Inventories and Strikes

Economica (1997) 64, 645–67

Inventories and Strikes

By SIMON CLARK

University of Edinburgh

Final version received 20 January 1997.

This paper analyses a dynamic model of bargaining and strikes in which the reservation wageof union members in each period is private information. The model endogenizes the firm’sdecision to accumulate inventories of finished goods in order to enhance its bargaining posi-tion. With higher inventories, the firm makes a lower wage offer, thereby accepting anincreased probability of a strike. The model can help explain empirical results that have founda positive correlation between wages and inventory accumulation. It also shows that, unlessinventories are controlled for, strike incidence will exhibit a spurious negative statedependence.

INTRODUCTION

Inventories of finished goods allow a firm to keep on selling during a strike,thereby maintaining its revenue and avoiding any loss of consumer good will.This was dramatically illustrated in the United Kingdom during the coalminers’ strike of 1984–85; the failure of strikers to prevent stocks of coal beingdelivered at power stations is widely held to have been one of the main reasonsfor the failure of the strike. A number of studies have established a relationshipbetween strikes and inventories empirically (e.g. Bernstein and Lovell 1953;Christenson 1953, 1955; Neumann and Reder 1984; Gunderson and Melino1987; Paarsch 1990): increased production before and after a strike enablesinventories to be built up before a strike, run down during it, and built upagain afterwards.

The possibility of such intertemporal substitution suggests that there is astrategic advantage to a firm if it enters wage negotiations well endowed withinventories, and a negative effect of inherited inventories on wages has beenfound by Holden (1989) and Coles and Hildreth (1995). There is also someevidence that firms anticipate this strategic advantage. Using data for theUnited States, where the typical wage contract lasts three years, Currie andMcConnell (1992) find that in the year preceding the negotiation of a newcontract the level of inventories increases by 4%. Furthermore, if inventoriesreduce the costs of a strike, we might expect higher inventories to increase thelikelihood of a strike. This is found to be the case by Nicolitsas (1995), usingdata on UK manufacturing in the 1980s.1

These results suggest a complex set of interactions between inventories,strike incidence and bargaining outcomes. The aim of this paper is to providea theoretical framework to analyse these relationships. It takes as its startingpoint the recognition that labour contracts between a firm and a union aretypically for a fixed term, and that on expiry a new contract is negotiated,usually between the same parties. Thus, the initial conditions of one set ofnegotiations are the end conditions of the previous contract. If these end con-ditions are determined in part by the conduct or outcome of the earlier nego-tiations, this induces a dynamic relationship between negotiations. Formal

The London School of Economics and Political Science 1997

ECONOMICA646 [NOVEMBER

models of strike behaviour, based on the theory of bargaining with privateinformation, focus almost exclusively on the analysis of a single trade or con-tract. Interpreting a strike as a form of delay, they emphasize the sequence ofoffer and counter-offer, and the working of screening, signalling and infor-mational updating.2 Very little attention has been paid to the relationshipsbetween negotiations. Although there are a number of papers that recognizethat inventories might play an important role in bargaining, none provides amodel of strategic accumulation. Hart (1989) suggests that during a strike theexistence of inventories can create a ‘crunch-date’, after which a firm’s profit-ability declines dramatically, the inventories having been sold. The effect is toincrease strike duration. His approach is taken up and extended in Cramtonand Tracy (1994). Coles and Hildreth (1995) analyse a Rubinstein bargaininggame with inventories, but in their model strikes do not occur as agreement isimmediate. Papers by Jones and McKenna (1988), Leach (1992) and Clark(1993) model the dynamic interaction between inventories and successive wagebargains, but in a complete information framework.3

In order to analyse the relationship between strike behaviour and inventoryaccumulation, I set up a dynamic model in which the only link between periodsis the level of inventories the firm carries over. I impose an incomplete infor-mation structure which, combined with a bargaining protocol, provides acoherent model of strike behaviour. The focus of interest is not the process ofbargaining per se, so I adopt a simplified representation of bargaining underconditions of asymmetric information. The object of bargaining in each periodis the wage rate; employment and any inventory accumulation or decumulationare determined by the firm; i.e., we have a version of the ‘right to manage’model. The ‘black box’ of union–firm bargaining is modelled by assuming thatin each period the firm makes a single take-it-or-leave-it offer to the union,with a strike occurring if the offer is rejected. I have chosen to focus on therelationship between negotiations, and so adopt the simplest model of bar-gaining that still allows for disagreement. It would, in principle at least, besimple to allow the firm to make a sequence of wage offers, enabling it toscreen the union. This would have the advantage of admitting strikes of uncer-tain length, so that strike duration as well as incidence could be analysed, andwould effectively endogenize the ‘crunch-date’ in Hart (1989). In the final sec-tion of this paper I briefly discuss the consequences of allowing a sequence ofoffers.

I assume that the firm does not know the reservation wage of union mem-bers. The model treats this as a random variable whose distribution is knownto both parties, but whose realization in each period is known only by theunion. It is this asymmetry of information that drives the bargaining processand gives rise to the possibility of strikes. One interpretation of the reservationwage is that it reflects conditions in the local labour market that union mem-bers are obliged to enter if they are not employed by the firm. It seems reason-able to assume that they know more about these local conditions than the firm.With one-sided private information, it is perhaps more common to assumethat the firm has superior knowledge about its profit opportunities. If the unionscreens a better informed firm by making a decreasing sequence of offers, thenthere should be a negative relationship between strike duration and wage settle-ments. On the other hand, if the firm screens the union, there should be a


INVENTORIES AND STRIKES1997] 647

positive relationship. The evidence is remarkably ambiguous on this point.(See e.g. Card (1990b) and the discussion in Kennan and Wilson (1989).) Theconsequences of allowing the union to make wage offers to a better informedfirm are discussed in the final section of the paper.4

I have decided to model the reservation wage as private informationbecause this also allows me to examine the cost-smoothing role of inventories.Since the reservation wage can vary between periods, there is the potential foran efficiency gain from inventory accumulation. An unusually low reservationwage would, in an efficient outcome, be exploited by a high level of employ-ment, the extra production going towards an increase in inventories; similarly,when the reservation wage is high, sales can be maintained by selling fromstock. The overall effect of such cost-smoothing is that production will tendto be more variable than sales. Empirically, this is indeed the case in mostindustries (see Table 4 in Blinder and Maccini 1991). However, it also seemsto be established that cost shocks provide only an incomplete explanation ofthis stylized fact. For example, Miron and Zeldes (1988) find little evidencethat input prices play any role in determining the timing of production.Maccini and Rossana (1984) find that inventory investment is affected by rawmaterials prices, but not by wage rates. More recently, Rossana (1993) hasfound some evidence of cointegration between inventory levels and inputprices; but in a number of industries the relevant coefficient has the wrongsign, and in others it has the right sign but an implausible magnitude. Eichen-baum (1989) finds in favour of cost-smoothing, but his results are based onunobservable and serially correlated shocks to technology, not factor costs.

