WAGE-TENURE CONTRACTS WITH

HETEROGENEOUS FIRMS1

Ken Burdett and Melvyn Coles

June 2006

1This is a preliminary draft. Comments most welcome. (Burdett�s email is Ken-

nethb@ssc.upenn.edu). We would like to thank Ben Lester for research assistance.

WAGE-TENURE CONTRACTS WITH HETEROGENEOUS

FIRMS

BY KEN BURDETT ANDMELVYN COLES1

1 Introduction

There have been signi�cant developments in the last few years in the study of equi-

librium in labor markets where �rms post wages and workers, both unemployed and

employed, search for better jobs. Perhaps the most used framework in this area was

developed by Burdett and Mortensen (1998) They showed that even if both workers

and �rms are homogeneous that at the unique equilibrium the distribution of wage

o¤ers is non-degenerate. It turns out that generalizations to the cases where �rms

or workers are heterogeneous is relatively straightforward - all yield an equilibrium

where the distribution of wage o¤ers is non-degenerate. Three restrictions play a

crucial role in Burdett and Mortensen framework: (a) employed workers receive job

o¤ers as well as unemployed workers, (b) the o¤ers made by any employee can be

completely described by a positive real number - the wage it is willing to pay any

employee, and (c) employers do not respond to o¤ers from other �rms received by

employees. These implies that in equilibrium a worker employed at a particular �rm

always receives the same wage. These imply the only reason the wage of an employed

worker increases is when this worker changes jobs.

Recently Burdett and Coles (2002) (hereafter, B=C) and Stevens (2004) have

extended the above analysis by allowing �rms to o¤er enforceable wage/tenure con-

tracts instead of a single wage. The basic idea behind this approach can be explained

as follows. Suppose for the moment that workers are indi¤erent to risk. Further,

assume a �rm makes an o¤er that yields any worker who accepts it a particular ex-

pected discounted lifetime income. It can be shown that, given workers search while

employed, there exists a two-step wage contract (a low wage for workers with low

1

tenure at the �rm, and then a high wage later on) that yields a �rm a greater ex-

pected pro�t �ow than the single wage o¤er which yields the same expected return.

Essentially, �rms prefer to back-load wages when workers search on-the-job. Indeed,

it was shown by Stevens that if �rms can post a two-step wage contracts, there exists

an equilibrium where all �rms o¤er the same two-step wage contract. In this case

there is no turnover as workers always prefer to stay at the employer who hired them

from unemployment but at each �rm those with low tenure are paid a low wage and

the others are paid greater wage.

In B=C a more general setting was considered. Here, �rms can post a general

wage/tenure contract. Further, it is assumed that workers are risk-averse. This set

up implies there is a tension between the desire of �rms to back-load wages and

the insurance incentive in that risk averse workers, ceteris paribus, prefer constant

wages through time. It was demonstrated that there exists an equilibrium where the

contracts o¤ered by �rms (contacts that maximize a �rm�s pro�t �ow) imply that

wages strictly increase with tenure. Further, Firms do not o¤er the same contract.

The relationship between the di¤erent contracts o¤ered can be described by what

we termed a baseline salary scale. Let w(:) denote the wage/tenure function o¤ered

by the �rm o¤ering the lowest initial wage, and as wages increase de�ne w(0) = w

and limt!1w(t) = w: In equilibrium it was shown that the optimal wage/tenure o¤er

made by any �rm in the market can be written as w(:jt0) for some t0 � 0 such that

(a) w(t0) = w(0jt0); :and (b) w(t0+ t) = w(tjt0) for any t � 0: Hence, w(:) is termed

the baseline salary scale and t0 a starting time. In equilibrium can be characterized

by a baseline salary scale and a distribution of starting times, (w(:); H(:)). It was

shown that in equilibrium the distribution of starting times is continuous except at

t = 0 and has support [0;1]:

As the baseline salary scale is increasing, a �rm that o¤ers a greater initial starting

time than another makes less pro�t per employee per unit of time. Firms o¤ering

a greater initial starting time, however, keeps its employees longer (on average) and

2

attracts more workers. To see this more clearly, let V (0jt0) indicates a worker�s

expected lifetime utility from accepting a job o¤er with starting time t0 (and then

following an optimal quit strategy). Clearly, V (tjt00) > V (tjt0) for any tenure t � 0;

where t00 > t0: On-the-job search by workers implies a �rm o¤ering a greater starting

time obtains a larger steady-state labor force. In equilibrium, of course, all �rms

make the same expected pro�t �ow which equals the expected pro�t per employee

multiplied by the expected number of employees. These results were established

assuming workers and �rms were homogeneous. This implied our results were not

easy to empirically implement. Here, we drop this restriction and assume that some

�rms are more productive than others.

Assuming there are n di¤erent productivity types among employers, yields many

new insights. Let p1 < p2 < :::: < pn denote the productivities of the di¤erent types

of �rms. It will be shown that the concept of a baseline salary scale also plays an

important role in this case in that it will be shown that when there are n productivity

types there are n baseline salary scales - one for each productivity type. The type i

�rm baseline salary scale is the wage/tenure o¤er made by the type i �rm o¤ering the

lowest initial wage, wi(:);where wi(0) = wi and limt!1wi(t) = w1i : In equilibrium

the o¤ers made by type i �rms can be fully characterized by (wi(:); Hi(:));whereHi(:)

is the distribution starting times for type i �rms.

The above implies for any type i �rm there exists a t0 � 0 such that wi(t0+ t) =

w(tjt0) describes the optimal wage/tenure contact made by that �rm. It will be

shown that not all points on the type i baseline salary scale need be a starting time.

In particular, it will be shown the support of the distribution of starting time, Hi(:);

is an interval [0;bti]:If for some type i; bti < 1; then (as the baseline salary scale is

increasing, wage) wi(bti + t) for t > 0 are paid by type i �rms but only to employeeswith large enough tenure - such a wage is not o¤ered as initial o¤ers by type i �rms.

If bti =1; then all points on the baseline salary are initial o¤ers for some type i �rm.

As will be shown this has important consequences.

3

Given the o¤ers made by �rms can be fully described by (wi(:); Hi(:)); i =

1; 2; 3; ::n; the objective becomes to establish the equilibrium relationship between

the o¤ers made by the di¤erent types of �rm. It will be shown that in equilibrium

the initial o¤ers made by type i employers can be described by an interval [V i; V i]

such that V 2 [V i; V i]; where V indicates a worker�s expected lifetime payo¤ from

accepting a particular job o¤er. An important result established is that in any mar-

ket equilibrium it will be shown that V i = V i+1; i = 1; 2; n� 1; i.e., more productive

�rms make more desirable job o¤ers than less productive �rms. Although more pro-

ductive �rms make more desirable o¤ers this does not imply the initial wage o¤ered

by such �rms are greater than less productive �rms. Indeed, it will be shown that an

employee of a type i �rm, will willingly move to a type i+1 �rm (if the opportunity

arrises) although the initial wage o¤ered is lower than that currently faced.

It will be shown that the above and other results lead to a reasonably complete

characterization of the equilibrium structure of the labor market. To illustrate

essentials, a detailed characterization of equilibrium is provided in the special case

where there are only two types of �rms. This turns out not to be simple task.

The reason for this can be explained as follows. If there are two types of �rms

in equilibrium a workers expected return to accepting an o¤er, V 2 [V ,V ], where

V = V 1 < V 1 = V 2 < V 2 = V ;i.e., the more productive type of �rms (type 2) make

more desirable o¤ers. It is shown, however, there exists a bt < 1 such that w1(bt1)is the greatest initial wage o¤er made by type 1 �rms, i.e., w1(bt1) < w1i ; whereas

with type 2 �rms bt2 = 1: This implies some type 1 employees enjoy an expected

lifetime utility greater than V 1:Let V 11 denote the maximum expected return a type

1 employee can enjoy. It follows that V 1 = V 2 < V 11 � V 2. Suppose a type

1 employee with positive tenure faces expected return bV 2 (V 2; V11 ): This implies

that if such an employee obtains an o¤er from a type 1 �rm it may be rejected. This

is not only leads to an ine¢ ciency but also much complicates the wage dynamics.

In the nest Section the basic framework is outlined and the optimal search strat-

4

egy of a worker is described. In Section 3 the optimal wage/tenure contract of

a representative type i �rm is derived. We then show that in equilibrium the wage

tenure/contracts o¤ered by all type i �rms can be fully described by a baseline salary

scale. As many elements of these two Sections repeats, or are reasonably straightfor-

ward extension of the analysis provided in B=C; we only outline essentials. Section 4

provides a description of a market equilibrium. From Section 5 onwards we provide

a detailed characterization of possible equilibria when there are two types of �rms.

Section 6 provides a simulation of the possible equilibria.

2 BASIC FRAMEWORK

Time is continuous and only steady states are considered. A unit mass of both

workers and �rms participate in a labor market. Workers are homogeneous but

there are n types of �rms. A type i �rm generate revenue pi per unit of time from

each worker it employs, i = 1; 2; ::n, where p1 < p2::: < pn: Proportion �i of all �rms

are type i. Workers are either unemployed or employed and obtain new job o¤ers

at Poisson rate �; independent of their employment status.

Any job o¤er is fully described by the wage contract o¤ered by the �rm. This

speci�es the wage the worker receives as a function of tenure at that �rm, i.e., an

o¤er is a function w(:) � 0 de�ned for all tenures t � 0: All new hires are o¤ered

the same contract and this contract is binding. There is there is no recall should a

worker quit or reject a job o¤er. For simplicity assume �rms and workers have a zero

rate of time preference. The objective of any �rm is to maximize steady state �ow

pro�t. Workers are strictly risk averse and are �nitely lived, where any worker�s life

is described by an exponential random variable with parameter � > 0: Any worker

who dies is instantly replaced by a new unemployed worker. The objective of any

worker is to maximize total expected lifetime utility. Unemployed workers obtain b

per unit of time and that p1 > b > 0: There are no �nancial markets.

5

2.1 Workers.

Given the lack of �nancial markets, a worker who obtains income w � 0 at any

instant of time obtains �ow utility u(w) by immediately consuming it: As in B=C

we utilize the following restriction on this utility function.

A1: Assume u is strictly increasing, strictly concave, twice di¤erentiable and

limw!0+

u(w) = �1:

This is restrictive but simpli�es the analysis (see B=C for further discussion).

