UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics. Note on a generalized wage rigidity result

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1 UNIVERSITY OF NOTTINGHAM Discussion Papers in Economics Discussion Paper No. 07/04 Note on a generalized age rigidity result By Arijit Mukherjee June DP 07/04

2 Note on a generalized age rigidity result Arijit Mukherjee University of Nottingham and The Leverhulme Centre for Research in Globalisation and Economic Policy, UK June 007 Abstract: Considering Cournot competition, this note shos that, if the firms differ in labor productivities, the equilibrium age rates under a centralized labor union are not independent of the number of firms and product differentiation if the labor union charges a uniform age rate. Hoever, if the centralized labor union can discriminate age rate beteen the firms, the equilibrium age rates do not depend on the number of firms and product differentiation. Hence, the age rigidity result of Dhillon and Petrakis (00) may not be robust ith respect to the asymmetry in firms, hich is perhaps more common than symmetry considered by them. The effects of the number of firms and product differentiation on the equilibrium age rate are also shon. Key Words: Asymmetry; Labor productivity; Labor union; Wage rigidity JEL Classification: J3; J5 Correspondence to: Arijit Mukherjee, School of Economics, University of Nottingham, University park, Nottingham, NG7 RD, UK arijit.mukherjee@nottingham.ac.uk Fax:

3 Note on a generalized age rigidity result. Introduction The empirically observed age rigidity has attracted attention of the researchers orking on Industrial Organization and Labor Economics. Recently, Dhillon and Petrakis (00) sho that, under fairly general conditions, if there is a centralized labor union, the equilibrium ag rates do not depend on the number of firms and product differentiation. Considering Cournot competition, this note extends this line of research ith asymmetry in firms. We sho that if the firms differ in labor productivities, the equilibrium age rates under a centralized labor union are not independent of the number of firms and product differentiation if the labor union charges a uniform age rate. Hoever, if the centralized labor union can discriminate age rate beteen the firms, the equilibrium age rates do not depend of the number of firms and product differentiation. Hence, the age rigidity result of Dhillon and Petrakis (00) may not be robust ith respect to the asymmetry in firms, hich is perhaps more common than symmetry considered by them. It is ell-knon that an upstream agent ill prefer price discrimination over uniform pricing if there are differences in the donstream agents (Yoshida, 000). Hence, it is arguable that a labor union may prefer to charge different ages to different firms if there is asymmetry beteen the firms. Hoever, empirical evidences suggest that in many situations a labor union charges uniform age rate irrespective of the differences beteen the firms. As discussed in Haucap et al. (000 and 00), a The irrelevance of the number of donstream firms on the upstream input price can also be found in the earlier orks by Greenhut and Ohta (976) and Tyagi (999).

4 common feature of many labor markets in continental Europe is coverage extension rules, hich implies that some or all employment terms are made generally binding for all industry participants and not only for the members of unions and employers associations. In Germany, for example, collective age agreements beteen a union and an employers association can be made compulsory even for independent employers through so-called Allgemeinverbindlicherklärung (AVE) The Ministry of Labor can, on application of either unions or employers associations, use an AVE to make some or all terms of a collectively negotiated employment contract generally binding for an entire industry, here otherise only those unions, employers and employers 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 number of AVEs almost continuously increased from 448 in 975 to 588 in 998. Thus, it justifies our analysis ith uniform age setting by a centralized labor union. 3 The remainder of the paper is organized as follos. The next section describes the model and shos the results. Section 3 concludes.. The model and the results Since, in presence of asymmetry beteen the firms, the calculations for shoing the effects of the number of firms and product differentiation are cumbersome, e sho these effects separately. In section., e consider the case of a homogeneous Haucap et al. (00) also sho hen the labor union may prefer a uniform age over discriminatory age. 3 The age bargaining by the labor union for the UK Universities may also support the uniform age setting by a centralized labor union in presence of productivity differences. While the people orking in different Universities may differ in productivities, the national labor union bargains for similar age rates for all the UK Universities.