Not surprisingly, there has been a wide variety of attempts to providealternative explanations of why sales are less volatile than production, includ-ing non-convex production costs (e.g. Ramey 1992), stock-out avoidance (e.g.Kahn 1992) and non-convex acquisition costs leading to the well known (S, s)model of inventory behaviour.5 But the bargaining model presented in thispaper is one involving cost-smoothing, although this would not be picked upin econometric studies misspecifying the informational and bargaining setup.In their paper, Miron and Zeldes estimate a model with competitive inputmarkets, so that the wage rate faced by a firm equals the alternative oppor-tunities available to suppliers of labour. But suppose that variations in thisreservation wage rate are not reflected in the wage rate at which the firm buyslabour, perhaps as a result of some market imperfection; there may still be aneffect on how much labour is employed. This is precisely the situation of themodel presented here, in which the reservation wage rate is private infor-mation; thus, a take-it-or-leave-it offer cannot reflect any shock to the trueopportunity cost of labour. But if an offer is accepted only if it is high relativeto the reservation wage, then production is zero if the reservation wage is high(because there is a strike), although sales can be maintained by running downinventories. Similarly, production is positive if the reservation wage is low(because the offer is accepted), and the firm can use the opportunity to buildup or restore inventory levels. Thus, there is some degree of intertemporalsubstitution. Furthermore, if the wage offer made by the firm is negativelyrelated to the level of inventories that it inherits, as suggested by the results ofHolden (1989) and Coles and Hildreth (1995), then, since inventories are a



substitute for current employment and production, one would observe a posi-tive relationship between wages and changes in inventories: a high level ofinherited inventories leads to a low wage offer, a low level of employment anda tendency to decumulation. This paper shows that the cost-smoothing modeldoes not necessarily imply a negative relationship between wages and inventorychanges. Although input prices may appear to play little role in determiningthe timing of production, this is not necessarily evidence against costsmoothing.

The model presented in this paper also has implications for our understand-ing of the state dependence of strike incidence. A strike will cause inventoriesto be run down; if the firm starts the next round of negotiations with lowerinventories, this will induce it to make a higher wage offer, which has a greaterprobability of acceptance. Although the probability of a strike does not dependdirectly on previous strike incidence, an empirical model that did not controlfor inventory levels would conclude the opposite. There have been a numberof econometric studies of strike behaviour which claim to have found negativestate dependence, none of which has included inventory levels as control vari-ables. The papers by both Schnell and Gramm (1987) and by Mauro (1982)interpret their results as evidence that strikes are a learning process: if strikesare the result of faulty negotiations, then the experience of striking offers thefirm and the union the opportunity to learn from their mistakes. In an extens-ive empirical study of the dynamics of strike behaviour, Card (1988) foundthat strike incidence was reduced by a long strike during previous negotiations,but was increased by a short strike. His results were confirmed by Montgomeryand Benedict (1989). There are a number of studies that claim to have foundpositive state dependence (e.g. Swidinsky and Vanderkamp 1982; Olson 1984)and others that have found no significant effect of previous strike history(Godard 1992; Ingram et al. 1993). These results suggest that neither a modelof learning nor one of strategic inventories is able to explain all the data; bothmodels predict a finding of negative state dependence if strike experience andyor inventories are excluded as control variables. Naturally, in some industriesinventory accumulation may not be possible: both Olson (1984) and Montgom-ery and Benedict (1989) looked at strikes by public schoolteachers in Pennsyl-vania: school boards cannot accumulate stocks of unused lessons.6 Even inindustries where inventory accumulation is feasible, the role of strategic inven-tories may be swamped empirically by other effects. This paper does not claimthat inventories are the only factor inducing a dynamic in the behaviour ofstrike incidence. But it does show that ignoring inventories may lead investi-gators to underestimate the strength of any positive state dependence, even tothe extent of inferring that there is negative state dependence when in fact it ispositive.

The plan of the paper is as follows. Section I sets up a simple model andbriefly analyses the efficient outcome when there is no informational asym-metry, i.e. when the firm knows the reservation wage rate in each period. InSection II, an asymmetry of information is imposed, and this generates thepossibility of strikes if bargaining breaks down. The model is then used toanalyse the dynamic behaviour of inventories and strikes, and the relationshipsbetween them. Section III presents the results of some numerical simulations,and Section IV concludes.



I. THE MODELLING FRAMEWORK

In setting up the model, I simplify the analysis in two important ways. First, Ido not set up an infinite-horizon model. Rather, I analyse a three-period frame-work of inventory accumulation (tG0, 1, 2), representing the past, the presentand the future, with the focus on events in period 1. In the last period (tG2)inherited inventories have a current but not a future value to the firm. Thus,the outcome in period 2 isolates the effect of inherited inventories, andabstracts from any forward-looking or strategic considerations. By con-structing the functions representing the value to the firm and the union ofentering period 2 with a certain level of inventories, we can then examine howthe players’ actions in period 1 (the present) are influenced by their anticipatedeffect on future payoffs. Period 0 is not fully analysed, but the level of inven-tories carried into period 1 is the channel by which previous strike and inven-tory decisions affect the present.

Second, I adopt linear or quadratic forms for all the main functions. Thisallows me explicitly to construct the value functions referred to above, and toderive clear results (although further simplifications are made at certainpoints). The three-period linearyquadratic model is not of course completelygeneral, and an analysis with an infinite horizon and general functional formscan be found in Clark (1994).

Inventories may be held for a wide variety of reasons. In order to concen-trate on the effect of variations in the cost of labour, I adopt a deliberatelyrestrictive model of inventories. I assume that production uses only labour,and that it has a constant average and marginal product. Furthermore, Iassume that the firm’s demand in each period is known before productionoccurs. Thus, in the absence of changes in the cost of labour, there is noefficiency gain to smoothing or bunching production, nor any need to useinventories as a buffer in the face of uncertain demand. I assume that the firm’scustomers cannot hold inventories, and that the firm cannot be prevented fromselling from stock. Furthermore, the firm is not involved strategically with anyother firms: this rules out a collusive motive for holding inventories, as ana-lysed by Rotemberg and Saloner (1989).

I assume that the union is utilitarian with risk-neutral members and a con-stant membership M. Union members not employed by the firm receive a reser-vation wage c, which behaves each period as a random variable drawn from auniform distribution on the interval [cL , cU ]. Let w and L denote the wage rateand the level of employment. Then in period t the union gets utility

UtGwtLtCct(MALt),

and the period t value of the union’s utility stream is

WtG ∑2

sGt

ϕsAtUs ,

where ϕ is the union’s discount factor (0YϕF1).Each period the firm faces the same downward-sloping inverse demand

curve, ptGaAbqt , where p and q stand for price and sales. Production usesonly labour; it has a constant average product, and for convenience I set this



equal to one. Output in period t is therefore Lt ; StA1 is the level of inventoriesat the end of period tA1 and is carried into period t, so that

qtGLtCStA1ASt .

The cost of storing StA1 is borne by the firm in period t, and equals eS2tA1 .

The firm’s profit in period t is therefore

∏tG(aAbqt)qtAwtLtAeS2tA1 ,

and the period t value of its profit stream is

VtG ∑2

sGt

ψ sAt∏s ,

where ψ is the firm’s discount factor (0YψF1).For this model to make sense and to be easily solvable, certain restrictions

on the parameters are necessary. First, I assume that aHcU . This implies that,if inventory accumulation is not possible, it is always efficient to employ labourin each period. Second, for the firm to want to engage in intertemporal substi-tution, it must be sufficiently forward-looking, and there must be enough vari-ation in c. I therefore also assume that ψc̄HcL , where c̄G(cLCcU )y2, the meanof c. Finally, to avoid certain corner solutions and to ensure the tractability ofthe model in the case of asymmetric information, I assume that ∆H4(aAcL)y9, where ∆GcUAcL , the inverse of the density of c. An example of a parameterset that satisfies these three assumptions is that used in the numerical simu-lations: aG3, cUG2, cLG1, and ψG0·9.

Throughout the paper, I assume that both the firm and the union knowthe form and parameters of the demand and cost-of-storage functions, thedistribution of c, and the values of ψ and ϕ. In addition, any change in inven-tory levels can be observed by the union as soon as they occur. The firm makesa single take-it-or-leave-it wage offer to the union in each period. If the unionis indifferent between rejecting and accepting the offer, it accepts. If the offeris accepted, the firm then decides on the levels of employment and inventoryaccumulation or decumulation. If the offer is rejected, the firm can choose alevel of decumulation.

