RESEARCH PAPER SERIES ENTERPRISE BEHAVIOR AND ECONOMIC REFORMS: A COMPARATIVE STUDY IN CENTRAL AND EASTERN EUROPE AND INDUSTRIAL REFORM AND PRODUCTIVITY IN CHINESE ENTERPRISES RESEARCH PROJECTS OF THE WORLD BANK 21950 March 1993 EAST EUROPE AND FORMER SOVIET UNION NUMBER EE-RPS #27 SEPTEMBER 1991 (REVISED MARCH 1993) Worker Power, Surplus Sharing and the Wage- Employment Outcome in Transitional Economies Frans Spinnewyn Catholic University of Leuven and CORE and Jan Svejnar University of Pittsburgh and CERGE-EI, Prague Transition and ,Macro Adjustment Division Policy Research Department World Bank Washington, D.C. CONTENTS Acknowledgem ent ........................................i I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. The M odel .................................... ...... 4 The Coordination of Insiders' Objectives ..................... 6 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Technology ....................................... 9 III. A Union Monopoly Model of Bargaining: The Non-Cooperative Equilibrium .............................10 The Setting of the Average Wage and Employment ............... 10 Models of Labor-Management Bargaining: Cooperative Equilibria ................................16 Nash Bargaining Solutions ..............................16 Management v. the EntireUnion ..........................20 Management v. the Core of the Union ....................... 23 IV. Concluding Remarks ....................................26 A ppendix A ..........................................27 ACKNOWLEDGEMENT The research projects on "Enterprise Behavior and Economic Reforms: A Comparative Study in Central and Eastern Europe", and "Industrial Reforms and Productivity in Chinese Enterprises" are research initiatives of the Transition and Macro Adjustment Division (PRDTM) of the World Bank's Policy Research Department and managed by I.J. Singh, Lead Economist. These projects are being undertaken in collaboration with the following institutions: for the project in China, The Institute of Economics of the Chinese Academy of Social Sciences (IE of CASS), The Research Center for Rural Development of the State Council (RCRD), and The Economic Systems Reform Institute (ESRI), all in Beijing; and for the projects in Central and Eastern Europe The London Business School (LBS); R6forme et Ouvertures des Systemes Economiques (post) Socialistes (ROSES) at the University of Paris; Centro de Estudos Aplicados da Universidade Cat6lica Portuguesa (UCP) in Lisbon; The Czech Management Center (CMC) at delikovice, Czech Republic; The Research Institute of Industrial Economics of the Janus Pannonius University, Peds (RHE) in Budapest, Hungary; and the Department of Economics at the University of L6di, in Poland. The research projects are supported with funds generously provided by: The World Bank Research Committee; The Japanese Grant Facility; The Portuguese Ministry of Industry and Energy; The Ministry of Research and Space; The Ministry of Industry and Foreign Trade, and General Office of Planning in France; and the United States Agency for International Development. The Research Paper Series disseminates preliminary findings of work in progress and promotes the exchange of ideas among researchers and others interested in the area. The papers contain the views, conclusions, and interpretations of the author(s) and should not be attributed to the World Bank, its Board of Directors, its management or any of its member countries, or the sponsoring institutions or their affiliated agencies. Due to the informality of this series and to make the publication available with the least possible delay, the papers have not been fully edited, and the World Bank accepts no responsibility for errors. The authors welcome any comments and suggestions. Request for permission to quote their contents should be addressed directly to the author(s). For additional copies, please contact the Transition and Macro Adjustment Division, room N-11065, World Bank, 1818 H Street, N.W., Washington, D.C. 20043, telephone (202) 473-1442, fax (202) 676-0083 or 676-0439. The series is also possible thanks to the contributions of Donna Schaller, Vesna Petrovic, Cecilia Guido-Spano and the leadership of Alan Gelb. i I. INTRODUCTION There is now a large literature dealing with wage-employment outcomes in unionized and labor-managed firms. Some, though not all, strands of this literature argue that worker power leads to inefficient allocation of resources. In the context of the transitional economies, some writers (notably Manuel Hinds, 1991) have cogently argued that, as central controls are lifted, workers are able to exert significant control over enterprise decisionmaking and affect negatively the economic performance of firms. More generally, authors such as Manuel Hinds point to the economies of Poland, Hungary and former Yugoslavia and attribute their macroeconomic problems in part to workers' power in enterprises. The relevance of these claims has been enhanced in recent years as free trade unions have been established in most transitional economies. With unemployment rising to a two digit