5 product and sho the effects of the number of firms. In section., e consider a duopoly market structure and sho the effects of product differentiation... The effects of the number of firms Let us consider an economy ith ( n + m) firms producing a homogeneous product. Assume that production requires only labor. For notational convenience arrange the firms as,,... n, n +, n +,..., n + m. Without loss of generality, assume that the each of the firms in [, n ] requires one labor to produce one unit of output, hile each of the firms in [ n +, n + m] requires λ orkers to produce one union of output, here λ. Hence, e consider asymmetry in labor productivities of the firms. Also, to make asymmetry meaningful in our analysis, assume that n and m. Assume that the inverse market demand function for the product is P a q, () here the notations have usual meanings. To provide counter examples to Dhillon and Petrakis (00), it ill be enough to focus on a particular demand function. We assume that there is a centralized labor union that sets the age rates for the firms. We ill consider to possibilities: (i) here the labor union sets a uniform age rate for all firms, and (ii) here the labor union can charge different age rates to different firms. Folloing Haucap and Wey (004), e can call the former age setting behavior as centralization and the latter as coordination. As a simplification, e assume that the reservation age rates of the labors are zero. We consider the folloing game. At stage, the labor union sets the age rates. At stage, the firms produce like Cournot oligopolists and the profits are realized. We solve the game through backard induction. 3

6 Hence, e consider the case of a monopoly 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 equilibrium age rates can depend on the number of firms and product differentiation in presence of asymmetric firms, it is enough for us to consider a monopoly labor union. Hoever, note that our qualitative results holds even if there is bargaining beteen the labor union and the firms. Bargaining beteen the labor union and the firms ill only complicate the calculations ithout adding much to the main purpose of the paper. Further, to make our point, e concentrate on the right-to-manage model of labor union, hich is perhaps the most idely used model of labor union in the Industrial Organization literature Uniform age setting by the labor union In this subsection e assume that the labor union charges a uniform age rate to all firms. Given the demand function and the uniform age rate, the equilibrium output of each of the firms in [, n ] is q i ( a ( m λm + )), and the equilibrium ( n + m + ) output of each of the firms in [ n +, n + m] is q j ( a ( λ + λn n)). ( n + m + ) the age rate: 5 Therefore, the labor union maximizes the folloing expression to determine 4 See, Layard et al. (99) for arguments in favor of right-to-manage models. 5 It is important to note that, due to asymmetry among the firms, the labor union may not ish to serve all the firms. Hence, the labor union may charge a sufficiently high age rate that encourages only the relatively efficient firms to hire labors and producing the product. Hoever, it can be shon that the labor union prefers to serve all the firms if λ is not sufficiently different from. Since, the age determination in presence of asymmetry is the main element of this paper, e do our analysis under the assumption that λ is sufficiently close to so that the labor union serves all the firms and solves the maximization problem (). 4

7 Max n( a ( m λm + )) + λm( a ( λ + λn n)). () ( n + m + ) The equilibrium age rate is a( n + λm), (3) ( n( + m) + λm( λ + λn n)) hich clearly shos that equilibrium age rate depends on the number of firms. Note that is independent of the number of firms if either λ or n 0 or m 0. Hence, the folloing proposition is immediate. Proposition : If there is a centralized labor union that sets a uniform age rate for all firms, the equilibrium age rate depends on the number of firms if the firms differ in labor productivities. The reason for the above result follos easily from Dhillon and Petrakis (00). They sho that, if the firm s equilibrium output and profit are log-linear in the age rate and the market features such as the number of firms and product differentiation, the equilibrium age rate is independent of the market features. If the firms are asymmetric in labor productivities, it is immediate from the above analysis that the firm s equilibrium output and the profit 6 is not log-linear in the age rate and the number of firms, and therefore, the equilibrium age rate depends on the number of firms. It is important to note that, like Dhillon and Petrakis (00), e have also considered that the production technologies of the firms are log-linear and the union utility is log-linear in age rate and aggregate employment. Hoever, the asymmetry beteen the firms does not satisfy that the firm s equilibrium output and the profit are log-linear in the age rate and the number of firms. 5

8 Let us no see the effects of the number of firms on the equilibrium age rate. Proposition : (i) Assume λ > 0. If λ, e get 0 n. Hoever, if λ, e get 0 n for ( λ ) > m. 7 (ii) If λ >, e get 0. Hoever, if λ, e get 0 m m for λ ( λ) n. 8 Proof: (i) We find from (3) that if λ > 0 and λ, then 0 n provided Hence, if m. (4) ( λ ) λ, e get 0 n, since m. Hoever, if λ >, e get > ( λ ) provided λ. Hence, for λ, e get 0 n for m. ( λ ) > (ii) We find from (3) that if λ, then 0 m provided 6 It is easy to check that the profit of the i th firm is 6 q i, i,,..., n + m. 7 Though, it ill be immediate from the proof of this proposition that 0 if λ >, e ill not n focus on the higher values of λ, since as mentioned in footnote 5, the labor union ould serve only the relatively efficient firm for higher values of λ. For example, it is easy to sho that if λ >, n m and the centralized labor union charges a uniform age rate to all firms, then the labor union prefers to serve both firms instead of serving only the firm ith higher labor productivity provided λ. When n, m >, the relevant values of λ for hich the labor union serves all the firms are less than λ ( n) that is loer than. So, for n, m >, e need to restrict λ beteen and λ ( ). n 8 Though, it ill be immediate that 0 for m attention to λ >. λ, folloing footnotes 5 and 7, e restrict our