The competitive outcome under symmetric information

Suppose that the actual value of ct becomes known to both parties at thebeginning of period t. Thus, there is no asymmetry of information, althoughthere is uncertainty concerning the future values of c. With symmetric infor-mation, it is clearly optimal for the firm to make a wage offer in each periodequal to the current reservation wage, so union utility in each period is ctM.By maximizing expected profits the firm also maximizes the expected value ofWtCVt . Since wtLt affects profits and union utility in an equal and oppositeway, this generates efficient levels of employment and inventory change.7

Allowing the firm to make a take-it-or-leave-it wage offer ensures that its mar-ginal cost of labour is the true opportunity cost. This generates the same out-comes as a competitive labour market, so the model with symmetricinformation is broadly the same as that estimated by Miron and Zeldes (1988),and for convenience I refer to the firm in this section as a competitive firm.8

This approach allows me to move on to a model of strikes simply by imposing



an extra informational constraint on the firm, namely that in each period itknows the distribution of ct but not its realization. I shall then refer to the firmas a bargaining firm.

The competitive firm in period 2

In period 2, the firm inherits S1 and faces a wage rate c2 . There is clearly noloss of generality in assuming that S1 is less than ay2b (the level of sales atwhich marginal revenue is zero), nor is there any point in choosing to end thisfinal period with positive inventories; the firm thus sets S2 equal to zero, andits profit-maximizing choice of L2 is given by

L2G5aAc2

2bAS1 if c2FaA2bS1

0 if c2XaA2bS1 ,

with a maximized value of period 2 profits of

V*2(S1 , c2)G5

(aAc2)2

4bCc2S1AeS2

1 if c2FaA2bS1

(aAbS1)S1AeS21 if c2FaA2bS1 ,

Thus, a high level of inherited inventories or a high wage will induce a zerolevel of employment, in which case the marginal value of extra inventories isaA2bS1A2eS1; but if employment is positive, extra inventories displace labourone-for-one and have a marginal value of c2A2eS1 . Of course, at the time thatS1 is chosen, the value of c2 is not known, and the firm cannot know for surewhether L2 will be positive or not. Define

α (S )G#cU

cL

V*2(S, c)∆−1 dc,

where ∆−1 is the density of c. Then α (S ) gives the expected value of carryingforward into period 2 a level of inventories S. It is possible, but unnecessaryfor our purposes, to derive α (S ) explicitly. More importantly, it is straightfor-ward to show that it is continuously differentiable and strictly convex. Notethat, since aHcU , if S1G0, employment in period 2 will certainly be positive,so that α′(0)Gc̄G(cLCcU )y2, the mean of c.

The competitive firm in period 1

In period 1 the firm inherits S0 and faces a wage c1 . It chooses L1 and S1 tomaximize

(1) [aAb(L1CS0AS1)](L1CS0AS1)Ac1L1AeS20Cψα (S1),

with q1GL1CS0AS1 being the resulting level of sales. This problem has a lotin common with that faced by the firm in period 2. In particular, if S0 andyorc1 are high, then the firm may find it unprofitable to employ any labour at all.In this case, sales and inventory accumulation depend only on S0 . But if thechosen value of L1 is positive, then the marginal cost of extra sales or inventoryaccumulation is c1 , since this requires extra employment. Any marginal



increase in S0 merely displaces employment, with no effect on q1 or S1 . To seethis consider the function f (c):

f (c)GargmaxqX0

{(aAbq)qAcq}CargmaxSX0

{ψα (S )AcS}.

Since α is concave, f is continuous and strictly decreasing over the interval[cL , cU]. f (c) gives the firm’s preferred level of sales plus future inventorieswhen each has a marginal cost of c; this total must be met from the sumof inherited inventories plus current production (employment) so that L1G

max [0, f (c)AS0]. In short, the concavity of α combined with the assumptionof constant returns to scale allows the firm to decouple the sales and inventorydecisions when employment is positive. The solution and comparative staticsof maximizing (1) are fairly straightforward, so I present them without proof.

Proposition 1

(i) If f (c1)XS0 , then L1Gf (c1)AS0 , q1G(aAc1)y2b, and S1G

argmaxSX0 {ψα (S )Ac1S}, which is positive only if c1Fψc̄.(ii) If f (c1)FS0 , then L1G0 and S1GargmaxSX0 {[aAb(S0AS )]

(S0AS )Cψα (S )}, which is positive only if aA2bS0Fψα′(0).

Note that case (ii) cannot occur if S0G0: since aHcU , even at the highestpossible wage the firm would still want to produce some output for currentsale.

Corollary

(i) If L1H0 then ∂S1y∂S0G0; if in addition S1G0, then ∂L1y∂c1G∂q1y∂c1F0,whereas if S1H0, then ∂S1y∂c1F0 and ∂L1y∂c1F∂q1y∂c1F0.

(ii) If L1G0, ∂L1y∂c1G∂S1y∂c1G∂q1y∂c1G0; if in addition S1H0, then 0F∂S1y∂S0F1.

Let us now take seriously the notion that period 1 of the three-period modelis representative of any period in a more general model. More precisely, wepursue the implications of treating the determinants of L1 and S1 as those thatwould apply over a sustained period of time. To emphasize this, I now writeLt , St and ct instead of L1 , S1 and c1 . Then the corollary to Proposition 1shows that the responsiveness (in absolute terms) of employment, and thereforeof output, to the current wage rate is always at least as great as that of sales.The inequality is strict when Lt and St are both positive (and given that cLFψc̄,this must sometimes occur), implying that in the long run we will observe agreater variance of output than of sales. This is the familiar sales-smoothingresult. Empirically, this is indeed the case, so that on this score the model withsymmetric information is not at odds with the data. However, the corollaryalso asserts that as long as employment is positive then not only will outputand sales be negatively related to the current wage rate, so too will inventoryaccumulation, StAStA1 . This is because the competitive model treats wt as anexogenous variable, completely unrelated to inherited inventories, StA1 . If Lt

and St are both positive, the only connection between StAStA1 and wt is thenegative effect of wt on St . As I have described in the Introduction, it is thisprediction of a negative relationship between wages and inventory accumu-lation that has stubbornly resisted empirical verification. There are many



studies that find that wage shocks do little to explain the timing of pro-duction and the pattern of inventory investment, and some authors haveeven found that in certain industries production and accumulation appearto be positively affected by wage levels (see e.g. Table II in Miron and Zeldes1988). In the next section I present a model that can account for theseanomalies.

II. A MODEL WITH INCOMPLETE INFORMATION

I now analyse the central model of the paper. It differs from that of the pre-vious section in one respect only: in each period, the union’s reservation wageis private information. Wage-setting continues to be represented by a singleoffer by the firm in each period, but there is now a chance that the firm’s offerwill be rejected. If so, there is a strike and no employment in that period, butif it wishes the firm can sell from stock in order to maintain revenue. If theoffer is accepted, the firm determines the level of employment and the level ofinventory accumulation or decumulation. As in the case of symmetric infor-mation, we solve the model backwards, but in addition we now have to analysenot only the firm’s decisions following rejection of its offer, but also what thatoffer will be.

