holdouts, backdating and wage negotiations

Holdouts backdating and wage negotiations

Harold Houba a and Wilko Boltb

a Department of Econometrics Free University De Boelelaan 1105 1081 HV

Amsterdam Netherlands b Econometric Research and Special Studies Department

De Nederlandsche Bank PO Box 98 1000 AB Amsterdam Netherlands

Received 1 June 1997

accepted 1 October 1998

Available online 25 September 2000

Abstract

The Haller and Holden (JET 1990) wage bargaining model is extended to

incorporate holdouts with and without work-to-rule inefficient holdouts and

backdating of new contracts The unions most effective action inflicts the highest

costs upon the firm among the credible actions Necessary and sufficient conditions

for equilibria with lengthy holdouts are derived Backdating does not affect the

bargaining positions of the parties The settlement wage negatively depends upon

the length of the holdout and this dependence does not disappear as the time

between bargaining rounds vanishes This result has implications for empirical work

Moreover this negative effect is small and confirms empirical evidence for the

Netherlands

Author Keywords Strategic bargaining Holdouts Backdating Settlement wages

JEL classification codes C78 J50

Article Outline

1 Introduction

2 A motivation of the basic assumptions

3 A model of wage bargaining

4 Work-to-rule as substitute for strike

5 Equilibria with lengthy work-to-rule

6 Backdating

7 Concluding remarks

Acknowledgements

References

1 Introduction

Collective bargaining in the Netherlands has a large impact on wage formation It

covers the employment terms of 70ndash80 of the labour force in the private sector

while employer coverage is about 90 of all firms Although strike incidence is very

low compared to other European countries (eg Layard et al 1991 p 98) wage

formation in the Netherlands is a time-consuming process with lengthy negotiations

between unions and employersrsquo associations Holdouts ie the negotiation periods

between expiration of an old contract and the signing of a new contract take an

average of 7ndash8 months (see Van de Wijngaert 1994) whereas in the US the average

holdout period only takes 2 months (see Cramton and Tracy 1992) During a holdout

production continues and the terms of the old contract apply

In the empirical literature on industrial disputes the main focus is on strikes as means

to convey private information see eg Kennan and Wilson (1993) for a survey Not

much attention however is paid to the economic content of holdouts in wage

bargaining except Cramton Cramton and Cramton Gu and Kuhn (1998) Holden

Holden and Holden and Moene (1988) Data on labour disputes indicate that for

instance in the US Canada and the Netherlands holdouts are more frequent than

strikes (see Cramton and Tracy 1992 Gu and Kuhn 1998 Van de Wijngaert 1994)

Furthermore it is well-known that unions often use other weapons than strikes eg

work-to-rule go-slow overtime bans etc which may reflect that a holdout takes

place1 In a recent article Van Ours and Van de Wijngaert (1996) present an

exploratory empirical analysis of the relationship between holdouts and wage

bargaining in the Netherlands Their estimations show that holdouts have a

significant negative effect of 01 per two months of holdout on the negotiated wage

increase However Van Ours and Van de Wijngaert (1996) do not offer a theoretical

explanation for this finding but merely conclude that lsquoto some extent holdouts are

substitutes for strikes Our article contributes to the above-mentioned literature by

addressing the (endogenous) choice between various types of industrial action and

the way this affects equilibrium behaviour and settlement wages

The aim of our analysis is threefold First within an extended version of the wage

bargaining model as proposed in Fernandez and Glazer (1991) Haller (1991) and

Haller and Holden (1990) we investigate to which extent holdouts are substitutes for

strikes For that purpose it is assumed that the union has three strategic actions

during negotiations that differ with respect to the costs inflicted upon the parties

These actions are strike and whether or not to work-to-rule during a holdout In this

article it is shown that the unions most effective action is the action that inflicts the

highest costs upon the firm among the unions options that are credible An option is

credible for the union if its cost do not exceed its benefits ie the wage increase

Second we show that the extended model is able to capture the time-consuming

wage negotiations with lengthy holdouts observed in the Netherlands Since the

unions actions may inflict costs upon both parties during these lengthy holdouts

such industrial action may not be something the union wants to choose just of itself

Two opposing effects are at play here The equilibrium conditions for the firm induce

an upper bound upon the settlement wage that is independent of the length of the

holdout period while the equilibrium conditions for the union induce a lower bound

upon the settlement wage that is increasing in the length of the holdout period So

the two opposing forces cannot unambiguously explain the negative relation between

length of the holdout period and wage increases as observed in Van Ours and Van

de Wijngaert (1996) Hence this result indicates that an important feature is still

lacking the model

The third aim is to identify this lacking feature which we link to a practice commonly

observed in the Netherlands namely backdating new wage contracts to the

expiration date of the old wage contract In this article it is shown that the length of

the holdout period has an unambiguously negative but small effect upon the

settlement wage when wage contracts are backdated confirming the main finding in

Van Ours and Van de Wijngaert (1996) Furthermore backdating does not affect the

bargaining position of each party in terms of utilities which strengthens common

wisdom that backdating is a minor detail of wage negotiations Although backdating

is easy to deal with our results imply that in empirical work a clear distinction should

be made between utility levels and wage levels in case holdouts and backdating are

observed in the data

The paper is organized as follows In Section 3 the wage bargaining model is formally

specified In Section 4 the maximum and minimum wage contract the union can

subtract from the firm are derived Section 5 contains the characterization of the limit

set of equilibrium payoffs as the time between bargaining rounds vanishes

corresponding to equilibria with lengthy work-to-rule before agreement is reached

