Labour market signalling and job turnover revisited Banerjee y Gaston 2003

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Labour Economics 11 (2004) 599–622

Labour market signalling and job turnover revisited

Dyuti S. Banerjee, Noel Gaston*

School of Business, Bond University, Gold Coast, Queensland 4229, Australia

Received 14 August 2002; received in revised form 20 June 2003; accepted 9 October 2003

Available online 17 December 2003

Abstract

Turnover by both low and high productivity workers occurs in all jobs. However, the reasons for

turnover by workers of different abilities differ. When the probability that better workers transmit

accurate signals to the labour market increases, we show that incumbent employers are more likely to

adopt a separating wage offer strategy. This reduces the likelihood of turnover by more productive

workers. We also show that turnover depends on a firm’s willingness to match outside wage offers.

Hence, the personnel and wage policies adopted by employers are important for understanding the

turnover of workers of all productivity types.

D 2003 Elsevier B.V. All rights reserved.

JEL classification: J31; J63; J40

Keywords: Worker turnover; Signalling; Counter-offers; Market Perfect Bayesian Equilibrium

1. Introduction

This paper is concerned with worker turnover. In particular, we investigate why it is that

workers of different abilities or levels of productivity are ‘‘poached’’ by a firm’s

competitors. In practice, we observe that workers of both high and low productivity leave

for ‘‘greener pastures’’. For instance, Baker et al. (1994a) find that workers who exit were

employed in all types of jobs and at all organisational levels; there is almost no effect of

position within the firm’s internal labour market on the likelihood of exit. They also note

that the firm’s administrative procedures as well as the external labour market have an

impact on the exit of workers.

We provide a general model in which workers send signals of their productivity to

the external labour market. When signals are not perfectly informative, we show that

0927-5371/$ - see front matter D 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.labeco.2003.10.001

* Corresponding author. Tel.: +61-7-5595-2220; fax: +61-7-5595-1160.

E-mail address: [email protected] (N. Gaston).

D.S. Banerjee, N. Gaston / Labour Economics 11 (2004) 599–622600

there may be some scope for employers to strategically exploit the asymmetry of

information to increase their profits. More specifically, employers choose different

wage strategies depending on how informative the signals are to competitors and how

costly it is to match outside wage offers. For example, if signals are sufficiently

‘‘noisy’’, employers adopt pooling wage strategies which ‘‘underpay’’ good workers.

By doing so, employers risk turnover by their more productive workers as well as

their less productive workers. However, this risk is attenuated when employers can

match their workers’ outside wage offers at low cost. Notwithstanding, firms that

successfully poach workers from rivals will sometimes find that they get ‘‘less than

they bargained for’’.

The existing theoretical literature on labour market signalling and job turnover does

not address the possibility of turnover by workers of more than one level of

productivity. Existing models essentially fall into three categories. There are models

in which there is no turnover (e.g., Waldman, 1984, 1990), numerous models in which

only the less productive workers turnover (e.g., Kahn and Huberman, 1988; Greenwald,

1986; Bernhardt and Scoones, 1993) and a far smaller number of models in which only

the most productive workers turnover (e.g., Lazear, 1986). As mentioned, taken in

isolation these models do not square with the observation that turnover of both high

productivity and low productivity workers often occurs within the same firm. The model

that we develop has each of these three scenarios embedded as possible outcomes, as

well as the more relevant scenario in which both less and more productive workers

turnover.

In this paper, we show that job turnover may result if a firm’s wage offer to a worker

depends on the information that firms in the external labour market receive. There is a

considerable body of empirical evidence that indicates that relative wage opportunities are

one of the primary determinants of ‘‘on-the-job’’ search and the likelihood of turnover by

employees.1 However, it is also the case that ‘‘. . .those who are well paid in their jobs, as

well as those who are not, exit. . . . they believe that they are likely to do as well in the

external labor market; i.e., that whatever talents made them well paid at this firm are not

very firm-specific’’ (Baker et al., 1994a, p. 913). By extension, the personnel and wage

policies adopted by employers are therefore crucial for understanding the retention or

turnover by workers of all productivity types.

We offer the following story to motivate our model. A firm, which we will refer to as

the initial firm, currently employs a worker. The worker may or may not actively seek

employment at another firm. However, a firm in the external labour market, which we refer

to as the raiding firm, may have an incentive to ‘‘poach’’ the worker. We assume that a

raiding firm is never able to directly observe the applicant’s productivity. However, the

raiding firm receives a signal about the worker that may be directly related to productivity.

For example, the signal may take the form of a job referral or, alternatively, a curriculum

vita (or educational qualifications, a list of publications or teaching awards). In the latter

case, academia provides an example of the type of labour market setting that we have in

1 Table 8.1 of Devine and Kiefer (1991, pp. 254–255) provides a comprehensive summary of the key

findings of this literature.

D.S. Banerjee, N. Gaston / Labour Economics 11 (2004) 599–622 601

mind.2 However, it should be stressed that our model has far wider application than the

academic labour market alone. For instance, the poaching of senior managers and CEO’s

by competitors based on profit or share price signals also fits nicely into the framework

that we describe below.

Obviously, the external labour market may receive ‘‘noisy’’ signals of worker

productivity. For example, it may be difficult to ascertain from a vita or number of

journal publications whether or not a worker will be good teacher or colleague. In addition,

it may be the case that a worker may strategically choose not to disclose some information

on their vita, because he believes that it may jeopardise his chances of getting outside

wage offers. Consequently, vitae are likely to be less than perfect signals of labour

productivity. We show that it is the noisiness of signals, in conjunction with the initial

wage and counter-offer wage strategy adopted by employers that generates job turnover,

irrespective of a worker’s type.

The paper is organised as follows. In Section 2, we outline and describe our model set-

up. In Section 3, we introduce the concept of Market Perfect Bayesian Equilibrium that is

used to determine the wage offers of the initial and the raiding firms. Section 4 contains the

key findings. In Section 5, we provide an extended discussion of the sensitivity of the

findings to the assumptions of our model of job turnover. Section 6 contains the

concluding comments.

2. Model preliminaries

There are three players: the initial firm, F1; a raiding firm, F2; and a worker. A firm can

hire at most one worker. The worker lives for two periods and supplies labour inelastically

in each period. In period 1, F1 learns what the worker’s productivity, Q, will be in period 2.

In particular, Qa{L, H}, where H (high)>L (low). The production technology is assumed

to be linear and the price of Q is normalised to one. There are no long-term contracts, i.e.,

the worker cannot irrevocably bind himself to any firm.

F2 is a representative firm in a perfectly competitive output market. After receiving a

signal of the worker’s productivity, F2 may choose to ‘‘poach’’ him from F1. Our interest

in this paper focuses on the circumstances under which the worker, either a high or low

productivity type, will accept F2’s wage offer. The game played between the three players

is specified in extensive form as follows.

Stage 1: F1 observes the worker’s type, Qa{L, H}, the cost of making a counter-offer, zz 0, and the worker’s cost of switching employers, cz 0. It then makes aninitial wage offer to the worker, w1(Q)

I.

2 Siow (1998) argues that academia has a rather peculiar combination of personnel practices (e.g., tenure and

up-or-out contracts) that are, in large part, explained by informational asymmetries. In addition, publications in

professionally refereed journals signal the ability of professors to transmit the most current knowledge to students.

Siow argues that the peer review and monitoring role provided by journals is rather unique to academia.

Moreover, publications are crucial for generating outside offers that, in turn, drive the current compensation for

academics.