The bargaining firm in period 2

The firm inherits S1 , and mades an offer w2 to the union. As with the competi-tive firm, there is no point in ending this final period with positive inventories,so if this offer is accepted the firm sets S2 equal to zero and the profit-maximiz-ing level of L2 is given by

L2G5aAw2

2bAS1 if w2FaA2bS1

0 if w2XaA2bS1 ,

with a maximized value of period 2 profits (conditional on acceptance) of

Va2(S1 , w2)G5

(aAw2)2

4bCw2S1AeS2

1 if w2FaA2bS1

(aAbS1)S1AeS21 if w2XaA2bS1 .

If w2 is rejected, the firm’s only course of action is to sell from inventory; thisyields profits (conditional on rejection) of

Vr2(S1)G(aAbS1)S1AeS2

1 .

How is w2 chosen? The firm makes a once-and-for-all offer to the union, sowhen considering the offer the union compares (w2Ac2)L2Cc2M with c2M. Ittherefore accepts if w2Xc2 . Since c2 is uniformly distributed on the interval[cL , cU], and there is clearly no point in making an offer outside this interval,the probability of acceptance is (w2AcL)y∆. In the bargaining model the wage



offer determines both the probability of acceptance and the firm’s profitabilityif the offer is accepted; the firm’s choice of w2 is a compromise between thesetwo. The chosen value of w2 therefore maximizes expected profits:

V2(S1 , w2)Gw2AcL

∆Va

2(S1 , w2)CcUAw2

∆Vr

2(S1)

G

w2AcL

∆ 1 (aAw2)2

4bCw2S1A(aAbS1)S12

C(aAbS1)S1AeS21 if w2FaA2bS1

(aAbS1)S1AeS21 if w2XaA2bS1 ,

subject to cLYw2YcU . Note that, since Va2(S1 , w2) is convex in w2 , the function

V2(S1 , w2) is not in general globally concave in w2 . This creates a complicationthat does not appear in the compeitive model. We can expect non-concavityto be a generic problem because the convexity of Va

2(S1 , w2) in w2 is a funda-mental property of such profit functions. In the present linearyquadratic setup,V2(S1 , w2) is either a cubic function of w2 (for w2FaA2bS1) or constant, andit is relatively straightforward to derive the firm’s optimal wage offer as afunction of its inherited inventories:

w2(S1)G5aC2cLA2bS1

3if S1F

aAcL

2b

cL if S1XaAcL

2b.

(The possibility that w2GcU is ruled out by the assumption that ∆H4(aAcL)y9.) Employment in period 2 following acceptance of this offer is nowgiven by

L2(S1)G5aAcLA2bS1

3bif S1F

aAcL

2b

0 if S1XaAcL

2b.

These results have a clear intuitive explanation. The higher the level ofinherited inventories, the less important it is for the firm to secure acceptanceof its wage offer, since in the event of rejection it can still generate a reasonablelevel of revenue. It therefore makes a lower wage offer. Even if this is accepted,employment is low, since the lower wage rate does not offset the effect on thefirm’s demand for labour of a high level of inventories. The lower wage offercarries with it a higher risk of rejection, so a high level of inherited inventoriesis associated with a high probability of a strike.



The maximized value of V2(S1 , w2) is the value to the firm of entering period2 with a level of inventories S1 , and is given by

β(S1)G5aAcLA2bS1)3

27b∆C(aAbS1)S1AeS2

1 if S1FaAcL

2b

(aAbS1)S1AeS21 if S1X

aAcL

2b.

This function plays a central role in what follows. The assumption that∆H4(aAcL)y9 ensures that it is concave at all positive levels of S1 . Initially, itis certainly increasing, but if eH0 high storage costs eventually cause it toreach a maximum before S1Gay2b.

Analogous to β, we can define the union’s value function, giving itsexpected utility in period 2 before it knows c2:

γ (S1)G#w2(S1)

cL

(w2(S1)Ac)L(S1)∆−1 dcCc̄M

G5(aAcLA2bS1)3

54b∆Cc̄M if S1F

aAcL

2b

c̄M if S1XaAcL

2b.

This function is decreasing and convex. Once inventories have been accumu-lated, they are bad news for the union: the higher is S1 , the lower will be thewage offer and the lower will the employment level. But union members willnever get less than their reservation wage.

The bargaining firm in period 1

The analysis of period 1 differs from that of period 2 in a number of respects.As in the case of complete information, the firm now has to decide how toallocate inventories inherited from period 0 plus any production between cur-rent sales and inventories to be carried into period 2. In doing so, it assessesthe future value of inventories using the function β. It also has to determineits offer w1 . But the union must now take into account that acceptance ratherthan rejection in period 1 allows the firm to carry more inventories into period2; this has a harmful effect on the union’s expected utility, which it can assessusing the function γ .

Let us begin by analysing the firm’s decisions if it inherits S0 and the unionhas accepted an offer w1 . The firm chooses L1 and Sa

1 to maximize

(2) [aAb(L1CS0ASa1)](L1CS0ASa

1)Aw1L1AeS20Cψβ(Sa

1),

with qa1GL1CS0ASa

1 being the resulting level of sales. The maximized value of(2) is denoted by V a

1(w1 , S0). Clearly, this problem has a lot in common withthat of the competitive firm when facing a wage c1 ; we can therefore use Prop-osition 1 and its corollary, mutatis mutandis. Of course, there would be littlepoint in the bargaining firm choosing a wage w1 which if accepted was so highthat the optimal employment level was zero. Analogous to f (c), define the



function

g(w)GargmaxqX0

{(aAbq)qAwq}CargmaxSX0

{ψβ(S )AwS},

which, like f, is continuous and strictly decreasing over the interval [cL , cU ].Then there is no loss of generality in assuming that g(w1)XS0 , and hence thatL1Gg(w1)AS0 . (We return later to the question of whether this implies thatw1 is less than cL .) Given the strict concavity of β, it is then straightforwardto establish the following proposition.

Proposition 2. If g(w1)XS0 , then

(i) L1Gg(w1)AS0 , qa1G(aAw1)y2b, and Sa

1GargmaxSX0 {ψβ(S )Aw1S}, whichis positive only if

w1Fψ1aA2(aAcL)2

9∆ 2;(ii) ∂Sa

1y∂S0G0; if in addition Sa1G0, then ∂L1y∂w1G∂q1y∂w1F0, whereas if

Sa1H0, then ∂Sa

1y∂w1F0 and ∂L1y∂w1F∂q1y∂w1F0;(iii) qa

1 and Sa1 are unaffected by marginal changes in S0 , but ∂L1y∂S0G−1;

(iv) ∂Va1y∂S0Gw1: ∂V a

1y∂w1G−L1 .

These results appear to imply that inventory accumulation, Sa1AS0 , is nega-

tively related to the wage rates. But w1 is not a random variable; in effect,Sa

1AS0 and w1 are jointly determined by S0 , and their correlation is not clear.In period 2, this was a trivial issue because S2 is always zero. We focus on theperiod 1 relationship in order to get away from the distortions of being at theend of the horizon.

If w1 is rejected, then the firm still has to decide how to allocate its inheritedinventories between current sales and future inventories. It chooses Sr

1 tomaximize

(3) [aAb(S0ASr1)](S0ASr

1)AeS20Cψβ(Sr

1)

with qr1GS0ASr

1 being the resulting level of sales. The maximized value of (3)is denoted by Vr

1(S0). It is straightforward to establish the followingproposition.

Proposition 3. If g(w1)HS0 , i.e. if L1H0, then

(i) qa1Hqr

1H0;(ii) Sa

1HSr1 , and Sr

1H0 if and only if S0H(aAψβ′(0))y2b, in which case0FdSr

1ydS0F1 and d2Sr1ycd(S0)2F0;

(iii) ∂Va1y∂S0FdVr

1ydS0Gg−1(S0).