The role of backdating is analyzed in Section 6 Finally Section 7 contains some

concluding remarks First the key assumptions of the model are discussed in Section

2


An essential ingredient of any wage bargaining model is that the union may use

different types of industrial action that in principle may inflict costs upon both parties

Several explanations of these costsrsquo sources are mentioned in Cramton Cramton

and Cramton Holden Holden and Holden Moene (1988) and Van Ours and Van de

Wijngaert (1996) Here we discuss how these explanations are reflected in our

model

In economic literature a holdout is the period in between the expiration date of the old

contract and the date a new contract is signed During this period production

continues under the terms of the old contract and meanwhile the parties negotiate

During holdouts the union may carry out strategic threats such as work-to-rule or go-

slow Work-to-rule in Holden (1997) means that workers deliberately follow the work

rules in an inflexible manner without breaking the expired contract in order to reduce

profits Crucial to work-to-rule is that there are no verifiable violations of the old

contract and therefore workers are paid the full wage as specified by the old

contract However in Holden (1997) it is argued that the pay system may allow for

some flexibility and could include for instance bonus payments which can be

suspended under a holdout In addition costs of organizing work-to-rule may exist

Defined in this manner the union bears some costs in adopting work-to-rule2 Strike

on the other hand disrupts production and implies a complete work stoppage

In our extended wage bargaining model the union has three options and these

actions are ranked with respect to the costs the union has to bear Strike has the

highest cost holdout with work-to-rule has lsquointermediatersquo costs and holdout without

work-to-rule has lowest costs which will be normalized to zero With three strategic

options for the union competition among these options enters the analysis There is

no loss of generality because the results for three options can be easily extended to

allow for more options

What about the costs the firm has to bear From the discussion above the answer

seems simple Work-to-rule and strike reduce labour productivity and therefore

reduce profitability Indeed the empirical studies in Cramton Cramton and Cramton

and Van Ours and Van de Wijngaert (1996) mention this possibility However

another possibility is also mentioned in Cramton and Tracy (1994a) namely due to a

technological change that is already implemented production under the old contract

is inefficient and a new contract is needed in order to improve efficiency Another

explanation could be an efficiency wage argument A wage increase boosts the

workersrsquo motivation and therefore a new contract increases labour productivity So

even without a work-to-rule policy the firm may already suffer opportunity costs from

not having reached a new contract during a holdout These costs are captured in our

extended model by assuming that holdout without work-to-rule is inefficient

Furthermore if the union adopts a work-to-rule policy then profitability is lower than

in case the union would not work-to-rule So we explicitly distinguish two sources of

inefficiency mentioned in Cramton and Tracy (1994a) As for the union the three

strategic options are ranked with respect to the costs the firm has to bear Holdout

without work-to-rule inflicts the lowest costs holdout with work-to-rule inflicts

intermediate costs and strike inflicts the highest costs

To summarize In the wage bargaining model in Fernandez and Glazer (1991) Haller

(1991) and Haller and Holden (1990) holdouts are simply treated as production under

the old contract that do not inflict any costs upon either party ie holdout is efficient

In our extended model there are three types of industrial actions ie strike holdout

with work-to-rule and holdout without work-to-rule and all three are inefficient So

our model captures several important aspects mentioned in the empirical literature

For convenience we will refer to holdouts with respectively without work-to-rule as

work-to-rule and holdouts throughout the remainder


The wage bargaining model studied in this paper extends the wage bargaining model

introduced in Fernandez and Glazer (1991) Haller (1991) and Haller and Holden

(1990) in order to incorporate on the one hand inefficient holdout and work-to-rule

and on the other hand backdating of new wage contracts We assume that both the

firm and the union discount the stream of payoffs with a common discount factor δ

[0 1) This assumption is made in order to avoid the technical problems reported in