9 λ n. (5) (λ ) Hence, if > λ, e get 0 m. Hoever, if λ λ, e get that > ( λ) provided λ >. Hence, for λ, e get 0 m for λ ( λ) n. Q.E.D. The reason for the above result is easy to understand. The presence of asymmetric firms makes the total labor demand curve as a kinked function, and the kink occurs at that age rate here the firms 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 firms in [, n ] are relatively inefficient, and none of them demand labor if a > ( m λm + ). An increase in n implies that the number of inefficient firms increases, hich makes the segment of the labor demand curve, here all firms find production profitable, more elastic, but does not affect. This is shon in Figure. Figure Assume that the total labor demand curve for a given λ, n and m is given by the kinked curve ABC. The age rate is the age rate at hich the labor demand by the loer productive firms is zero. The labor demand is coming from all the firms on the segment BC, hile, on the segment AB, the labor demand is coming only from the firms ith higher labor productivities, i.e., from the firms in [ n +, n + m]. No, if n increases, it rotates BC to the equilibrium age rate reduces ith n. BC ', and makes this segment more elastic. As a result, 7

10 Hoever, if λ >, then, the firms in [, n ] are relatively efficient, and these firms stop producing if a > ( λ + λn n) '. If n increases, it shifts the segment of the labor demand curve above ' outard. Hoever, higher n also implies that ' falls, since, no the inefficient firms are facing competition from more efficient firms. Hence, in this situation, higher n not only shifts the labor demand curve outard, it also reduces the critical level of the age rate at hich kink occurs. This is shon in Figure. Figure Assume that the labor demand curve for a given λ >, n and m is given by the kinked curve MNP. The age rate ' is the age rate at hich the labor demand by the loer productive firms is zero. No, if n increases, it shifts the labor demand curve to MN ' P', and the kink occurs at ". So, hile the outard shift of the labor demand curve tends to increase the equilibrium age rate, the fall of ' to " tends to reduce the equilibrium age rate. Hence, the net effect of a change in n depends on the number of inefficient firms and the labor productivities of the inefficient firms. Similar argument follos for the case of Proposition (ii).... Wage discrimination We have shon that the age rigidity result of Dhillon and Petrakis (00) does not hold if the centralized labor union sets a uniform age rate and the firms differ in labor productivities. Hence, the previous section provides a counter example to their results hen the centralized labor union needs to charge a uniform age rate to all firms (may be due to the institutional set up as mentioned in the introduction). 8

11 In this section e sho that if the centralized union keeps the flexibility of charging different age rates to different firms, the age rigidity result of Dhillon and Petrakis (00) holds in presence of the asymmetric firms. Let us no consider age discrimination beteen the firms. Given the age rates, the equilibrium output of each of the firms in [, n ] is q i ( a ( n + m) + i n k i k k ( n + m + ) + λ n+ m j n+ j ) and the equilibrium output of each of the firms in [ n +, n + m] is q j ( a λ( n + m) j + n i ( n + m + ) i + λ n+ m s n+ j s s ). Therefore, the labor union maximizes the folloing expression to determine the age rates for each firm: n n+ m iqi + λ jq i j n+ Max,...,,,..., + ( n + m n n + ) + n m j. (6) The equilibrium age rates are a i, i,..., n, and a j, j n +,..., n + m. λ Hence, the folloing proposition is immediate. Proposition 3: If a centralized labor union can discriminate age rate beteen the firms, the equilibrium age rates are independent of the number of firms even if the firms differ in labor productivities. If the union discriminates age rate beteen the firms, it considers the labor demand of different firms separately. Hence, it is important to see hether, in the firm s output and the profit, the firm s on age rate and market features such as the 9