level in most transitional economies, the distinction between worker-insiders, who are insulated and codetermine the policies of the firms, and worker outsiders, who would like to enter the firms, should become a focal point of the analysis. In this paper, we re-examine the theoretical premises underlying the literature on unions and labor-managed firms and draw conclusions with respect to the transitional economies. The traditional view has been that unions raise the wage above the competitive level and employers reduce employment so as to achieve the highest profit, given the higher wage. In terms of Figure 1, the union moves the outcome from point A, corresponding to a competitive wage W* and employment L*, to point B on the marginal product curve of labor. A revisionist view, pioneered by Leontief,(1946) and resurrected by McDonald and Solow (1981) holds that outcomes on labor's marginal product curve above point A are inefficient because of unexploited gains from trade. This view maintains that for all wages above W* an efficient outcome lies to the right of labor's marginal product curve at a tangency of a downward sloping union indifference curve and a management iso-profit curve (e.g. point C). The traditional view rebuts that efficient outcomes at points such as C are unsustainable because in practice the management is only bound by its agreement on wages rather than both wages and employment. If the management agreed to wage Wc, corresponding to point C, it would move horizontally leftward to the marginal product curve of labor in order to increase its profit at the given wage. The revisionists' response has been that work (feather bedding) rules might reduce the management's ability to cut down on employment. The debate has remained open and become more intense in recent years. In examining the vast literature, it is striking that most models are static, assuming that union membership is exogenously given at a hif h level (e.g., L1 in Figure 1) and that all equally productive workers receive the same wage. Moreover, the few studies that have 1 See for instance MacDonald and Solow (1981), Brown and Ashenfelter (1986), Card (1986), MaCurdy and Pencavel (1986) and Svejnar (1986). 1 2 Worker Power, Surplus Sharing and the Wage-Employment Outcome allowed for an endogenously determined membership have not tackled the issue of surplus sharing in a systematic dynamic framework.2 Recently, the weakness of the assumption of fixed membership has been dealt with in an influential set of writings by Layard and Nickell (1990) and Layard, Nickell and Jackman (1991). These authors have taken into account attrition in union membership and argued that all steady state outcomes have to lie on labor's marginal revenue product curve. In terms of Figure 1, their insight is that union indifference curves are horizontal to the right of the membership point and that attrition results in the membership shifting leftward. The process continues until the kink in the indifference curve reaches labor's marginal product curve at the given wage (e.g., point D). Since with further attrition the tangency between the union indifference curve and the firm's iso-profit curve occurs in the horizontal region of both curves at point D, the authors conclude that in the steady state the "contract curve lies (locally) along the demand curve," ... "the union is happy to allow employment to be determined by the firm," ...the union "behaves as if it only cares about wages," ... and "hiring occurs."3 Our point of departure is the realization that, given attrition, the above analysis fails to explain why the remaining insiders do not try to restrict new hires and move up the demand curve from point D in order to raise their wages. Moreover, the literature fails to explain why the newly hired workers should receive the same wage as the insiders within a consistent dynamic framework. Take the widely used union monopoly model, with the union setting the wage and management employment.4 As attrition brings about full employment of the union membership on labor's marginal product curve, the union will find it desirable to raise the wage along the marginal product curve with further attrition. In fact, assuming that the firm would go out of business with negative profits, the union will try to raise the wage along the marginal product curve of labor until profit is zero and the wage equals labor's net average product (point E, with employment Lmin). This is the case of Cartter's (195 ) "all powerful union" or Ward's (1958) labor-managed firm. Further attrition beyond 1nin would force the shrinking membership to move leftward along the zero profit curve and accept lower wages. This provides an incentive to bring in new members to restore Ivin- In fact, a stronger point can be made. Union members receiving a uniform wage and facing equal probability of layoffs will not push excessively for higher wages. However, 2 See e.g. Grossman (1983), Booth (1984), Ulph and Ulph (1982, 1983), Grout (1983), Kidd and Oswald (1983), and Spinnewyn and Svejnar (1990). 3 Layard, Nickell and Jackman (1991, pp.114-115). 