a
firm destruction shock, where all employees exit into unemployment. The analysis goes through directly.
with (vG) = X =+1 ∗() [ln −ln−1] (19) (vG) = + X =1 ∗() [ln−ln−1] (20)
Proof of Lemma 3is in an online appendix.
The equilibrium vector valuev(G) =(1() ())is a particular
solu-tion to this differential equasolu-tion system, given an arbitrary initial distribusolu-tion of workers G0 and the transversality condition (4) for each type
Theorem 1 A steady state equilibrium exists.
Proof. Any vector of steady state values satisfies
= −(vG) + max≥0{−()}+ P (−) +++(vG) (21) for each = 1 The steady state vector of employment measures solves
= ++0(Γ0−1) + P =+1 £ Σ≤ ¤ (−−1) −P=1 £ Σ ¤ (−−1) ++(vG) (22) 0 = +−0 ++0(vG) and = 1
As (vG) and (vG), = 1 defined in (19) and (20) are real
con-tinuous functions, the RHSs of (21) and (22) jointly form a concon-tinuous map from the bounded real vector space[0 ]
×[01]+1 into itself. Hence a fixed
point exists by Brouwer’s fixed point theorem.
Establishing that there is only one steady state equilibrium is a more difficult task. Nevertheless, it is unique in the case of permanent productivity
shocks and a large number of types = 0. As
lim −−1→0 ∙ ln −ln−1 −−1 ¸ = 0() ()
the continuous analogues of the steady state conditions above can be written as () = −() + max≥0{()−()} +++() (23) () = ++0(Γ()−1) ++() (24)
for all ∈[ ] where Γ0() is the distribution of productivity over firms at
birth and the quit rate function is () =
Z
∗(())()
() (25)
and the wage rate function is () =+
Z
()∗(())()
() (26)
A steady state is a collection of functions, (), (), (), and (), that satisfy these equations.
Theorem 2 In the case of a continuum of types, a unique steady state equi-librium exists when firm productivity is permanent.
Proof of Theorem 2 is in the online appendix.
In any numerical example, one can find steady state equilibria by
iter-ating equations (19) and (20) constrained by equations (21) and (22) and then compute the eigen values of ODE system at the steady state. If the computed equilibrium is a saddle, then the stable trajectories generated by a linear approximation to the system near the steady state is the locally unique equilibrium function ().
Although a general analytic proof of existence of a non-steady state is not currently available, analogous to existence results found in Robin (2011) and Moscarini and Postel-Vinay (2010), we can formally establish that a unique non-steady state equilibrium exists in the case in which the hire rate =b
is exogenous This case holds exactly when the hire cost function takes the
form () = 0 for all ≤ b and () = ∞ for all b which lies at the boundary of the set of convex cost functions of interest. Given =b for all
· = Γ0+(1−)−(−bln) + X =+1 £ Σ≤ ¤ (−−1)− X =1 £ Σ ¤ (−−1)
for = 0 −1 with = 1. As one can easily show that ∈(01) for
all , the initial condition G0 uniquely determines the equilibrium
em-ployment dynamic. The associated value vector v(G) is the unique forward
solution to the equations of (17) along the equilibrium employment path con-ditional on G0 = G for all of its feasible values. That the forward solution exists follows by Lemma 2. Note that this solution approximates the case in which the generic hire function ∗() is highly inelastic.
5
Conclusion.
This paper has identified a new tractable equilibrium model of job and labor
flows that seems ideal for both macro-policy applications and micro-empirical analysis. Novel features include a hiring cost structure which exhibits
con-stant returns to scale, heterogeneous firm productivity that is subject to
shocks and a wage renegotiation mechanism consistent with private infor-mation. Firm size dynamics are correspondingly rich as small start-up
com-panies enter the market over time, some older, existing firms exit and
em-ployees keep searching on-the-job for preferred employment prospects. Coles and Mortensen (2012) discuss the new micro-empirical implications of this framework. Future research will extend the framework to stochastic aggre-gate shocks and endogenous start-ups.
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Appendix [to be put online].
Proof of Lemma 2.
Below we establish the result where the upper bound isas defined in (11) provided that the caseflow−−(∗())0at∗()≡arg max
≥h−()i. Equivalently, = max ≥0 −−() ++− subject to −−()0
Now, if −−(++)0then the criterion function is positive at= 0
given that , is strictly convex on the interior of the interval[0 ++]
and converges to negative infinity as →++. Hence, a unique interior solution exists. However, if −−(++)0, the criterion function is increasing throughout the interval and converges to infinity at the boundary so no solution exists. Finally, if −−(++) = 0, then the criterion function is increasing in the interior and
= lim
→++
−−()
++− =
0(++)
by L’Hospital’s Rule at the boundary. Consequently,∗()≤++follows
as a corollary.