Bolt (1995) in case the firm is less patient than the union Furthermore even if we

would assume that the firm is more patient than the union then the analysis with

different discount factors would follow our analysis However formulas in case of

different discount factors are rather cumbersome

The firms gross profits are normalized to 1 in each period Hence the set of feasible

payoff vectors in every period is given by where s1

denotes the unions payoff and s2 denotes the firms payoff The expired wage

contract specifies the per period expired wage w0 0ltw0lt1 If the union decides to

strike in case of disagreement then the vector with per period disagreement payoffs

of strike is normalized to (0 0) Alternatively the union may also choose to holdout or

to work-to-rule The vector with per period payoffs under holdout is given by (w0

αminusw0) with αlt1 an efficiency parameter Similarly the vector of per period

disagreement payoffs of work-to-rule are ((1minusγ)w0 βminusw0) with 0ltγlt1 the per period

costs of work-to-rule measured as a fraction of the expired wage and βleα the

efficiency parameter of work-to-rule We assume that production under either holdout

or work-to-rule is profitable for the firm ie w0ltβleα

As already discussed in Section 2 holdout respectively work-to-rule induce some

inefficiency which are captured by 1minusα and 1minusβ Note that the inefficiency of work-to-

rule consists of two parts namely the inefficiency 1minusα due to holdout and on top of

that the inefficiency αminusβ due to deliberately work-to-rule In the empirical literature no

distinction is made between holdouts and work-to-rule in the estimations but lsquothersquo

efficiency parameter is estimated to be 098 for the Netherlands (eg Van de

Wijngaert 1994) and 094 for the US (eg Cramton and Tracy 1992) Although we

assume βleαlt1 and γgt0 we will also discuss the case α=β=1 and γ=0 because we

regard the latter case as the model in Fernandez and Glazer (1991) Haller (1991)

and Haller and Holden (1990)

Bargaining begins just after the expiration of the old contract at time t=0 with the

union making the initial proposal As long as no agreement is reached the parties

alternate in making wage offers with the union making offers in even periods and the

firm in odd periods In each period of disagreement the union selects its threat that

is decides to strike or to adopt a work-to-rule policy or to holdout If a proposed

wage is accepted then negotiations are over and the new wage contract is assumed

to hold thereafter Thus implicitly it is assumed that only a single new wage contract

is negotiated

The total payoffs of the firm and the union depend upon the disagreement payoffs

before an agreement is reached (if reached at all) and the wage of the new

agreement Consider negotiations that are concluded at time with

agreement upon w w [0 1] and the sequence of vectors xtTminus1t=0 that denote the

payoff vector at period t xt (0 0) (w0 αminusw0) ((1minusγ)w0 βminusw0) and 0letleTminus1 The

corresponding vector of normalized discounted payoffs is given by

The second innovative feature in our model is that the new wage contract is

backdated This means that the firm pays once an additional one-period lump-sum

transfer to the workers on top of the newly agreed wage contract at the time the new

agreement is reached The size of this sum is equal to the foregone difference

between the new and old wage contract times the number of periods the contract is

backdated Formally if w is the new wage contract agreed upon at time T and this

contract is backdated for hT 0lehTleT periods then the firm pays w+hT (wminusw0) at time

T and w at time t tgeT+1 The unions utility of such an agreement at time T is given

by

(31)

Similarly the present value of the firms profit at time T is given by

Backdating is not considered until Section 6 where it is assumed that hT=T Different

assumptions for instance when backdating only applies to periods in which

production takes place would not qualitatively change our results

Finally the wage bargaining model is a multi-stage game of complete information

and consequently we will focus on subgame perfect equilibria (SPE)


In this section we characterize the minimum and maximum equilibrium wage as a

function of the discount factor under the assumption that no backdating takes place

The aim is to derive conditions under which work-to-rule can be a substitute for strike

Similar as in Fernandez and Glazer (1991) Haller (1991) and Haller and Holden

(1990) the minimum equilibrium wage corresponds to strategies in which the union

chooses the least costly option ie holdout as long as no agreement is reached

Thus the union refrains from work-to-rule or strike Since holdout is also the action

that inflicts the lowest costs upon the firm holdout is the unions action with the

lowest efficiency loss Therefore the Pareto improvement of any new contract is

limited to 1minusα and consequently the wage increase has to be modest

Whenever strike is credible then the maximum equilibrium strategies are identical to

those in Fernandez and Glazer (1991) Haller (1991) and Haller and Holden (1990)

and the union alternates between holdout and strike in case of disagreement such

that the costs it inflicts upon the firm are as large as possible This is accomplished if

the union strikes just after the firm has rejected a demand made by the union and it

should holdout just after it rejected an offer made by the firm However a strike does

not only inflict costs upon the firm but also on the union Therefore for a strike threat

to be credible the union must nevertheless gain from carrying out this threat This is

ensured by the equilibrium strategies which prescribe an immediate switch to the

equilibrium that induces the lowest equilibrium wage whenever the union fails to carry

out such a strike threat So at the first occasion in which the union does not carry out

its threat of strike the minimum wage equilibrium strategies prescribe the

continuation in the game from that point in time onwards If strike is not considered

credible ie δ2ltw0α below then the union can use the threat of work-to-rule

similarly as just described with respect to strike (read work-to-rule instead of strike

every time strike is mentioned) The results in Haller (1991) can be applied directly in

order to determine the highest equilibrium wage that can be obtained by the threat of

work-to-rule

The next theorem precisely characterizes the minimum and maximum wage at period

t denoted by wmin(t) respectively wmax(t) for t is even The economic interpretation is

that the maximum equilibrium wage is achieved if the union adopts the option that

inflicts the highest costs upon the firm among the options that are credible We do not

explicitly state the equilibrium wages at t is odd because it consists of w0 plus δ

times the equilibrium wage increases at t is even

Theorem 41 Let t be even The wage wmin(t) at period t as function of δ is given by