4 The argument with respect to a right-to-manage model, where the union bargains with management over the wage and management then sets employment, is a little more complicated but similar. Worker Power, Surplus Sharing and the Wage-Employment Outcome 3 suppose layoffs are carried out in the inverse order of seniority and a core Lin of the most senior members has the decisionmaking power.5 These senior members have an incentive to drive the uniform wage up to the zero profit point on the demand curve (point E) without waiting for attrition. As this discussion implies, outcomes to the right of Lin might be feasible if wage differentials across different generations of equally skilled and productive workers were permitted to occur. Indeed, the assumption of a uniform wage generates unexploited gains from trade between worker-insiders and worker-outsiders since outsiders would be willing to join the firm at a wage that falls short of the insider wage. The insiders would in turn want to admit lower paid workers if their contribution to the total wage bill exceeded their own wage bill, since this would increase the insiders' wage. In this paper we address these issues in a consistent dynamic framework. We allow for endogenous membership and the realization of gains from trade between the insiders and the firm on the one hand and the insiders and incoming outsiders on the other hand. Undertaking a dynamic analysis requires us to recast the static models of union monopoly, right-to-manage and efficient bargaining in the dynamic context. We show that the outcomes depend on institutional settings and we present results for three main settings: (a) the noncooperative case when the union has an effective control over the admission of new members (the closed shop), (b) the noncooperative case when the management controls the inflow of workers and hence union membership (the open or union shop),6 and (c) the cooperative case of joint union-management determination of wages and employment (efficient bargaining), with noncooperative equilibria serving as threat points. In all cases we allow for competition among outsiders, which results in the dissipation of rents and equal lifetime income in the unionized and nonunionized sectors in equilibrium. Even with equal lifetime incomes in the two sectors, in the noncooperative case employment in the unionized sector is below the level in an otherwise identical non-unionized sector and the outcome lies on labor's marginal product curve. Employment expands along the marginal product curve as one moves from the closed to open shop. Within the cooperative framework, contrary to the claim by Layard, Nickell and Jackman (1991), even with attrition the parties prefer cooperative outcomes which lie to the right of the marginal product of labor curve. The interests of the union and the management are congruent and in many cases it does not matter which of the players has the right to manage the firm. In these cases the employment level coincides with the one of the non- unionized sector. 5 A similar argument can be constructed for another core decisionmaking group of senior insiders. 6 Whether the insiders entering the firm formally become union members is unimportant for our analysis so long as they are covered equally with union members by the seniority scheme. 4 Worker Power, Surplus Snaring and the Wage-Employment Outcome The dynamic model reveals a particular effect of the threat of unionization on wage differentials. As we pointed out earlier, if a firm paid its workers a competitive wage over their lifetime, it would be in the interest of the insiders to increase their wage by imposing a wage differential (an increasing wage profile) on the incoming generations. In this sense the first generation of insiders could reap lifetime rents. In the expectation of this behavior, the firm will have to impose the steep lifetime wage profile on its work force in order to prevent this rent seeking activity. The format of the paper is as follows. In Section 2 we show that fixing wage differentials coordinates the interests of insiders and the newly admitted outsiders and we analyze its distributive effects. In Section 3 we present the results of a non-cooperative dynamic union-monopoly model of bargaining between the union and the management both for the closed and open union shop. In Section 4 we analyze the possibilities of cooperation in determining the wage bill when the non-cooperative outcome is used for deriving the disagreement payoffs should the cooperation fail. Conclusions are drawn in Section 5. II. THE MODEL We start by defining the actors and their instruments (control variables), identifying instruments that can be committed within a contract, and outlining the sequencing of decisions. With the management, insiders and outsiders being the actors, the control variables are g(t)dt = the wage differential between class t and t + dt of insiders, w = the average wage of all the insiders employed in the firm, I = the admission rate of the outsiders into the union (the hiring rate), and Le = the employment level in the firm. The union monopoly model is