12 number of firms behave like a log-linear function. It follos from the above analysis that, in the firm s equilibrium output and profit, the firm s on age rate and the number of firms behave like a log-linear function. As a result, the equilibrium age rate of each firm is independent of the number of firms... The effects of product differentiation Let us no consider a horizontally differentiated duopoly market structure. Assume that firm requires one labor to produce one unit of output, and firm requires λ labors to produce one unit of output. Like section., e assume that the reservation age rates for the labors are zero. Assume that the inverse market demand function for firm i is P i a q γq, i j, (7) i j here γ [0,] shos the degree of product differentiation. γ 0 implies that the products are isolated, hile γ implies that the products are homogeneous.... Uniform age setting Let us first consider the situation here the centralized labor union sets a uniform age rate for both firms. Given the demand specification and the uniform age rate, the equilibrium outputs of firm and firm are respectively ( a( γ ) + γλ) q and (4 γ ) ( a( γ ) λ + γ) q. (4 γ ) The centralized labor union determines the uniform age rate by maximizing the folloing expression: 9 Max ( a( γ ) + γλ) + λ( a( γ ) λ + γ). (8) (4 γ ) 0

13 The equilibrium age rate is ( γ )( + λ) a. We also find that 0. 4( + λ λγ ) γ Hence the folloing proposition is immediate. Proposition 4: If the firms differ in labor productivities and a centralized labor union charges uniform age rate to the firms, the equilibrium age rate depends on the degree of product differentiation. As the degree of product differentiation increases (i.e., γ falls), the equilibrium age rate increases.... The intuition of this result is similar to the intuition provided in subsection... Wage discrimination Let us no consider the situation here the labor union discriminates age beteen the firms. Given that the labor union charges and to firms and respectively, the equilibrium outputs of firms and are respectively q ( a( γ ) + γλ (4 γ ) ) and ( a( γ ) λ + γ ) q. (4 γ ) The labor union determines the age rates by maximizing the folloing expression: ( a( γ ) + γλ ) + λ ( a( γ ) λ Max, (4 γ ) + γ ). (9) The equilibrium age rates are a and a. λ Hence, the proposition is immediate. 9 The qualification made in footnote 5 also holds for the subsection..

14 Proposition 5: If a centralized labor union can discriminate age beteen the firms, the equilibrium age rates are independent of the degree of product differentiation even if the firms differ in labor productivities.... The intuition of this result is similar to the intuition provided in subsection 3. Conclusion Considering Cournot competition, this note shos that if the firms differ in labor productivities and the centralized labor union needs to charge a uniform age rate to the firms, the age rigidity result of Dhillon and Petrakis (00) does not hold. The asymmetry beteen the firms does not satisfy that the firm s output and profit is loglinear in the age rate and the market features such as the number of firms and product differentiation. As a result, the equilibrium age rate depends on the number of firms and product differentiation. Hence, the age rigidity result of Dhillon and Petrakis (00) may not be robust ith respect to the asymmetry in firms, hich is perhaps more common than symmetry considered by them. Hoever, if the centralized labor union can discriminate age beteen the firms, the age rigidity result of Dhillon and Petrakis (00) holds even if the firms differ in labor productivities. In this situation, in the firm s equilibrium output and profit, the firm s on age rate and the market features such as the number of firms and product differentiation behave like a log-linear function.

15 References Dhillon, A. and E. Petrakis, 00, A generalized age rigidity result, International Journal of Industrial Organization, 0: Dunlop, J. T., 944, Wage determination under trade union, Ne York, Macmillan. Greenhut, M. L. and H. Ohta, 976, Related market conditions and interindustrial mergers, The American Economic Revie, 66: Haucap, J., U. Pauly and C. Wey, 000, The incentives of employers associations to raise rivals costs in the presence of collective bargaining, in de Gijsel, Olthoff and Zick (Eds.), The unemployment 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 Economics, : Haucap, J. and C. Wey, 004, Unionisation structures and innovation incentives, Economic Journal, 4: C Layard, R., S. Nickell and R. Jackman, 99, Unemployment, macroeconomic performance and the labour market, Oxford University Press, Oxford. Osald, A. J., 98, The microeconomic theory of the trade union, Economic Journal, 9: Tyagi, R. K., 999, On the effects of donstream entry, Management Science, 45: Yoshida, Y., 000, Third-degree price discrimination in input market, The American Economic Revie, 90:

16 Figure : The effect of a higher n hen λ. Figure : The effect of a higher n hen λ >. 4