Let V (tjw(:)) denote a worker�s expected lifetime payo¤ when employed with

tenure t at a �rm o¤ering w(:); given an optimal quit strategy is followed in the future

An unemployed worker�s maximum expected lifetime payo¤ is indicated by Vu: Let

F (V0) denote the proportion of �rms in the market whose o¤er, if accepted, yields

a worker an expected lifetime income no greater than V0; i.e., F is the distribution

of V (0jw(:)) in the market. Random search implies F (V0) denotes the probability

a worker contacts a �rm o¤ering an expected return no greater than V0; given a

contact is made. Let V and V denote the in�mum and supremum of this distribution

function. Using standard arguments imply V (tjw(:))can be written as

(1) �V (tjw(:))� dV (tjw(:))dt

= u(w(t)) + �

Z V

V (tj:w())[V0 � V (tjw(:))]dF (V0):

If V (tj:) < Vu for some tenure t; the worker�s optimal strategy is to quit into unem-

ployment. Although this event never occurs in equilibrium, we need to allow for this

possibility and so de�ne T = infft � 0 : V (tj:) < Vug: Hence, T denotes the tenure

at which a worker optimally quits into unemployment: If V (tj:) � Vu for all t; then

de�ne T =1.

Assume an unemployed worker accepts a job o¤er if indi¤erent to doing so,

whereas employed workers only quit if the outside job o¤er is strictly preferred.

This implies optimal job search implies the following worker strategies: First, if un-

employed, the worker accepts a job o¤er if it has starting value V0 � Vu: Second,

6

if employed with contract w(:) and tenure t < T; the worker quits if and only if a

job o¤er is received with starting value V0 > V (tjw(:)): Finally, if employed with

contract w(:) and tenure t = T <1; the worker quits into unemployment.

Given wage contract w(:) and tenure s < T; a worker leaves a �rm�s employment

at rate � + �(1� F (V (sj:))): For tenures t < T; the survival probability

(2) (tjw(:)) = e�R t0 [�+�(1�F (V (sjw(:)))]ds

describes the probability a newly employed worker does not leave the �rm before

tenure t: Of course if T <1; then = 0 for all t � T:

2.2 Firms.

Let G(V ) denote the steady-state number of workers who are either unemployed or

employed currently enjoying an expected lifetime utility strictly less than V: Suppose

a type i �rm posts w(:) and let V0 = V (0jw(0)): If V0 < Vu; the �rm attracts no

workers and so obtains zero pro�t. If V0 � Vu; then �G(V0) describes this �rm�s

steady-state hiring rate and therefore its steady state �ow pro�t can be written as

i = [�G(V0)]

�Z 1

0

(tjw(:))[pi � w(t)]dt

�;

Note, the �rm�s steady state �ow pro�t equals its hiring rate �G(V0) multiplied by

the expected pro�t per new hire.

First, we determine a �rm�s wage/tenure contract that maximizes it�s expected

pro�t �ow conditional on o¤ering a new hire lifetime payo¤ V0 � Vu:Such a contract

is termed an optimal contract. Assuming an optimal contract exists for a type i �rm,

let w�pi(tjV0) denote it and let ��pi(0jV0) denote the �rm�s maximized payo¤ per new

hire. A type i �rm�s steady-state �ow pro�ts can then be written as

�i (V0) = �G(V0)��pi(0jV0):

7

and the �rm�s optimization problem reduces to choosing a starting payo¤ V0 to

maximize �i (V0):

Let Fi denote the o¤er distribution of �rms of type i with support denoted [V i; V i];

and note F =P

i �iFi: To simplify the exposition, we impose the following restriction

on F .

A2: F is continuously di¤erentiable almost everywhere and has a connected support.

We now de�ne a market equilibrium.

De�nition: A market equilibrium is:

(ME1) a distribution of starting payo¤s F satisfying A2;

(ME2) a set of optimal wage tenure contracts w�p(:jV0) with p = pi; and indexed by

V0 � Vu;

(ME3) optimal job search by unemployed workers, where optimality implies Vu sat-

is�es

�Vu = u(b) + �

Z V

Vu

[x� Vu]dF (x);

(ME4) optimal quit behavior by employees, where V (tjw�pi) describes the worker�s

expected lifetime payo¤ at tenure t given wage contract w�pi(:);

(ME5) a distribution of expected lifetime payo¤s G consistent with steady state

turnover, and

(ME6) a steady state pro�t condition where for all �rm types:

(3)�i (V0) = i � 0 if dFi(V0) > 0;

�i (V0) � i; otherwise,

so that any o¤ered contract (i.e., an optimal contract w�p(:) with �rm productivity

p = pi and starting value V0 which lies in the support of Fi); maximizes steady state

�ow pro�t.

8

3 Optimal Wage/Tenure Contracts

Given productivity pi = p; a �rm�s optimal contracting problem is formally de�ned

as

(4) maxw(:)

Z 1

0

(tjw(:))[p� w(t)]dt

subject to (a) w(:) � 0 and (b) V (0jw(:)) = V0; where is de�ned by (2).

Before solving this problem it is useful to note that as the arrival rate of o¤ers

is independent of a worker�s employment status, an unemployed worker accepts a

contract which o¤ers w(t) = b for all t: Further, as b < pi by assumption, a �rm can

always obtain strictly positive pro�t by o¤ering this contract. It is now straightfor-

ward to show that if an equilibrium exists, (a) all �rms make strictly positive pro�t;

i > 0; (b) V � Vu, (c) V < u(pn)=� (no �rm pays a worker more than the worker�s

expected value), and (d) T (w�p) =1 for all p = pi (an employed worker never quits

to unemployment): Hence, without loss of generality, we may assume that F not only

satis�es A2, but also Vu � V and V < u(pn)=�:

For a given F and corresponding V ; de�ne w by u(w) = �V : In equilibrium, w;

is the highest wage paid in the market. We also de�ne for each type i �rm:

w1i = min[pi; w];

(5) �V 1i = u(w1i ) + �

Z V

V i

[V0 � V 1i ]dF (V0);

�1i = (pi � w1i )=�;

where w1i is the wage paid by type i �rms at arbitrarily long tenures, while V 1i

describes the value of being employed at that wage.

Hence, for given F (and therefore given V and w) it is possible to partition the

di¤erent types of �rms into two - those types of �rms with pi < w and those with

pi � w. If pi � w; type i �rms o¤er limiting wage w1i = w at long tenures: As

9

w is the largest wage in the market, a worker employed at such a wage never quit

and enjoy expected lifetime payo¤ V = u(w)=� � V : Further; �1i = (pi � w)=� > 0

describes the �rm�s limiting (continuation) pro�t. In contrast, if pi < w; type i

�rms pay limiting wage w1i = pi; i.e. they pay marginal product to employees with

(arbitrarily) long tenures. Note that pi < w implies V 1i < V ; and so its employees

always have a strictly positive quit rate. Paying marginal product also implies zero

(continuation) pro�t �1i = 0 at arbitrarily long tenures:

Claim 1 now establishes these limiting contracts are jointly e¢ cient.

Claim 1

Fix an F satisfying A2. For any type i �rm o¤ering starting payo¤ V0 = V 1i ; the

optimal wage contract is w(t) = w1i for all t � 0:

Proof:

See Appendix.

Let V �p (tjV0) denote the expected lifetime utility of an employee with tenure

t given the optimal contract, i.e. V �p (tjV0) � V (tjw�p(:jV0)):We now describe an

optimal contract. As �rms make strictly positive pro�t, for each p = pi we only

consider starting payo¤s V0 < V 1i :

Theorem 1

Fix an F satisfying A2 with Vu � V and a productivity p = pi > b: Given any

starting payo¤V0 2 [V ; V 1i ); the optimal wage-tenure contract w

�p and corresponding

worker and �rm payo¤s fV �p ;�

�pg are solutions to the di¤erential equation system

fw; V;�g:

(6)�u00(w)u0(w)2

dw

dt= �F 0(V )�;

(7)dV

dt= �u0(w)d�

dt

(8) [� + �(1� F (V ))]�� d�

dt= [p� w];

10

subject to the boundary conditions: (a) limt!1fw(t); V (t);�(t)g = (w1i ; V 1i ;�1i );

and (b) the initial condition V (0) = V0

As the proof is almost identical to the one given in B=C, we omit it here.1 An

important di¤erence, however, is the boundary condition (a). The homogeneous

�rms case considered in B=C implies w < p for all �rms and so (w1i ; V1i ;�1i ) �

(w; V ; (p � w)=�): With heterogeneous �rms it can happen that pi < w for some i

and so these �rms pay limiting wage pi < w: In both cases, however, Claim 1 implies

the limiting contract is jointly e¢ cient. By promising higher earnings in the future,

the �rm reduces employee quit rates and so increases joint surplus (noting that a

quit is jointly ine¢ cient when the outside o¤er has value V0 < V 1i ).

The limit point (w1i ; V1i ;�1i ) is a stationary point of the di¤erential equation

system (6)-(8). It follows that the optimal wage contract corresponds to the saddle

path converging to that stationary point. For �rms with productivities pi > w;

limV!V F0(V ) > 0 would imply the optimal wage contract w�pi(:) reaches w at a

�nite tenure. In a steady state, however, this would imply a positive mass of workers

with lifetime utility V = V : Claim 2 below establishes that this cannot occur in a

market equilibrium. Anticipating that result, equilibrium requires limV!V F0(V ) = 0

and w�pi(:) then only converges to w asymptotically.

Given an F satisfying A2, the type i baseline salary scale is de�ned as the optimal

wage/tenure contract, wi(:); of the type i �rm o¤ering the lowest initial wage. As the

baseline salary scale is a increasing function of t which converges (asymptotically)

to V 1i ; the o¤er made by any type i �rm, w�pi(:jV0); is related to the baseline salary

scale by a t0 � 0 such that w�pi(tjV0) � wi(t + t0) for. all t0 � 0: Note, this also

implies �rm i�s continuation payo¤ ��pi(tjV0) � �i(t+ t0):

The next section now characterizes a market equilibrium where �rms not only

choose an optimal wage tenure contract given V0; but also choose V0 to maximize

1A proof is available in ken4.tex which we should post on the web

11

steady state pro�t �i (V0).

4 MARKET EQUILIBRIUM

Claim 2 establishes there cannot be mass points in F and G at V = V and therefore

a market equilibrium implies wi(:) only converges to w asymptotically (for pi > w).

Claim 2

At a market equilibrium 1 � G and F do not have a mass point at V ; and so

limV!V F0(V ) = 0:

Proof:

See Appendix.

Lemma 1 establishes an essential property implied by the conditions of Theorem

1.

Lemma 1.