traditionally defined as one in which the union selects w and the firm responds by choosing Le. In the dynamic framework one also has to specify the rules about who selects the wage differential g and the hiring rate f. We consider two polar cases. In one, which we call the closed shop, g and f are determined by the union. In the other one, called open shop, they are determined by the management. In both cases the union selects w and management Le. Upon being hired, an outsider receives an open ended contract that specifies his wage conditional on employment, the wage differential vis a vis the more senior incumbents when employed, and the inverse seniority clause with respect to layoffs. As will become evident, in the dynamic framework the difference between the unionized and nonunionized cases depends strongly on the presence of the seniority clause. At each time t, the production and wage payment are preceded by the following sequence of decisions (steps): 1) The management in the open shop and the union in the closed shop set the wage differential g and the hiring rate f. Worker Power, Surplus Sharing and the Wage-Employment Outcome 5 2) The outsiders who were offered a contract in step 1 either accept it and acquire seniority rights or reject it. 3) The union determines the average wage w. 4) The management determines employment Le- At each step of the sequence, the relevant decisionmaker selects the optimal strategy conditional on the decisions made in previous stages. As a result, in the earlier stages the decisionmakers take into account the effect of their decisions on those made later. The union for instance knows that by setting the average wage w in step 3 it will influence the employment level Le in management's profit maximizing calculations in step 4. In the open shop, it may be unprofitable for the firm in step 4 to employ all the hired workers if the union sets a high average wage. Futhermore, when the wage differential g is low, the employed insiders' wages are increased by increasing the average wage w above its full employment level. But then, anticipating unemployment, the outsiders may be unwilling to accept an offer in step 2. Changing the allocation of instruments or the sequence of decisions naturally changes the outcome. For instance, if the union could set the wage differential but not the hiring rate, or if the union could commit itself to an average wage before the contracts were offered, then the management would lose most of the control it derives from an open shop. For the sake of maximizing insight while maintaining brevity, we present the two polar cases outlined above. In addition, we show that when a fifth step is introduced, allowing for a dynamic cooperative solution, the effect of the first four steps may be overturned and an efficient outcome is possible. We consider a stationary economic environment with full employment in the economy and the nonunionized sector characterized by a constant wage 1 over a worker's lifetime. We assume that workers compete for union and nonunion jobs in a frictionless environment, and in equilibrium they hence become indifferent between employment in either sector and voluntary unemployment as competitive forces eliminate any rents. As a result, a person's utility in unemployment and while employed in the nonunionized sector are identical. If incoming outsiders expect lifetime employment in the unionized sector, the discounted lifetime incomes in the unionized and nonunionized sectors are also equal. With seniority induced wage growth in the unionized sector, the starting union wage is lower than the competitive one but it becomes higher later in one's lifetime. Should an insider be unexpectedly laid off, he would expect to receive his rent and would hence wait to be recalled. This waiting behavior is justified for two reasons: (a) with the seniority clause, laid off insiders must be recalled before the firm employs outsiders and (b) because of the agreed upon wage differentials, the laid off insiders with positive seniority must have a larger remaining lifetime income than outsiders who will join the firm. 6 Worker Power, Surplus Sharing and the Wage-Employment Outcome The equilibria which we consider below are hence characterized by the following properties: [P1] Laid off insiders wait to be recalled. [P2] Outsiders accept offers from unionized firms only if they anticipate immediate employment. [Pl] also applies to outsiders who have just acquired seniority by accepting an offer in step 2. Because of competition, outsiders joining the firm will not suffer a loss in lifetime utility if they are not immediately employed and they are thus indifferent between membership and continuing search. [P2] simply provides a tie breaker in this case. A. The Coordination of Insiders' Objectives Let yt(c) be the wage at time t of workers who were admitted into the unionized firm at time c, r be the individual discount rate and m the uniform, exogenously given attrition rate. Moreover, let members of any seniority class c who expect lifetime employment have access to perfect capital markets of the life insurance