Stage 2: F2 can observe neither the worker’s type, nor the worker’s cost of switching

employers. A priori, F2 believes that Q can be H or L with probabilities p and (1� p),respectively. F2 receives a signal, sa{0,1}, of the worker’s productivity and makesa wage offer, w2(s).Stage 3: The worker reveals F2’s offer to the initial firm only if w2(s)� c>w1(Q)

I. F1

receives a ‘‘message’’ ma{0, s = 0, s = 1}. F1 may or may not choose to make a

counter-offer exceeding w1QI to a worker. The counter-offer, which depends on the

worker’s type, the initial wage offer and the message that F1 receives, is denoted as

w1(w1(Q)I, m, Q)R. For notational simplicity, we define w1(w1(Q)

I, m, Q)Ruw1(Q)R.

The payoff to a Q-type worker from remaining with F1 is either w1(Q)I or w1(Q)

R. The

payoff from moving to F2 is w2(s)� c. The worker chooses the wage offer yielding the

highest payoff.

If the worker receives an outside wage offer, we assume that it is credible and costlessly

verifiable by F1. That is, if the worker’s payoff from the outside offer exceeds F1’s wage

offer, then the worker reveals the outside offer to F1. Hence, if F2’s wage offer to the

worker is better than F1’s original wage offer, then F1 learns the signal received by F2.

Note that the initial employer has the option to make counter-offers to workers who

receive wage offers from the external labour market. We assume that F1’s counter-offer

cannot be lower than its initial offer to the worker.3 If a worker does not approach F1 for a

counter-offer, then F1’s initial wage offer must be at least as good as the outside wage

offer, net of the switching cost.

The cost of making counter-offers, z, affects the wage strategy of the initial

employer. The variable z captures the cost of renegotiating contracts with employees

who receive outside offers or alternatively, the cost associated with a firm changing its

wage structure. Baker et al. (1994a,b) provide empirical findings consistent with the

firm and organisational environment that we have in mind. For example, they find that

firms have rigid hierarchical structures and wage policies that tend to be simple, easy-

to-understand and extremely stable across time.4 In addition, there is what they term a

‘fast track exit effect’ in which the fastest promotees (i.e., most productive workers)

exit more frequently. ‘‘Perhaps some very high ability workers are not optimally

employed at the firm, or the firm is not paying wages equal to expected marginal

products for these employees. The latter interpretation is consistent with administrative

constraints on pay levels’’ (Baker et al., 1994a, p. 903). In effect, the firm is

administratively constrained by being unable to give the best performers large pay

increases. These wage rules and administrative constraints create a ‘‘wedge between an

employee’s pay and what pay would be in an external spot market’’ (Baker et al.,

1994a, p. 913).

3 We exclude the possibility, suggested by Lazear (1986), that the absence of a counter-offer gives the firm a

pretext to reduce a worker’s wage.4 This suggests that firms choose their wage strategies so that they are ex ante optimal. The strategies are

codified into rules and relatively fixed wage and salary structures. Naturally, there may be ex post regret, if the

better workers depart.


If signals are perfectly informative to a raiding firm, we show below that, for all but

prohibitive switching costs, F1 would employ a separating wage strategy, i.e., it offers

w1(H)I =H� c and w1(L)

I = L� c to high productivity and low productivity workers,

respectively. F1 would never pay less than these wages because counter-offers are

costly and the retention of workers is profitable. If a worker’s signal to the external

labour market truly reflected productivity, then, in a similar fashion to Waldman’s

(1984) job assignment model, any worker would always remain with their initial

employer.

In Waldman (1984), the worker’s wage and the job assignment pair are both publicly

observable. The job assignment is related to the output of the worker. Hence, even though

the productivity of the worker is private information to the initial firm, the external

market gets a signal about worker productivity from the job assignment. Since job

assignments are publicly observable, the initial firm cannot conceal information about the

productivity of the worker. However, inefficient outcomes may result due to the initial

firm’s private information about the worker’s productivity. Specifically, inefficiency is

manifested by the employer’s strategic choice of a threshold level for assigning workers

to the higher-level jobs.

In real life, signals of worker productivity are unlikely to be perfectly informative. For

example, whether by design or not, it seems reasonable to expect that signals provide noisy

information about worker productivity. The information structure we consider is as

follows. For simplicity, we assume that the signal, s, takes two values, 0 or 1. If the

worker’s type is H then the signal takes the values 1 and 0, with probabilities q and (1� q),

respectively. If the worker’s type is L, then the signal takes the values 1 and 0 with

probabilities r and (1� r), respectively. That is,

Probðs ¼ 1 j Q ¼ HÞ ¼ q

Probðs ¼ 0 j Q ¼ HÞ ¼ 1� q

Probðs ¼ 1 j Q ¼ LÞ ¼ r

Probðs ¼ 0 j Q ¼ LÞ ¼ 1� r: ð1Þ

Without loss of generality, we assume that q>r. This means that the probability of

receiving s = 1 is higher if the worker’s type is H and the probability of receiving s = 0 is

higher if the worker’s type is L. Accordingly, we refer to s = 1 as the ‘‘good’’ signal and to

s = 0 as the ‘‘bad’’ signal.5

5 If r>q is assumed, then (1� q)>(1� r), i.e., the probability of getting s = 0 is higher if the worker’s type is

H and the probability of getting s = 1 is higher if the worker’s type is L. So, in this case, s = 0 is the good signal

and s= 1 is the bad signal. Hence, irrespective of whether q>r or r>q is assumed, it is always the case that the

probability of receiving a good signal is higher if the worker’s type is H.


After receiving a signal, F2 updates its belief about the expected productivity of the

worker and then decides on its wage offer. We let EQ(s) denote F2’s expectation of the

worker’s productivity after it receives the signal, s, i.e.,

EQðsÞ ¼ ProbðH j sÞH þ ProbðL j sÞL: ð2Þ

The expected productivity of the worker as perceived by the raiding firm,

corresponding to each of the signals, is given by Bayes’ rule, i.e.,

ProbðQ ¼ X j s ¼ iÞ ¼ rðQ ¼ X j s ¼ iÞ ¼ ProbðQ ¼ X ; s ¼ iÞProbðs ¼ iÞ ; ð3Þ

where Xa{L, H} and ia{0, 1} The draw of the signal and the draw of productivity are

independent. The conditional probabilities are given by

rðQ ¼ H j s ¼ 1Þ ¼ pq

pqþ ð1� pÞr ¼ d ð4Þ

rðQ ¼ L j s ¼ 1Þ ¼ ð1� pÞrpqþ ð1� pÞr ¼ 1� d ð5Þ

rðQ ¼ H j s ¼ 0Þ ¼ pð1� qÞpð1� qÞ þ ð1� pÞð1� rÞ ¼ c ð6Þ

rðQ ¼ L j s ¼ 0Þ ¼ ð1� pÞð1� rÞpð1� qÞ þ ð1� pÞð1� rÞ ¼ 1� c: ð7Þ

Note that r(Qjs) denotes F2’s updated belief about the worker’s type after it has

received the signal. Using Eqs. (4)–(7), the expected productivities associated with the

different signals are

EQðs ¼ 1Þ ¼ dH þ ð1� dÞL ð8Þ

EQðs ¼ 0Þ ¼ cH þ ð1� cÞL: ð9ÞSince c< p < d we have EQ(s = 1)>EQ(s= 0). Thus, expected output of a worker who

provides a good signal unambiguously exceeds that for a worker sending a bad signal.

3. Equilibrium of the signalling game

We now discuss the payoffs to F1 and F2 and define the equilibrium solution concept.

Before discussing the wage offer strategies for the two firms, we impose some restrictions

and introduce some further notation. Both firms’ wage offers are non-negative and less

than H. That is, the offer strategy space for each firm is [0, H]. We divide [0, H] into two


sub-intervals, W l=[0, L] and W h=(L, H]. We let wil (or wi

h) denote the wage offer of firm

Fi, i= 1, 2, which is an element of W l (or W h).