(41)

If γlt(αminusβ)(αminusw0) then the wage wmax(t) at period t as function of δ is given by

(42)

Similarly if γge(αminusβ)(αminusw0) then the wage wmax(t) at period t is given by wmin(t) if

δ2ltw0α and w0+(1minusw0)(1+δ) otherwise

Proof First consider wmin(t) Since the union chooses the least costly option ie

holds out the union has no incentive to deviate Then wmin(t) is identical to player 1s

unique SPE proposal in round t of the standard alternating offer model in which one

dollar is disputed utility functions are δtsi i=1 2 and disagreement point (w0 αminusw0)

Second as in Haller (1991) and Haller and Holden (1990) the maximum equilibrium

wage under the threat of strike is given by w0+(1minusw0)(1+δ) at t even and

w0+δ(1minusw0)(1+δ) if t is odd The only relevant equilibrium condition requires that

strike is credible in case of disagreement at t even ie

(43)

where w0+δ(1minusα)(1+δ) is wmin(t) at t odd This condition reduces to δ2gew0α Third if

strike is not credible then in terms of Haller (1991) we have that a=βminusw0 b=(1minusγ)w0

1minusr=w0 and the union demands 1minusα=1minus1(1+δ) [r+δa] and the firm offers

1minusβ=1minus1(1+δ)[a+δr] The only relevant equilibrium condition requires that work-to-

rule is credible in case of disagreement at t is even ie

which yields δ2geγw0(αminusβ+γw0) Finally the interval [γw0(αminusβ+γw0) w0α) is empty iff

γge(αminusβ)(αminusw0)

The results in Fernandez and Glazer (1991) Haller (1991) Haller and Holden (1990)

ie α=β=1 and γ=0 belong to the case γge(αminusβ)(αminusw0) which shows that these

results are robust if the standard model is extended Furthermore strike (work-to-

rule) is credible if the unions costs w0 (γw0) of this action do not exceed the net gain

of this action that comes in the form of a future wage increase ie investment in such

an action should be profitable Note that γ does not enter wmax(t) because work-to-

rule is only used in every even period in which only the firms disagreement payoff

βminusw0 matters

Theorem 41 makes it possible to answer the question to what extent work-to-rule

can be used as a substitute for strike It is easy to see that the maximum wage

increase corresponding to work-to-rule is a factor λ=(1minusβ)(1minusw0) times the wage

increase associated with strike Obviously β=1 corresponds to λ=0 Furthermore

work-to-rule is an imperfect substitute for strike ie λlt1 iff βminusw0gt0 The latter

inequality should be read as Production under the work-to-rule yields a higher profit

than strike does or equivalently the firms costs of strike exceed those of strike

However there is a situation in which work-to-rule serves as a substitute for strike

namely in case the unions costs of work-to-rule are small and work-to-rule is credible

while the more effective strike is not available as a credible option ie γ [0

(αminusβ)(αminusw0)) and δ2 [γw0(αminusβ+γw0) w0α)

The results in this section enable us to briefly comment on a closely related issue of

independent interest namely the special case in which the union fails strike as a

strategic weapon and it has to resort to holdout or work-to-rule This is the relevant

case for professions such as the police the army customs and firemen for which

strike is simply forbidden by law Also in the Netherlands strike is forbidden by law if

the coverage of workers that are willing to strike is too low Finally this is the relevant

case if there are other compelling non-economic reasons as for instance ideological

reasons for why it is simply taboo for individual employees to go on strike From

Theorem 41 it immediately follows that for this special case wmin(t) is not affected

and that wmax(t) at t even is simply given by


Dutch wage negotiations often feature lengthy delay without strike activity before

agreement is reached The question arises whether this pattern of wage

determination can be supported within the bargaining model under investigation In

this section an affirmative answer to this question is given Since holdout can be

regarded as a special case of work-to-rule ie β=α and γ=0 only equilibria with

lengthy work-to-rule are considered First we will derive necessary and sufficient

equilibrium conditions for lengthy work-to-rule before the negotiations are concluded