Let us now consider the union’s response to a wage offer w1Fg−1(S0). Theunion knows that accepting will allow the firm to put a part of any productioninto inventory accumulation. This has consequences for the wage offer anddemand for labour in period 2. More precisely, the union must consider theimplications for its future welfare if it accepts w1 and enters period 2 with thefirm holding inventories Sa

1 (bearing in mind the wage offer and employmentprospects that this will entail); the alternative is to reject w1 and forgo any



employment this period, thereby inducing the firm to sell from stock to main-tain revenue and enter the next period with lower inventories Sr

1 . Then theunion will accept the offer w1 if

w1L1Cc1(MAL1)Cϕγ (Sa1)Xc1MCϕγ (Sr

1),

i.e. if

(4) c1Fw1Aϕ [γ (Sa

1)Aγ (Sa1)]

L1

.

Since γ (S ) is monotonically decreasing, γ (Sr1)Hγ (Sa

1), and to induce accept-ance the firm has to offer a margin over and above the union’s reservationwage in order to compensate for the loss of future utility caused by the higherinventories carried into period 2. The probability that w1 is accepted is givenby

(5) w1Aϕ [γ (Sr

1)Aγ (Sa1)]

L1

AcL@∆,

which is less than (w1AcL)y∆. Since L1 and Sa1 depend on w1 , this probability

changes with w1 in a very complex way. In order to get the analysis off theground, I shall assume that ϕG0. Although there are arguments to suggestthat the union might well be more myopic than the firm (e.g. as a result ofcredit rationing, or turnover within the union), the assumption is made onlytemporarily and its purpose is to make the model analytically tractable, ratherthan realistic.

If ϕG0, then w1 is chosen to maximize

(6) V1(S0 , w1)Gw1AcL

∆Va

1(S0 , w1)CcUAw1

∆Vr

1(S0)

subject to cLYw1YcU . A formal analysis of the firm’s profit-maximizing choiceof w1 (which I now denote by w1(S0)) and the associated comparative statics isgiven in Appendix A as the proof to Proposition 4, but here I give a moreintuitive account.

In order to understand how this problem differs from that faced by thefirm in period 2, let us exploit the fact that, if w1 is accepted and L1 is positive,we can separate the firm’s sales and inventory decisions, in which case we canexpress V1(S0 , w1) as the sum of three components,

V1(S0 , w1)G5A(S0 , w1)CB(w1)CC(S0 , w1) if w1Fg−1(S0)

Vr1(S0) if w1Xg−1(S0),

where

A(S0 , w1)Gw1AcL

∆ 1maxqX0

{(aAbq)qAw1qCw1S0}2C

cUAw1

∆(aAbS0)S0AeS2

0Cψβ(0),



B(w1)Gw1AcL

∆ 1maxSX0

{ψβ(S )Aw1S}Aψβ(0)2 ,

C(S0 , w1)GcUAw1

∆ 1maxSX0

{[aAb(S0AS )](S0AS )Cψβ(S )}

A(aAbS0)S0Aψβ(0)2.

The term A(S0 , w1) gives the firm’s expected profits in period 1 if it were to setSa

1GSr1G0 and confine itself to a choice of L1 (and hence qa

1), with qr1 identically

equal to S0 . Thus, A(S0 , w1) is the firm’s period 1 maximand if ψG0. Indeed,if w1 were chosen to maximize A(S0 , w1), the firm would set w1(S0)Gw2(S0)Gmax {cL , (aC2cLA2bS0)y3). We may take this as a benchmark with which tocompare the actual value of w1(S0), and put down any difference to the for-ward-looking behaviour of the firm embodied in the terms B(w1) and C(S0 , w1).B(w1) represents the gain to the firm from being able to choose a non-zerolevel of Sa

1 , weighted by the probability of acceptance, while C (S0 , w1) rep-resents the gain from being able to choose a non-zero level of Sr

1 , weighted bythe probability of rejection.

To fix ideas, suppose that ψG1. Then it is possible to show that B reachesa unique local (and hence global) maximum at some value wB less than(aC2cL)y3 and greater than cL . Now, if S0G0, then clearly Sr

1G0 and henceC(S0 , w1)G0. Thus, for S0G0 the firm’s choice of w1 will lie between wB and(aC2cL)y3. Put briefly, the maximizer of (ACB) lies between the respectivemaximizers of A and B. For S0G0, this implies that the forward-lookingbehaviour of the firm induces it to make a lower wage offer than w2(0). Sinceβ′(0) is greater than w2(0), this also means that a sufficiently forward-lookingfirm that inherits zero inventories in period 1 will, if its offer is accepted, enterperiod 2 with positive inventories.9

As S0 increases, the function B is unaffected; but A shifts up and it ismaximized at a lower value, (aC2cLA2bS0)y3. Thus, there is a downwardeffect on the firm’s wage offer.10 For a high enough level of S0 , it becomesworthwhile for the firm to set aside inventories to carry into period 2 even ifits wage offer is refected. If Sr

1 is positive, C is a linear function of w1 with anegative slope that increases in absolute size with S0 . It therefore has nointerior maximum on the interval [cL , cU], but it acts as a further downwardinfluence on the firm’s choice of w1 .

The higher the firm’s inherited level of inventories, the lower its wage offerand the greater the risk of a strike it is prepared to accept. For what value ofS0 would the firm set w1 equal to cL , an offer that is bound to be rejected? Itwould do so if there were no advantage in being able to employ workers at arate equal to or marginally above cL , i.e. if g(cL)GS0 . To take a simple case,let ψG1 and eG0. Then if w1GcL , qa

1GSa1G(aAcL)y2b, and g(cL)G(aAcL)yb,

which is exactly twice the level of S1 that would induce the firm to setw2GcL .11

The following proposition gives a more precise statement of thesearguments.



Proposition 4. Let w1(S0) maximize V1(S0 , w1); then(i) for 0YS0Fg(cL), w1(S0) is a continuous and strictly decreasing function;(ii) as S0 tends to g(cL), w1(S0) tends to cL;(iii) if ψH(aC2cLA2bS0)y3β′(0), then Sa

1H0.

For a proof, see Appendix A. If S0Hg(cL), there is no offer the firm could makewhich if accepted would induce it to choose L1 greater than zero; effectively, astrike will then occur for sure in period 1. Increases in S0 beyond this levelincrease S1GSr

1 and hence the probability of a strike in period 2. At somecritical level, S*0 , there is a strike for sure in both periods; this occurs whenthe condition determining the choice of S1 , aA2b(S0AS1)Gψβ′(S1), is satisfiedwith S1G(aAcL)y2b.

Proposition 4 is the key to understanding why the model predicts a positiveassociation between wages and inventory accumulation. Suppose ψH(aC2cL)y3β′ (0); then if S0G0, Sr

1G0, but Sa1 is positive; since there is a positive

probability of acceptance of the firm’s wage offer, the expected level of inven-tory accumulation is positive. By contrast, if S0Xg(cL ), then, whatever thevalues of ψ and e, employment in period 1 is zero and all sales are met entirelyfrom inventory decumulation. In short, a low level of inherited inventories isassociated with a high wage offer and positive expected accumulation, and ahigh level of inventories is associated with a low wage offer and negativeexpected accumulation.

The model with a forward-looking union

I now consider the effects of relaxing the assumption that the union is com-pletely myopic. This introduces considerable complications: the expression in(5) shows that the probability of acceptance in period 1 is determined not onlyby w1 (directly and via L1 and Sa

1), but also through the effect of S0 on Sr1 .

The approach taken here is not to give a generalization of Proposition 4, butto give a more informal analysis. The numerical simulations of the next sectionsuggest that quantitatively the effects of allow ϕ to be positive are not substan-tial, and do not reverse the conclusions of the previous section in which ϕ isassumed to be zero.