type, so that a given member's lifetime utility is an increasing function of the average discounted wage stream 4t(c): Let Lx be the number of members admitted between the founding date of the firm b and time cx and surviving at t: CZ L(t) = fe -m(-c)9(c)dc + e (2) b bsc{scz ci , i EI, are the points in time when a discontinuity in membership (e.g., lumpy hiring) occurs, I+ is the jump in the membership at such points and fx + < f + is the number belonging to Lx. By seniority, fx+ (ci )= + (ci +) if ci+ t. As a result, fixing the wage differentials will ensure that the union's intertemporal optimizing strategy is subgame perfect, despite the fact that the union's composition changes over time. B. Feasibility The maximization of the common objective 4(t) by all the generations of insiders with lifetime employment is subject to a feasibility constraint. In particular, in the context of equation (3), feasibility requires that at any given moment in time the base wage and the wage differential be chosen so that the sum of the wages paid out to all the generations of employed insiders equals the wage bill o: Ct) q () fe n(`c)f(c)y,(c)dc + e AKt-c,*)i (cy- c (10) b bIC,SC,(T) Worker Power, Surplus Sharing and the Wage-Employment Outcome 9 Upon substitution of equation (3) into (10) one obtains a (t) =1 e-m(t-c)G(c,c(t))P(c)dc + (e1)(t-c)G(c ,tc ) b bic Pc+ ) Note that 1/a is the seniority weighted membership, when the base class is weighted by 1.0. It follows upon differentiation of (11) that for ce(t) # ci + a = [m - (g(Ce) - ap(Ce))]a, (12) where dce =1 if the hiring rate is positive. In equation (12), income is released by attrition at the rate m, while through the wage differential income is appropriated by the insiders at the rate g and shared with the incoming outsiders at the rate at. In the stationary state a is constant and m = g + at: the rate at which income is released by attrition equals the rate at which it is appropriated by the surviving insiders and outsiders. Then, from (8), t = mL and a = [m-g]/mL it follows that for G,=1 r+m m-g -r (13) m r+m-g L C. Technology To keep technological issues simple, we consider a Cobb-Douglas production function and a fixed cost. In particular, with employment Le, the value added to be distributed among profits and wages can be defined as Le -H, a Cobb-Douglas function of employment with a fixed cost H incurred with positive production and all other fixed costs being sunk. In what follows, we compare the unionized firm with an otherwise identical nonunionized firm paying wage 0. Since 0 is both the competitive wage and the output elasticity of labor in the production function, it follows that employment in the non-unionized firm is measured so that it equals 1 and the fixed cost satisfies H = (1-fl)Lmin . Hence, as in the literature on labor- managed firms, the fixed cost plays an important part in determining the scale of operation of the firm. In order to derive sharp results, we invoke the following two assumptions: [Al] The core members Li have decisionmaking power. 7 See e.g., Svejnar (1982). 10 Worker Power, Surplus Sharing and the Wage-Employment Outcome From the assumption it follows that the core members of the union have the power to set the average wage w at any level they wish (e.g., so that only core members are employed). With a majority voting scheme, for example, the core members have decisionmaking power only if the fixed cost is sufficiently high so that Lmin L/2. When the interests of the decisionmaking core are congruent with those of the full union, [Al] is of course not needed.8 [A2] All members of class b have lifetime employment. Since all members of class b ("the founding fathers") have the same seniority, it is inappropriate to use the maximization of lifetime income as an objective if any of these members were to face unemployment risk. We do not model the strategies for this case but we note that the problem could only arise in the starting phase. As a result of attrition class b eventually becomes a subclass of the core, with all members of b having lifetime employment by the nonnegative profit constraint. m. A UNION MONOPOLY MODEL OF BARGAINING: THE NON-COOPERATIVE EQUILIBRIUM In this section we derive Markov Perfect Equilibria (MPE) for the sequence of decisions given by steps 1-4. The strategies corresponding to a MPE depend only on the state variables, i.e. the membership L, the share a of a member of the base class, the cumulated wage differential G1, and the date of entry of the lowest class with lifetime employment cf. A. The Setting of the Average Wage and Employment As in the static union monopoly model, the setting of w and Le remains a Stackelberg game in a MPE. By [Pl] all laid off union members wait to be recalled and future membership hence cannot be changed by either an excessive wage claim of the union or layoffs by the firm. Neither the union's nor the management's actions have lasting effects on the future state variables and, in a MPE these actions are determined within a given four step sequence for each r 2 t. In step 4, the profit maximizing management chooses Le on the marginal product curve of labor yielding the wage bill p = flLe '. In step 3, the decisionmaking core whose objective is t(t) selects w (leading to Le in step 4) such that Uf =Gfa(p, the wage of the least senior member with lifetime employment, is maximized. In step 2, outsiders, if any, accept the contract of step 1 if by [Pl] they expect employment in step 3 and the participation constraint is satisfied. 