Second we derive limit results for such equilibria if the time between proposals

vanishes

Loosely stated the strategies with work-to-rule for the first T periods (without loss of

generality we assume T is even) are as follows at an even period t tltT the union

demands a wage equal to 1 the firm (obviously) rejects such offer after which the

union works to rule At time T the union demands w and the firm accepts every wage

not exceeding w At an odd period t tltT the firm offers the wage w0 which the union

rejects followed by work-to-rule As soon as the union does not make the prescribed

demand at even periods t tleT this party is punished by an immediate switch to the

minimum wage equilibrium of Theorem 41 Similar if the firm does not make the

prescribed offer at odd periods before T this party is punished by an immediate

switch to the maximum-wage equilibrium of Theorem 41 Obviously these strategies

induce T periods of work-to-rule followed by agreement upon w The associated

continuation payoff vector at the start of round t tleT is denoted by s(Tminust w δ) and

given by

(51)

Note that the firms continuation payoff strictly decreases in t if and only if 1minuswltβminusw0

ie work-to-rule generates higher profits than the new wage

The presence of decreasing continuation payoffs is the more interesting case from

both a theoretical as from an empirical point of view From a theoretical point of view

this case includes α=β=1 and γ=0 which is loosely speaking assumed in the standard

wage bargaining model (eg Fernandez and Glazer 1991 Haller and Holden 1990)

From an empirical point of view this case reflects the estimate of the efficiency

parameter of 098 for the Netherlands (eg Van de Wijngaert 1994) and 094 for the

US (eg Cramton and Tracy 1992)

In principle in deriving strategies which support delay in equilibrium in a full-

information framework two opposing forces are at play First during a delay the

union must be willing to forego additional income available from immediate

agreement by expecting a sufficient high settlement wage after the delay This

determines a lower bound on the settlement wage Second the firm must not have

an incentive to make an offer that the union cannot reject ie by offering the union

the maximum equilibrium wage This determines an upper bound on the settlement

wage profits afterwards must be sufficient to make up for the loss suffered during the

delay In order to support an equilibrium the settlement wage must at least offset

these two opposing effects

Theorem 51 Suppose βgt(1+δw0)(1+δ) and δ2gew0α Then for Tge2 and T even the

vector s(T w δ) is a vector of equilibrium payoffs at t=0 iff w and T satisfy

Moreover is a vector of equilibrium payoffs at t=0 iff

Proof Consider T is even The relevant equilibrium conditions are s1(Tminust w

δ)gewmin(t) and s2(Tminust w δ)ge1minuswmax(t) for all t=0hellipT First for t=T we obtain w

[wmin(T) wmax(T)]=[wmin(0) wmax(0)] because T is even Second wgewmin(0)gew0

implies that the unions utility s1(Tminust w δ) increases in t and therefore the most

profitable deviation for the union is at t=0 Rewriting yields

Third strictly decreases in t if and only if wgtw0+1minusβ The presence of

either decreasing or increasing payoffs makes it necessary to distinguish two cases

Case 1 wlew0+1minusβ Then increases in t and the most profitable

deviation for the firm is at t=0 Rewriting yields

(52)

and βge(1+δw0)(1+δ)gt(w0+δ)(1+δ) implies that the right-hand side is larger than

w0+1minusβ Therefore (52) is not binding

Case 2 wgtw0+1minusβ Then strictly decreases in t and therefore the

most profitable deviation for the firm is at t=Tminus1 Rewriting

yields

Then the interval

is not empty iff βgt(1+δw0)(1+δ) The latter is assumed

The two conditions in this theorem are only imposed for explanatory reasons

Condition

is the necessary and sufficient condition that ensures equilibria with decreasing

continuation payoffs for the firm are present Without this condition only Case 1 in the

proof has to be considered and nothing changes if

and for βlt(w0+δ)(1+δ) condition (52) in the proof becomes the upper bound upon w

Condition δ2gew0α is imposed in order to restrict the number of cases to be

considered because the analysis in case of

would be similar to the one in Case 1 in the proof and only a minor modification is

needed with respect to the relevant maximum equilibrium wage

The upper bound upon the settlement wage is independent of the length of the

holdout period while the lower bound upon the settlement wage is increasing in the

length of the work-to-rule period So these bounds cannot unambiguously explain

the negative relation between length of the holdout period and wage increases

observed in Van Ours and Van de Wijngaert (1996) Of course the multiplicity of

equilibria implies that it is not hard to find two pairs (w T) and (wprime Tprime) such that TltTprime

and wgtwprime However doing so is not convincing because the opposite ie TltTprime and

wltwprime can also easily be achieved

Finally we mention that the interval of wages is not empty if and only if

(53)

ie the length of the equilibrium work-to-rule cannot become too large

We continue by characterizing the limit set of equilibrium payoffs corresponding to

equilibria with lengthy work-to-rule as time between proposals vanishes This limit set

is denoted as S and it is given by

(54)

where

and Cohellip refers to the convex hull Denote Δ Δgt0 as the time between every two

consecutive bargaining rounds r as the rate of time preference and l lge0 as the

length of the work-to-rule phase measured in continuous time It is standard to take