The initial firm never makes a wage offer that exceeds the productivity of the worker;

otherwise, it would make a negative profit. Hence, F1 offers a wage w1haW h, if at all, only

to an H-type worker. Therefore, F1’s strategy could be to offer w1h to an H-type worker and

w1l to an L-type worker. Alternatively, it could offer w1

l to both types of workers. The initial

firm’s wage offer strategy is a contingent plan of action, i.e., an initial wage and a wage

counter-offer pair, w1(Q)=(w1(Q)I, w1(Q)

R), Qa{L, H}.

We now analyse the equilibrium wage offers of the two firms. Let p1 and p2 denote thepayoffs to F1 and F2. Since the external market is perfectly competitive, all firms in the

external market receive the same signal and F2’s wage offer equals the expected output of

the worker. The initial firm’s equilibrium strategy is to maximise its expected profit, given

the external market’s strategy. We call the equilibrium concept the Market Perfect

Bayesian Equilibrium (MPBE), which is formally defined as follows.

Definition 1. The MPBE of the game consists of F2’s updated belief, s(Qjs) formed by

Eqs. (4)–(7), F2’s equilibrium wage offer, w2(s)*, and F1’s equilibrium wage offer

strategy, w1(Q)*=(w1(Q)I*, w1(Q)

R*), where w1(Q)I is the initial wage offer and w1(Q)

R

is the wage counter-offer, satisfying the following conditions for Qa{L, H}.

C1. r(Qjs)z 0.

C2. r(H js) + r(Ljs) = 1, for sa{0, 1}.

C3. For each signal, s, F2’s equilibrium offer, w2(s)*, is the solution toPQafL;Hg rðQ j sÞp2ðQ; s;w2Þ ¼ 0 . In particular, w2(s)*=EQ(s), where EQ(s)

is the expected productivity of the worker corresponding to signal s.

C4a. A counter-offer is profitable for F1 if w1(Q)I <w2(s)*� c =w1(Q)

R and p1(w1

(Q)R; Q, z) =Q�w1(Q)R + c� z>0. The equilibrium wage offer strategy is w1

(Q)*=(w1(Q)I*,w1(Q)

R*, where w1(Q)R*arg max p1(w1(Q)

R; Q, z) and w1(Q)I*=

arg max Ep1(w1(Q); Q, z) = xp1(w1(Q)I; Q)+(1� x)p1(w1(Q)

R*; Q, z), where

p1(w1(Q)I; Q) =Q�w1(Q)

I + c and x is the probability of retaining the worker at

the initial wage offer.

C4b. If a counter-offer is not profitable, then F1’s equilibrium wage offer strategy is

w1(Q)*=w1(Q)I*= arg max Ep1(w1(Q); Q) = xp1(w1(Q)

I; Q).

The worker’s actual productivity and the cost of moving are private information to the

initial firm and the worker. This asymmetry of information prevents the movement of a

worker, i.e., whether he remains or stays with his initial employer, from providing any

information about his actual productivity. Moreover, the competitive output market

ensures that any wage offer made by F2 satisfies the zero profit constraint. It is the

asymmetric information about the worker’s actual productivity and the competitiveness of

the outside market which ensures that w2(s)* =EQ(s).6

6 Waldman (1984) introduced the concept of Restricted Market Nash Equilibrium (RMNE), in which the

original employer’s wage offer is chosen to maximise its profit and the outside market’s strategy is consistent with

the zero expected profit constraint. While the RMNE has equilibrium strategies, unlike the MPBE, there is no

updating of beliefs and therefore no equilibrium beliefs.


If the external labour market can perfectly observe a worker’s type, as revealed by his

movement, then as in Waldman (1990), only the low productivity workers would ever

move to other firms. Accordingly, the labour market would completely breakdown as in

the market for ‘‘lemons’’ (see Greenwald, 1986). In our model, there is no job assignment

related to the output of the worker that can prevent the initial firm from lying. We use the

MPBE concept in order to capture the idea of credible threats and promises that form the

foundation of the concept of Perfect Bayesian Nash Equilibrium.7

4. The equilibrium analysis

In this section, for given costs of switching to a new employer, worker turnover is

shown to be driven by two important factors: the noisiness of the signals that workers

convey to the external labour market and the willingness of incumbent employers to match

outside wage offers. We first show that asymmetric information about a worker’s

productivity is a necessary, but not sufficient, condition for worker turnover. Specifically,

when labour market signals are perfectly informative, we show that employers will

implement a separating wage strategy. This yields zero worker turnover, independent of

the cost of making counter-offers and the cost of worker mobility.

We then investigate the second of the two factors by considering the case in which

counter-offers are made by F1. When the costs of worker mobility are not prohibitive, we

show that there may be job turnover, albeit only by low productivity workers. Finally, we

consider the case in which making counter-offers to workers is unprofitable. In this case,

there can be job turnover by workers of both productivity types. Thus, it is only when both

assumptions, low costs of making counter-offers and imperfectly informative labour

market signals, are simultaneously satisfied that we observe turnover by both high and

low productivity workers in equilibrium.

To determine the equilibrium wage offers of the initial firm and raiding firm, we specify

three ranges of the switching cost.8

Definition 2.

a. Low switching costs: ca[0, EQ(s = 0));

b. Moderate switching costs: ca[EQ(s = 0), EQ(s = 1));

c. Prohibitive switching costs: czEQ(s = 1).

To highlight the importance of noisy information for job turnover, we first consider the

simplest case in which there are perfectly informative signals. When q = 1 and r= 0, a good

signal is always received if the worker is an H-type and a bad signal is always received if

7 The MPBE concept differs from a Perfect Bayesian Equilibrium in that only one player maximises his

payoff, while the firms in the external labour market choose their wage offer to satisfy the zero expected profit

constraint.8 Clearly, it is trivially the case that workers with prohibitive switching costs will never turnover. However,

we present the analysis for this cost range for completeness.


the worker is an L-type. The equilibrium is characterised in Proposition 1. (All proofs are

relegated to Appendix A).

Proposition 1. Perfectly informative signals. Suppose that the signals give perfect

information about the worker’s type, i.e., r(Q =Hjs = 1) = 1, r(Q = Ljs = 1) = 0,

r(Q=Hjs = 0) = 0 and r(Q= Ljs = 0) = 1, then the MPBE is given by w2(s = 1)*=EQ(s =

1) =H and w2(s = 0)*=EQ(s = 0) = L and

(a) Low switching costs: ca[0, L)

w1ðQÞI¼ Q� c;QafL;Hg:

There is no turnover by any type of worker.

(b) Moderate switching costs: ca[L, H)

ðw1ðHÞI;w1ðLÞI

Þ ¼ ðH � c; 0Þ:


(c) Prohibitive switching costs: czH

w1ðQÞI¼ 0:


Unless switching costs are prohibitive, the initial employer adopts a separating wage

offer strategy. The worker, irrespective of type, remains with the initial firm for all c.

Clearly, making a counter-offer is unnecessary. (Obviously, w2(s)� cVw1(Q)I). As the

moving cost rises, F1’s wage offer falls. The switching cost acts like a hold-up cost, of

course. Therefore, the initial firm can retain the worker by offering lower wages. In the

limit, with prohibitive moving costs, F1’s wage offer is zero.9

The results for the perfectly informative signalling case are qualitatively similar to

thinking of the wage offers made by F1 as a promotion or a better job assignment.

Therefore, the sub-optimal task assignment result in Waldman (1984, 1990) pertains.