For any i; j with b < pi < pj; and for any V0 2 [V ; V 1i ) :

(i)

w�pi(0jV0) > w�pj(0jV0)

(ii)@��pi(0jV0)

@V0<@��pj(0jV0)

@V0< 0:

Proof:

See Appendix.

For a given starting payo¤ V0, Lemma 1(i) establishes that a lower productivity

�rm o¤ers a higher starting wage given such an o¤er yields the same expected return.

Lemma 1(ii) establishes that for any starting payo¤ V0; the marginal loss in pro�t

per new hire by increasing V0 is smaller for the higher productivity �rm. This re�ects

that a worker with a higher expected lifetime payo¤ has a lower quit rate and the

marginal return to retaining that worker is greater for the high productivity �rm.

12

Armed with Lemma 1(ii), Proposition 1 now establishes that in equilibrium, higher

productivity �rms o¤er higher expected payo¤ to workers.

Proposition 1

A market equilibrium implies V i = V i+1; i.e. the support of F can be partitioned

into n sets, where type i �rms o¤er V0 2 [V i�1; V i] where V � V 1 � V 2 � :::: �

V n = V :

Proof:

A �rm with productivity p = pi chooses V0 to maximize

�i (V0) = �G(V0)��(0; pijV0)

Lemma 1(ii) establishes that the marginal return to increasing V0 is strictly greater

for a higher productivity �rm. Hence for a given G(:); optimality implies a higher

productivity �rm chooses a higher V0: Connectedness of F (assumption A2) then

implies the Proposition.

Having identi�ed the basic equilibrium market structure, for ease of exposition

the remainder of the paper focusses on the two types case.

5 Two Types

Assume now only two types of �rms - type 1 �rms and type 2 �rms with p1 < p2:

From the above it follows that in any equilibrium the distribution of initial o¤ers

made by type 1 �rms has support [V 1; V 1], whereas the support of the distribution

of initial o¤ers made by type 2 �rms is [V 2; V 2]:Proposition 2 implies V= V 1 �

V 1 = V 2 � V 2 = V :

Assume for the moment an equilibrium implies that p1 < w < p2; where u(w) =

�V From (5) it follows limt!1w1(t) = w11 = p1: Hence, o¤ering initial o¤er w11

yields zero pro�t for a type 1 �rm and therefore cannot be part of an equilibrium.

13

This implies not all points on the the type 1 baseline salary scale are initial o¤ers

in equilibrium. This implies V 1 = V 2 < V 11 : Hence, although type 1 �rms do

not make initial o¤ers that yield V 2 (V 1; V11 ], type 1 employees. with tenure

great enough can obtain such a return. It is essentially the same if the equilibrium

implies w < p1: In this case (5) implies limt!1wk(t) = w; k = 1; 2: This implies

V 1 = V 2 < V 11 = V 2:Again, although no type 1 �rm makes an initial o¤er that

yields return V 2 (V 1; V11 ]; some type 1 employees with large enough tenure obtain

this return. In what follows we term V 2 [V 11; V ]; Phase A; V 2 [V 1; V

11); Phase

B; and V 2 [V ; V 1); Phase C: .

In what follows the following objects play an important role. For i = 1; 2 :

(i) bti(V ) is the inverse function of Vi(t); i.e. bti describes the salary point t at whichan employed worker enjoys lifetime payo¤ V = Vi(t) along baseline salary scale i: Of

course, bti is a positive and strictly increasing function of V for V 2 [V i; V1i );

(ii) bwi(V ) = wi(bti(:)) is the wage paid on baseline salary scale i when an employee�slifetime payo¤ is V ;

(iii) b�i(V ) = �i(bti(:)) is a type i �rm�s expected pro�t when any employee�s lifetimepayo¤ is V ;

(iv) bNi(V ) is the number of employed workers with payo¤ strictly less than V on

baseline salary scale i and note G(V ) � U + bN1(V ) + bN2(V ):These functions are central to the equilibrium analysis as workers compare lifetime

payo¤V when deciding whether to quit or not. Claim 3 now characterizes (bti; bwi; b�i)and Claim 4 characterizes bNi:Claim 3

Fix an F satisfying A2 with Vu � V : The type i baseline salary scale implies

( bwi; b�i) are solutions to the di¤erential equation systemd bwidV

=

�u0( bwi)�u00( bwi)

��F 0(V )b�ih

pi � bwi � [� + �(1� F (V ))]b�ii

14

db�idV

= � 1

u0( bwi)subject to the boundary condition ( bwi; b�i) = (w1i ;�1i ) at V = V 1

i :

Further, bti is given bydbtidV

=1

u0( bwi)[pi � bwi � [� + �(1� F (V ))]b�i]and bt(V i) = 0:

Proof :

This follows directly from Theorem 1 and the de�nitions of (bti; bwi; b�i).Claim 4.

Optimal job search in a Market Equilibrium implies:

(i) U = �=(�+ �);

(i) During Phase A (where V 2 [V 11; V ]); bN2 satis�es

d bN2dV

=h�(N2 � bN2)� ��2(1� F2(V ))(1�N2 + bN2)i dbt2

dV

where N2 = bN2(V );(iii) During Phase B (where V 2 [V 1; V

11));

d bN1dV

=

��1�U � � bN1 � �2�

Z V

V

[1� F2(x)]d bN1(x)� dbt1dV

;

d bN1dV

=dbt1dV

+d bN2dV

=dbt2dV

= �U [�1 + �2F2(V )]� [� + ��2[1� F2(V )]]h bN1 + bN2i ;

subject to bN2(V 1) = 0:

(iv) During Phase C (where V 2 [V ; V 1)); bN2 = 0 and bN1 satis�esd bN1dV

=hU��1F1(V )� [� + �(1� �1F1(V ))] bN1i dbt1

dV

subject to bN1(V ) = 0;15

Proof

See Appendix.

In Phase A only type 2 �rms o¤er contracts that yield such a payo¤ . No type

1 �rm o¤er yields such an expected return - no matter what the tenure. In Phase

B (where V 2 (V 1; V11)):only type 2 �rms make starting o¤ers that yield such a

return. Although all initial o¤ers by type 1 �rms yield an expected lifetime payo¤

V � V 1; type 1 �rms increase wages with tenure and this implies some employees

of type 1 �rms with signi�cantly long tenure that have a such lifetime payo¤. This

has an important implication that some employees of type 1 �rms may receive an

o¤er from a type 2 �rms that they reject. This generates a complicated interaction

between the two types of �rms. Claim 4 (iv) describes these dynamics. Finally,

Phase C where V 2 [V ; V 1]: Here only type 1 �rms make o¤ers that yield such an

expected payo¤ to workers. Claim 4 (ii) describes turnover in this phase.

As F has connected support, Proposition 1 and the constant pro�t condition

[ME6] requires

�G(V )b�1(V ) = 1 > 0 for all V 2 [0; V 1]

�G(V )b�2(V ) = 2 > 0 for all V 2 [V 1; V ](9)

where �G(V ) is the hiring rate of a �rm that o¤ers starting payo¤ V: Note that

at V = V ; Claim 2 implies G = 1 while Claim 3 implies b�2 = �12 : Hence 2 =

�(p2 � w)=� and so the constant pro�t condition for type 2 �rms implies

(10) G(V )b�2(V ) = (p2 � w)=� for all V 2 [V 1; V ]:

Proposition 2 now establishes that V = Vu; i.e., type 1 �rms o¤ering the least gen-

erous wage contract in the market extract all the surplus from unemployed workers:

This leads to the following characterization of a market equilibrium.

Proposition 2

16

Necessary and su¢ cient conditions for a market equilibrium are a vector of

functions fbti; bwi; b�i; bNi; Fg for i = 1; 2 satisfying the conditions of Claims 3, 4 andthe constant pro�t condition (9), and

(a) F (V ) = G(V ) = 1 with V < u(p2)=�;

(b) V 1 = V 2;

(c) Vu = V where

�Vu = u(b) + �

Z V

Vu

[x� Vu]dF (x):

Further, bti; bNi must be positive and increasing functions, G must have the propertiesof a distribution function and F must have properties A2:

Proof

See Appendix

5.1 Identifying a Market Equilibrium.

We now solve the conditions of Proposition 2 in two steps. The �rst step �xes a

trial value w 2 [b; p2] and corresponding V = u(w)=�: Ignoring for the moment the

equilibrium search condition (c), that Vu = V where

�Vu = u(b) + �

Z V

Vu

[x� Vu]dF (x);

we �rst solve the other conditions of Proposition 2 recursively, starting at V =

V with bw2 = w: Conditional on a trial value w; backward induction identi�es a

candidate equilibrium and corresponding distribution function F; which we might

denote F c(:jw): If this distribution function in addition satis�es

�V = u(b) + �

Z V

V

[x� V ]dF c(xjw);

then Vu = V and we have also identi�ed a Market Equilibrium. Hence the second

step searches over w 2 [b; p2] for an implied F c(:jw) which satis�es this equilibrium

criterion.

17

5.2 A Candidate Equilibrium (given w)

Suppose w � p1: As V 11 = V this case implies Phase A is degenerate. For ease

of exposition, however, and noting that the simulations only report cases where a

Market Equilibrium implies w > p1; we shall focus on the case w 2 (p1; p2) and hence

V 11 < V : Thus type 2 (high productivity) �rm pay limiting wage w12 = w while

type 1 (low productivity) �rms pay limiting wage w11 = p1: Given w 2 (p1; p2) �xed;

the (candidate) equilibrium functions f bwi; b�i; bNi; Fg are identi�ed using backwardinduction, starting at w = w and V = V = u(w)=�:

Phase A. V 2 [V 11; V ]:

During this phase, type 1 �rms are not active and equilibrium determines ( bw2; b�2; F;G)where G = U + bN1 + bN2 with bN1 = bN1(V 1

1 ). Theorem 2 in B=C describes a closed

form solution for this phase. Consistent with those equations, it can be shown that

( bw2; b�2; F ) evolve in Phase A according to the di¤erential equations for bw2; b�2 givenin Claim 3 (with i = 2) and

(11)dF

dV=�u00( bw2)u0( bw2)2

2(p2 � bw2) hp2 � bw2 � [� + �(1� F )]b�2i�b�22

subject to initial values ( bw2; b�2; F ) = (w; (p2 � w)=�; 1) at V = V : We discuss the

derivation of dF=dV below. Note this initial value problem implies ( bw2; b�2; F ) arefully determined in Phase A given the trial value w: G is then determined by the

constant pro�t condition written as G = (p2 � w)=(�b�2):Phase A ends, and Phase B begins, at V = V 1

1 : The de�nition of V 11 implies

�V 11= u(p1) + �

Z V

V11

[x� V 11 ]dF (x):

Now at salary point t on the type 2 baseline salary scale, an optimal quit strategy

implies equation (1) holds, which is rewritten here as

18

�V2 �dV2dt

= u(w2) + �

Z V

V2

[x� V2]dF (x):

Using (7) and (8) in Theorem 1 to substitute out dV2=dt, then setting V2(t) = V 11

implies

(12) u(p1) = u(w2) + u0(w2)[p2 � w2 � [� + �(1� F )]�2]:

Hence using backward iteration from ( bw2; b�2; F ) = (w; (p2�w)=�; 1); phase A stopswhen ( bw2; b�2; F ) satis�es (12) and V2(t) = V 1

1 at that point. It can be shown this

termination point always exists and is unique and we let bwA2 ; b�A2 ; FA; and GA

denote the corresponding values of bw2; b�2; F; and G at the termination point. The

(candidate) solution is relevant only if it implies FA > �1 (some type 2 �rms make

o¤ers V 2 [V 1; V11 ]) and GA > U (some workers are employed in type 1 �rms). In

that case we proceed to Phase B. This completes the description of Phase A.