8 An extension of the analysis carried out in this paper would be to consider the case where core members try to prevent the expansion of membership so as not to lose decisionmaking power. Worker Power, Surplus Sharing and the Wage-Employment Outcome 11 Lemma 1: Under [P1],P2],[A1] and [A2], a MPE employment level for all Tk t will be set so as to maximize the wage of the least senior insider with lifetime employment at T: argmax (U,(u,c,,P') Ici.(t) ce T , e+--f(er (14) where U, = G,aj3L' and the effect of increasing ce or fe (ce) on Uf is of the same sign as f-ei(ce)L(ce). Proof: See Appendix A. To ensure full employment of members (L = Le Lmin), it is necessary from Lemma 1 that Uj be non-decreasing or that P3 ; aL (15) It follows that the marginal contribution of the incoming outsiders to the wage bill, 02L-1, is at least as large as the base wage aLO = y. Otherwise the senior members could increase their current wage by increasing the average wage, since at the higher wage, the management would not employ the outsiders and the wage bill for the senior members would increase. Therefore, if management would increase membership by additional hiring above L, the entrants would realize that they face unemployment and would not accept management's offer. Even if the union cannot control hiring, it can check the expansion of the membership by setting the wage. It follows that w - fLO-1 > 02LO-1 > y. To appreciate this result, recall that in the static union monopoly framework the marginal worker's contribution to the wage bill is less than his contribution to the firm's product since the hiring of an additional worker is contingent on the union lowering the average wage. If the insiders and incoming outsiders were to be paid a uniform wage w, the cost of the marginal worker to the union would exceed the marginal contribution to the wage bill. No hiring would take place and full employment would be impossible for any membership exceeding Lmin. Such a situation presents unexploited gains from trade, since outsiders would be willing to work at a lower wage. As the above condition indicates, in the dynamic model this issue will be resolved by setting the base wage y below the average wage w, thus creating wage differentials across generations. 12 Worker Power, Surplus Sharing and the Wage-Employment Outcome The open shop The management's objective is P(d-z (16) which is maximized subject to the wage setting constraint (14), the participation constraint, the laws of motion for membership, and the share of the incoming outsiders. From the previous discussion, (14) can be replaced by (15) when I > 0 and L > Lmin. Since the management's profits are increasing in L, the hiring policy is completely determined by the constraints (8) and (15) when hiring takes place. Proposition 1 The stationary state of an open shop MPE is characterized by P = mL a m-g mL L = max[{p r+m -p (17) r+mp g = m min[1 -P,] Proof: Immediate. O At the time of the founding of the firm, the management sets the variables such that (17) is satisfied. However, if initial membership is too high or/and the past wage differentials are too low, the union in a MPE sets high wages and precludes employment of outsiders. A transition period without hiring (but with attrition of the membership) is needed before entering the stationary state. Such a situation might arise if an expansionary policy with employment to the right of the marginal product curve has failed or if a non-unionized firm is unionized and the continuation of the game is a MPE. Lemma 2: In an open shop MPE, the time t* at which the stationary state is entered satifies the condition that the wage of the least senior employed member with lifetime employment when no hiring takes place falls below his wage in the stationary state of the open shop MPE: Worker Power, Surplus Sharing and the Wage-Employment Outcome 13 t* = mini I t , Le(t) 0 and 1(L,a) is differentiable, the values of the control variables are optimal if they maximize the current value Hamiltonian H* = aT - (r+m-g)# + 1L(O-mL) + Xea(m-g-t) + where (19) )6= p X., - alfffix Proof: See Appendix A. In the open shop, the base wage of the new member is equal to his marginal contribution to the current wage bill. Since the wage is growing over time, the income received by a new member over his lifetime therefore exceeds his marginal contribution to the wage bill. In the closed shop, the union will restrict the membership so that lifetime income is equal to the marginal contribution to the wage bill over the life time: (p /(L) - (20) r+m With positive wage growth, the annuity income (r+m)t exceeds the base wage and (15) is satisfied with inequality. The forward looking union will not exploit the