δ=eminusrΔ Every s S uniquely determines a wage and a delay l (s) measured in

real time (to made precise later) Hence given s S and Δgt0 the number of periods

featuring work-to-rule is which goes to infinity as Δ goes to 0

Note that and in the definition of S

The following theorem states that S is the limit set of equilibrium payoffs and

specifies the wage and length of work-to-rule l (s) for every s S

Theorem 52 Every payoff vector s S is an equilibrium payoff vector

corresponding to an equilibrium with work-to-rule for

(55)

length of time and agreement upon the wage

(56)

Proof Fix s S Then for any Δgt0 there exists a unique real number of periods T(s

Δ) with work-to-rule and wage w(s Δ) such that

where is defined in (51) Solving for and δT(sΔ) and making use

of s S yields where is given in (56) and

δT(sΔ)=(s2+s1minusβ+γw0)(1minusβ+γw0)le1 Making use of δ=eminusrΔ and

yields the expression for given in (55) Next given and we have to

show that the equilibrium conditions in the proof of Theorem 51 hold for sufficiently

small Δs By definition of S and

we have that every s S is a convex combination of and

where both points also belong to S Therefore

lies on the Pareto frontier in between and Hence

and Consider Case 2 in the proof of Theorem 51 The two relevant

equilibrium conditions for Case 2 are

The first condition holds for sufficiently small Δgt0 because and

converges to as Δ goes to 0 The second condition also holds for sufficiently small

Δgt0 because

and as Δ goes to 0 For Case 1 in the proof of Theorem 51 similar

arguments apply

Note that condition δ2gew0α which is imposed in Theorem 51 is automatically

satisfied for sufficiently small Δgt0 As is the case in Theorem 51 the condition

is the necessary and sufficient condition that ensures equilibria with

decreasing continuation payoffs for the firm are present For completeness we

mention that this theorem also holds for For the special case α=β=1

and γ=0 considered in Fernandez and Glazer (1991) and Haller and Holden (1990)

the set S is a line piece on the Pareto frontier with endpoints

3 The length of l (s) is a measure of the degree of

inefficiency if s is relatively close to the Pareto-frontier then l (s) is relatively close to

0

6 Backdating

In this section we first show that the unions minimum and maximum utility of

Theorem 41 are not affected if backdating is incorporated into the model Therefore

the aspect of backdating does not effect the parties strategic opportunities in terms of

utilities which confirms the commonly held point of view that backdating is only a

minor detail of wage negotiations However this theorem also states that lengthy

work-to-rule in the presence of backdating has a dampening effect on the equilibrium

wage Denote respectively as the unions maximum equilibrium

utility respectively the maximum equilibrium wage at period t after ht periods of

production under the old contract Similarly and refer to the

minimum equilibrium values

Theorem 61 Let and be given as in Theorem 41 Then

and and the corresponding wages are

given by

and

Proof It is without loss of generality to assume δ2gew0α and consider

only The unions problem at t even is given by

st

because hT=T implies that ht+1=t+1=ht+1 Solving yields the boundary solution

Substitution into the unions objective function and rewriting yields

Similar at t+1 odd under ht+2=ht+1+1 the firms problem given by

st

yields

Substitution of into and rewriting yields

which admits even as its solution Substitution into

even yields the expression stated for t+1 odd Finally follows from

The dampening effect of holdouts on the wage increase is relatively small4 This can

be seen as follows Rewriting the expression for yields

(61)

and the term is relatively small for lsquorealisticrsquo values of δ and ht For

example if Δ=1 (one bargaining round lasts a day) ht=210

(roughly 7 months) and δ=eminusrΔasympr with r=14times10minus5 (an annual rate of 511) Thus

neglecting backdating yields a prediction of the maximum wage increase

that overshoots the prediction of the model with backdating (by about 29 in the

example) Empirical evidence for this theoretical small effect is reported in Van Ours

and Van de Wijngaert (1996) who report a 01 negative effect on new wages per

two months of production under the expired wage contract for the Netherlands

The equilibria of the previous section can be easily extended to incorporate

backdating Backdating simply means that we have to distinguish between utilities

and wages The relation between wage w and utility s1 after T periods of holdout is

straightforward

Hence backdating has a dampening effect This result also holds in the limit as Δ

goes to 0 provided the length of the holdout in real time is kept constant Let s S

then given by (56) has to be interpreted as the unions utility of the agreement

that includes backdating after time of work-to-rule where is given in (55)

Denote the settlement wage including backdating as The following

theorem states that the negative relation between the wage and the

length of work-to-rule l (s) Hence backdating unambiguously explains the empirical

findings in Van Ours and Van de Wijngaert (1996)

Theorem 62 Every s S is a vector of equilibrium utilities and the limit wage

where respectively are given in (56) and (55)

