Note on a generalized wage rigidity result. Abstract

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1 Note on a generalized age rigidity result Ariit Mukheree University of Nottingha Abstract Considering Cournot copetition, this note shos that, if the firs differ in labor productivities, the equilibriu age rates under a centralized labor union are not independent of the nuber of firs and product differentiation if the labor union charges a unifor age rate. Hoever, if the centralized labor union can discriinate age rate beteen the firs, the equilibriu age rates do not depend on the nuber of firs and product differentiation. Hence, hether the age rigidity result of Dhillon and Petrakis (00 holds ith asyetric firs depends on the age setting behavior of the labor union. The effects of the nuber of firs and product differentiation on the equilibriu age rate are also shon. I ould like to thank an anonyous referee of this ournal for helpful coents. The usual disclaier applies. Citation: Mukheree, Ariit, (007 "Note on a generalized age rigidity result." Econoics Bulletin, Vol. 0, No. pp. -9 Subitted: August 7, 007. Accepted: October 9, 007. URL:

2 . Introduction The epirically observed age rigidity has attracted attention of the researchers orking on Industrial Organization and Labor Econoics. Recently, Dhillon and Petrakis (00 sho that, under fairly general conditions, if there is a centralized labor union, the equilibriu ag rates do not depend on the nuber of firs and product differentiation. Considering Cournot copetition, this note extends this line of research ith asyetric firs. We sho that if the firs differ in labor productivities, the equilibriu age rates under a centralized labor union are not independent of the nuber of firs and product differentiation if the labor union charges a unifor age rate. Hoever, if the centralized labor union can discriinate age rate beteen the firs, the equilibriu age rates do not depend of the nuber of firs and product differentiation. Hence, hether the age rigidity result of Dhillon and Petrakis (00 extends to the case of asyetric firs depends on the age setting behavior of the labor union. Epirical evidence suggests that, in any situations, centralized labor unions charge unifor age rates irrespective of the differences beteen the firs. As discussed in Haucap et al. (000 and 00, a coon feature of any labor arkets in continental Europe is coverage extension rules, hich iplies that soe or all eployent ters are ade generally binding for all industry participants and not only for the ebers of unions and eployers associations. In Gerany, for exaple, collective age agreeents beteen a union and an eployers association can be ade copulsory even for independent eployers through so-called Allgeeinverbindlicherklärung (AVE The Ministry of Labor can, on application of either unions or eployers associations, use an AVE to ake soe or all ters of a collectively negotiated eployent contract generally binding for an entire industry, here otherise only those unions, eployers and eployers associations that have actually negotiated and signed the contract ould be directly bound by it ( 3 I TVG (Haucap et al., 00. It is also noted in Haucap et al. (00 that the nuber of AVEs alost continuously increased fro 448 in 975 to 588 in 998. Thus, it ustifies our analysis ith unifor age setting by a centralized labor union. 3 The reainder of the paper is organized as follos. The next section describes the odel and shos the results. Section 3 concludes.. The odel and the results Since, in the presence of asyetric firs, the calculations for shoing the effects of the nuber of firs and product differentiation are cubersoe, e sho these effects separately. In section., e consider the case of a hoogeneous product and sho the effects of the nuber of firs. In section., e consider a duopoly arket structure and sho the effects of product differentiation. The irrelevance of the nuber of donstrea firs on the upstrea input price can also be found in the earlier orks by Greenhut and Ohta (976 and Tyagi (999. Haucap et al. (00 also sho hen the labor union ay prefer unifor age over discriinatory age. 3 The age bargaining by the labor union for the UK Universities ay also support the unifor age setting by a centralized labor union in the presence of productivity differences. While the people orking in different Universities ay differ in productivities, the national labor union bargains for the siilar age rates for all the UK Universities.

