Documents de Travail du Centre d Economie de la Sorbonne

Transcription

1 Documents de Travail du Centre d Economie de la Sorbonne On equilibrium payos in wage bargaining with discount rates varying in time Ahmet OZKARDAS, Agnieszka RUSINOWSKA Maison des Sciences Économiques, boulevard de L'Hôpital, Paris Cedex 13 ISSN : X

2 On equilibrium payos in wage bargaining with discount rates varying in time AHMET OZKARDAS 23 and AGNIESZKA RUSINOWSKA 1 1 Paris School o Economics - CNRS Université Paris I Panthéon-Sorbonne, Centre d Economie de la Sorbonne Bd de l Hôpital, Paris, France Phone: , Fax: agnieszka.rusinowska@univ-paris1.r 2 Turgut Ozal University Faculty o Economics and Administrative Sciences, Department o Economics Ayvali Mh Cd. Etlik Kecioren/Ankara, Turkey aozkardas@turgutozal.edu.tr 3 Université Paris I Panthéon-Sorbonne, Paris, France Abstract. We provide an equilibrium analysis o a wage bargaining model between a union and a irm in which the union must choose between strike and holdout in case o a disagreement. While in the literature it is assumed that the parties o wage bargaining have constant discount actors, in our model preerences o the union and the irm are expressed by sequences o discount rates varying in time. First, we describe necessary conditions under arbitrary sequences o discount rates orthesupremumotheunion spayosandtheinimumotheirm spayosundersubgameperect equilibrium in all periods when the given party makes an oer. Then, we determine the equilibrium payos or particular cases o sequences o discount rates varying in time. Besides deriving the exact bounds o equilibrium payos, we also characterize the equilibrium strategy proiles that support these extreme payos. JEL Classiication: J52, C78 Keywords: union - irm bargaining, varying discount rates, subgame perect equilibrium, equilibrium payos Corresponding author: Agnieszka Rusinowska 1 Introduction There are numerous works in the literature that generalize Rubinstein s bargaining model o alternative oers (Rubinstein (1982; or surveys, see e.g. Osborne and Rubinstein (1990, Muthoo (1999. One extension o that model concerns incorporating the choice o going on strike in union-irm negotiations. Such a generalized model with the same discount rate δ or both parties is presented e.g. in Haller and Holden (1990 and Holden (1994. It is assumed that in each period until an agreement is reached, the union must decidewhetheritwillstrikeorholdoutinthatperiod.thesamewagebargainingbutwith the union and the irm having dierent discount rates δ u and δ is studied in Fernandez and Glazer (1991. In this model, reerred here as the F-G model, the union achieves the maximum-wage contract by threatening an alternating strike strategy (to go on strike when the irm rejects an oer but to continue working at the old contract wage when the irm makes an unacceptable oer. As shown by Bolt (1995, this subgame perect equilibrium (SPE only holds i δ u δ. I, however, δ u > δ, then the irm can increase

3 its payo by playing the so called no-concession strategy (reject all oers o the union and make always unacceptable oers. In this case, the SPE is restored by modiying the alternating strike strategies. Houba and Wen (2008 apply the method o Shaked and Sutton (1984 to derive the exact bounds o equilibrium payos in the original F-G model and characterize the equilibrium strategy proiles that support these extreme equilibrium payos or all discount actors. While several works in the literature concern the F-G model, it is usually assumed that the parties have constant discount rates. However, usually in real lie discount rates o the parties are not really constant. Patience o bargainers may be changing over time due to many circumstances (e.g., political, economic, inancial, social, environmental. To the best o our knowledge, our previous work on wage bargaining (Ozkardas and Rusinowska (2014 is the irst one in which the issue o changing patience o the union and the irm in the F-G model is addressed. More precisely, in Ozkardas and Rusinowska (2014 we generalize the F-G model to the union-irm wage bargaining in which both parties have preerences expressed by sequences (δ u,t t N and (δ,t t N o discount actors varying in time. We determine SPE or three cases when the strike decision o the union is exogenous: the case when the union is committed to go on strike in each period in which there is a disagreement, the case when the union is committed to go on strike only when its own oer is rejected, and the case when the union is supposed to go never on strike. We show that while in the original F-G model the exogenous always-strike case coincides with Rubinstein s model, it is not the case in the generalized ramework. We present the unique SPE or this case and also or each o the remaining two cases. Furthermore, we consider the general model with no assumption on the commitment to strike and ind subgame perect equilibria or some particular cases. The aim o the present paper is to continue our research on the wage bargaining initiated in Ozkardas and Rusinowska(2014 and to ind the extreme payos under SPE in the wage bargaining with discount rates varying in time. In order to achieve that, we apply to our generalized model the method used in Houba and Wen (2008. First, we describe necessary conditions under arbitrary sequences o discount rates or the supremum o the union s SPE payos and the inimum o the irm s SPE payos in all periods when the given party makes an oer. Then, we determine the extreme payos under SPE or particular cases o sequences o discount rates varying in time. Apart rom deriving the exact bounds o the equilibrium payos, we also characterize the equilibrium strategy proiles that support these extreme payos. Our indings or the model with varying discount rates generalize the results o Houba and Wen (2008 obtained or the model with constant discount rates. The remainder o the paper is structured as ollows. In Section 2 we briely present the generalized wage bargaining model with discount rates varying in time. In Section 3 necessary conditions or the supremum o the union s SPE payos and the inimum o the irm s SPE payos are determined. In Section 4 we calculate the extreme payos or particular cases o the sequences o discount rates varying in time. We also present equilibrium strategy proiles that support these payos. Section 5 contains some concluding remarks. Proos o all results are presented in the Appendix. 2