Phase B: V 2 [V 1; V11 ]

Phase B jointly determines {bw1; bw2; b�1; b�2; bN1; bN2; Fg: A major complication isa singularity in the di¤erential equations at V = V 1

1 : Appendix B provides a formal

analysis of the equilibrium conditions. Here we provide an intuitive interpretation.

De�ne � = d bN1=dV which is the density of type 1 employees with expected

lifetime value V: Appendix B establishes that equilibrium in Phase B can be reduced

to an autonomous di¤erential equation system {bw1; bw2; b�1; b�2; Fg where bw1; bw2; b�1;and b�2 evolve according to the di¤erential equations stated in Claim 3,

(13)dF

dV=

2H[u( bw1)�u( bw2)]u0( bw1)[p1� bw1�H b�1]�+ 2(p2� bw2)(p2�w)

�b�32u0( bw2)[

hu0( bw1)2�u00( bw1)

i�b�1[u( bw1)�u( bw2)]

u0( bw1)[p1� bw1�H b�1]2 +

hu0( bw1)2�u00( bw1)

i�b�1

[p1� bw1�H b�1]�hu0( bw2)2�u00( bw2)

i�b�2

[p2� bw2�H b�2] ]�+�(p2�w)

hu0( bw2)�u00( bw2)

i�b�2[p2� bw2�H b�2]

where � is given by

(14) � =�

u( bw1)� u( bw2)"(p2 � bw2)(p2 � w)

�2b�22 � 1#:

19

and H = [�+�(1�F )]: Note that F 0 has the simple structure F 0 = (a+ b�)=(c+d�)

and so is monotonic in �: Furthermore setting � = 0 (no type 1 employees) yields F 0

as described in Phase A.

Appendix B derives (14) by using Claim 4 to substitute bN2 out of the constantpro�t condition (U + bN1+ bN2)b�2 = (p2�w)=�: Claim 4 also implies the employmentdensity � evolves according to

(15)d�

dV= �

u0( bw1)d bw1dV� 2[� + �(1� F )]

u0( bw1)[p1 � bw1 � [� + �(1� F )]b�1]The denominator in (15) is always strictly positive (as dV1=dt > 0 along the type 1

baseline salary scale). Note, a high density of outside o¤ers (F 0 large) implies type 1

�rms raise wages relatively quickly with tenure ( d bw1=dV large) and (15) then impliesthe density of type 1 employees (�) falls less quickly with V: Indeed, (15) implies the

employment density � decreases with V if and only if the density of outside o¤ers

F 0(V ) is not too high.

Note, (14) determines � consistent with the constant pro�t condition for type 2

�rms, while (15) describes how � evolves with V with steady state turnover and op-

timal wage setting by type 1 �rms. Equilibrium requires these objects are consistent.

Hence di¤erentiating (14) with respect to V to obtain d�=dV; setting this equal to

d�=dV given by (15), and then using Claim 3 to substitute out the d bw1=dV; d bw2=dV;and db�2=dV yields (13) and so determines the equilibrium F 0(V ).

It is important to notice that (13) is singular at V = V 11 (where bw1 = p1; b�1 = 0)

and there is a continuum of solutions to the above di¤erential equation system.

Indeed Phase A with � � 0 describes such a solution but does not describe a Phase

B equilibrium (where � > 0): To characterize the limiting equilibrium in Phase B as

V ! V 11 ; de�ne

fA =

��u00( bwA2 )u0( bwA2 )2

� 2(p2 � bwA2 ) hp2 � bwA2 � [� + �(1� FA)]b�A2 i�[b�A2 ]2

20

which is F 0(V ) according to Phase A at V = V 11 (i.e. put � = 0 and (bw2; b�2; F ) =

( bwA2 ; b�A2 ; FA) in (13)). Also de�ne the critical density f c = f c(F ) :

f c =2

�

��u00(p1)u0(p1)2

�[� + �(1� F )]2:

Note that if �=(p1 � bw1)!1 as V ! V 11 ; then (13) implies F 0 ! f c.2

Appendix B establishes there are three possible types of limiting equilibrium in

Phase B where existence of each type depends on the magnitude of the density fA

relative to f c:When fA=f c is small we say there is low poaching intensity, meaning

that there are relatively few type 2 �rms trying to attract type 1 employees at

V = V 11 :

Case 1: Low Poaching Intensity (fA=f c < 3=8):

If the poaching intensity of type 2 �rms for type 1 employees is small (i.e. fA is

small), then a Phase B equilibrium potentially exists where F 0(V )! fA as V ! V 11 :

Given this low density of outside o¤ers, type 1 �rms raise wages relatively slowly with

tenure and steady state turnover implies �! 0 as V ! V 11 : Appendix B establishes

this equilibrium exists only if fA=f c < 3=8 as this ensures �=(p1 � bw1) ! 0 as

V ! V 11 (i.e. � falls very quickly with V ) and (13) then implies F 0(V ) ! fA as

required: Hence for fA=f c < 3=8; a Phase B equilibrium (potentially) exists where

{bw1; bw2; b�1; b�2; Fg evolve according to the di¤erential equations described abovewith initial values

limV!V11

( bw1; bw2; b�1; b�2; F ) = (p1; bwA2 ; 0; b�A2 ; FA);and F 0 is continuous across V 1

1 :

Case 2. Intermediate Poaching Intensity (fA=f c < 1):

A second type of limiting equilibrium is that F 0(V ) ! f c(FA) as V ! V 11 :

Relative to case 1, the higher density of outside o¤ers (f c > fA) implies type 1 �rms

2this requires using l�Hopital�s rule and (32) in the appendix.

21

raise wages more quickly with V: The di¤erence is that � now converges to zero

slowly and instead �=(p1 � bw1) ! 1 as V ! V 11 : (13) then implies F 0 ! f c(FA)

as required. Note that this equilibrium does not exist if fA > f c(FA) (otherwise

(14) and F 0(V 11 ) = f c < fA implies � < 0 at V = [V 1

1 ]�): Also note there may

be multiple equilibria for fA < (3=8)f c(FA) as both cases 1 and 2 apply. The

multiplicity arises as a higher density of outside o¤ers imply type 1 �rms raise wages

more quickly with V which implies a higher density of type 1 employees around V 11 .

This region then becomes a more pro�table place for type 2 �rms to try to poach

type 1 employees.

Hence for fA < f c(FA); a Phase B equilibrium (potentially) exists where {bw1; bw2; b�1; b�2; Fgevolve according to the di¤erential equations described above with initial values

limV!V11

( bw1; bw2; b�1; b�2; F ) = (p1; bwA2 ; 0; b�A2 ; FA)and limV!V11 F 0(V ) = f c(FA) > fA:

Case 3. High Poaching Intensity (fA=f c > 1):

Neither Case 1 nor 2 holds when fA=f c > 1: A high poaching intensity instead

implies �! � > 0 as V ! V 11 and equilibrium now implies a mass point m > 0 in

F at V 11 : The mass point m > 0 and Claim 3 imply bw2 is not continuous at V 1

1 - it

increases by a discrete amount across V 11 : Let bwB2 = limV![V11 ]� bw2(V );which is the

limiting wage bw2 according to phase B: Appendix B establishes that(16) u( bwB2 ) + u0( bwB2 )[p2 � bwB2 � [� + �(1� F )]b�A2 ] = u(p1)

with F = FA �m: Contrasting with (12) which determines bwA2 at the end of PhaseA, it can be shown that m > 0 implies bwB2 < bwA2 :To understand (16), note that at any salary point t on a type i baseline salary

scale, an optimal quit strategy implies (1) which is rewritten here as

(17) �

Z V

Vi(t)

[x� Vi(t)]dF (x) = �Vi(t)�dVi(t)

dt� u(wi(t)):

22

Now Theorem 1 describes the optimal (pro�t maximizing) baseline salary scale. Us-

ing (7) and (8) in Theorem 1 to substitute out dVi=dt in (17) yields:

(18) �Z V

Vi(t)

[x�Vi(t)]dF (x) = �Vi(t)�u0(wi(t))[pi�wi� [�+�(1�F )]�i]�u(wi(t)):

The left hand side of this expression is the expected surplus from receiving outside

job o¤ers. Note that this surplus, S(Vi) = �R VVi[x � Vi]dF (x); is continuous across

any mass points. Hence at salary point t = t2 on baseline salary scale 2 where

V2(t) = V 11 ; the right hand side of the above expression must be continuous in t;

even with a mass point m > 0: Note further that V = V 11 implies w1 = p1 and

�1 = 0 on baseline salary scale 1. Hence at Vi(t) = V 11 ; the above expression implies

u(p1) = u(w2) + u0(w2)[p2 � w2 � [� + �(1� F )]�2];

which determines both bwA2 (at t = t+2 where F = FA) and bwB2 (at t = t�2 with

F = FA �m):

Given m > 0 and bwB2 given by (16), (14) implies(19) � =

�

u(p1)� u( bwB2 )"(p2 � bwB2 )(p2 � w)

�2[b�A2 ]2 � 1#:

Note that m = 0 implies bwB2 = bwA2 and Theorem 2 in B/C then implies � = 0:

Hence a Phase B equilibrium with � > 0 necessarily requires m > 0: Given such an

m; {bw1; bw2; b�1; b�2; Fg evolve according to the di¤erential equations described abovewith initial values

limV!V11

( bw1; bw2; b�1; b�2; F ) = (p1; bwB2 ; 0; b�A2 ; FA �m)

and limV![V11 ]� F0(V ) = f c(FA �m):

The above describes three possible types of limiting equilibrium in Phase B as

V ! V 11 : It is explicit in case 3 that there is a continuum of possible solutions:

there is a free choice of m: There is a continuum of solutions in the �rst two cases

23

as well - the equilibrium dynamics imply � ! 0 with limiting solution of the form

� = A[V 11 � V ]a where A is the constant of integration and is not determined.