3 .. The effects of the nuber of firs Let us consider an econoy ith ( n + firs producing a hoogeneous product. Assue that production requires only labor. For notational convenience arrange the firs as,,... n, n +, n +,..., n +. Without loss of generality, assue that each of the firs in [, n ] requires one labor to produce one unit of output, hile each of the firs in [ n +, n + ] requires λ orkers to produce one unit of output, here λ. Hence, e consider asyetry in labor productivities of the firs. Also, to ake asyetry eaningful in our analysis, assue that n and. Assue that the inverse arket deand function for the product is P a q, ( here the notations have usual eanings. We assue that there is a centralized labor union that sets the age rates for the firs. We ill consider to possibilities: (i here the labor union sets a unifor age rate for all firs, and (ii here the labor union can charge different age rates to different firs. Folloing Haucap and Wey (004, e can call the forer age setting behavior as centralization and the latter as coordination. As a siplification, e assue that the reservation age rates of the labors are zero. We consider the folloing gae. At stage, the labor union sets the age rates. At stage, the firs produce like Cournot oligopolists and the profits are realized. We solve the gae through backard induction. Hence, e consider the case of a onopoly labor union as in Dunlop (944 and Osald (98. Since, the purpose of this paper is to sho that, under a centralized labor union, the equilibriu age rates can depend on the nuber of firs and product differentiation in presence of asyetric firs, it is enough for us to consider a onopoly labor union. Hoever, note that our qualitative results holds even if there is bargaining beteen the labor union and the firs. Bargaining beteen the labor union and the firs ill only coplicate the calculations ithout adding uch to the ain purpose of the paper. Further, to ake our point, e concentrate on the right-to-anage odel of labor union, hich is perhaps the ost idely used odel of labor union in the Industrial Organization literature Unifor age setting by the labor union In this subsection e assue that the labor union charges a unifor age rate to all firs. Given the deand function and the unifor age rate, the equilibriu ( a ( λ + output of each of the firs in [, n ] is q i, and the equilibriu ( n + + ( a ( λ + λn n output of each of the firs in [ n +, n + ] is q. ( n + + Therefore, the labor union axiizes the folloing expression to deterine the age rate: 5 4 See, Layard et al. (99 for arguents in favor of right-to-anage odels. 5 It is iportant to note that, due to asyetry aong the firs, the labor union ay not ish to serve all the firs. Hence, the labor union ay charge a sufficiently high age rate that encourages only the relatively efficient firs to hire labors and producing the product. Hoever, it can be shon that the labor union prefers to serve all the firs if λ is not sufficiently different fro. Since, age deterination in the presence of asyetric firs is the ain eleent of this paper, e do our analysis

4 n( a ( λ + + λ( a ( λ + λn n Max. ( ( n + + The equilibriu age rate is a( n + λ, (3 ( n( + + λ( λ + λn n hich clearly shos that the equilibriu age rate depends on the nuber of firs. Note that is independent of the nuber of firs if either λ or n 0 or 0. Hence, the folloing proposition is iediate. Proposition : If there is a centralized labor union that sets a unifor age rate for all firs, the equilibriu age rate depends on the nuber of firs if the firs differ in labor productivities. The reason for the above result follos easily fro Dhillon and Petrakis (00. They sho that, if the fir s equilibriu output and profit are log-linear in the age rate and the arket features such as the nuber of firs and product differentiation, the equilibriu age rate is independent of the arket features. If the firs are asyetric in labor productivities, it is iediate fro the above analysis that the fir s equilibriu output and the profit 6 is not log-linear in the age rate and the nuber of firs, and therefore, the equilibriu age rate depends on the nuber of firs. It is iportant to note that, like Dhillon and Petrakis (00, e have also considered that the production technologies of the firs are log-linear and the union utility is log-linear in the age rate and aggregate eployent. Hoever, the asyetry beteen the firs does not satisfy that the fir s equilibriu output and the profit are log-linear in the age rate and the nuber of firs. Let us no see the effects of the nuber of firs on the equilibriu age rate. Proposition : (i Assue λ > 0. If e get 0 for ( λ >. 7 λ, e get 0 (ii If λ >, e get 0. Hoever, if λ, e get 0 for. Hoever, if λ, λ ( λ n. 8 under the assuption that λ is sufficiently close to so that the labor union serves all the firs and solves the axiization proble (. 6 It is easy to check that the profit of the i th fir is 3 q i, i,,..., n +. 7 Though it ill be iediate fro the proof of this proposition that 0 if λ >, e ill not focus on the higher values of λ, since as entioned in footnote 5, the labor union ould serve only the relatively efficient fir for higher values of λ. For exaple, it is easy to sho that if λ >, n and the centralized labor union charges a unifor age rate to all firs, then the labor union prefers to serve both firs instead of serving only the fir ith higher labor productivity provided λ. When n, >, the relevant values of λ for hich the labor union serves all the firs are less than λ ( n that is loer than. So, for n, >, e need to restrict λ beteen and λ (. n