However, the absence of turnover in the present case is efficient, since all firms equally

value workers and turnover costs are a deadweight loss. Of course, the complete absence

of turnover is counterfactual.

In the next two Propositions, we show that imperfectly informative signals, low costs of

mobility and high costs of making counter-offers are sufficient conditions for job turnover.

Note that F1 may or may not make a counter-offer. Recall that F1 makes a counter-offer, if

and only if w2(s)� c>w1(Q)I and p1(w1(Q)

RjQ; z) =Q�w1(Q)R� z>0 both apply.

Otherwise, if either w2(s)� cVw1(Q)I or p1(w1(Q)

RjQ; z) < 0 hold, then F1 leaves the

initial wage offer unchanged. In other words, if the worker’s outside wage offer exceeds

9 Ransom (1993) argues that the negative wage premium for additional years of seniority in academia is

largely explained by moving costs. Ceteris paribus, higher moving costs contribute to both increased worker

immobility and to lower wages for the non-movers. Farber (1999, p. 2468) notes that workers may change type

overtime by becoming less mobile on average. ‘‘This could reflect maturation of young workers as they acquire

families and settle into their careers’’.


F1’s initial wage offer, an initial employer will only make a counter-offer if it is profitable

to match the outside offer. Proposition 2 contains the results for the case in which a

counter-offer strategy is optimal. (The proof is in Appendix A).

Proposition 2. Imperfectly informative signals and counter-offers. F2’s equilibrium

strategy is given by w2(s = 1)*=EQ(s = 1) and w2(s = 0)*=EQ(s = 0). If w2(s)� c>w1(Q)I

and z <Q+ c�w2(s) both apply, then F1’s equilibrium wage offer strategy is

i . w1ðQÞ ¼ ðw1ðQÞI¼ EQðs ¼ 0Þ � c;w1ðQÞR

¼ EQðs ¼ 1Þ � cÞ; if EQðs ¼ 0Þ �

c > 0 and ð1� nÞ½EQðs ¼ 1Þ � EQðs ¼ 0Þ� � nz > 0;

ii . w1ðQÞ ¼ ðw1ðQÞI¼ 0;w1ðQÞR

¼ EQðs ¼ 1Þ � cÞ; if EQðs ¼ 1Þ � c > 0zEQ

ðs ¼ 0Þ � c and ð1� nÞ½EQðs ¼ 1Þ � c� � nz > 0;

where Qa{L,H} and na{r,q}. Specifically, w1(L)* and w1(H)* are as follows.

Case 1: Q=L and s = 1. w1ðLÞ* ¼ ð0;EQðs ¼ 1Þ � cÞ; where ca[d(H� L), EQ(s = 1))

and (1� r)[EQ(s = 1)� c]� rz>0

Case 2: Q=H and s = 1.

a. w1ðHÞ* ¼ ðEQðs ¼ 0Þ � c;EQðs ¼ 1Þ � cÞ; where ca[0, EQ(s = 0)) and

(1� q)[EQ(s = 1)�EQ(s = 0)]� qz>0.

b. w1ðHÞ* ¼ ð0;EQðs ¼ 1Þ � cÞ; where ca[EQ(s = 0), EQ(s = 1)) and

(1� q)[EQ(s = 1)� c]� qz>0.

The most obvious feature of Proposition 2 is that the initial employer only needs to

make a counter-offer if the worker sends a good signal to the external market.

Consequently, if making counter-offers is sufficiently inexpensive, it may be profitable

to offer workers a low wage initially and then to simply match an outside offer if

workers happen to send a good signal. (However, it is interesting to note that paying an

initial wage equal to zero is not always the optimal strategy because of the cost of

making counter-offers). The implications for turnover probabilities are easily

illustrated.

Corollary 1. Suppose that the worker receives an outside wage offer w2(s)� c>w1(Q)I

and that z <Q+ c�w2(s) both apply. Then, in an MPBE, the turnover probabilities, t i,

i = l, h, for each worker type are given by

H-type worker:

t h ¼ 0; for all cz0:

L-type worker:

If c < c0; then tl ¼ 1; if ca½c0; c1Þ; then t l ¼ r; otherwise tl ¼ 0;

where c0 = c(H� L) and c1 = d(H� L).

The initial firm has the incentive to retain workers because by doing so it makes the

non-negative profit, c. In this sense, c is qualitatively equivalent to a match-specific rent.


Note that an H-type worker is always retained. For an L-type worker who has sent a good

signal and received a wage offer of EQ(s = 1), the wage needed to retain this worker may

be ‘‘too’’ high for low values of c. Hence, low productivity workers may turnover.

Finally, we turn to the case in which no counter-offers are possible. Irrespective of the

cost range, from the definition of MPBE, we know that F2 will always make one of two

wage offers to a worker, viz., EQ(s = 0) or EQ(s = 1). F1 knows the worker’s productivity

and chooses its wage offer to maximise profits. If it adopts a pooling wage strategy, F1

risks losing its high productivity workers; if it adopts a separating wage strategy, F1 retains

its high productivity workers, but has higher wage costs. Proposition 3 contains the results.

(The proof is provided in Appendix A).

Proposition 3. Imperfectly informative signals and no counter-offers

Suppose that F2 receives a signal, s, of the worker’s productivity. If either of the

following two conditions hold:

i. w2(s)� cVw1(Q)I, for any z, or

ii. w2(s)� c>w1(Q)I and z>Q+ c�w2(s),then a counter-offer will not be made. F2’s wage

offers are given by w2(s = 1)*=EQ(s = 1) and w2(s = 0)*=EQ(s = 0). The equilibrium

wage offer strategy for F1, w1(Q)*=w1(Q)I*, Qa{L,H}, is given by

(a) . Low switching costs: ca[0,EQ(s = 0))

ðw1ðHÞ;w1ðLÞÞ ¼

ðEQðs ¼ 0Þ � c;wl1Þ if ca½0; cAÞ

ðEQðs ¼ 1Þ � c;wl1Þ if ca½cA; c0Þ

ðEQðs ¼ 1Þ � c;EQðs ¼ 0Þ � cÞ if ca½c0;EQðs ¼ 0ÞÞ;

8>>>><>>>>:

where c0 = c(H� L) and cA ¼ ðd�cÞ�qð1�cÞq

ðH � LÞ;(b) . Moderate switching costs: ca[EQ(s = 0), EQ(s = 1))

ðw1ðHÞ;w1ðLÞÞ¼

ð0; 0Þ if ca½EQðs ¼ 0Þ; cBÞ

ðEQðs ¼ 1Þ � c; 0Þ if ca½cB; cCÞ

ðEQðs ¼ 1Þ � c;EQðs ¼ 1Þ � cÞ if ca½cC;EQðs ¼ 1ÞÞ;

8>>>><>>>>:

where cB=(d� q)H+(1� d)L and cC= d(H� L)+(1� r)L;

(c) . Prohibitive switching costs: czEQ(s = 1)

ðw1ðHÞ;w1ðLÞÞ ¼ ð0; 0Þ:

Within each cost range, there is a critical cost level that indicates when it is more

profitable for F1 to pay its H-type worker a wage the same as, or different from, the

wage offered to an L-type worker. To illustrate, consider the critical cost level cA within

the low cost range. For moving costs less than cA, F1 pays the H-type worker


EQ(s= 0)� c and risks losing the worker with probability q. For moving costs greater

than cA, it is more profitable for F1 to pay the H-type worker EQ(s = 0)� c and retain

him with certainty.10

The characterisation of the turnover probabilities for the various switching cost ranges

is summarised in Corollary 2. Before doing so, in Lemma 1, we describe some

properties of the critical switching cost levels that determine which wage strategy the

initial employer finds most profitable to employ.