Let us denote the right-hand side of (4) by w̃1 , and call it the effective wageoffer. It incorporates the effect on union welfare per member employed offigure inventory accumulation. If ϕH0, then w1 is chosen to maximize

(7) V1(S0 , w1)Gw̃1AcL

∆Va

1(S0 , w1)CcUAw̃1

∆Vr

1(S0)

subject to cLYw̃1YcU . Since w̃1Fw1 if L1 is positive, we might expect the firmto react to a less myopic firm with a higher wage offer, but not necessarily sohigh that it increases the probability of acceptance. But if ϕH0, then as w1

rises there is not only a direct, one-for-one, effect on w̃1 , but also an indirecteffect as both L1 and Sa

1 fall. If w1 is high enough, L1G0 and Sa1GSr

1 , in whichcase w̃1Gw1 .12 On average, then, these indirect effects reinforce the directeffect, so that changes in w1 will have a magnified effect on w̃1 . More generally,it is possible to show that if L1 and Sa

1 are positive, then dw̃1ydw1H1. Now,



consider the first-order condition associated with the maximization of (7):

(8)dw̃1

dw1

(Va1AVr

1)A(w̃1AcL)L1G0.

Suppose the firm chose w1Gw1(S0), the same offer it would make to a myopicunion; if the offer were accepted, the firm would then choose the same level ofL1 and achieve the same Va

1 . But since dw̃1ydw1H1 and w̃1Fw1 , the right-handside of (8) must be positive when w1Gw1(S0). Indeed, moving from ϕG0 toϕH0 imparts an upward shift to ∂V1y∂w1 , implying that (7) is maximized at ahigher value of w1 than w1(S0).13 If the firm does offer a higher wage, then onacceptance it optimizes by choosing lower levels of qa

1 and Sa1 . Thus, both sales

and inventories following acceptance will tend to be lower, and unless theprobability of acceptance increases substantially, expected sales, and henceaverage employment, will fall.

III. NUMERICAL SIMULATIONS

A central theme of this paper is that the relationship between inventories andwages depends critically on the informational environment in which wage bar-gaining takes place. Because the model with asymmetric information and aforward-looking union is complex, simple qualitative propositions are not easyto establish. The purpose of the numerical simulations is to act as a comp-lement rather than a substitute for qualitative analysis. The limited numberof simulations presented here should be interpreted as counterexamples to aconjecture (for instance that variations in factor opportunity costs alwaysinduce a negative covariance between observed factor prices and accumulationor output).

All the simulations have the following in common:

1. The demand curve is ptG3Aqt .2. The cost of storage parameter, e, equals 0.1.3. The firm’s discount factor ψ equals 0.9.4. The union’s reservation wage, c, is distributed uniformly on the interval

[1, 2].

They differ as follows.14

1. Simulation 1: information about c is symmetric.2. Simulation 2: information about c is asymmetric, ϕG0.3. Simulation 3: information about c is asymmetric; ϕG0·9.

Details of how the simulations were performed are given in Appendix B.Briefly, they are as follows.

Step 1. The relevant value functions are computed: α(S ) in the symmetricinformation case, β(S ) and γ (S ) in the asymmetric case.

Step 2. The model is then run for 50,000 periods. The firm is given aninitial inventory endowment of 1. In each period the union is given a reser-vation wage drawn from the uniform distribution on [1, 2], and the firm inheritsinventories from the previous period. With the value functions from step 1, thepayoff-maximizing decisions of the two parties are then determinate. These



TABLE 1

SAMPLE DATA FROM SIMULATION 1

Mean

StA1 ct , wt Lt qt St StAStA1

Covariance 0·266 1·501 0·769 0·769 0·266 0·000

StA1 0·141ct , wt 0·000 0·083Lt −0·116 −0·125 0·333qt 0·013 −0·035 0·046 0·018St 0·012 −0·089 0·171 0·041 0·141StAStA1 −0·130 −0·089 0·287 0·028 0·130 0·259

value functions are those relevant to period 1 of the three-period model, so ineach period the firm and the union are behaving as if the next period were thelast. Thus, the simulations are consistent with the theoretical models of Sec-tions I and II, in the sense that period 1 is taken to be representative of anyperiod in a more general model. Simulations based on infinite horizon valuefunctions are reported in Clark (1994). It is remarkable how little the two setsof results differ.

Step 3. In each period all data are recorded, allowing them to be analysedas if they have been generated from the real world.

Details of the results of the simulations are presented in Tables 1–3. Thisgives means, variances and covariances of the most important variables (thevariable strike takes the value 1 if there is a strike and 0 otherwise), as well asthe empirical transition matrix showing the proportion of strikes that werefollowed by a further strike. There are a number of important points.

1. All of the simulations exhibit sales smoothing: the variance of output, L, isgreater than that of sales, q.

TABLE 2


Mean

Covari- StA1 ct wt STRIKE Lt qt St StAStA1

ance0·265 1·500 1·431 0·569 0·501 0·501 0·265 0·000

StA1 0·083ct 0·000 0·084wt −0·027 0·000 0·009STRIKE 0·027 0·119 −0·009 0·245Lt −0·050 −0·135 0·016 −0·285 0·342qt 0·037 −0·065 −0·012 −0·118 0·134 0·099St −0·003 −0·070 −0·001 −0·140 0·158 0·072 0·083StAStA1 −0·087 −0·070 0·028 −0·167 0·208 0·035 0·087 0·173

Strikes Non-strikes

% followed by a strike in the next period 49.0 67.3

% followed by no strike in the next period 51.0 32.7



TABLE 3


Mean

Covari- StA1 ct wt STRIKE Lt qt St StAStA1

ance0·216 1·499 0·492 0·613 0·440 0·440 0·216 0·000

StA1 0·068ct 0·000 0·083wt −0·022 0·000 0·007STRIKE 0·027 0·113 −0·009 0·237Lt −0·043 −0·127 0·014 −0·270 0·310qt 0·032 −0·065 −0·010 −0·117 0·131 0·098St −0·006 −0·062 0·002 −0·125 0·139 0·065 0·068StAStA1 −0·074 −0·062 0·024 −0·153 0·182 0·033 0·074 0·149

Strikes Non-strikes

% followed by a strike in the next period 53.1 74.2

% followed by no strike in the next period 46.9 25.8

2. With symmetric information (simulation 1), both the covariance of wagesand output and that of wages and inventory accumulation are negative,whereas with asymmetric information (simulations 2 and 3) they are bothpositive. However, in all of the simulations increases in c, the true oppor-tunity cost of labour, result in lower employment, sales and inventoryaccumulation. Even in the case of asymmetric information there is cost-smoothing, although this would not be picked up empirically if the actualwage rate were taken as a measure of the true cost of labour.

3. With asymmetric information, inherited inventories depress the firm’s wageoffer (cov(StA1 , wt )F0) and increase the probability of a strike (cov(StA1 ,strike)H0).

These points suggest strongly that a model with asymmetric information ismore in accord with established empirical results than one with symmetricinformation. There are two further points of interest.

4. The transition matrices show that a strike in one period reduces the prob-ability of a strike in the next, as predicted by Proposition 4. This suggeststhat, if empirical investigators do not control for inventories, they mayunderestimate the strength of any positive state dependence.

5. The assumption of a myopic union does not have a great effect on thenumbers (compare simulations 2 and 3). Moving to a forward-lookingunion has the effect of raising the average wage offer, but not enough toreduce strike incidence. Thus, average levels of employment, sales andinventories are all lower.