5 Proof: (i We find fro (3 that if λ > 0 and λ, then 0 ( λ Hence, if provided. (4 λ, e get 0, since provided λ. Hence, for λ, e get 0 for (ii We find fro (3 that if λ, then 0 provided λ (λ Hence, if >. Hoever, if λ >, e get > ( λ. ( λ > n. (5 λ, e get 0. Hoever, if λ >. Hence, for λ, e get 0 for λ λ, e get that > provided ( λ λ ( λ n. Q.E.D. The reason for the above result is easy to understand. The presence of asyetric firs akes the total labor deand curve as a kinked function, and the kink occurs at that age rate here the firs ith relatively loer labor productivities do not find it profitable to produce. Let us no consider Proposition (i. If λ, then, in our analysis, the firs in [, n ] are relatively inefficient, and none of the deand labor if a >. An increase in n iplies that the nuber of inefficient firs ( λ + increases, hich akes the segent of the labor deand curve, here all firs find production profitable, ore elastic, but does not affect. This is shon in Figure. Figure : The effect of a higher n hen λ. 8 Though it ill be iediate that 0 attention to λ >. for λ, folloing footnotes 5 and 7, e restrict our 4

6 Assue that the total labor deand curve for a given λ, n and is given by the kinked curve ABC. The age rate is the age rate at hich the labor deand by the loer productive firs is zero. The labor deand is coing fro all the firs on the segent BC, hile, on the segent AB, the labor deand is coing only fro the firs ith higher labor productivities, i.e., fro the firs in [ n +, n + ]. No, if n increases, it rotates BC to BC ', and akes this segent ore elastic. As a result, the equilibriu age rate reduces ith n. Hoever, if λ >, then, the firs in [, n ] are relatively efficient, and these a firs stop producing if > '. If n increases, it shifts the segent of ( λ + λn n the labor deand curve above ' outard. Hoever, higher n also iplies that ' falls, since, no the inefficient firs are facing copetition fro ore efficient firs. Hence, in this situation, higher n not only shifts the labor deand curve outard, it also reduces the critical level of the age rate at hich kink occurs. This is shon in Figure. Figure : The effect of a higher n hen λ >. Assue that the labor deand curve for a given λ >, n and is given by the kinked curve MNP. The age rate ' is the age rate at hich the labor deand by the loer productive firs is zero. No, if n increases, it shifts the labor deand curve to MN ' P', and the kink occurs at ". So, hile the outard shift of the labor deand curve tends to increase the equilibriu age rate, the fall of ' to " tends to reduce the equilibriu age rate. Hence, the net effect of a change in n depends on the nuber of inefficient firs and the labor productivities of the inefficient firs. Siilar arguent follos for the case of Proposition (ii.... Wage discriination We have shon in the previous section that the age rigidity result of Dhillon and Petrakis (00 does not hold if the centralized labor union sets a unifor age rate and the firs differ in labor productivities. In this section e sho that if the centralized union keeps the flexibility of charging different age rates to different firs, the age rigidity result of Dhillon and Petrakis (00 holds in the presence of the asyetric firs. Let us no consider age discriination beteen the firs. Given the age rates, the equilibriu output 5

7 ( a ( n + + of each of the firs in [, n ] is qi and the ( n + + equilibriu output of each of the firs in [ n +, n + ] is ( a λ( n + + n i + λ i s s s q. ( n + + Therefore, the labor union axiizes the folloing expression to deterine the age rates for each fir: n iqi + λ q i Max,...,,,..., + ( n + n n + n. (6 a a The equilibriu age rates are i, i,..., n, and, n +,..., n +. λ Hence, the folloing proposition is iediate. Proposition 3: If a centralized labor union can discriinate age rate beteen the firs, the equilibriu age rates are independent of the nuber of firs even if the firs differ in labor productivities. If the union discriinates age rate beteen the firs, it considers the labor deand of different firs separately. Hence, it is iportant to see hether, in the fir s output and the profit, the fir s on age rate and arket features such as the nuber of firs behave like a log-linear function. It follos fro the above analysis that, in the fir s equilibriu output and profit, the fir s on age rate and the nuber of firs behave like a log-linear function. As a result, the equilibriu age rate of each fir is independent of the nuber of firs... The effects of product differentiation Let us no consider a horizontally differentiated duopoly arket structure. Assue that fir requires one labor to produce one unit of output, and fir requires λ labors to produce one unit of output. Like section., e assue that the reservation age rates for the labors are zero. Assue that the inverse arket deand function for fir i is Pi a qi γq, i, (7 here γ [0,] shos the degree of product differentiation. γ 0 iplies that the products are isolated, hile γ iplies that the products are hoogeneous.... Unifor age setting Let us first consider the situation here the centralized labor union sets a unifor age rate for both firs. Given the deand specification and the unifor age rate, the equilibriu outputs of firs and are respectively ( a( γ + γλ ( a( γ λ + γ q and q. (4 γ (4 γ i n i k k k + λ 6