Lemma 1. For qz r>0,

(i) If cA <EQ(s = 0)ZcB < EQ(s = 0);

(ii) If cAzEQ(s = 0)ZcBzEQ(s = 0);

(iii) EQ(s = 1)z cC>EQ(s = 0) and cCz cB.

Corollary 2. In an MPBE, for each worker type, the turnover probabilities, t i, i = l, h, are

given by

H-type worker:

(i) If cA = cB =EQ(s = 0), then t h = q for ca[0, EQ (s = 0)), otherwise t h = 0;

(ii) If cA < EQ(s = 0), then t h = q for c a[0,cA), otherwise, th = 0;

(iii) If cB>EQ(s = 0), then t h = q for ca[0, cB), otherwise t h = 0.

L-type worker:

If c < c0; then t l ¼ 1; if ca½c0; cCÞ; then t l ¼ r; otherwise; t l ¼ 0:

The intuition behind our results is readily conveyed by graphs of the profit

function for the initial employer for the various ranges of the worker’s switching

cost. Figs. 1 and 2 depict the initial firm’s profit functions for the L-type and H-type

workers, which we denote by p1l and p1

h. The graphs also summarise the relevant

information about the initial firm’s adoption of a separating or pooling wage strategy

as well as what the turnover probabilities for the various cost ranges and wage

strategies are.

First, consider Fig. 1 and the L-type worker. For switching costs up to c0, the raider’s

wage offer, net of moving costs, exceeds L� c and the initial firm never retains an L-type

worker for any signal that F2 receives (i.e., tl = 1). The initial firm’s profits are zero. For

costs greater than c0 in the low cost range, F1’s offer to an L-type worker is less than L.

The maximum offer that an L-type worker can get from the raiding firm is EQ(s = 1).

Since, F1’s equilibrium offer to an L-type worker is EQ(s = 0)� c, an L-type worker is

10 In general, our model implies that turnover declines with mobility costs. Farber (1999) provides a model in

which workers are differentiated by their turnover probabilities. He argues that heterogeneity in mobility rates can

largely account for the central facts about inter-firm worker mobility in modern labour markets (e.g., the

probability of a job ending declining with tenure) without resort to the existence of specific capital.

Fig. 1. Profit function: L-type worker.


retained by F1 only if the external market receives a bad signal, which occurs with

probability (1� r), hence, t l = r. Beyond critical cost level cC, the L-type is always

retained.

The three panels of Fig. 2 display the profit functions of the initial firm corresponding

to the three cases described in Lemma 1.11 In the range [0, cA) in Case 2, F1’s maximum

offer to an H-type worker is only EQ(s = 0)� c. For the initial employer, the expected

profits from exposing an H-type worker to some risk of turnover is higher than that that

would be obtained by paying the higher separating wage, EQ(s = 1)� c, which would

retain the worker with certainty. A raiding firm cannot offer less than EQ(s= 1), because

firms in the external labour market make wage offers consistent with zero expected

profits. Consequently, in the lowest cost range, the initial firm can retain the H-type

worker only if the external market receives s = 0. Hence, t h = q. In the range [cA,

EQ(s = 1)), F1 finds it more profitable to pay the higher wage to the H-type worker

and to retain the worker with certainty.

From Case 3 of Fig. 2, in the moderate cost range note that F1’s wage offers can be

either 0 or EQ(s = 1)� c. (In this cost range, note that EQ(s = 0)� c is less than 0). Hence,

if F2 receives a bad signal, the worker remains with F1 irrespective of their type, because

by switching the payoff would be negative. For the different sub-ranges within the

moderate cost range, the cost level cB represents the critical cost at which F1’s expected

11 In the perfect signalling case, we have cA= cB =EQ(s= 0) = 0, so that the profit function for the H-type

worker consists of the last two segments of the profit function in Case 1 of Fig. 2. For the L-type worker,

cC =EQ(s = 1) =H, so that the profit function consists of the last two segments of the profit function in Fig. 1.


profit from offering higher wages to an H-type worker, and retaining them with certainty,

is equal to the expected profit from making a zero wage offer. Wage offers of zero will

lead to turnover of H-type workers if the worker transmits a good signal, which occurs

with probability q.

Fig. 2. Profit Function: H-type worker.

Fig. 2 (continued).


Finally, from Case 3 of Fig. 2, note that initial firms are more likely to expose their

good workers to some risk of turnover as q falls.12 Intuitively, as q falls, the risk of losing

a good worker to poaching falls. Consequently, employers take a calculated ‘‘risk’’ by

underpaying their workers and adopting a pooling wage strategy.

Overall, the initial employer takes into consideration the signals that the external

market is likely to receive and the associated risk of worker turnover. Since the initial

firm can make non-negative profits by retaining a worker, irrespective of type, it has an

incentive to avoid turnover. However, an L-type worker’s maximum wage is L and for

an H-type worker the wage range constitutes the entire range of productivity. Our

analysis of the noisy signalling scenario shows that, except for prohibitive switching

costs, the retention of an H-type worker not only depends on the wage offer of the initial

firm, but also on the offer by the outside firm. In turn, this depends on the signal

received by the market. This occurs because, for some range of switching costs, it may

be profitable for an employer to ‘‘underpay’’ good workers and expose them to some

risk of being poached. This incentive does not arise for low productivity workers. In

general, retention of L-type workers depends purely on the signal received by the

market. Accordingly, L-type workers are always more likely to turnover than H-type

workers.

12 Note that cB =EQ(s = 1� qH ), so that cB!EQ(s = 1) as q! 0. Hence, while the actual turnover

probability is lower, the range of costs over which the initial firm will expose an H-type worker to some risk of

being poached increases. Further, as cB increases, the range of costs for which the initial firm adopts a pooling

strategy increases.


For the case in which counter-offers are possible, the initial employer has an

incentive to ‘‘underpay’’ initially and then to simply match an outside offer should it

be the case that the worker sends a good signal. Comparison of Corollaries 1 and 2

reveals that, when counter-offers are profitable, that an H-type worker never leaves and

that the cost range for which an L-type worker may turnover is smaller in the case when

counter-offers are possible (since c1 < cC). To rationalise observed turnover by H-type

workers in equilibrium, not only must it be the case that signals are noisy but, in

addition, the cost of making counter-offers must be significant.

5. Discussion

In this paper, the results about worker turnover rely on the assumption that the

external labour market receives noisy signals about worker productivity. In addition, it

is necessary that firms in the external market receive no other information or signals

that would help them to determine a worker’s actual productivity. To ensure that the

external market does not get any other signal about worker productivity that may be

sufficiently informative to make the original noisy signals of worker type irrelevant, we

assumed that switching costs are private information to the worker and the incumbent

employer. In effect, asymmetric information about worker switching costs prevents the

external market from deriving information about the initial firm’s wage offers to

workers of different types. To understand the sensitivity of our findings to the

assumptions of our model, in this section, we discuss some alternative information

structures. In particular, we show that, if switching costs were public knowledge, or

equivalently if wages were publicly observable, that there would be no turnover in

equilibrium.

Let us suppose that the initial employer’s wage offer is publicly observable. That is,

wages act as a signal. However, wages are an imprecise signal of productivity. A

priori, F2 knows that the worker is an H- or L-type with probabilities p and (1� p).

After F2 observes F1’s wage offer, it updates its belief about the productivity of the

worker and then makes its own wage offer.