IV. CONCLUDING REMARKS

The model of strikes and inventories analysed in this paper is very simple,perhaps excessively so, and there many ways in which it can be developed. The



most obvious is to allow for variation in strike duration by adopting a bar-gaining protocol that is less abrupt than the single take-it-or-leave-it offer.

Consider the following setup (which I admit is rather aritificial). The reser-vation wage changes in January of each year, and contracts, once agreed, lastuntil December. Starting at the beginning of Janaury, the firm makes asequence of weekly wage offers to the union, until one is accepted. Decisionsabout production, sales and inventories are made each week and last for aweek. From standard bargaining theory, we can expect the firm to make anascending sequence of offers (i.e. to screen the union), and a union with a highreservation wage to hold out, or stay on strike, for longer. Intuition suggeststhat, the higher the firm’s inventories are at the beginning of the year, thelower is its initial offer, the more likely is a strike, the longer any strike willlast, and the lower is the wage eventually agreed. The firm can run down itsinventories during the strike, but since it cannot predict with certainty theduration of any strike, this presents an interesting optimal depletion problem.Once a wage is agreed, the firm can restore its inventories. The higher theunion’s reservation wage, the longer any strike will be, and the lower will bethe firm’s inventories at the end of December.

Thus, the firm will make a higher initial offer in the year following a longstrike, making a further strike less likely andyor of shorter duration. This gen-erates negative state dependence of strike incidence and of strike duration. Inaddition, high initial inventories are associated with a lower agreed wage anda tendency to decumulation. The predictions of this extension of the bargainingmodel are therefore richer, but in most respects qualitatively similar. Quantitat-ively, however, a long contract length, e.g. a year, compared with the pro-duction and inventory period, e.g. a week, seems likely to reduce the strengthof negative state dependence, especially of very short strikes.

What if the firm has private information about its profitability, and theunion makes the offers? For simplicity, let us return to the assumption of asingle take-it-or-leave-it offer. Suppose that the intercept in the demand curve,at , is privately revealed to the firm at the beginning of period t, and that ct isconstant over time. The higher is at , the higher is the maximum wage offer thatthe firm will accept. On the other hand, the higher are the firm’s inventories atthe outset of the period, the lower will be the union’s perception of the prob-ability of acceptance of any given offer. Thus, higher inventories will induce theunion to make a lower offer, which will be less likely to be accepted, causing atendency to decumulation. In data, this would show up as a positive associ-ation between wages and inventory change. In addition, a strike in one periodmakes one in the next period less likely.

These predictions are the same as those of the model in which the reser-vation wage is private information.15 Taken with the arguments of the previousparagraph, this suggests that the main conclusions of the paper are robust withrespect to the assumptions about the timing of production and the distributionof information.

APPENDIX A

In this appendix, I provide a proof of Proposition 4. It is simpler to transformV1(S0 , w1) and consider w1(S0) to be the maximizer of ZG(w1AcL)X (S0 , w1), whereX (S0 , w1)GV1(S0 , w1)AVr

1(S0). Note that ∂Xy∂w1G−L1 (so X is strictly convex in w1 if



L1H0), and ∂Xy∂S0Gw1Ag−1(S0) which is strictly negative if L1H0. For w1GcL , ZG0, and also for w1Xg−1(S0), since then L1G0 and V1(S0 , w1)GVr

1(S0). To prove part (i),consider the case where cLHg−1(S0); then, over the interval [cL , g−1(S0)], Z behaves asfollows: ∂Zy∂w1GXA(w1AcL)L1 , which is positive at w1GcL and zero at w1Gg−1(S0);∂2Zy(∂w1)2G−2L1A(w1AcL )∂L1y∂w1 , which is negative at w1GcL and positive at w1Gg−1(S0); ∂3Zy(∂w1)3G−3∂L1y∂w1A(w1AcL )∂2L1y(∂w1)2, which is positive over the entireinterval [cL , g−1(S0)]. (If Sa

1G0, ∂2L1y(∂w1)2G0; if Sa1 is positive, then from the negative

coefficient on S3 in β(S ) it can be shown that ∂2L1y(∂w1)2 is negative). From this itfollows that Z is initially increasing and concave, reaches a maximum, decreases,becomes convex and eventually flattens out. For any value of S1Fg(cL ), Z therefore hasa unique local and hence global maximum. The first-order condition XA(w1AcL)L1G0has two solutions: that with L1H0 always characterizes the global maximum, w1(S0);the other with L1G0 is a minimum. The global comparative statics of changing S0 aretherefore the local comparative statics associated with the unique local maximum. Moreprecisely, in the neighbourhood of w1(S0), ∂2Zy(∂w1)2 is always negative, so the implicitfunction theorem asserts that w1(S0) is a continuous function, whose derivative ∂w1(S0)y∂S0 has the same sign as ∂2Zy(∂w1)(∂S0) evaluated at w1Gw1(S0). Now, ∂2Zy(∂w1)(∂S0)G∂Xy∂S0A(w1AcL)∂L1y∂S0 , and at w1Gw1(S0) L1H0, so ∂L1y∂S0G−1 and, using thefirst-order condition, ∂2Zy(∂w1)(∂S0)G∂Xy∂S0CXyL1 . Since (i) ∂Xy∂S0Gw1Ag−1(S0)F0 if L1H0; (ii) by definition X (S0 , g−1(S0))G0; and (iii) L1G∂Xy∂w1 evalu-ated at w1Gw1(S0), it follows from the strict convexity of X when L1 is positive that∂2Zy(∂w1)(∂S0) and hence ∂w1(S0)y∂S0 is strictly negative. This proves part (i) of Prop-osition 4.

To prove part (ii), note that cLFw1(S0)Fg−1(S0) if g−1(S0)HcL . It follows immedi-ately that as g−1(S0) tends to cL , so w1(S0) tends to cL .

To prove part (iii), recall that the function A has a maximum at wAG(aC2cLA2bS0)y3. But if ψβ′(0)HwA , then A must have a negative slope at w1Gψβ′(0).But B has zero slope at w1Gψβ′(0) and C has either zero or negative slope; since part(i) above establishes that ACBCC has a unique maximum at w1(S0), this impliesw1(S0)Fψβ′(0), and hence that Sa

1H0.

APPENDIX B

This appendix gives details of the simulations reported in Section III. A central part ofthe methodology is that all maximization decisions use a global grid search rather thanrelying on analytical results such as concavity. In all three reported simulations, theupper bounds to the choice of employment and inventories were both set at 2; in simu-lations not reported here and with higher upper bounds, it was checked and confirmedthat the firm would never chose L or S greater than 2. In what follows, grid (L)Ggrid (S )G{0, 0·02, 0·04, . . . , 2}, grid (w)Ggrid (c)G{1, 1·01, 1·02, . . . , 2}.

(i) The competitive model (simulation 1)

Step 1: Computation of α (S ). For each value of inherited inventories in grid (S ), andfor each value of c in grid (c), the firm searches over grid (L ) to maximize[3A(SCL )](SCL)AcLA0·1S2; i.e., bequeathed inventories are implicitly set to zero.For each value of S, α (S ) is then the average of the maximized profits over all possiblevalues of c, where each value has equal weight owing to the uniform distribution of c.

Step 2. Simulations. For tG1, . . . , 50,000, the firm chooses Lt from grid (L ) and St fromgrid (S ) to maximize [3A(StA1CLtASt)](StA1CLtASt)ActLtC0·9α (St)A0·1(StA1)2,subject to StA1CLtAStX0, where StA1 is inherited from the previous period’s decision(with S0G1), and ct is drawn from a uniform distribution on [1, 2].