opportunity to increase the wage bill in the current period by admitting more members and lowering the wage, since it realizes that the current gain will be undone by the future wage growth. Proposition 2: The stationary state of a closed shop MPE is characterized by p = r and P = mL a _ m-g mL 1 L = max[P1-P,] (21) g = m(m+r) min[- , ]___1 r+m1-3)r+m(1-4 ) Proof: See Appendix A. We now discuss the transition to the stationary state. Since in the closed shop the union controls the hiring, the lifetime contribution to the wage bill of the cohort of new entrants must compensate for the loss from recalling the redundant insiders. Worker Power, Surplus Sharing and the Wage-Employment Outcome 15 Lemma 4: In a closed shop MPE the time t* at which the stationary state is entered satifies where ce(r) for t ! T 5 t satisfies (13) and L,a and 1= V/a satisfy (22). i + Dt(t)G,(t)G(ce(t),t)emi(-()a(t)[1-a(L-L(t))]Y) Proof: See Appendix A. Discussion We close this section by comparing the two institutional setups for L > Lmin. We follow Layard and Nickell (1990) by distinguishing between partial equilibrium effects when the change in one firm does not affect the lifetime income of new applicants and general equilibrium effects when the firm considered is a representative firm and the lifetime income of applicants changes. In the closed shop g = [m(1-0) (m+r)]/[m(l-O) + r], in the open shop g = m(1-0) and in a non-unionized firm g = 0. The. higher wage differential in the closed than open shop results in a higher employment level when switching from the former to the latter. The switch to the non-unionized firm would increase employment even further. These partial equilibrium effects will be partially offset by the general equilibrium effect of the increased lifetime income required by the applicants to fill in the increased labor demand -- from Equation (13), increasing 4 decreases L. Finally, we note that any positive multiplicative shift in the production function (e.g., a price increase) has the opposite effect to a change in g. If follows that perverse price responses, frequently attributed to labor-managed firms, are ruled out. The decisionmaking core Lmn of insiders is unambiguously better off by expanding membership and employment beyond Lmin and in this sense the senior insiders earn a rent. However, this rent is foreseen and it induces the incoming members to accept an upward sloping earnings profile within the constraint 4' = 4. Without an additional rent-generating mechanism the union members' rents are dissipated over their lifetime. They merely earn period-by-period wage rents as senior insiders when compared to the wage earned concurrently by equally skilled and productive junior colleagues. However, the senior insiders use these rents to repay consumption smoothing debts which they incurred as junior 16 Worker Power, Surplus Sharing and the Wage-Employment Outcome members foreseeing future rents. Similarly, within the constraint 4) = 4), the junior members are currently accepting low wages in the expectation of receiving higher wages later on. The interest costs associated with an upward sloping wage profile increase the wage bill to give a predetermined lifetime income and therefore decreases employment. B. Models of Labor-Management Bargaining: Cooperative Equilibria In this section, we show that it is possible to eliminate the inefficiencies of the noncooperative MPE within a cooperative framework. We propose a dynamic solution for two issues that have been dealt with in the static literature on cooperative equilibria but have not been tackled successfully in a dynamic context. The first one deals with the division of the surplus generated by cooperation. To solve this problem, most static studies have used the nonsymmetric Nash bargaining solution in order to divide the gains from cooperation in accordance with the parties' relative bargaining power and alternative opportunities. We complement the four step sequence of decisions introduced in Section 2 with a bargaining process (step 5) that ends with a Nash bargaining solution to the division problem. The second issue concerns the decision variables to which the players can commit themselves and the credibility of their commitment. In the static literature one finds that the efficient bargaining outcome to the right of labor's marginal product curve is hard to sustain if management can unilaterally reneg and move back to the marginal product curve. In the last section we have shown that even in the dynamic context with strategies satisfying perfection (i.e., in a MPE) one finds the solution to lie inefficiently on the marginal product curve. However, we show presently that if one augments the four step decisionmaking process with a Nash bargaining solution as step 5, an efficient outcome can be credibly guaranteed if the strategies satisfy perfection. In addition, it turns out that in this context the identity of the player who proposes or sets the values of the relevant variables is irrelevant (in a sense which will be defined below). C. Nash Bargaining Solutions Suppose the management and union, realizing that