First, consider the separating equilibrium, i.e., F1 uses the strategy (w1(H),w1(L))=

(w1h,w1

l). F2 knows that a worker offered w1h(w1

l) by F1 is of H- (L-) productivity, i.e., r(Q =

Hjs =w lh) = 1 and r(Q = Ljs =w l

h) = 1. Consequently, F2 will offer H (or L) to an H- (or L-)

productivity worker. If F1 is using the separating strategy, it knows ex ante that F2 will make

these offers. Therefore, its strategy will be to offer w1(H) =H� c and w1(L) = L� c. Clearly,

a worker of either type would elect to stay with F1.

However, F1 has an incentive to deviate. Suppose that F1 offers w1l to an H-type worker.

If the raiding firm believes that the worker’s type is low it therefore offers a wage L.

Consequently, the H-type worker would stay with the initial employer. Thus, the separating

equilibrium does not exist. However, pooling equilibria do exist. That is, after observing

w1l , F2’s updated beliefs are r(Q =Hjs=w l

l) = p and r(Q = Ljs =w ll)=(1� p). Accordingly,

being unable to determine a worker’s type, F2 offers the worker’s expected output. The

results for the wage signalling case are summarised in Proposition 4. (Proofs are in

Appendix A).


Proposition 4. Publicly observable wages

If F2 has perfect information about the wage offer of F1, and this is common knowledge,

then there exists a continuum of pooling equilibria for cz p(H� L) and for all z>0. The

equilibrium wage offer strategies are given by

(i) w2(s =w1l )*= EQ(s) = L+ p(H� L) and

(ii) w1(Q)*=EQ(s)� c, if ca[p(H� L), L + p(H� L)],

where Qa{L,H}. There is no turnover by any type of worker.

The analysis implies that the existence of a pooling equilibrium in pure strategies, as

long as cz p(H� L). The maximum wage offer that guarantees the existence of a pooling

equilibrium cannot exceed L. Furthermore, since F2 perfectly observes the wage offer

made by F1, it is not possible for F1 to offer different wages to different types of workers

even in the range [0, L] because, once again, F2 would be able to infer a worker’s actual

productivity. The relative ‘‘underpayment’’ of H-type workers occurs because the initial

firm is able to exploit its private information.

In this paper, we have focussed on the cases in which F2 cannot perfectly observe the

wage offer of F1 but receives imperfectly informative signals of worker productivity.

Imperfect information about worker switching costs is therefore necessary. (Recall that

these ‘‘costs’’ represent the initial employer’s rents in equilibrium). F1 knows that the

raiding firm has received noisy information, but is never sure about the signal it has

received unless the worker returns to the initial employer for a counter-offer. The case in

which switching costs are public information is equivalent to the case in which the

raiding firm receives a signal about a worker’s type and perfectly observes F1’s wage

offer. In turn, both cases are associated with zero worker turnover, as described in

Proposition 4.

The point, of course, is that any meaningful equilibrium solution requires consistency

in the belief structure of F1 and F2 regarding each other’s wage offer strategies. If F1

offers anything greater than L, F2 knows with certainty that the worker is an H-type and

so offers H. If F2 believes that F1 uses a separating wage strategy, then F2 will always

offer L if it observes w1(L) = L. However, F1 may deviate from its separating strategy and

offer L to an H-type worker if it believes that F2 always offers L if it observes L. But,

then it is also possible that F2 also deviates from its strategy of offering L if it observes L

by offering EQ(s). In this case, beliefs are inconsistent. That is, any other types of

information or signals from which the raiding firm may ascertain the worker’s type

disturbs the equilibrium concept outlined in the definition of MPBE.13

13 Changing the staging of the game is an alternative way in which our turnover results may be replicated. For

example, if the initial employer and firms in the external market simultaneously offer wages, the possibility of

inconsistent beliefs in equilibrium is avoided. In this case, if there is asymmetric information about a worker’s

productivity and a sufficiently high cost of making counter-offers, then both types of workers may turnover in

equilibrium. The reason why good workers may turnover is that the initial employer may gamble onH-type workers

sending bad signals and paying a pooling wage. We are grateful to an anonymous referee for this observation.


It is also important to note that private information about the worker’s switching cost

together with the noisy signals is sufficient to guarantee the existence of equilibrium.

Informational asymmetry about the cost of making counter-offers is not necessary for the

equilibrium analysis. The fact that the switching cost is private information to F1 and the

worker makes it impossible for the raiding firm to gauge whether a counter-offer will be

made or not.14

6. Concluding comments

Most previous theoretical research has shown that it is the least productive workers

who are the more likely to change jobs. In contrast, empirical research has shown that

the most productive workers are equally, if not more, likely to change jobs. In this paper,

we have shown that turnover by both high productivity and low productivity workers is

possible when information about the productivity of workers is noisy and making

counter-offers is costly. The informational asymmetry prevents the firm’s competitors

from knowing the worker’s exact productivity type. In other words, the movement of the

worker, i.e., whether he remains with or leaves his current employer, does not convey

any information about actual productivity. Consequently, there may be turnover of both

types of workers.

In the case of low productivity workers, labour turnover depends upon the signal

received by the raiding firm. In contrast, the likelihood of turnover by more productive

workers depends upon the initial employer’s wage offer, the possibility of making a

counter-offer and the signal received by the raiding firm. Specifically, we showed that in

the case in which making counter-offers is too costly, it may be profitable for an

employer to adopt a pooling wage strategy and expose their better workers to some risk

of being poached by outside firms than to adopt a separating wage strategy that would

retain all workers with certainty.

While we find that less productive workers are more likely to turnover, as

discussed by Greenwald (1986), the labour market does not completely breakdown

as in the market for ‘‘lemons’’. In our model, inefficiencies arise because of

asymmetric information and switching costs. We have shown that there are efficient

outcomes where the least productive workers get a low wage and the most productive

workers get a high wage. However, we also showed that there are situations in which

inefficiencies may arise, in a similar fashion to Waldman (1984, 1990). Some of the

workers who would have been more efficiently employed in higher ranking or more

highly paid positions are not promoted, because the job assignment signals worker

14 To illustrate, consider the switching cost range, ca[0, EQ(s = 0)), which is private information to F1 and

the worker. F2 always offers EQ(s). In this cost range, F1 will never make any counter-offers to an L-type worker,

as shown in Proposition 2. Since F2 does not know the cost range, it therefore does not know whether a counter-

offer will be made or not. Therefore, asymmetric information about the cost of making counter-offers is not a

necessary condition in our analysis.


productivity to competitors. In the case of up-or-out contracts, the outcome is

efficient provided that the criteria for promotion are credible and enforceable. In

the case of academia, for example, promotion and salary decisions signal worker

productivity to poaching firms, but can often be subjective and highly discretionary.

In turn, this ‘‘noisiness’’ of signals implies that efficient outcomes may be not always

be achieved.

Acknowledgements

The paper has benefitted from the suggestions of the editor, an anonymous referee

and the participants in the seminars at which it was delivered. In particular, the

authors would like to thank Art Goldsmith, Hideshi Itoh, Jeff Kline, Hodaka Morita,

Dan Sasaki and Mike Waldman. Naturally, the usual disclaimers apply.

Appendix A

Proof of Proposition 1. First, note that the signals give perfect information about the

worker’s type, i.e., EQ(s = 0)=L and EQ(s = 1)=H. Therefore, it follows that F2 offers

w2(s = 0)*= L and w2(s = 1)*=H.

(a) Low switching costs: ca[0, L). Next, note that 0 < EQ(s = 0)� c V L and

EQ(s= 1)� c>0. It follows that F1’s equilibrium offers are w1(L)I*= L� c and

w1(H)I*=H� c. The worker, irrespective of type, remains with F1.