Proof Minor modification is the arguments of the proof of Theorem 51 show that

every s S is a vector of equilibrium utilities Furthermore for every s S and Δgt0

the backdated wage satisfies

where Thus

Finally application of LHopitacircls rule yields

For every s S it holds that the limit discrepancy between the unions utility and the

level of the settlement wage level is given by

(62)

which increases the larger l(s) becomes The implication for empirical work is evident

If production under the old contract and backdating are observed in the data then the

unions utility and the level of the wage should be clearly distinguished and a

modification is necessary

The bargaining model can easily be extended in order to let the parties propose

whether or not to backdate wage contracts ie endogenous backdating From above

we have that both the firm and the union are indifferent between the wage

without backdating and the wage at every period t But then all the

equilibrium strategies derived thus far constitute one of the SPEs in the extended

model with endogenous backdating Furthermore the (limit) set of equilibrium payoffs

will not change Thus a richer model can explain the equilibrium behaviour derived in

this section ie lengthy work-to-rule and backdating

The interesting case is the extension to different discount factors ie δUneδF First

suppose the firm is more patient than the union ie δFgtδU Then the reduction in

future wage level that the union will require in order to obtain backdating is less than

what the firm would be willing to offer This means that there is room for Pareto

improvement by backdating Formally consider the wage contract wBgtw0 after T

periods of production then the sum of the parties utilities is equal to

and the parties will backdate new wage contracts Recursive relations for the unions

maximum equilibrium and can easily be given simply by

replacing δ by either δU or δF in the proof of Theorem 61 but its solution is very

cumbersome Therefore it remains an open question whether the immediate

agreement result in the unions best and worst SPE found for δU=δF also holds for

δFgtδU because backdating and lengthy production under the old contract (which

causes delay) enlarge the surplus For the opposite case neglecting the problems

reported in Bolt (1995) we do not expect backdating because it reduces the size of

the surplus


One remark should be made with respect to equilibria in which the union strikes in all

periods before a new settlement wage is agreed upon Since backdating only applies

to periods in which the union held out and these equilibria do not involve holdouts it is

obvious that an analysis of such equilibria in our model simply boils down to the by

now well-known analysis of these equilibria given in Fernandez and Glazer (1991)

Haller (1991) and Haller and Holden (1990) Therefore we feel that there is no loss in

generality by not investigating these equilibria in this paper although a minor

modification is needed in order to take into account the efficiency parameter of

holdout

One essential variable that is absent in the modified wage bargaining model is

employment If the wage bargaining model with backdating would be further modified

such that the firms employment adjusts to wage increases and the union cares about

wages and employment then the maximum wage increase in such an extended

model would be lower than the maximum wage increase in Theorem 41 The

intuition is simple The union faces a trade off between a higher wage and a lower

level of employment and it therefore sacrifices some of the wage increase in order to

make the deterioration of employment less Thus the absence of employment

considerations in our model leads to a systematic bias toward higher wage increases

and consequently toward a systematic higher prediction of the dampening effect of

holdouts on wage increases

Acknowledgements

The authors thank Gerard van der Laan Steinar Holden and the anonymous referees

for valuable suggestions and critical comments The usual disclaimer applies

References

Bolt W 1995 Striking for a bargain between two completely informed agents

Comment American Economic Review 85 pp 1344ndash1347

Cramton P and Tracy J 1992 Strikes and holdouts in wage bargaining Theory

and data American Economic Review 82 pp 100ndash121

Cramton P and Tracy J 1994 The determinants of US labour disputes Journal of

Labor Economics 12 pp 180ndash209 Full Text via CrossRef

Cramton P and Tracy J 1994 Wage bargaining with time-varying threats Journal

of Labor Economics 12 pp 594ndash617 Full Text via CrossRef

Fernandez R and Glazer J 1991 Striking for a bargain between two completely

informed agents American Economic Review 81 pp 240ndash252

Gu W and Kuhn P 1998 A theory of holdouts in wage bargaining American

Economic Review 88 pp 428ndash449 View Record in Scopus | Cited By in Scopus (4)

Haller H and Holden S 1990 A letter to the editor on wage bargaining Journal of

Economic Theory 52 pp 232ndash236 Article | PDF (299 K) | View Record in Scopus

| Cited By in Scopus (49)

Haller H 1991 Wage bargaining as a strategic game In Selten R Editor 1991

Game Theoretic Equilibrium Models III Strategic Bargaining Springer Berlin pp

230ndash241

Holden S 1989 Wage drift and bargaining Evidence from Norway Economica 56

pp 419ndash432 Full Text via CrossRef | View Record in Scopus | Cited By in Scopus

(18)

Holden S 1994 Wage bargaining and nominal rigidities European Economic

Review 38 pp 1021ndash1039 Abstract | PDF (1188 K) | View Record in Scopus |

Cited By in Scopus (22)