This degree of freedom is tied down by the following boundary condition. Noting

that Phase B begins with FA > �1 and GA > U; using backward iteration from

V = V 11 ; Phase B ends when either F = �1 (there are no more type 2 �rms and

G � U) or G = U (there are no more type 1 employees and F � �1): In either case,

Claim 4 implies � = �B where

�B =� � (� + �2�)G

u0( bw1)[p1 � bw1 � [� + �(1� F )]b�1]and G = (p2 � w)=(�b�2):Assuming a Phase B equilibrium can be found satisfying the initial values de-

scribed above and the boundary condition � = �B at V = V 1; we let bwC1 ; b�C1 ; FC ;andGC denote the corresponding values of bw1; b�1; F;and G at the termination point andproceed to Phase C.

Phase C. V 2 [V ; V 1]:

If GC = U then we are done - there is a mass of �rms FC o¤ering a contract with

starting value V = V = V 1; where FC � �1 type 2 �rms o¤er the type 2 baseline

salary scale while all type 1 �rms o¤er the type 1 baseline salary scale.

Suppose instead FC = �1 and GC > U which then implies Phase C is non-

degenerate. The constant pro�t condition for type 1 �rms implies Gb�1 = GC b�C1 : Itthen follows that (bw1; b�1; F ) evolve according to the di¤erential equations given inClaim 3 and

dF

dV=

��u00( bw1)u0( bw1)2

�2�

�

hp1 � bw1 � [� + �(1� F )]b�1i

GC b�C1with initial values (bwC1 ; b�C1 ; �1):3

3Di¤erentiate the constant pro�t condition Gb�1 = GC b�C1 wrt V to get � = G=(u0( bw1)b�1) where

24

Iterating backwards, Phase C ends when G = U (there are no more type 1

employees): At this termination point, de�ne V = V and let bw1; b�1;F denote the

corresponding values of (bw1; b�1; F ):5.3 Equilibrium (determining w).

For a given w; a solution to phases A-C yields end values (bw1; b�1; F ). The free choiceof w is now tied down by the optimal job search condition which requires Vu = V

where

�Vu = u(b) + �

Z V

Vu

[x� Vu]dF (x):

Using (1), the same trick as described in Case 3 above establishes that V1(0) = V =

Vu occurs if and only if

(20) u( bw1) + u0( bw1)[p1 � bw1 � [� + �(1� F )]b�1] = u(b):

In other words, given end values (bw1; b�1; F ); we have identi�ed a Market Equilibriumif b satis�es (20). Alternatively, given an arbitrary value of b, identifying a Market

Equilibrium requires �nding a w which implies end values bw1; b�1;F satisfying (20).

B/C formally establish the existence of such a w for the homogeneous �rm case.

Given the complexity of Phase B with heterogeneous �rms, we do not provide an ex-

istence proof. Instead we use numerical examples to illustrate how �rm heterogeneity

a¤ects equilibrium market behavior.

� � dG=dV in this phase. Claim 4 also implies

� =� � [� + �(1� F )]G

u0( bw1)[p1 � bw1 � [� + �(1� F )]b�1] :Setting these two expressions equal and simplifying yields

p1 � bw1 = �b�1G:

Now di¤erentiate wrt V: Using Claim 3 for d bw1=dV implies the stated di¤erential equation for F 0.

25

6 Simulations

The object here is to present simulations of equilibria in the model presented above.

In Appendix C a brief description of the program constructed to achieve this goal is

presented. Here, we report on two experiments in the �rst we consider changes in

the productivity of low productivity �rms. In the second experiment we consider

how equilibria changes as the proportion of low productivity �rms changes.

In the �rst experiment we set �=� = 20 so that a worker expects to receive 20 job

o¤ers over a working lifetime. Assuming a working career of 40 years, this implies an

outside job o¤er on average every two years. This implies a sensible unemployment

rate U = 4:7%. The implied unemployment durations though are very high, being

two years. Discounting between job o¤ers is therefore high, �=� = 5%: with � = 0:025

and � = 0:5. We assume b = 43:75 and �1 = 0:50:A CRRA utility function is used

u(w) = w1��=(1� �) with � = 1:2:

In the experiment we vary p1 holding p2 = 100: The results are presented in Table

1 and Figures 1 and 2. Only when p1 = 55 is a Case 1 equilibrium possible. A Case

2 equilibrium is feasible when p1 = 55; 65; and 75:When p1 = 85 or p1 = 95;only a

Case 3 equilibrium is possible

In the second experiment we vary the proportion of low productivity �rms. In this

case we assume the value of the parameters are b = 60; � = 0:5; � = 0:025, p1 = 90;

p2 = 100; Again we assume a CRRA utility function with parameter � = 1:2: The

results are presented in Table 2 and Figures 3 and 4. In each case only a Case 3

equilibrium is feasible.

Appendix A - Proofs.

Proof of Claim 1

Suppose �rst pi � w, which implies w1i = w and V 1i = V : However, pi � w

implies it is jointly e¢ cient for the �rm and worker that the worker never quits

26

Now consider wage contract w(t) = w for all t: The Bellman equation (1) implies

V (tjw(:)) = V for all t; and so the worker never quits. As this contract also o¤ers full

insurance and not quitting is incentive compatible, this contract is therefore jointly

e¢ cient. As this contract also extracts maximal employee rents - it o¤ers expected

lifetime payo¤ V0 = V 1i as required - it describes the �rm�s optimal contract.

Consider now pi < w which implies w1i = pi and V 1i < V . The wage contract

w(t) = pi for all t; implies the �rm makes zero pro�t and so the worker enjoys the

full value of the match. Hence, the worker has e¢ cient quit incentives - the worker

will quit only if the value of an employment o¤er elsewhere exceeds the value of the

current match. As side payments conditional on a quit are ruled out by assumption,

this contract is therefore jointly e¢ cient. As it also extracts maximal employee rents

(given V0 = V 1i ) it describes the �rm�s optimal contract. This completes the proof

of Claim 1.

Proof of Claim 2

1�G(V ) describes the steady state number of workers employed whose current

lifetime utility is at least V: The proof is by contradiction - suppose in equilibrium

that 1 � G has a mass � > 0 at V : Strictly positive pro�t requires that any �rm

which in equilibrium o¤ers starting payo¤ V0 = V must have productivity pi > w:

Further, Claim 1 implies that any such �rm enjoys steady state pro�t �ow �i (V ) =

�(1 � �)[(pi � w)=�]: Consider instead a �rm of that type but which instead o¤ers

w = w+ " for some " > 0 and so obtains �ow pro�t = �[(pi�w� ")=�]: As w < pi

then � > 0 implies > �i (V ) for " small enough which contradicts the de�nition of

a market equilibrium. As 1�G cannot have a mass point at V ; then Claim 1 implies

F cannot have a mass point at V : Finally, no mass point in G at V and Theorem 1

requires the stated restriction on F: This completes the proof of Claim 2.

Proof of Lemma 1.

Part (i). Fix a p > b and a V0 2 [V ; V 1p ): Now consider a V 2 [V0; V 1

p ) and de�nebt(V; pjV0) as the tenure point t where V �(t; pjV0) = V ;i.e. bt is the tenure point27

where the employed worker has lifetime value V given the optimal wage contract:

V 2 [V0; V 1p ) implies bt � 0 and is strictly increasing with V . For such V we can

de�ne bw(V; pjV0) = w�(bt(:); pjV0)which is the wage paid at tenure t = bt(:); i:e: bw(:) is the wage paid at the tenure pointwhere the worker obtains lifetime value V: Also de�ne b�(V; pjV0) = ��(bt(:); V jV0):As V0 is �xed throughout this proof, we simplify notation by subsuming reference to

V0 in these functions.

For given p and V0; the time pro�les of w�; V � are determined by (6) and (7) in

Theorem 1. In particular,

(21)@ bw@V

=

�dw=dt

dV=dt

�t=bt =

�F 0(V )b�(:);h� u00( bw(:))u0( bw(:))2

i hp� bw(:)� [� + �(1� F (V ))]b�(:)i ;

describes how wages paid bw change as the worker�s expected lifetime value V increaseswith tenure along the optimal contract. Hence for given p; a �rst order di¤erential

equation describes bw; where Claim 1 implies the boundary condition bw(V 1p ; p) = w1p :

Note (7) in Theorem 1 also implies @b�=dV = �1=(u0( bw)):The proof uses the following single crossing property. Given productivities p0 >

p > b; consider functions bw(:; p); bw(:; p0) de�ned above. Suppose a V 2 [V0; V1p )

exists where bw(V; p) = bw(V; p0):We now establish that at any such point (should oneexist), then

@ bw@V(V; p0) >

@ bw@V(V; p);

i.e. these functions can only cross once.

Hence suppose a V 2 [V0; V 1p ) exists where bw(V; p) = bw(V; p0) and �x w = bw at

that point. Given (w; V ); partial di¤erentiation of the right hand side of (21) with

respect to p yields

@

@p

24 �F 0(V )b�(V; p);h� u00(w)u0(w)2

i hp� w � [� + �(1� F (V ))]b�(V; p)i

35 > 028

if and only if

(p� w)@b�(V; p)

@p� b�(V; p) > 0:

Now the de�nition of b� implies:b�(V; p) � �(bt; p) = Z 1

�=bt(� jw�)(btjw�) [p� w�(� ; p)]d� :

Optimality of w� (for given V ) and the Envelope Theorem yield

@b�(V; p)@p

=

Z 1

s=bt(� jw�)(btjw�) d�

as any variations in the optimal choice of w�(:) yield only second order pro�t e¤ects.