4 2 Wage bargaining with discount rates varying in time The point o departure o our study is the ollowing bargaining procedure between the union and the irm, presented in Fernandez and Glazer (1991, and Haller and Holden (1990. There is an existing wage contract, that speciies the wage that a worker is entitled to per day o work, which has come up or renegotiation. Two parties (union and irm bargain sequentially over discrete time and a potentially ininite horizon. They alternate in making oers o wage contracts that the other party can either accept or reject. When a party rejects a proposed wage contract, the union must decide whether or not to strike in that period. Under the previous contract w 0, where w 0 [0,1], the union and the irm gets w 0 and 1 w 0, respectively. By a new contract W [0,1], the union will receive W and the irm 1 W. Figure 1 presents the irst three periods o this wage bargaining. ABOUT HERE FIGURE 1 More precisely, the bargaining procedure is the ollowing. The union proposes x 0. I the irm accepts the new wage contract, the agreement is reached and the payos are (x 0,1 x 0. I the irm rejects it, then the union can either strike, and then both parties obtain (0, 0 in the current period, or continue with the previous contract with payos (w 0,1 w 0. I the union goes on strike, it is the irm s turn to make a new oer y 1, where y 1 is assigned to the union and (1 y 1 to the irm. This procedure continues until an agreement is reached, where x 2t denotes the oer o the union made in an even-numbered period2t,andy 2t+1 denotestheoerotheirmmadeinanodd-numberedperiod(2t+1. Our generalization o the original F-G model concerns preerences o the union and the irm and, as a consequence, the payo unctions o both parties. While Fernandez and Glazer (1991 assume stationary preerences described by constant discount rates δ u and δ, we analyze a wage bargaining in which preerences o the union and the irm are described by sequences o discount actors (rates varying in time, (δ u,t t N and (δ,t t N, respectively, where δ u,t = discount actor o the union in period t N, δ u,0 = 1, 0 < δ u,0 < 1 or t 1 δ,t = discount actor o the irm in period t N, δ,0 = 1, 0 < δ,0 < 1 or t 1 The result o the wage bargaining is either a pair (W,T, where W is the wage contract agreed upon and T N is the number o proposals rejected in the bargaining, or a disagreement denoted by (0, and meaning the situation in which the parties never reach an agreement. We introduce the ollowing notation. Let or each t N and or 0 < t t δ u (t := t δ u,k, δ (t := k=0 t δ,k (1 δ u (t,t := δ u(t t δ u (t 1 = δ u,k, δ (t,t := δ (t t δ k=t (t 1 = 3 k=0 k=t δ,k (2

5 The utility o the result (W,T or the union is equal to U(W,T = δ u (tu t (3 where u t = W or each t T, and i T > 0 then or each 0 t < T u t = 0 i there is a strike in period t N u t = w 0 i there is no strike in period t. t=0 The utility o the result (W,T or the irm is equal to V(W,T = δ (tv t (4 where v t = 1 W or each t T, and i T > 0 then or each 0 t < T v t = 0 i there is a strike in period t v t = 1 w 0 i there is no strike in period t. The utility o the disagreement is equal to t=0 U(0, = V(0, = 0 (5 We assume that the ininite series in (3 and (4 are convergent. A suicient condition or the convergence o (3 and (4 is to assume that the sequences o discount rates (δ u,t t N and (δ,t t N are bounded by a certain number smaller than 1, i.e., there exist a < 1 and b < 1 such that δ u,t a and δ,t b or each t N (6 Let u (t and (t denote the generalized discount actors o the union and the irm in period t, respectively. They take into account the sequences o discount rates varying in time and the act that the utilities are deined by the discounted streams o payos. They are deined as ollows, or every t N + : u (t := k=t δ u(t,k 1+ k=t δ u(t,k, (t := k=t δ (t,k 1+ k=t δ (t,k (7 and consequently, or every t N + 1 u (t = 1 1+ k=t δ u(t,k, 1 (t = 1 1+ k=t δ (t,k (8 Note that or every t N + δ (t,k k=t δ u (t,k i and only i (t u (t k=t For the special case o constant discount rates, i.e., i δ u,t = δ u and δ,t = δ or every t N +, u (t = δ u and (t = δ. 4