(ii) The bargaining model (simulations 2 and 3)

Step 1: Computation of β (S ) and γ (S ). For each value of S in grid (S ) and for eachvalue of w in grid (w), the firm searches over grid (L) to maximize[3A(SCL )](SCL )AwLA0·1S2, with bequeathed inventories implicitly equal to zero.This generates Va

2(S, w). For each value of S, the firm then searches over grid (w) to



find w2(S ); this maximizes prob Va2(S, w)C(1Aprob)[(3AS )SA0·1S2], where probG

(wAcL )y∆. The maximized value is β(S ). Given w2(S ), L2(S ) can then be determinedfrom the results of the earlier search over grid (L ). γ (S ) is then computed as(max {w2(S )Ac, 0})L2(S )Cc̄M, averaged over all values of c in grid (c); the last termin γ (S ) is effectively an arbitrary constant and for convenience I set MG0.

Step 2: Simulations. There is no need to conduct 50,000 grid searches. It is only neces-sary to compute the firm’s choices of employment, inventories, wage offer and theimplied effective wage, purely as functions of its inherited inventories S. This is similarto step 1 above, with the addition of forward-looking behaviour by the firm and poss-ibly by the union. Thus, for each value of inherited inventories S in grid (S ) and foreach value of w in grid (w), the firm chooses L from grid (L) and Sa from grid (S ) tomaximize [3A(SCLASa)](SCLASa)AwLA0·1S2C0·9β(Sa). This generatesVa

1(S, w). By setting LG0, Vr1(S, w) can be similarly derived. We then have the firm’s

choices of L, Sa and Sr as functions of inherited inventories S and w. Given the functionγ derived in step 1, the effective wage and hence the firm’s maximand can be con-structed. Thus, for each value of S in grid (S ), the firm searches over grid (w) to maxim-ize prob Va

1(S, w)C(1Aprob)Vr1(S ), where prob is given by the formula in (5). Note

that, if ϕH0 and SY2, the optimal choice of w lies in the interval [1, 2]; this is simpleto confirm theoretically and was checked and confirmed by further simulations notreported here. From the results of this maximization we can derive L (S ), Sa(S ), Sr(S ),w(S ) and w̃(S ), i.e. the firm’s choices and the implied effective wage offer, purely asfunctions of its inherited inventories S. Generating the numbers is now very simple. Inperiod t, the firm inherits StA1 (with S0G1) and the union’s reservation wage ct isdrawn from a uniform distribution on [1, 2]; if w̃(StA1)Xct , then w(StA1) is accepted,STRIKEG0, LtGL(StA1), and StGSa(StA1); if w̃(StA1)Fct , then w(StA1) is rejected,STRIKEG1, LtG0, and StGSr(StA1).

ACKNOWLEDGMENTS

I would like to thank John Hardman Moore and Donald A. R. George for manyhelpful comments during the preparation of this paper. Needless to say, they are notresponsible for any errors that remain.

NOTES

1. This effect is more pronounced on strikes about wages. Nicolitsas (1995) finds a positive effectof inventories on non-pay disputes, but the relevant coefficient is poorly determined.

2. For a survey of the theory of bargaining with asymmetric information, see Kennan and Wilson(1993). On applying this theory in order in order to explain strike behaviour, see the surveysin Kennan and Wilson (1989) and Card (1990a). For an assessment of a number of striketheories in the light of recent British evidence, see Ingram et al. (1993).

3. It is only in Leach (1992) that anything resembling a strike occurs; this is when the firm,having received a take-it-or-leave-it wage offer from the union, sets its demand for labour tozero. Since employment is restricted to a zero–one choice, labelling such an outcome as astrike is rather artificial. If employment can take on intermediate values, high inventory levelswould result in low employment, but it would be a distortion to describe the limiting case ofzero employment as a strike.

4. A referee has made the very good point that it is easier for a firm to investigate local labourmarket conditions than it is for a union to find out about the profitability of the firm. I acceptthis. But the reservation wage may reflect other factors, especially if we consider the appli-cation of this model to the situation of a single seller, not necessarily of labour but perhapsof some intermediate product. Then the reservation wage should be interpreted more broadlyas an opportunity cost, determined by privately known personal or technological factors. Moregenerally, it might be wiser to admit that we have little direct evidence on the distribution ofinformation, and that in any event it is likely to vary greatly from one situation to another.

5. Some authors have denied that production is more volatile than than sales—e.g. Fair (1989),Beason (1993).

6. However, in Pennsylvania school boards carry over a legal obligation to reschedule lost teach-ing days: the costs and benefits of not meeting that obligation are the focus of Olson’s paper.



7. Efficiency here relates to the trade-off between the firm’s expected profits and the union’sexpected utility. The firm’s monopoly power in its output market clearly generates an inef-ficiency, but this is present whatever the bargaining or informational setup.

8. If wt were determined by Nash bargaining, as in Clark (1993), this would result in an efficientoutcome only if both Lt and St were included in the Nash bargain.

9. Were this not the case, the model would have little to offer as an account of inventory behav-iour over many periods, since it would predict that with probability one inventories wouldeventually fall to zero and remain there permanently.

10. If Sa1 is positive (aC2cLA2bS0)y3 is a local but not necessarily a global maximizer of A. This

is because if Sa1H0 then the non-negativity constraint on L1 , a condition for the decomposition

of V1 into ACBCC to be valid, cuts in at a higher level of w1 than aA2bS0 (recall that L2G0 if w2XaA2bS1), and A, if defined, is increasing if w1HaA2bS0HcL . However, the proof ofpart (i) of Proposition 4 can easily be adapted to show that if CG0 it is appropriate to thinkof w1(S0) as a weighted average of wB and this local maximizer of A.

11. In this case there would also be a strike for certain in period 2. More generally, if ψF1 oreH0, then g(cL)F(aAcL )yb; then for S0Gg(cL ), Sa

1GSr1F(aAcL )y2b. However, g(cL) cannot

fall below (aAcL )y2b, even if ψG0 or e is infinite. Whereas w2 falls from (aC2cL )y3 to cL asS1 ranges from 0 to (aAcL)y2b, w1 falls from a level less than (aCcL)y3 to cL as S0 ranges from0 to a level greater than (aAcL )y2b. This implies that the overall sensitivity of the firm’s choiceof w1 to S0 is less than that of w2 to S1 .

12. More precisely, as w1 tends to g−1 (S0) from below, w̃1 also tends to g−1 (S0).13. The proof of part (i) of Proposition 4 shows that the equation ∂V1y∂w1G0 has two solutions,

of which w1(S0) is the lower. Thus, if γ H0, no solution of (8) can be lower than w1(S0).14. More extensive simulations are reported in Clark (1994). These vary the support of the uni-

form distribution of c. All the major conclusions of the paper appear to be robust to suchchanges. Other parameters could also be varied, but from a simple homogeneity argument wecan see that a reduction in the constant in the linear demand curve (here set at 3) would havethe same effect as an equiproportional increase in cL and cU . Similarly, an iso-elastic increasein demand can be engineered by reducing the slope of the demand curve (here set equal to 1),and, given the technology and the simple union utility function, this would have an effectequivalent to increasing the coefficient in the cost of storage function (here set equal to 0.1).This in turn takes us closer to the static case where inventories cannot be stored, which seemsnot such an interesting part of the parameter space to explore. But the main argument againstrunning many simulations, each with a different parameter configuration, is that it is tootempting to infer general patterns and results from what are, after all, only extended numericalexamples.

15. But if the firm’s profitability is private information, extension to a sequence of offers is notso straightforward. If the extent of decumulation during a strike depends on the firm’s currentdemand, then this acts as a signal about its profitability and gives the union a huge infor-mational advantage. I am grateful to a referee for making this point.

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Inventories and Strikes