8 The centralized labor union deterines the unifor age rate by axiizing the folloing expression: 9 ( a( γ + γλ + λ( a( γ λ + γ Max. (8 (4 γ ( γ ( + λ The equilibriu age rate is a. We also find that 0. 4( + λ λγ γ Hence the folloing proposition is iediate. Proposition 4: If the firs differ in labor productivities and a centralized labor union charges unifor age rate to the firs, the equilibriu age rate depends on the degree of product differentiation. As the degree of product differentiation increases (i.e., γ falls, the equilibriu age rate increases.... The intuition of this result is siilar to the intuition provided in subsection... Wage discriination Let us no consider the situation here the labor union discriinates age beteen the firs. Given that the labor union charges and to firs and respectively, ( a( γ + γλ the equilibriu outputs of firs and are respectively q (4 γ ( a( γ λ + γ and q. (4 γ The labor union deterines the age rates by axiizing the folloing expression: ( a( γ + γλ + λ ( a( γ λ + γ Max, (4 γ. (9 a a The equilibriu age rates are and. λ Hence, the proposition is iediate. Proposition 5: If a centralized labor union can discriinate age beteen the firs, the equilibriu age rates are independent of the degree of product differentiation even if the firs differ in labor productivities.... The intuition of this result is siilar to the intuition provided in subsection 3. Conclusion Considering Cournot copetition, this note shos that if the firs differ in labor productivities and the centralized labor union charges a unifor age rate to the firs, the age rigidity result of Dhillon and Petrakis (00 does not hold. The asyetry beteen the firs does not satisfy that the fir s output and profit is loglinear in the age rate and the arket features such as the nuber of firs and 9 The qualification ade in footnote 5 also holds for the subsection.. 7

9 product differentiation. As a result, the equilibriu age rate depends on the nuber of firs and product differentiation. Hoever, if the centralized labor union can discriinate age beteen the firs, the age rigidity result of Dhillon and Petrakis (00 holds even if the firs differ in labor productivities. In this situation, the fir s on age rate and the arket features such as the nuber of firs and product differentiation behave like a loglinear function in the fir s equilibriu output and profit. Hence, hether the age rigidity result of Dhillon and Petrakis (00 extends to the case of asyetric firs depends on the age setting behavior of the labor union. References Dhillon, A. and E. Petrakis, 00, A generalized age rigidity result, International Journal of Industrial Organization, 0: Dunlop, J. T., 944, Wage deterination under trade union, Ne York, Macillan. Greenhut, M. L. and H. Ohta, 976, Related arket conditions and interindustrial ergers, The Aerican Econoic Revie, 66: Haucap, J., U. Pauly and C. Wey, 000, The incentives of eployers associations to raise rivals costs in the presence of collective bargaining, in de Gisel, Olthoff and Zick (Eds., The uneployent debate: current issues, 85 6, Marburg, Metropolis Verlag. Haucap, J., U. Pauly and C. Wey, 00, Collective age setting hen ages generally are generally binding: an antitrust perspective, International Revie of La and Econoics, : Haucap, J. and C. Wey, 004, Unionisation structures and innovation incentives, Econoic Journal, 4: C Layard, R., S. Nickell and R. Jackan, 99, Uneployent, acroeconoic perforance and the labour arket, Oxford University Press, Oxford. Osald, A. J., 98, The icroeconoic theory of the trade union, Econoic Journal, 9: Tyagi, R. K., 999, On the effects of donstrea entry, Manageent Science, 45:

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