6 3 Necessary conditions in the generalized wage bargaining Houba and Wen (2008 apply the method o Shaked and Sutton (1984 to the F-G model to derive the supremum o the union s SPE payos in any even period and the inimum o the irm s SPE payos in any odd period. We generalize their method to the wage bargaining with sequences o discount rates varying in time. Let or t N u = supremum o the union s SPE payos in any even period 2t where the union makes an oer m 2t+1 = inimum o the irm s SPE payos in any odd 2t+1 period where the irm makes an oer Mu 2t and m 2t+1 depend on the sequences (δ u,t t N, (δ,t t N, and 0 w 0 1. Since in this model w 0 is the union s worst SPE payo (see Ozkardas and Rusinowska (2014, we have or each t N w 0 u 1 and w 0 1 m 2t+1 1 Inthissection,wedeterminenecessaryconditionsor,wheret N.The analysis is practically the same as the one given in Houba and Wen (2008 or constant discount rates, except that we consider periods 2t and 2t+1, and the generalized discount actors i (t in a given period t N instead o constant discount rates δ i or i = u,. For coherence o the exposition, we present this analysis or the generalized case. We have or all (δ u,t t N, (δ,t t N, 0 w 0 1 and t N u andm 2t+1 w 0 (1 (2t+1+(1 m 2t+1 (2t+1 (9a u max w 0 (1 u (2t+1+(1 m 2t+1 u (2t+1 (9b 1 m 2t+1 (2t+1 subject to (1 m 2t+1 u (2t+1 w 0 (9c To see that, consider an arbitrary even period 2t, t N. First o all, note that or i = u, (9 1 (1 w 0 (1 i (2t+1 m 2t+1 i (2t+1 = w 0 (1 i (2t+1+(1 m 2t+1 i (2t+1 (1 I the union holds out ater its oer is rejected, the irm will get at least (1 w 0 (1 (2t+1+m 2t+1 (2t+1 by rejecting the union s oer. Hence, the union s SPE payos must be at most 1 (1 w 0 (1 (2t+1 m 2t+1 (2t+1 (10 rom making the least acceptable oer, or rom making an unacceptable oer. w 0 (1 u (2t+1+(1 m 2t+1 u (2t+1 (11 5

7 (2 The union may threaten to strike in period 2t i the irm rejects its oer, which is credible i and only i (1 m 2t+1 k=2t+1 δ u(2t+1,k w 0 +w 0 k=2t+1 δ u(2t+1,k, i.e., i and only i (1 m 2t+1 u (2t+1 w 0 In this case, the union s SPE payos must be smaller than or equal to rom making the least acceptable oer, or 1 m 2t+1 (2t+1 (12 (1 m 2t+1 u (2t+1 (13 rom making an unacceptable oer. Note that we have always which is equivalent to 1 u (2t+1 m 2t+1 ( (2t+1 u (2t+1 1 m 2t+1 (2t+1 (1 m 2t+1 u (2t+1 This means that i the union threatens to strike, it will not make an unacceptable oer in period 2t. Hence, the union s SPE payos cannot be greater than the maximum o the three cases (10, (11 and (12, and thereore we obtain (9. From (9 we get the necessary conditions or the supremum o the union s SPE payos in an even period: Proposition 1 We have or all (δ u,t t N, (δ,t t N, 0 w 0 1 and t N 1 m 2t+1 (2t+1 i (15 Mu 2t w 0 (1 (2t+1+(1 m 2t+1 (2t+1 i (16 w 0 (1 u (2t+1+(1 m 2t+1 u (2t+1 i (17 where (14 (1 m 2t+1 u (2t+1 w 0 (15 (1 m 2t+1 u (2t+1 < w 0 and (2t+1 u (2t+1 (16 (1 m 2t+1 u (2t+1 < w 0 and (2t+1 < u (2t+1 (17 For the proo o Proposition 1, see the Appendix. Similarly, we have or all (δ u,t t N, (δ,t t N, 0 w 0 1 and t N m 2t+1 { max min 1 w 0 (1 u (2t+2 +2 u 1 w 0 (1 (2t+2 +2 u 1 +2 u u (2t+2 subject to +2 u (2t+2 (18a (2t+2 (18b u u (2t+2 w 0 (18c (18 Consider an arbitrary odd period 2t+1, t N. 6