Hence

[p� w]@b�(V; p)

@p� b�(V; p) = Z 1

�=bt(� jw�)(btjw�) [w�(� ; p)� w�(bt; p)]d� ;

as w = bw(V; p) � w�(bt; p): But w� is strictly increasing in tenure and as (:) > 0 forall tenures (the worker never quits into unemployment), we have that

[p� w]@b�(V; p)

@p� b�(V; p) > 0 for all V 2 [V0; V 1

p ):

The above now yields the desired single crossing property: if a V 2 [V0; V 1p ) exists

where bw(V; p) = bw(V; p0); then at that point, (21) implies(22)

@ bw@V(V; p0) >

@ bw@V(V; p):

Now use backward induction from V = V 1p ; where Claim 1 implies bw(V 1

p ; p) =

w1p : Suppose �rst that p < w which implies w1p < w and V 1p < V : As p0 > p implies

V 1p0 > V 1

p then, as wages are increasing in tenure, we have bw(V 1p ; p0) < w1p : Hencebw(V; p0) < bw(V; p) at V = V 1

p : (22) now implies these two functions cannot meet at

any V 2 [V0; V 1p ); and so bw(V0; p0) < bw(V0; p): This establishes part (i) for p < w:

Suppose now that p � w which implies w1p = w and V 1p = V : Hence for p0 > p we

have bw(V 1p ; p0) = bw(V 1

p ; p) = w: Noting that Claim 2 implies F 0(V ) = 0; L�Hopital�s

29

rule at V = V implies

(23)@ bw@V

=�F 00(V )�;h

� u00(w)u0(w)2

i �� @ bw@V+ �=u0(w)

� :Solving implies a quadratic for @ bw=dV: There are two solutions but the only relevantone, the one which implies @ bw=dV > 0 (and so corresponds to the stable manifold),

yields

u0(w)@ bw@V(V ; :) =

1

2[� +

q�2 + 4�(p� w)u0(w)4(�F 00(V ))=[��u00(w)]]

where given F 0(V ) = 0; A2 requires F 00(V ) < 0: Hence for p0 > p � w; we have

@ bw(V ; p0)=@V > @ bw(V ; p)=@V: (22) now implies bw(V0; p0) < bw(V0; p) for V0 < V 1p

which completes the proof of part (i).

Part (ii). Fix p > b: Theorem 1 describes the optimal time pro�les of V �;��; and so

@b�(V; p)@V

��d�=dt

dV=dt

�bt

where pro�t b� changes as the worker�s expected lifetime value V increases with

tenure along the optimal contract, and d�=dt; dV=dt are de�ned by (7) and (8) in

Theorem 1. Putting V = V0 and using (7) implies

@b�(V0; p)@V

=

�d�=dt

dV=dt

�bt=0 = �1=u

0(w�(0; pjV0)):

Part (i) above now completes the proof.

Proof of Claim 4.

(i) As positive pro�t implies Vu � V ; an unemployed worker accepts the �rst job

o¤er received, which implies U = �=(�+ �) in a steady state.

(ii) For V < V 1; steady state turnover on baseline salary scale 1, over arbitrarily

short period of time � > 0; implies

U���1F1(V ) = [ bN1(V )� bN1(V � [dV1(bt1)=dt]�)]+ bN1(V )[� + �[1� �1F1(V )]]� + o(�):

30

where the left hand side describes the in�ow of workers who earn no more than

V;and the right hand side describes out�ow - the �rst term describes those whose

tenure rises above bt1(V ); the second is the out�ow due to exiting the labor marketor receiving a better outside o¤er. Taking the limit � ! 0 implies the condition

stated.

(iii) for V 2 [V 1p1; V ];steady state turnover on baseline salary scale 2, over arbitrarily

short period of time � > 0; implies

[N2 � bN2]�� = [ bN2(V )� bN2(V � [dV2(bt2)=dt]�)]+[U +N1 + bN2(V )]���2[1� F2(V )] + o(�)

where the left hand side describes the out�ow of workers with payo¤ greater than

V; and the right hand side describes the in�ow - those whose tenure rises abovebt2(V );and those who receive an outside o¤er greater than V: Rearranging and letting�! 0 implies the condition stated.

(iv) For V 2 (V 1; V11 ); steady state turnover on baseline salary scale 1, over arbi-

trarily short period of time � > 0; implies

U��1� = [ bN1(V )� bN1(V � [dV1(bt1)=dt]�)]+ bN1(V )��+ Z V

V

d bN1(x)��[�2(1� F2(x))] + o(�):

where the left hand side describes the in�ow of workers onto baseline salary scale

one with payo¤ less than V , and the right hand side describes the out�ow due to

increasing tenure, exiting the labor market or receiving a better outside o¤er from a

type 2 �rm. Rearranging and letting �! 0 implies the �rst condition stated.

For V 2 (V 1; V11 ); steady state turnover implies the total number of employees

with payo¤ less than V satis�es

U�[�1 + �2F2(V )]� = [ bN1(V )� bN1(V � [dV1(bt1)=dt]�)]+[ bN2(V )� bN2(V � [dV2(bt2)=dt]�)]+[ bN1(V ) + bN2(V )][� + ��2(1� F2(V ))]�

31

where the left hand side describes the in�ow, the right hand side describes the out�ow

due to tenure e¤ects (both types), exiting the labor market and receiving an outside

o¤er greater than V: Rearranging and letting � ! 0 implies the second condition.

This completes the proof of Claim 4.

Proof of Proposition 2

Note, given the equilibrium functions f bwi; b�i; bNi;btig; then Vi(t) is identi�ed asthe inverse function of bti: The baseline salary scale wi(t) corresponds to bwi(Vi(t)) andpro�t �i(t) is b�i(Vi(t)): We now prove these conditions are necessary and su¢ cientfor a Market Equilibrium.

Necessary. Strictly positive pro�t for type 1 �rms implies Vu � V : Suppose for

the moment Vu < V : Now consider the conditions of Theorem 1 for a type 1 �rm

(p = p1), but with starting payo¤V0 2 [Vu; V ): The optimal contract implies an early

tenure phase for some � > 0; where the �rm pays a �xed wage w0 2 (0; w1(0)] for

tenures t < �; and wages w = w1(t � �) for tenures t � � : It is straightforward to

argue that this contract generates strictly greater pro�t per hire.4 As the hiring rate

is the same as for a type 1 �rm o¤ering V = V ; this implies �1(V0) > �1(V ) which

contradicts the de�nition of a market equilibrium. Hence Vu = V is a necessary

condition of equilibrium.

Proposition 1 and Claims 2-4 now imply Proposition 2 describes necessary con-

ditions for a market equilibrium.

Su¢ cient. By construction of the type i baseline salary scales, any type i �rm o¤ering

4Using the arguments in the next section (see equation (16)), w0 is in fact given by:

u(w0) + u0(w0)[p1 � w0 � [� + �]�1(0)](24)

= u(w1) + u0(w1)[p1 � w1 � [� + �(1� F (V )]�1(0)](25)

where w1 = w1(0) and F (V ) � 0 is the mass of type 1 �rms o¤ering V = V : As the conditions

of Theorem 1 imply the continuation pro�t �(:) is strictly decreasing along the optimal contract,

then this earlier phase must yield greater pro�t (conditional on a hire).

32

V 2 [V i; Vi] implies the �rm is o¤ering an optimal wage contract, and the constant

pro�t condition (9) guarantees that each optimal contract o¤ered generates the same

steady state pro�t i > 0: Further for type 1 �rms, o¤ering V0 < V = Vu generates

zero pro�t, while Lemma 1(ii) and the constant pro�t condition for type 2 �rms

implies o¤ering V0 > V 1 yields strictly lower pro�t. Similarly for type 2 �rms.

Lemma 1(ii) and the constant pro�t condition for type 1 �rms implies o¤ering V0 <

V 1 yields strictly lower pro�t, while o¤ering V0 > V attracts no more workers than

a contract which o¤ers V = V and pays a strictly higher wage. Hence a solution to

the conditions of Proposition 2 imply conditions (ME1)-(ME6) are satis�ed and so

describes a market equilibrium.

33

Appendix B - Equilibrium Phase B dynamics.

During Phase B, Claim 3 establishes that (bw1; bw2; b�1; b�2) evolve according to(26)

d bw1dV

=

�u0( bw1)�u00( bw1)

��F 0b�1h

p1 � bw1 � [� + �(1� F )]b�1i

(27)d bw2dV

=

�u0( bw2)�u00( bw2)

��F 0b�2h

p2 � bw2 � [� + �(1� F )]b�2i

(28)db�1dV

= � 1

u0( bw1)

(29)db�2dV

= � 1

u0( bw2)Claim 4 implies ( bN1; bN2) evolve according to(30)

d bN1dV

=

��1�U � � bN1 � �2�

Z V

V

[1� F2(x)]d bN1(x)� dbt1dV

;

(31)d bN1dV

=dbt1dV

+d bN2dV

=dbt2dV

= �U [�1 + �2F2(V )]� [� + ��2[1� F2(V )]]h bN1 + bN2i ;

where dbti=dV is given by Claim 3. The constant pro�t condition requires

(constant pro�t) Gb�2 = �2where �2 = (p2 � w)=� and G = U + bN1 + bN2:The objective here is to show this system reduces to the one described in the

text. We do this in three steps. The �rst step derives the reduced form dynamics for

34

(bw1; bw2; b�1; b�2; F ) during Phase B. The second step considers the limiting dynamicsas V ! [V 1

1 ]� and the third step considers the limiting dynamics as V ! [V 1]

+:We

start with two useful algebraic conditions.

Claim B1.

During Phase B:

(32) (i)d

dV

hu0( bwi)[pi � bwi � [� + �(1� F (V ))]b�i]i = � + �(1� F )� u0( bwi)d bwi

dV

(ii) u( bw1)+u0( bw1)[p1� bw1�[�+�(1�F )]b�1] = u( bw2)+u0( bw2)[p2� bw2�[�+�(1�F )]b�2]Proof :

Claim 3 and straightforward algebra establish (i). Claim B1(ii) is a direct corol-

lary of (i). This completes the proof.

Step 1. Characterizing the equilibrium dynamics in Phase B for V 2

(V 1; V11 ):

Di¤erentiating the (constant pro�t) condition Gb�2 = �2 with respect to V yieldsGdb�2dV

+ [d bN1dV

+d bN2dV

]b�2 = 0;which can be rearranged as

� =�2

u0( bw2)b�22 � d bN2dV

where � = d bN1=dV: Now solve (31) for d bN2=dV: Noting that bN1 + bN2 � G � U;

U = �=(�+ �) and G = �2=b�2;(31) implies(33)

d bN2dV

=� � [� + �(1� F )] �2b�2 � u0( bw1)[p1 � bw1 � [� + �(1� F )]b�1]�

u0( bw2)[p2 � bw2 � [� + �(1� F )]b�2] :

Using this condition to substitute out d bN2=dV in the previous expression, simplifyingusing Claim B1(ii) then yields

� =�

u( bw1)� u( bw2)"(p2 � bw2)(p2 � w)

�2b�22 � 1#

35

which is (14) stated in the text.