The forward solution to (3) that satisfies the transversality condition (4) along any arbitrary future time path for the state {}∞0 is the fixed point
of the following transformation
( )( 0) = Z ∞ 0 max ≥0 ¿ ()−+( )−() + Z 1 0 ( )Γ1(|) À ×exp µ − Z 0 (++++(b( ) )) ¶ As (b( ) ))≥0 in general and ≥ by Lemma 1, it follows by (11)
( )( 0) ≤ Z ∞ 0 max ≥0 h()−+−() +i −(+++) ≤ max≥0h−+−() +i +++ =
for any ( ) ≤ . Because () is increasing in and Γ1(|) is
in . Thus the transformation maps the set of uniformly bounded func-tions that are increasing in into itself. Further, the transformation is increasing and (( 0) +) = ( 0) +|| Z ∞ 0 (∗() +) ×exp µ − Z 0 (++++(b( ) )) ¶ ≤ ( 0) + ∗() + +++|| for all0
because (b( ) )) ≥ 0 and ∗(( )) ≤ ∗() for any ( ) ≤ . As
∗()≤++the map thus satisfies Blackwell’s condition for a contraction
map which guarantees that a unique fixed point exists in the set of bounded
functions increasing in. This completes the proof of Lemma 2.
Proof of Proposition 1. To show (13) is sufficient, let ( ) denote the solution to the initial value problem defined in Proposition 1. As(0) =
0 this solution is continuous and strictly increasing in . Now consider anyfirm∈(01]and let
(0 ) =0+( ) Z 1 (0) ( )() ()
describe the minimand in (10)If thefirm sets lower wage 0 =(0 )with 0 ∈ [0 ] its employees believe b = 0 By marginally increasing the wage 0differentiation implies 0 = 1−( ) (0 )(0) (0) b 0 But equation (12) implies
1−(0 )(
0 )0(0) (0)
b 0 = 0 at 0 and combining yields
0 = 1−
( )
by lemma 2. Thus for 0 ( ) an increase in 0 decreases cost () The same argument applies for 0 = (0 ) with 0 ∈ [1] - this time it is cost decreasing to cut the wage Finally note for wages 0 (1 ) the worker’s belief is fixed at b = 1 and so higher wages are even further cost increasing. Hence given all firms offer wages according to Proposition 1, the cost minimizing wage for firm is to offer =( ) This completes the proof of Proposition 1.
Proof of lemma 3.
As each type firm ∈ (−1 ] enjoys the same value optimal
re-cruitment effort ∗(
) is the same for all such firms. Equation (8) then
implies [1−b(G] = Z 1 ( )() () = X =+1 ∙ ∗() ln −1 ¸ (27) Equal profit at all suchfirms implies the equilibrium wage equation
(G) +[1−b(G)] =−1+ X = ∙ ∗() ln −1 ¸
for all ∈(−1 ] Putting = and using (27) one obtains
=−1+∗() ln
−1
.
As0 =by Lemma 1, this recursion yields(vG)as stated in the lemma.
The equation for(vG)follows from (27). The dynamics for follow from
standard turnover arguments. This completes the proof of Lemma 3.
Proof of Theorem 2:
Note that the envelope theorem and the fact that the equilibrium wage function satisfies 0() =−()0() by equations (25) and (26) imply
0() = 1
+++()−∗(()) (28)
Further-more, a differentiation of equations (25) and (24) yields 0() = −∗(()) 0() () (29) = −∗(()) µ 0Γ00() ++0(Γ()−1) − 0() ++() ¶ = − ∗(()) (++()) ++()−∗(()) 0Γ00() ++0(Γ0()−1)
Any solution to these two differential equations consistent with the boundary conditions () = 0 and () =0()where
0() =
−+ max≥0{0()−()}
+++ (30) identifies a steady state. As this is a boundary value problem, existence and uniqueness of a solution are not guaranteed in general.
Figure 2 represents the phase diagram for the ODE system. As 0()
=
0()
+++−(0())
(31)
is the slope of the locus of point that satisfy equation (30), the curve labeled 0() in the Figure converges asymptotically to zero as becomes large.
Although solutions trajectories shift with (except in the case of a uniform distribution of productivity), they do not cross and their relative positions are as represented in the Figure where the arrows indicate the "direction of motion" as increase from to .
Let the pair ( ) and ( ) represent the particular solution to the ODE system associated with a specific choice ofA steady state equilibrium, then, corresponds to a choice of that satisfies
( ) =+
Z
0( )= 0 (32)
Obviously, = 0is too small given and =∞is too large given ∞ as 0( ) is negative and finite. As ( ) is continuous in , at least one solution exists by the mean value theorem.
For any choice of suppose there exists somee (b)and e such that e= (e ) = ( e +) for any 0. As + the supposition implies that ( ) ( +)and( ) ( +)for allewhere eis the smallest solution to (eb) = (eb+). These facts and that ∗() is
increasing imply 0(b) 0(b+) by equation (28). Therefore,
( e ) =0() + Z 0( ) 0(+) + Z 0 0( +)=( e +)
which contradicts the supposition. Hence, an inspection of the phase diagram implies(b) (b+)for all∈[ ]As the same argument also yields (b) (b−), equation (32) has a unique solution.