Holden S 1997 Wage bargaining holdout and inflation Oxford Economic Papers

49 pp 235ndash255 View Record in Scopus | Cited By in Scopus (12)

Kennan Wilson 1993 Bargaining with private information Journal of Economic

Literature 31 45ndash104

Layard R Nickell S and Jackman R 1991 Unemployment Macroeconomic

Performance and the Labour Market Oxford University Press Oxford

Moene K 1988 Unionsrsquo threats and wage determination Economic Journal 98 pp

471ndash483 Full Text via CrossRef

Salamon M 1987 Industrial Relations Theory and Practice Prentice-Hall

London

Van Ours J and Van de Wijngaert R 1996 Holdouts and wage bargaining in the

Netherlands Economics Letters 53 pp 83ndash88 Article | PDF (561 K) | View

Record in Scopus | Cited By in Scopus (5)

Van de Wijngaert R 1994 Trade Unions and Collective Bargaining in the

Netherlands PhD Thesis

Corresponding author email hhoubaeconvunl

1 Salamon (1987 p 331) reports that in the US around 25 of industrial disputes are

due to work-to-rule and go-slow

2 In Moene (1988) go-slow is distinguished from work-to-rule where the latter is

without cost for the union Go-slow also refers to situations in which labour

productivity is deliberately reduced but it involves verifiable violations of the old

contract which reduces the wage to be paid

3 A minor modification in the proof is needed if α=β=1 and γ=0 Then we first choose

s S such that and next arbitrarily choose

Then

suffices to obtain

4 We thank Steinar Holden for bringing this point to our attention and suggesting

formula (61)


6 Backdating


Acknowledgements

References

1 Introduction

Collective bargaining in the Netherlands has a large impact on wage formation It

covers the employment terms of 70ndash80 of the labour force in the private sector

while employer coverage is about 90 of all firms Although strike incidence is very

low compared to other European countries (eg Layard et al 1991 p 98) wage

formation in the Netherlands is a time-consuming process with lengthy negotiations

between unions and employersrsquo associations Holdouts ie the negotiation periods

between expiration of an old contract and the signing of a new contract take an

average of 7ndash8 months (see Van de Wijngaert 1994) whereas in the US the average

holdout period only takes 2 months (see Cramton and Tracy 1992) During a holdout

production continues and the terms of the old contract apply

In the empirical literature on industrial disputes the main focus is on strikes as means

to convey private information see eg Kennan and Wilson (1993) for a survey Not

much attention however is paid to the economic content of holdouts in wage

bargaining except Cramton Cramton and Cramton Gu and Kuhn (1998) Holden

Holden and Holden and Moene (1988) Data on labour disputes indicate that for

instance in the US Canada and the Netherlands holdouts are more frequent than

strikes (see Cramton and Tracy 1992 Gu and Kuhn 1998 Van de Wijngaert 1994)

Furthermore it is well-known that unions often use other weapons than strikes eg

work-to-rule go-slow overtime bans etc which may reflect that a holdout takes

place1 In a recent article Van Ours and Van de Wijngaert (1996) present an

exploratory empirical analysis of the relationship between holdouts and wage

bargaining in the Netherlands Their estimations show that holdouts have a

significant negative effect of 01 per two months of holdout on the negotiated wage

increase However Van Ours and Van de Wijngaert (1996) do not offer a theoretical

explanation for this finding but merely conclude that lsquoto some extent holdouts are

substitutes for strikes Our article contributes to the above-mentioned literature by























lacking the model



















2







model























































































is negotiated















by

(31)



































work-to-rule








(41)


(42)











(43)















βminusw0 matters































vanishes













given by

(51)
































(52)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)























lacking the model



















2







model























































































is negotiated















by

(31)



































work-to-rule








(41)


(42)











(43)















βminusw0 matters































vanishes













given by

(51)
































(52)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)










2







model























































































is negotiated















by

(31)



































work-to-rule








(41)


(42)











(43)















βminusw0 matters































vanishes













given by

(51)
































(52)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)










































































is negotiated















by

(31)



































work-to-rule








(41)


(42)











(43)















βminusw0 matters































vanishes













given by

(51)
































(52)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)



































work-to-rule








(41)


(42)











(43)















βminusw0 matters































vanishes













given by

(51)
































(52)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)









(43)















βminusw0 matters































vanishes













given by

(51)
































(52)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)































vanishes













given by

(51)
































(52)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)





















(52)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)





yields

Then the interval



Condition


















(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)






(53)





(54)

where












(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)

(55)


(56)
















Δgt0 because


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)


arguments apply










0

6 Backdating













given by

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)

and



st




st

yields






(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)



(61)












straightforward














where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)





where Thus




(62)



























the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)











the surplus










holdout












Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)



Acknowledgements



References


















230ndash241



(18)













London















Then

suffices to obtain


formula (61)










London















Then

suffices to obtain


formula (61)


formula (61)

holdouts, backdating and wage negotiations

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