8 (1 I the union holds out ater rejecting the irm s oer, the union will get at most Hence, the irm could get at least w 0 (1 u (2t+2+Mu 2t+2 u (2t+2 1 w 0 (1 u (2t+2 +2 u u (2t+2 (19 rom making the least irresistible oer. The irm could receive at least (1 w 0 (1 (2t+2+(1 Mu 2t+2 (2t+2 = 1 w 0 (1 (2t+2 Mu 2t+2 (2t+2 (20 rom making any unacceptable oer. The irm will make either the least irresistible oer or an unacceptable oer, depending on whether (19 or (20 is greater. For the union, it is always credible to holdout ater rejecting a irm s oer, as the union gets at most u (2t+2 when it strikes ater the irm s oer is rejected. +2 u (2 I the union strikes ater rejecting the irm s oer, then the irm will get at least rom making the least irresistible oer, or 1 Mu 2t+2 u (2t+2 (1 Mu 2t+2 (2t+2 rom making an unacceptable oer. Since +2 u which is equivalent to 1, note that Mu 2t+2 ( u (2t+2 (2t+2 1 (2t u u (2t+2 (1 Mu 2t+2 (2t+2 This implies that the irm will never make an unacceptable oer i the union threatens to strike ater rejecting the irm s oer. Strike in period 2t +1 is credible i and only i k=2t+2 δ u(2t+2,k w 0 +w 0 k=2t+2 δ u(2t+2,k, i.e., i and only i +2 u Hence, we obtain ( u u (2t+2 w 0 From (18 we get the necessary conditions or the inimum o the irm s SPE payos in an odd period: Proposition 2 We have or all (δ u,t t N, (δ,t t N, 0 w 0 1 and t N 1 Mu 2t+2 u (2t+2 i (22 m 2t+1 1 w 0 (1 (2t+2 Mu 2t+2 (2t+2 i (23 1 w 0 (1 u (2t+2 Mu 2t+2 u (2t+2 i (24 where Mu 2t+2 ( u (2t+2 (2t+2 w 0 (1 (2t+2 (22 Mu 2t+2 ( u (2t+2 (2t+2 < w 0 (1 (2t+2 and u (2t+2 > (2t+2 (23 u (2t+2 (2t+2 (24 (21 7

9 For the proo o Proposition 2, see the Appendix. Note that our Propositions 1 and 2 generalize the corresponding results on necessary conditions or M u and m or the model with constant discount rates presented in Houba and Wen (2008 (Propositions 2 and 1. From Propositions 1 and 2 we can write the ollowing act which will be useul or determining Mu 2t and m 2t+1 or particular cases o the generalized discount actors. Fact 1 Let t N. (i I u (2t+1 (2t+1, then u { 1 m 2t+1 (2t+1 i (1 m 2t+1 u (2t+1 w 0 w 0 (1 (2t+1+(1 m 2t+1 (2t+1 i (1 m 2t+1 u (2t+1 < w 0 (ii I u (2t+1 > (2t+1, then u { 1 m 2t+1 (2t+1 i (1 m 2t+1 u (2t+1 w 0 w 0 (1 u (2t+1+(1 m 2t+1 u (2t+1 i (1 m 2t+1 u (2t+1 < w 0 (iii I u (2t+2 (2t+2, then m 2t+1 (iv I u (2t+2 > (2t+2, then where m 2t+1 1 w 0 (1 u (2t+2 Mu 2t+2 u (2t+2 { 1 +2 u u (2t+2 i ( u (2t+2 w 0 (1 (2t+2 i (26 +2 u ( u (2t+2 (2t+2 w 0 (1 (2t+2 (25 +2 u ( u (2t+2 (2t+2 < w 0 (1 (2t+2 (26 Since there exist ininitely many cases o the relation between the generalized discount actors o the union and the irm in even and odd periods, and consequently, ininitely many combinations o the necessary conditions, we cannot ully determine Mu 2t and m 2t+1 or all possibilities. However, given the sequences o discount rates, the corresponding necessary conditions can be used to ind Mu 2t and m 2t+1, i they exist. 4 Maximum wage contract in the generalized model From the necessary conditions presented in the previous section, we now determine Mu 2t and m 2t+1 or t N or some particular cases o the discount rates varying in time. Let u (t and (t or t N be the generalized discount rates o the union and the irm, respectively, as deined in (7. In order to simpliy the presentation o the results, irst we introduce the notation or dierent sums o the generalized discount rates. We have or each t N: 8