(b) Moderate switching costs: ca[L, H). F1 offers w1(L)I*= 0 and w1(H)I*=H� c. Once

again, the worker, irrespective of type, remains with F1.

(c) Prohibitive switching costs: czH. F1’s offer is w1(H)I*=w1(L)I*= 0. The worker’s

payoff is non-positive if he switches to F2, irrespective of his type. Hence, the worker

remains with F1. 5

Proof of Proposition 2. Fig. 3 gives a diagrammatic representation of the game with

counter-offers. The game begins with F1 making an initial wage offer to a Q-type worker,

w1(Q)Ia{0,EQ(s = 0)� c, EQ(s = 1)� c}. F2 receives a signal s. Prob(s = 1jQ) = n and

Prob (s = 0jQ) = 1� n, n = r, q corresponding to Q = L, H. F2 updates its belief according to

Eqs. (4)–(7), determines the expected productivity of the worker, EQ(s) and makes a wage

offer, w2(s)*=EQ(s).

First, assume that EQ(s)� c>0. The worker may or may not approach F1 for a counter-

offer. The expressions at the end of each branch of the game tree are the realised profits for

F1.

(i) w1(Q)I = 0. In this case, the worker approaches F1 and discloses the outside offer,

which is EQ(s = 1) with probability n and EQ(s = 0) with probability 1� n. If F1

zcsEQQ += )1(

csEQQ += )0(

zcsEQQ += )0( zcsEQQ += )1(

s=0 with s=1 with s=0 with s=1 with

prob. 1-n prob. n prob. 1-n prob. n

! ! csEQQ += )1(

csEQ = )0(

0 csEQ = )1(

!I

Qw )(1

Fig. 3.


matches the outside offer, then its profit is Q�EQ(s=1)+ c� z withprobability n or Q�EQ(s=0)+ c� z with probability 1� n. So F1’s expected

profit from offering w1(Q)I = 0 is

Ep1ðQ;w1ðQÞI ¼ 0Þ ¼ np1ðQ;w1ðQÞI ¼ 0;w1ðQÞR ¼ EQðs ¼ 1Þ � cÞþ ð1� nÞp1ðQ;w1ðQÞI ¼ 0;w1ðQÞR ¼ EQðs ¼ 0Þ � cÞ

¼ Q� nEQðs ¼ 1Þ � ð1� nÞEQðs ¼ 0Þ þ c� z: ðA1Þ

(ii) w1(Q)I =EQ(s = 0)� c. In this case, the worker approaches and discloses the outside

offer only if it is EQ(s = 1), which occurs with probability n. If F1 matches the outside

offer (net of the mobility cost), then the expected profit of F1 corresponding to

w1(Q)I =EQ(s = 0)� c is

Ep1ðQ;w1ðQÞI ¼ EQðs ¼ 0Þ � cÞ ¼ np1ðQ;w1ðQÞI ¼ EQðs ¼ 0Þ;w1ðQÞR

¼ EQðs ¼ 1Þ � cÞ þ ð1� nÞp1ðQ;w1ðQÞI ¼ w1ðQÞR

¼ EQðs ¼ 0Þ � cÞ ¼ Q� nEQðs ¼ 1Þ � ð1� nÞEQðs ¼ 0Þþ c� nz: ðA2Þ

(iii) w1(Q)I =EQ(s = 1)� c. In this case, no counter-offers are necessary. The expected

profit of F1 is

Ep1ðQ;w1ðQÞI ¼ EQðs ¼ 1Þ � cÞ ¼ Q� EQðs ¼ 1Þ þ c: ðA3Þ


Comparing Eqs. (A1) and (A2) yields

Ep1ðQ;w1ðQÞI ¼ EQðs ¼ 0Þ � cÞ � Ep1ðQ;w1ðQÞI ¼ 0Þ ¼ ð1� nÞz > 0;

bz > 0: ðA4Þ

It follows that w1(Q)I = 0 cannot be an equilibrium, when EQ(s)� c>0. Hence, the

equilibrium initial offer is w1(Q)I*=EQ(s = 0)� c.

Comparing Eqs. (A2) and (A3) yields

Ep1ðQ;w1ðQÞI ¼ EQðs ¼ 0Þ � cÞ � Ep1ðQ;w1ðQÞI ¼ EQðs ¼ 1Þ � cÞ

¼ ð1� nÞ½EQðs ¼ 1Þ � EQðs ¼ 0Þ� � nz: ðA5Þ

If Eq. (A5) is positive, then w1(Q)I*=EQ(s = 0)� c and w1(Q)

R*=EQ(s = 1)� c, if

the outside offer is EQ(s= 1).

Now consider the case in which EQ(s = 1)� c>0zEQ(s = 1)� c. The two possible

strategies are: (w1(Q)I = 0, w1(Q)

R =EQ(s = 1)� c) and (w1(Q)I =EQ(s= 1)� c). (In

the latter case, no counter-offers are necessary).

Ep1ðQ;w1ðQÞI ¼ 0Þ ¼ np1ðQ;w1ðQÞI ¼ 0;w1ðQÞR ¼ EQðs ¼ 1Þ � cÞþ ð1� nÞp1ðQ;w1ðQÞI ¼ 0Þ ¼ Q� nEQðs ¼ 1Þþnc� nz:

ðA6ÞComparing Eqs. (A3) and (A6), we get

Ep1ðQ;w1ðQÞI ¼ 0Þ � Ep1ðQ;w1ðQÞI ¼ EQðs ¼ 1Þ � cÞ¼ ð1� nÞ½EQðs ¼ 1Þ � c� � nz: ðA7Þ

If Eq. (A7) is positive, then w1(Q)*=(0, EQ(s = 1)� c) is the equilibrium strategy.

Case 1: Substituting n = r and Q = L into Eq. (A7) yields the required expressions.

When ca[0, d(H� L)), note that paying the worker EQ(s = 1)� c would yield strictly

negative profit for all z>0. When c>EQ(s= 1), high mobility costs guarantee retention

of both worker types.

Case 2: Substituting n = q and Q =H into Eqs. (A5) and (A7) yields the required

expressions for the two cost ranges. For any cz 0, note that it is always profitable to

retain the H-type worker. 5

Proof of Proposition 3. First, note that F2 offers expected output based on the signal

received, i.e., w2(s = 1)*=EQ(s = 1) or w2(s = 0)*=EQ(s = 0).

(a) Low switching costs: ca[0, EQ(s= 0)). Note that EQ(s= 1)� c>EQ(s = 0)� c>L, for

ca[0, c0). Since F1’s wage offer never exceeds the productivity of the worker, then

w1(L)*=w1la[0, L]. Consequently, F1 can never retain an L-type worker. For ca[c0,

EQ(s= 0)), EQ(s = 1)� c>L>EQ(s = 0)� c. In this cost range, w1(L)*=EQ(s = 0)� c


and F1 retains an L-type worker only if F2 receives s= 0, which occurs with probability

(1� r).