(30) describes how � changes with V and can be rearranged as

(34)

�[u0( bw1)[p1� bw1�[�+�(1�F (V ))]b�1]] = ��1�U � � bN1 � �2�

Z V

V

[1� F2(x)]d bN1(x)� ;Di¤erentiating with respect to V (use Claim B1(i) or it will take you all day)) then

yields (15).

Di¤erentiating (14) with respect to V to obtain d�=dV; setting this equal to

d�=dV given by (15) and using claim 3 to substitute out the d bw1=dV; d bw2=dV; anddb�2=dV terms yields (13). Hence (13) with � given by (14) and (26)-(29) are neces-

sary conditions describing the equilibrium evolution of bw1; bw2; b�1; b�2; F during PhaseB.

Step 2. The Limiting Equilibrium as V ! [V 11 ]

�:

In the limit as V ! [V 11 ]

�; suppose F 0(V ) ! f1; F (V ) ! F1 � FA and that

a �rst order Taylor expansion for � implies

�(V ) ! A(V 11 � V )a

d�

dV! �aA(V 1

1 � V )a�1

where A > 0 and a; f1; F1 are to be determined.

Note that (26) is singular at V 11 : Let [d bw1=dV ]1 denote the limit value of d bw1=dV

as V ! V 11 : Using Claim B1(i) and l�Hopital�s rule in (26) imply

[d bw1dV

]1 = limV![V11 ]�

d bw1dV

= limV![V11 ]�

�F 0hu0( bw1)2�u00( bw1)

idb�1dV

� + �(1� F )� u0( bw1)d bw1dV

(35)

=��f1

hu0(p1)�u00(p1)

i� + �(1� F1)� u0(p1)[

d bw1dV]1

(36)

Rearranging as a quadratic for [d bw1=dV ]1 and noting that d bw1=dV > 0 along the

baseline salary scale implies

(37) [d bw1dV

]1 =

[� + �(1� F1)] +

r[� + �(1� F1)]2 + 4

hu0(p1)2�f1

�u00(p1)

i2u0(p1)

:

36

A �rst order Taylor expansion about V 11 now implies

bw1 ! p1 + (V � V 11 )[

d bw1dV

]1

as V ! V 11 ; while Claim B1(i) implies

u0( bw1)[p1 � bw1 � [� + �(1� F (V ))]b�1]! (V � V 11 )[� + �(1� F1)� u0(p1)[

d bw1dV

]1]:

Now consider (15) written as

1

�

d�

dV�

u0( bw1)d bw1dV� 2[� + �(1� F )]

u0( bw1)[p1 � bw1 � [� + �(1� F (V ))]b�1] = 0:The above �rst order Taylor expansions yield:

�aA(V 11 � V )a�1

A(V 11 � V )a

�u0( bw1)d bw1dV

� 2[� + �(1� F )]

(V � V 11 )[� + �(1� F1)� u0(p1)[

d bw1dV]1]

! 0

as V ! [V 11 ]

�: Hence

a =2[� + �(1� F1)]� u0(p1)[

d bw1dV]1

u0(p1)[d bw1dV]1 � [� + �(1� F1)]

and so a depends on f1 (via the [d bw1=dV ]1 term).

Now consider f1 as implied by (13). a > 1 implies �=(p1� bw1)! 0 as V ! V 11

and (13) then implies

(38) f1 =2(p2 � bw2)� hp2 � bw2 � [� + �(1� F1)]b�2i

�b�22� h u0( bw2)2�u00( bw2)i :

A little work using Theorem 2 in B/C establishes that if (bw2; b�2; F )!(bwA2 ; b�A2 ; FA)in this limit, then f1 = fA which is the density of o¤ers according to Phase A at

V = V 11 :

If instead a < 1; then �=(p1 � bw1) ! 1 as V ! V 11 : (13) and l�Hopital�s rule

then imply

(39) f1 =2[� + �(1� F1)][u0(p1)[

d bw1dV]1 � [� + �(1� F1)]]

�hu0(p1)2

�u00(p1)

i37

Substituting out [d bw1=dV ]1 and solving for f1 yields

f1 =2

�

�u00(p1)u0(p1)2

[� + �(1� F1)]2:

This critical density is denoted f c(F1) in the text. Further, f1 = f c(F1) implies

a = 0:

The knife-edge case a = 1 implies �=(p1 � bw1)! A � 0 and monotonicity of F 0

wrt � in (13) implies f1 takes an intermediate value between f c(F1) and fA: But

a = 1 also requires2[� + �(1� F1)]� u0(p1)[

d bw1dV]1

u0(p1)[d bw1dV]1 � [� + �(1� F1)]

= 1

and substituting out [d bw1=dV ]1 then implies f1 = (3=8)f c(F1): We are now in a

position to characterize the set of possible limiting equilibria as V ! V 11 :

(i) First suppose there are no mass points in F: This implies (bw2; b�2; F )! ( bwA2 ; b�A2 ; FA)as V ! [V 1

1 ]�: (14) and Theorem 2 in B/C then imply � ! 0 in this limit. Note

also that A > 0 requires f1 � fA (otherwise (14) implies � < 0 at V = [V 11 ]

�):

Now any equilibrium with a > 1 implies f1 = fA: As a is given by

a =2[� + �(1� FA)]� u0(p1)[

d bw1dV]1

u0(p1)[d bw1dV]1 � [� + �(1� FA)]

;

then f1 = fA and a > 1 requires fA < (3=8)f c(FA): Hence a limiting solution

exists when fA < (3=8)f c(FA) where F (V ) ! FA; f1 = fA and �=(p1 � bw1) ! 0

as V ! V 11 :

Any equilibrium with a < 1 implies f1 = f c(FA) which automatically implies

a = 0: Hence, a limiting solution exists when fA < f c(FA) where F (V ) ! FA;

f1 = f c(FA) and �=(p1 � bw1)!1 as V ! V 11 :

Limiting solutions with a = 1may exist but will not generically describe a Market

Equilibrium. To see why, suppose a = 1 and so � ! A(V 11 � V ) as V ! V 1

1 : But

a = 1 requires f1 = (3=8)f c(FA) and (13) then uniquely determines A: As there is

no degree of freedom, this solution will not generically satisfy the boundary condition

� = �B at V = V 1 as discussed in Step 3.

38

(ii) Now consider equilibria with a mass point m > 0 in F at V 11 : Claim 3 impliesbw2 is not continuous across V 1

1 ; but b�2 is continuous and equals b�A2 : Note that (27)implies

d bw2dV

=

�u0( bw2)�u00( bw2)

��b�A2h

p2 � bw2 � [� + �(1� F )]b�A2 idF

dV

at V = V 11 : Integrating across V = V 1

1 ; this di¤erential equation implies

d bw2 = � u0( bw2)�u00( bw2)

��b�A2h

p2 � bw2 � [� + �(1� F )]b�A2 idFwhere F 2 [FA �m;FA] across the corresponding mass point: Inspection establishes

this di¤erential equation has solution:

u( bw2) + u0( bw2) hp2 � bw2 � [� + �(1� F )]b�A2 i = constant.The termination criteria for Phase A implies this constant equals u(p1) at F = FA:

Hence at F = F1 = FA �m; bw2 = bwB2 is given byu( bwB2 ) + u0( bwB2 ) hp2 � bwB2 � [� + �(1� F1)]b�A2 i = u(p1)

and m > 0 implies bwB2 < bwA2 : (14) now implies �! �(m) > 0 where

� =�

u(p1)� u( bwB2 )"(p2 � bwB2 )(p2 � w)

�2[b�A2 ]2 � 1#:

Note all such equilibrium imply �=(p1 � bw1) ! 1 as V ! V 11 : (13) then implies

f1 = f c(F1), which in turn implies a = 0: Hence given m > 0; a limiting solution

exists where F (V )! FA �m; f1 = f c(FA �m) and �! � as V ! V 11 :

Step 3. The boundary condition at V = V 1:

For V 2 (V 1; V11 ); (30) implies � evolves as:

�(V ) =

��1�U � � bN1 � �2�

Z V

V

[1� F2(x)]�(x)dx

�dbt1dV

:

As F2 = 0 for all V < V 1; taking the limit V ! [V 1]+ implies

39

�(V )!h�1�U � [� + �2�] bN1(V 1)

i dbt1dV

:

But bN2(V 1) = 0 and so G(V 1) = U + bN1(V 1): Noting U = �=(� + �); we therefore

have the boundary condition �! �B as V ! [V 1]+ where

�B =� � (� + �2�)G

u0( bw1)[p1 � bw1 � [� + �(1� F )]b�1]at V = V 1: This completes the characterization of the equilibrium conditions for

Phase B.

Appendix C - Simulation Program

We used an adaptive step size Runge-Kutta methods to solve for the equilibria

described in the body of the paper Subroutines: odeint, rkqs, and rkck from Nu-

merical Recipes in Fortran 90, along with the necessary modules nr, nrutil, nrtype,

and nrerror.1 (See http://www.library.cornell.edu/nr/bookf90pdf/chap16f9.pdf)

The program works in the following way

1) The program takes as given the parameters b; p1; p2; �1; �; �; and the constant

relative risk aversion parameter �:

2) For a candidate w, the program solves Phase A according to the appropriate

system of di¤erential equations and initial conditions. It uses a bisection routine to

identify the values of w2, 2, F, and V that coincide with the end of Phase A. The

bisection stops when these computed values are within 1*10-7 of the truevalues.

3) The program then identi�es the case if a case 1 or a case equilibrium is possible.

4) For cases 1 or 2, we assign the appropriate value for dF=dV at V 11 and solve

Phase B according to the appropriate system of di¤erential equations with initial

conditions given by the end of Phase A. For case 3, we search across all values of m

(where F = FA �m) between 0 and FA for a candidate Phase B that satis�es the

given condition on . Note here that each m implies a unique starting value for w2

40

and . Again, we identify the values of w1, w2, 1, 2, F , G, , and V at the end

of Phase B using a bisection routine.

5) Finally, we solve Phase C according to the appropriate system of di¤erential

equations and initial conditions given by the end of Phase B. It uses a bisection

routine to identify the values of w1, 1, F , G, and V that coincide with the end of

Phase C.

6) Given this candidate equilibrium, we calculate the di¤erence between the left and

right side of equation (20) in the text. Again, we use a bisection routine (here with

accuracy 1E-6) to identify the w which minimizes the di¤erence between the left and

right side of equation (20).