10 m (t := 1 (2t+1+ (1 (2m+3 u (2j +2 (2j +1 (27 m=t (t := 1 u (2t+2+ ( m (1 u (2m+4 u (2j +2 (2j +3 (28 m=t (t := w 0 +(1 w 0 1 (2t+1+ m (1 (2m+3 (2j +1 (2j +2 m=t (29 (t := (1 w 0 ( 1 (2t+2+ m (1 (2m+4 (2j +2 (2j +3 m=t When we consider the model with constant discount rates, i.e., δ u,t = δ u and δ,t = δ or each t N, we get or every t N (t = 1 δ 1 δ u δ, (t = 1 δ u 1 δ u δ, (t = 1+w 0 δ 1+δ, (t = 1 w 0 1+δ Our irst results present the supremum o the union s SPE payos in any even period and the inimum o the irm s SPE payos in any odd period or the particular cases with u (2t+2 (2t+2 or every t N: when either the strike is always credible (Proposition 3(i or the strike is never credible (Proposition 3(ii. Proposition 3 Let u (2t+2 (2t+2 or every t N. (i I or every t N [ w 0 +(1 w 0 u (2t+2 (t+1 ] u (2t+1 w 0 (31 (30 then u = w 0 +(1 w 0 (t [ ] (32 m 2t+1 = (1 w 0 1 u (2t+2 (t+1 (33 The SPE strategy proile that supports these u and m 2t+1 deined in (32 and (33 is given by the ollowing generalized alternating strike strategies : In period 2t the union proposes w 0 +(1 w 0 (t, in period 2t+1 it accepts an oer y i and only i y w 0 +(1 w 0 u (2t+2 (t+1, it goes on strike ater rejection o its own proposals and holds out ater rejecting irm s oers. In period 2t+1 the irm proposes w 0 +(1 w 0 u (2t+2 (t+1, in period 2t it accepts x i and only i x w 0 +(1 w 0 (t. 9

11 I, however, at some point, the union deviates rom the above rule, then both parties play thereater according to the ollowing minimum-wage strategies : - The union always proposes w 0, accepts y i and only i y w 0, and never goes on strike. - The irm always proposes w 0 and accepts x i and only i x w 0. (ii I or every t N [ w 0 +(1 w 0 u (2t+2 (t+1 ] u (2t+1 < w 0 (34 then u = w 0 and m 2t+1 = 1 w 0 (35 The SPE strategy proile that supports these Mu 2t and m 2t+1 by the minimum-wage strategies. deined in (35 is given For the proo o Proposition 3, see the Appendix. Note that our Proposition 3 generalizes the corresponding results on M u and m or the model with constant discount rates presented in Houba and Wen (2008 (Proposition 3. When we consider the model with constant discount rates, i.e., we put δ u,t = δ u and δ,t = δ or each t N, and we assume that δ u δ, we get or every t N u = w 0 + (1 w 0(1 δ 1 δ u δ, m 2t+1 = (1 w 0(1 δ u 1 δ u δ and the strike credibility condition (31 is equivalent to (1 w 0 δ 2 u +w 0 δ u w 0 δ u δ (δ u w 0 Our next results concern some particular cases when the the generalized discount rate o the union is always greater than the generalized discount rate o the irm in the same even period. Three particular cases are considered. Proposition 4 Let u (2t+2 > (2t+2 or every t N. (i I or every t N ( 1 (t u (2t+1 w 0 (36 and (t+1( u (2t+2 (2t+2 w 0 (1 (2t+2 (37 then u = (t and m 2t+1 = (t (38 The SPE strategy proile that supports these Mu 2t and m 2t+1 deined in (38 is given by the ollowing always strike strategies : In period 2t the union proposes (t, in period 2t+1 it accepts an oer y i and only i y 1 (t, it always goes on strike i there is a disagreement. In period 2t+1 the irm proposes 1 (t, in period 2t it accepts x i and only i x (t. I, however, at some point, the union deviates rom the above rule, then both parties play thereater according to the minimum-wage strategies. 10

12 (ii I or every t N ( 1 (t u (2t+1 w 0 (39 and (t+1( u (2t+2 (2t+2 < w 0 (1 (2t+2 (40 then u = (t and m 2t+1 = (t (41 The SPE strategy proile that supports these Mu 2t and m 2t+1 deined in (41 is given by the ollowing modiied generalized alternating strike strategies : In period 2t the union proposes (t, in period 2t+1 it accepts an oer y i and only i y (1 u (2t+2w 0 + u (2t+2 (t+1, it strikes in even periods and holds out in odd periods i no agreement is reached. In period 2t+1 the irm proposes 0, in period 2t it accepts x i and only i x (t. I, however, at some point, the union deviates rom the above rule, then both parties play thereater according to the minimum-wage strategies. (iii I or every t N and then or each t N Mu 2t+2 ( u (2t+2 (2t+2 < w 0 (1 (2t+2 (1 m 2t+1 u (2t+1 < w 0 u = w 0 and m 2t+1 = 1 w 0 (42 The SPE strategy proile that supports these Mu 2t and m 2t+1 by the minimum-wage strategies. deined in (42 is given For the proo o Proposition 4, see the Appendix. Note that our Proposition 4 generalizes the corresponding results on M u and m or the model with constant discount rates presented in Houba and Wen (2008 (Proposition 4. Consider the model with constant discount rates, i.e., let δ u,t = δ u and δ,t = δ or each t N, and assume that δ u > δ. Then or Proposition 4(i, we get or every t N Mu 2t = 1 δ, m 2t+1 = 1 δ u 1 δ u δ 1 δ u δ and the strike credibility conditions (36 and (37 are equivalent to the set C in Houba and Wen (2008: (δ u w 0 δ δ2 u w 0 δ u and δ δ u w 0 1 w 0 δ u respectively. For Proposition 4(ii, we get or every t N u = 1+w 0δ 1+δ, m 2t+1 = 1 w 0 1+δ and the conditions (39 and (40 are equivalent to the set B in Houba and Wen (2008: δ (δ u w 0 w 0 (1 δ u and δ > δ u w 0 1 δ u w 0 11