To determine F1’s wage offer to an H-type worker, it is necessary to compare the

expected profit for the wage offers EQ(s = 1)� c and EQ(s= 0)� c. If w1(H ) <

EQ(s= 0)� c, then F1 loses the H-type worker with certainty. The wage offer that

retains an H-type worker with certainty is w1(H ) =EQ(s= 1)� c. So w1(H )*a{EQ(s = 1)� c, EQ(s = 0)� c}. If F1 offers w1(H ) =EQ(s= 0)� c, it retains an H-type

worker only if F2 receives s = 0, which occurs with probability (1� q). Hence,

Ep1ðH ;wðHÞI ¼ EQðs ¼ 1Þ � cÞ ¼ H � EQðs ¼ 1Þ þ c ðA8Þ

Ep1ðH ;wðHÞI ¼ EQðs ¼ 0Þ � cÞ ¼ ð1� qÞðH � EQðs ¼ 0Þ þ cÞ: ðA9Þ

Equating the difference between Eqs. (A8) and (A9) to zero and solving for c, we get

cA ¼ ðd � cÞ � qð1� cÞq

ðH � LÞ: ðA10Þ

If ca[0, cA), then Ep1(H, w(H)I =EQ(s = 0)� c)zEp1(H, w(H)

I =EQ(s= 1)� c), so

that w1(H)*=EQ(s = 0)� c and F1 loses the H-type worker with probability q. If

ca[cA, EQ(s = 0)), then Ep1(H, w(H)I =EQ(s = 0)� c)VEp1(H, w(H)

I =EQ(s = 1)� c)

and w1(H)*=EQ(s = 1)� c and F1 retains the worker. Finally, note that if cAzEQ(s= 0), then w1(H)*=EQ(s = 0)� c and if cAV 0, then w1(H)*=EQ(s = 1)� c.

(b) Moderate switching costs: ca[EQ(s= 0), EQ(s = 1)). First, consider F1’s wage offer

to the L-type worker. Since EQ(s = 1)� c>L, for c < c1, and assuming that

c1 <EQ(s = 0), F1 offers w1(L)*= 0.15 It follows that an L-type worker remains with

F1 only if F2 receives s = 0, which occurs with probability (1� r).

For the cost range ca[c1, EQ(s= 1)), LzEQ(s = 1)� c>0. Therefore, the lowest

wage offer that will retain the L-type worker is 0, i.e., if F2 receives s = 0, which

occurs with probability (1� r). The maximum wage offer that will retain the L-type

worker with certainty is EQ(s = 1)� c. Therefore, w1(L)*a{0, EQ(s = 1)� c}. So

the analysis of F1’s equilibrium wage offer to an L-type worker involves comparing

the expected profit for the wage offers 0 and EQ(s = 1)� c.

Ep1ðL;w1ðLÞI ¼ EQðs ¼ 1Þ � cÞ ¼ L� EQðs ¼ 1Þ þ c ðA11Þ

Ep1ðL;w1ðLÞI ¼ 0Þ ¼ ð1� rÞL: ðA12Þ

Subtracting Eq. (A12) from Eq. (A11), equating to 0 and solving for c, we get

cC ¼ dðH � LÞ þ ð1� rÞL: ðA13Þ

15 Comparison of EQ(s = 0) and c1 is algebraically intractable. For simplicity, we assume that c1>EQ(s = 0).

Assuming that EQ(s = 0)>c1 adds to the number of equilibria, without yielding any further insights. Within each

cost range, there are a number of sub-intervals and, hence, a number of possible equilibria. The assumption that

EQ(s = 0) < c1 leads to fewer sub-intervals and fewer possible equilibria. The insights gained are virtually identical

and the results obtained are less repetitive.


Also, note that

EQðs ¼ 1Þ � cC ¼ rL > 0ZcC < EQðs ¼ 1Þ: ðA14Þ

cC � c1 ¼ L > 0ZcC > c1: ðA15Þ

By assumption, c1�EQ(s = 0)=(d� c)(H� L)� L>0, so EQ(s= 1)>cC>EQ(s = 0).From

Eqs. (A14) and (A15), we get cCa[c1, EQ(s = 1)). In the cost range ca[c1, cC), Ep1(L,w1

(L)I = 0)>Ep1(L,w1(L)I =EQ(s = 1)� c), so that w1(L)*= 0. In the range ca[EQ(s = 0),

cC),w1(L)*= 0. In the range ca[cC, EQ(s = 1)), Ep1(L,w1(L)I = 0)VEp1(L,w1(L)

I =EQ

(s = 1)� c and w1(L)*=EQ(s = 1)� c.

The analysis for F1’s wage offer to an H-type worker once again involves comparing

the expected profit for the wage offers EQ(s= 1)� c and 0. In the latter case, it retains

the H-type worker only if F2 receives s = 0, which occurs with probability (1� q).

Hence,

Ep1ðH ;w1ðHÞI ¼ EQðs ¼ 1Þ � cÞ ¼ H � EQðs ¼ 1Þ þ c ðA15Þ

Ep1ðH ;w1ðHÞI ¼ 0Þ ¼ ð1� qÞH : ðA16Þ

Subtracting Eq. (A16) from Eq. (A15), equating to 0 and solving for c, we get

cB ¼ ðd � qÞH þ ð1� dÞL: ðA17Þ

Also, note that cC� cB = qH� rL>0, because q>r and H>L. Therefore, we get

EQðs ¼ 1Þ > cC > cB: ðA18Þ

Comparisons of cB and c1 as well as cB and EQ(s = 0) are intractable. However, if

ca[EQ(s = 0), cB), then Ep1(H, w1(H)I = 0)>Ep1(H, w1(H)

I = EQ(s = 1)� c), so that

w1(H)*= 0. F1 retains the H-type worker with probability only if F2 receives s= 0, which

occurs with probability (1� q). If ca[cB, EQ(s= 1)), then it follows that Ep1(H, w1(H)I =

EQ(s = 1)� c)zEp1(H, w1(H)I = 0) and w1(H)*=EQ(s= 1)� c. Hence, F1 retains the

worker. Finally, note that if cB <EQ(s = 0), then w1(H)*=EQ(s = 1)� c and F1 retains

the H-type worker with certainty.

(c) Prohibitive switching costs: czEQ(s = 1). We have EQ(s = j)� cV 0, j = 0, 1, so that F1’s

F1’s wage offer is (w1(H)*, w1(L)*)=(0,0) and workers of both types are retained with

certainty. 5

Proof of Lemma 1. First, note that EQ(s= 0)� cA=(qH� (d� c)(H� L))/q and EQ

(s = 0)� cB = qH� (d� c)(H� L). It follows that qH� (d� c)(H� L)>0Z cA <EQ(s= 0)

Z cB <EQ(s = 0) and cB <EQ(s = 0)Z cA <EQ(s= 0). Next, note that if qH� (d� c)(H� L)V 0Z cAzEQ(s = 0)Z cBzEQ(s = 0) and cBzEQ(s = 0)Z cAzEQ(s = 0). The

proof of part (iii) is contained in the proof of Proposition 3. 5


Proof of Proposition 4. The separating equilibrium does not exist, so consider the pooling

equilibrium, i.e., w1(Q) =w1l, Qa{L,H}, w1

lawl. We have r(Q =Hjs =w1l) = p and

r(Q = Ljs =w1l)=(1� p). Let EQ(s =w1

l) =EQ(s) denote F2’s expected productivity of the

worker, given F1’s wage strategy. Clearly, F2’s equilibrium offer is w2(s =w1l)*=

EQ(s) = L + p(H� L). If the worker switches to F2, his payoff is EQ(s)� c. F1’s maximum

wage offer is L to any worker. So the minimum switching cost, cmin, that can guarantee a

pooling equilibrium satisfies EQ(s)� cmin = L, i.e., cmin = p(H� L). If c>EQ(s), then the

payoff to the worker from switching is 0; therefore, w1(Q)*= 0 and the worker stays with

F1. If ca[cmin, EQ(s)], then EQ(s)a[cmin, L]. In this case, F1 can retain the worker by

offering EQ(s)� c, i.e., w1(Q)*=EQ(s)� c. Finally, note that, since F2 offers EQ(s) to all

workers and since it is profitable for F1 to retain all workers for w1(Q)V LVQ, then F1’s

equilibrium wage offers are the same for all z>0. 5

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Labour market signalling and job turnover revisited Banerjee y Gaston 2003

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