REFERENCES

ACEMOGLU, D., and R. SHIMER (1999). �Wage and Technology Dispersion,�

Review of Economic Studies, 68, 585-608.

ALBRECHT, J. and B. AXELL (1984) �An Equilibrium Model of Search Un-

employment,�Journal of Political Economy 92, 824-840.

BONTEMPS, C., J.-M. ROBIN, and G.J. VAN DEN BERG (1999) �An Empirical

Equilibrium Job Search Model with Search on the Job and Heterogenous Work-

ers,�

International Economic Review 40, 1039-1075.

BONTEMPS, C., J.-M. ROBIN, and G.J. VAN DEN BERG (2000) �Equilibrium

Search with Continuous Productivity Dispersion: Theory and Non-Parameteric

Estimation,�International Economic Review 41, 305-358.

BURDETT, K and COLES, M.G. (2003) "Equilibrium Wage/Tenure Contracts,"

Econometrica, Vol. 71. No. 5 (Sept.) 1377-1404.

BURDETT, K. and MORTENSEN, D.T. (1989): �Equilibrium Wage Distributions:

Compensating and Otherwise,�MEDS Working Paper, Northwestern University.

� � � � � � � � � � � � � � � � � - (1998): �Wage Di¤erentials, Employer Size,

41

and Unemployment,�International Economic Review, 39: 257-273.

Cahuc, P., F. Postel-Vinay, and J-M Robin (2006) �Wage Bargaining with on-

the-job Search: Theory and Evidence�, Econometrica, (forthcoming)

DIAMOND, P.A. (1971), �A Model of Price Adjustment�, Jourrnal of Econornie

Theory, 3, 156-168.

ECKSTEIN, Z. and K.I. WOLPIN (1990) �Estimating a Market Equilibrium Search

Model from Panel Data on Individuals,�Econometrica 58, 783-808.

LEONARD, D and N.V. LONG (1992).Optimal Control Theory and Static Opt-

imization in Economics, Cambridge University Press.

MORTENSEN, D.T. (2002) Wage Dispersion: Why are Similar Workers Paid

Di¤erently, mimeo Northwestern University.

POSTEL-VINAY, F., and J.-M. ROBIN (2002a). �The Distribution of Earnings

in an Equilibrium SearchModel with State-Dependent O¤ers and Counter-O¤ers,�

forthcoming in International Economic Review

POSTEL-VINAY, F., and J.-M. ROBIN (2002b). �"Equilibrium Wage Dispersion

with Worker and Employer Heterogeneity.," forthcoming in Econometrica.

SHIMER, R. (1996) �Contracts in a Frictional Labor Market,�mimeo, MIT.

STEVENS, M. (2004) "Wage-Tenure Contracts in a Frictional LabourMarket: Firms�

Strategies for Recruitment and Retention", Review of Economic Studies, v. 71(2),

pp 535-551.

VAN DEN BERG, G.J. (1999) �Empirical Inference with Equilibrium Search Models

of the Labor Market,�Economic Journal 109, F283-F307.

VAN DEN BERG, G.J., and G. RIDDER (1998). �An Empirical Equilibrium Search

Model of the Labor Market,�Econometrica, 66, 1183-1221.

42

Table 1

p1 55.000 65.000 75.000 85.000 95.000 99.000

At V

V 3979.965 3981.757 3983.730 3980.707 3988.030 3991.552

w 99.499 99.544 99.593 99.518 99.701 99.789

� 20.035 18.243 16.270 19.293 11.970 8.448

At {V 11 g+

F 0.535 0.576 0.635 0.779 0.899 0.994

G 0.077 0.077 0.077 0.094 0.102 0.219

w2 15.685 22.073 30.920 45.463 71.230 95.634

At {V 11 g�

m 0.000(1) 0.000(2) 0.000(2) 0.112 0.176 0.064

At V 2

w1 46.581 48.990 50.561 59.304 59.135 41.736

w2 12.340 14.690 17.520 17.770 31.873 35.304

G 0.028 0.070 0.088 0.078 0.059

At V 1

w1 34.647 28.917 24.409 21.825 18.903 12.833

F 0.387 0.308 0.244 0.086 0.049 0.085

G 0.047 0.046 0.0460 0.047 0.047 0.047

43

Table 2

�1 0.100 0.200 0.300 0.400 0.500

At V

V 3993.061 3992.021 3990.482 3988.120 3984.406

w 99.827 99.801 99.762 99.703 99.610

� 6.938 7.979 9.518 11.880 15.593

At {V 11 g+

F 0.688 0.715 0.748 0.783 0.822

G 0.069 0.065 0.071 0.081 0.094

w2 52.392 52.880 53.504 54.310 55.533

At {V 11 g�

m 0.117 0.141 0.161 0.173 0.177

At V 2

w1 42.153 48.776 55.162 61.463 67.324

w2 14.741 17.642 20.165 22.483 24.605

G 0.049 0.055 0.062 0.072 0.086

At V 1

w1 39.585 39.393 38.929 38.301 36.772

F 0.065 0.082 0.107 0.148 0.190

G 0.470 0.048 0.046 0.048 0.048

44

�1 0.600 0.700 0.800 0.900

At V

V 3978.207 3966.893 3943.437 3883.157

w 99.455 99.172 98.586 97.079

� 21.792 33.107 56.563 116.843

At {V 11 g+

F 0.864 0.906 0.947 0.982

G 0.113 0.144 0.201 0.334

w2 60.142 64.907 73.775

At {V 11 g�

m 0.171 0.155 0.125 0.077

At V 2

w1 72.978 78.438 83.269 87.344

w2 26.922 29.869 34.517 44.468

G 0.106 0.137 0.194 0.326

At V 1

w1 34.501 31.829 27.543 24.571

F 0.232 0.273 0.293 0.312

G 0.048 0.048 0.048 0.048

45

Figure 1

p1 = 55

0

20

40

60

80

100

120

3977

3977

3977

3977

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3977

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3977

3978

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3978

3978

3979

3979

3979

3979

3979

3979

3979

3979

3979

3979

3979

V

w

Series1Series2

p1= 65

0

20

40

60

80

100

120

3979

3979

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3979

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3980

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3981

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3981

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V

w

Series1Series2

p1= 75

0

20

40

60

80

100

120

3981

3981

3981

3981

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3981

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3983

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3983

3983

V

w

Series1Series2

P1 = 85

0

10

20

30

40

50

60

70

80

90

100

3979

3979

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3981

V

w

Series1Series2

p1 = 95

0

10

20

30

40

50

60

70

80

90

100

3986

3986

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3987

3987

3987

3987

3987

3988

3988

3988

3988

3988

3988

3988

V

w

Series1Series2

p1 = 99

0

10

20

30

40

50

60

70

80

90

100

3992

3992

3992

3992

3992

3992

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3992

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3992

3992

3992

3992

3992

3992

3992

3993

3993

3993

V

Wag

es Series1Series2

46

Figure 2

p= .55

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

3977

3977

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V

F an

d G

Series1Series2

p=.65

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

3979

3979

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V

F an

d G

Series1Series2

p1 = 75

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

3981

3981

3981

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3983

3983

3983

V

G a

nd F Series1

Series2

p1 = 85

0.00E+00

1.00E­01

2.00E­01

3.00E­01

4.00E­01

5.00E­01

6.00E­01

7.00E­01

8.00E­01

9.00E­01

1.00E+00

3979

3979

3979

3979

3979

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3980

3980

3980

3980

3980

3980

3980

3980

3980

3981

V

F an

d G

Series1Series2

P=.95

0.00E+00

1.00E­01

2.00E­01

3.00E­01

4.00E­01

5.00E­01

6.00E­01

7.00E­01

8.00E­01

9.00E­01

1.00E+00

3986

3986

3986

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3986

3987

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3987

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3988

V

F an

d G

2.474717982113361E­002  4Series2

p1 = 99

0

0.2

0.4

0.6

0.8

1

1.2

3992

3992

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3993

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V

F Series1

47

Figure 3a = 0.1

0

10

20

30

40

50

60

70

80

90

100

3992

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3993

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3993

V

Wag

es Series1Series2

a = 0.2

0

10

20

30

40

50

60

70

80

90

100

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

V

Wag

es Series1Series2

a = 0.3

0

10

20

30

40

50

60

70

80

90

100

3989

3989

3989

3989

3989

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

3990

V

Wag

es Series1Series2

a = 0.4

0

10

20

30

40

50

60

70

80

90

100

3987

3987

3987

3987

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3987

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3987

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3988

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3988

3988

3988

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3988

3988

3988

3988

3988

3988

3988

V

Wag

es Series1Series2

a = 0.5

0

10

20

30

40

50

60

70

80

90

100

3983

3983

3983

3983

3983

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V

Wag

es Series1Series2

a = 0.6

0

10

20

30

40

50

60

70

80

90

100

3977

3977

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V

Wag

es Series1Series2

48

a = 0.7

0

10

20

30

40

50

60

70

80

90

100

3965

3965

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V

Wag

es Series1Series2

a = 0.8

0

10

20

30

40

50

60

70

80

90

100

3942

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V

Wag

es Series1Series2

a = 0.9

0

10

20

30

40

50

60

70

80

90

100

3881

3881

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Wag

es Series1Series2

49

Figure 4

a = 0.1

0.00E+00

2.00E­01

4.00E­01

6.00E­01

8.00E­01

1.00E+00

1.20E+00

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

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3992

3992

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3993

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3993

V

F Series1

a = 0.2

0.00E+00

2.00E­01

4.00E­01

6.00E­01

8.00E­01

1.00E+00

1.20E+00

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3991

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

3992

V

F Series1

a = 0.3

0

0.2

0.4

0.6

0.8

1

1.2

3989

3989

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3989

3990

3990

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3990

3990

3990

3990

3990

3990

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3990

3990

3990

3990

3990

3990

V

F Series1

a = 0.4

0

0.2

0.4

0.6

0.8

1

1.2

3987

3987

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3987

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3988

V

F Series1

a = 0.5

0

0.2

0.4

0.6

0.8

1

1.2

3983

3983

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3983

3983

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3984

V

F Series1

a  = 0.6

0

0.2

0.4

0.6

0.8

1

1.2

3977

3977

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V

F Series1

50

a = 0.7

0

0.2

0.4

0.6

0.8

1

1.2

3965

3965

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V

F Series1

a = 0.8

0

0.2

0.4

0.6

0.8

1

1.2

3942

3942

3942

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3942

3942

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V

F Series1

a = 0.9

0

0.2

0.4

0.6

0.8

1

1.2

3881

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F Series1

51