13 In Propositions 3 and 4, Mu 2t and m 2t+1 or every t N are determined or several cases where particular conditions on the discount rates o both parties are satisied. In order to calculate Mu 2t and m 2t+1 or an arbitrary case, we can proceed as ollows. Given the sequences o discount rates (δ u,t t N and (δ,t t N, we also obtain the sequences o the generalized discount rates ( u (t t N and ( (t t N. Depending on which conditions hold, we apply Fact 1 to determine the ininite sequence o necessary conditions or Mu 2t and m 2t+1 or every t N. Note that we get always an ininite regular triangular system o equations which has a unique solution, being the sequence ( t N. However, the solution does not always satisy the required conditions. To see that consider the case where or every t N, Then, solving or every t N u,m 2t+1 u (t > (t, (1 m 2t+1 u (2t+1 < w 0 and Mu 2t+2 ( u (2t+2 (2t+2 w 0 (1 (2t+2 Mu 2t = w 0 (1 u (2t+1+(1 m 2t+1 u (2t+1 and m 2t+1 = 1 Mu 2t+2 u (2t+2 leads to u = w 0 (1 u (2t+1+ m (1 u (2m+3 u (2j +1 u (2j +2 but this means that u < w 0, and thereore we get a contradiction. m=t Similarly, consider the case where or every t N, u (2t+1 (2t+1, u (2t+2 > (2t+2, (1 m 2t+1 u (2t+1 < w 0 and Then, solving or every t N Mu 2t+2 ( u (2t+2 (2t+2 w 0 (1 (2t+2 Mu 2t = w 0 (1 (2t+1+(1 m 2t+1 (2t+1 and m 2t+1 = 1 Mu 2t+2 u (2t+2 leads to u = w 0 (t < w0, and thereore we get again a contradiction. 5 Conclusion We calculated the equilibrium payos or the wage bargaining model between the union and the irm with preerences o the parties expressed by discount rates varying in time. We extended the analysis presented in Houba and Wen (2008 or the wage bargaining with constant discount rates. We ocused on determining the supremum o the union s payo and the inimum o the irm s payo under SPE in all periods when the given party makes its oer. While we described the necessary conditions or these payos or arbitrary sequences o discount rates, we determined them and the supporting equilibria only or some particular (representative cases o sequences o discount rates varying in 12

14 time. In all these equilibria, the agreement is reached immediately in period 0. In a ollowup research on the wage bargaining with sequences o discount rates varying in time it would be interesting to analyze the existence o ineicient SPE with a strike or some periods ollowed by an agreement. Another issue in our uture research agenda is to see how our results would be aected i not only the union could go on strike or holdout, but also the irm could lock out. We would like also to apply the model to one o the important economic issues pharmaceutical product price determination; see e.g. Jelovac (2005; Garcia-Marinoso et al. (2011. Appendix - Proos Proo o Proposition 1 Consider an arbitrary t N. (1 Suppose that strike is not credible, i.e., (1 m 2t+1 u (2t+1 < w 0. We have (9a (9b i and only i w 0 (1 (2t+1+(1 m 2t+1 (2t+1 w 0 (1 u (2t+1+(1 m 2t+1 u (2t+1 (1 m 2t+1 w 0 (2t+1 (1 m 2t+1 w 0 u (2t+1 As 1 m 2t+1 w 0 0, this establishes the second and the third cases o (14. (2 Suppose that strike is credible, i.e., (1 m 2t+1 u (2t+1 w 0. Then, (9c (9a. Moreover, (9c (9b, because (9c (9b i and only i 1 m 2t+1 (2t+1 w 0 (1 u (2t+1+(1 m 2t+1 u (2t+1 (1 w 0 (1 u (2t+1 m 2t+1 ( (2t+1 u (2t+1 which is always true, since m 2t+1 1 w 0. Then, we obtain the irst case o (14. Proo o Proposition 2 Consider an arbitrary t N. (1 Assume that u (2t+2 (2t+2. We have 1 w 0 (1 (2t+2 +2 u (2t+2 = 1 +2 u +(Mu 2t+2 w 0 (1 (2t+2 1 Mu 2t+2 +(Mu 2t+2 w 0 (1 u (2t+2 = 1 w 0 (1 u (2t+2 Mu 2t+2 u (2t+2 Hence, (18a (18b. Moreover, (18c > (18a, and we get the third case o (21. (2 Assume that u (2t+2 > (2t+2. Then (18b > (18a. Moreover, (18c > (18b i and only i w 0 (1 (2t+2+Mu 2t+2 (2t+2 > Mu 2t+2 u (2t+2 Mu 2t+2 ( u (2t+2 (2t+2 < w 0 (1 (2t+2 which gives the second case o (21. On the other hand, i Mu 2t+2 ( u (2t+2 (2t+2 w 0 (1 (2t+2 13

15 then (18c (18b and the strike is credible, because +2 u u (2t+2 w 0 +2 u (2t+2+w 0 (1 (2t+2 w 0 = We get then the irst case o (21. = (2t+2(+2 u w 0 0 Proo o Proposition 3 Let u (2t+2 (2t+2 or every t N. (i Consider the case when the strike is always credible, i.e., (1 m 2t+1 u (2t+1 w 0 or every t N. From Fact 1 we have or every t N: u +m 2t+1 (2t+1 = 1 and m 2t+1 +Mu 2t+2 u (2t+2 = 1 w 0 (1 u (2t+2 which is a regular triangular system AX = Y, with A = [a ij ] i,j N +, X = [(x i i N +] T, Y = [(y i i N +] T, where or each t,j 1 and or each t N a t,t = 1, a t,j = 0 or j < t or j > t+1 a 2t+1,2t+2 = (2t+1, a 2t+2,2t+3 = u (2t+2 x 2t+1 = Mu 2t, x 2t+2 = m 2t+1, y 2t+1 = 1, y 2t+2 = 1 w 0 (1 u (2t+2 Any regular triangular matrix A possesses the (unique inverse matrix B, i.e., there exists B such that BA = I, where I is the ininite identity matrix. The matrix B = [b ij ] i,j N + is also regular triangular, and its elements are the ollowing: b t,t = 1, b t,j = 0 or each t,j 1 such that j < t b 2t+1,2t+2 = (2t+1, b 2t+2,2t+3 = u (2t+2 or each t N and or each t,m N and m > t b 2t+1,2m+1 = b 2t+2,2m+2 = m 1 m 1 m 1 (2j+1 u (2j+2, b 2t+1,2m+2 = m 1 u (2j+2 (2j+3, b 2t+2,2m+3 = (2j+1 u (2j+2 (2m+1 u (2j+2 (2j+3 u (2m+2 Next, by applying X = BY we get Mu 2t as given in (32 and m 2t+1 as given in (33. The strike credibility condition (1 m 2t+1 u (2t + 1 w 0 or every t N is then written as in (31. In Ozkardas and Rusinowska (2014 (Proposition 4 we show that under an equivalently expressed condition (31 and u (2t+2 (2t+2 or every t N, the proposed strategy proile (ormed by the generalized alternating strike strategies is a SPE. 14

16 (ii Consider the case when the strike is never credible, i.e., (1 m 2t+1 u (2t+1 < w 0 or every t N. Then we have the ininite system or t N or and Mu 2t +m 2t+1 Mu 2t +m 2t+1 m 2t+1 +Mu 2t+2 (2t+1 = w 0 (1 (2t+1+ (2t+1 u (2t+1 = w 0 (1 u (2t+1+ u (2t+1 u (2t+2 = 1 w 0 (1 u (2t+2 which as a regular triangular system possesses a unique solution. This solution is given by (35. It is supported by the minimum-wage strategies proile which is a SPE as shown in Ozkardas and Rusinowska (2014 (Fact 3. Proo o Proposition 4 Let u (2t+2 > (2t+2 or every t N. (i Consider the case when or every t N, (1 m 2t+1 u (2t + 1 w 0 (i.e., strike is credible in period 2t and condition (25 holds. I (25 is satisied, then strike is credible in period 2t+1. From Fact 1 we get the ininite system or every t N u +m 2t+1 (2t+1 = 1 and m 2t u u (2t+2 = 1 which is is a regular triangular system AX = Y, with A = [a ij ] i,j N + and X = [(x i i N +] T the same as in the proo o Proposition 3, and with Y = [(y i i N +] T such that y 2t+1 = y 2t+2 = 1. The (unique inverse matrix B is the same as beore, and by applying X = BY we get Mu 2t and m 2t+1 as given by (38. The conditions (1 m 2t+1 u (2t+1 w 0 and (25 are equivalent to (36 and (37. In Ozkardas and Rusinowska (2014 (Proposition 3 we show that the proposed strategy proile (ormed by the always strike strategies is a SPE under an equivalently expressed condition (36. (ii Consider the case when or every t N, (1 m 2t+1 u (2t + 1 w 0 (i.e., strike is credible in period 2t and condition (26 holds. Then, we solve the ininite system or every t N u +m 2t+1 (2t+1 = 1 and m 2t+1 +Mu 2t+2 (2t+2 = 1 w 0 (1 (2t+2 which is a regular triangular system AX = Y. By applying X = BY we get Mu 2t and m 2t+1 as given by (41. The conditions (1 m 2t+1 u (2t+1 w 0 and (26 are equivalent to (39 and (40. In Ozkardas and Rusinowska (2014 (Theorem 3 we show that i u (2t + 2 > (2t + 2 or each t N, then the proposed strategy proile (ormed by the modiied generalized alternating strike strategies is a SPE under the ollowing condition: w 0 u (2t+1 ( (1 u (2t+2w 0 + u (2t+2W 2t+2 (43 where W 2t = 1+ m=t δ (2t+1,2m+2+w 0 m=t δ (2t+1,2m+1 1+ m=2t+1 δ (2t+1,m 15

17 isthespeoerotheunionproposedinperiod2t.onecanshowthatw 2t = Mu 2t = (t: ( 1+ W 2t m=t = w 0 +(1 w 0 δ (2t+1,2m+2 1+ m=2t+1 δ = (2t+1,m ( m=t = w 0 +(1 w 0 1 (2t+1+ δ (2t+1,2m+2 1+ m=2t+1 δ = (2t+1,m ( m = w 0 +(1 w 0 1 (2t+1+ (1 (2m+3 (2j +1 (2j +2 = (t m=t Moreover, note that (39 implies condition (43: ( w 0 u (2t+1 1 (t = = u(2t+1 = u(2t+1 [ [ w 0 +(1 w 0 w 0 +(1 w 0 (2t+2 ( (2t+2 ( ] m (1 (2m+4 (2j +2 (2j +3 m=t 1 (2t+3+ m=t+1 (1 (2m+3 m +1 ( = u (2t+1 w 0 + (2t+2( (t+1 w 0 < ( < u (2t+1 w 0 + u (2t+2( (t+1 w 0 = ( = u (2t+1 (1 u (2t+2w 0 + u (2t+2 (t+1 (iii Consider the case when or every t N, (1 m 2t+1 (2j +1 (2j +2 u (2t+1 < w 0 and Mu 2t+2 ( u (2t+2 (2t+2 < w 0 (1 (2t+2 Then, rom Fact 1 we get the ininite system or every t N u +m 2t+1 u (2t+1 = w 0 + u (2t+1(1 w 0 ] or and u +m 2t+1 (2t+1 = w 0 + (2t+1(1 w 0 m 2t+1 +Mu 2t+2 (2t+2 = 1 w 0 (1 (2t+2 which is is a regular triangular system AX = Y with the solution Mu 2t = w 0 and m 2t+1 = 1 w 0 or each t N. The SPE supporting this solution is the minimum-wage strategies proile. 16

18 Bibliography W. Bolt. Striking or a bargain between two completely inormed agents: Comment. The American Economic Review, 85: , R. Fernandez and J. Glazer. Striking or a bargain between two completely inormed agents. The American Economic Review, 81: , B. Garcia-Marinoso, I. Jelovac, and P. Olivella. External reerencing and pharmaceutical price negotiation. Health Economics, 20(6: , H. Haller and S. Holden. A letter to the editor on wage bargaining. Journal o Economic Theory, 52: , S. Holden. Bargaining and commitment in a permanent relationship. Games and Economic Behavior, 7: , H. Houba and Q. Wen. On striking or a bargain between two completely inormed agents. Economic Theory, 37: , I. Jelovac. On the relationship between the negotiated price o pharmaceuticals and the patients co-payment. WP 0208, A. Muthoo. Bargaining Theory with Applications. Cambridge University Press, M. J. Osborne and A. Rubinstein. Bargaining and Markets. San Diego, Academic Press, A. Ozkardas and A. Rusinowska. Wage bargaining with discount rates varying in time under dierent strike decisions. RAIRO-Operations Research, Forthcoming. A. Rubinstein. Perect equilibrium in a bargaining model. Econometrica, 50:97 109, A. Shaked and J. Sutton. Unemployment as a perect equilibrium in a bargaining model. Econometrica, 52(6: , 1984.

19 Period 0 Union: Propose x 0 Firm: Accept/Reject Game ends (x 0,1 x 0 Y N Union: Strike / No Strike (0,0 (w 0,1 w 0 1 Firm: Propose y 1 Union: Accept/Reject Game ends (y 1,1 y 1 Y N Union: Strike / No Strike (0,0 (w 0,1 w 0 2 Union: Propose x 2 etc. Figure 1: Non-cooperative bargaining game between the union and the irm 18