the noncooperative MPE outcomes are inefficient, engage in a cooperative effort to improve the outcome, without giving up the option to choose the instruments they control in a most advantageous way to each of them. Suppose this cooperative effort consists of of a fifth step in the decisionmaking sequence in which the parties agree to explore a Nash cooperative framework as a means of possibly finding a superior outcome. For example, the management might make a proposal {g(r),J(r),w(T)},, for future play that would result in lifetime payoffs (P(t),*(t)) for the management and union, respectively and where p(r) reflects the choice of w and Le in steps 3 and 4. If the proposal is considered unacceptable to the union because it does not reflect the Nash solution, the management may try to avoid conflict by offering a one-shot "conflict resolution transfer" to the union (e.g., a special contribution to a pension fund). We embed this decisionmaking process into a dynamic framework in order to see what outcome is sustainable. Worker Power, Surplus Sharing and the Wage-Employment Outcome 17 In general, let Tm(t) and Tu(t) be the one-shot conflict resolution transfers at time t to the management and union, respectively: (Tm(t),Tu(t) E T = {Tm,Tu ITm+Tu;0 }. The form of the transfers ensures that the future play remains unchanged and that the adjusted payoffs form a closed convex set P(t), as needed for a Nash solution: P(t) = {P,* I P=P(t)+Tm,l*=(t)+Tu,(Tm,Tu)GT }. For the choice of (P,*) E P(t), we select the nonsymmetric Nash bargaining solution with the management and union power shares (-ym'yu), respectively. The respective management and union disagreement payoffs are given by (Pd(t),*d(t)). In the dynamic context we require that the continuation payoffs, should cooperation fail, be the outcomes of a perfect equilibrium. The MPE of the previous section, not involving conflict resolution transfers T, is a natural candidate which we use. It follows that T(t) = Tu(t) = -Tm(t) is the maximizer of the Nash product. In line with our discussion, we complement the four-step sequence of decisions with an option to proceed to a fifth step: 5) The management pays a conflict resolution transfer T to the union. The one-shot nature of the conflict resolution payoffs in the dynamic context provides a strong incentive for the parties not to rely on these transfers since, once paid, they are sunk and hence invite a repetition of the transfer payment in the future. It is therefore preferable for the parties to formulate equivalent dynamic plans that do not entail the payment of one- shot transfers. We hence consider proposals which cannot be improved u on by the payment of such conflict resolution transfers and we call these proposals balanced. Definition: The proposal {g(),t(7),p(r)}7>, is balanced if for each 7 the continuation payoffs (P(r),t(7)) of the proposal and the continuation payoffs (Pd(t),kd(t)) of the MPE satisfy 0 = argmax[f(r) - T) + TJ'- [P(T) - PdT) - TJ'" (25) T 9 By [Al] only union membership classes with lifetime employment in the MPE bargain with the management. 18 Worker Power, Surplus Sharing and the Wage-Employment Outcome L]Let the surplus to be shared in agreement be S(t) = (t) + P(t) - YJt) -Pdt) (26) so that the payoffs can be rewritten as a balancing constraint Y = 'Pit) + yuS(t) (27) P(t) = Pd(t) + YmS(t) In order to see the part that the balancing constraint plays in the cooperative framework, assume that at t the management proposes to the union a plan for future play. The plan is acceptable to the union only if *(t) zt ird(t) + -YuS(t). This inequality restricts the state variables at the start of the play but the firm may be tempted to cash in early and disagree later when it could benefit from a conflict resolution transfer or from switching to another play. The management's plan will hence be credible only if at each future point of time it is in the management's interest to continue the play. The Nash transfers proposed by the firm are not profitable if P(r) Pd(,) + -ymS(T) Vr > t . In the absence of commitment, credibility of the management's plan requires perfection. We further require state dependent strategies and therefore derive Balanced Markov Perfect Equilibria (BMPE). The situation is of course reversed when the union proposes a cooperative play to the management. The way in which the balancing constraint binds therefore depends on the identity of the proposing party and on whether the restriction is on the initial values or on the future values. As we mentioned at the start of this paper, in the static literature on unionized firms there is an unsettled debate between those who argue that efficiency gains will be realized despite restrictions imposed by the institutional environment (e.g., the sequencing of decisions and the instruments of each player) and those who stress the players' unwillingness to give up their rights to manage (set variable values) which stem from the distribution of property and control rights. The Nash solution transfers in step 5 provide a way to resolve this dilemma. The effect of a strategic choice of instruments in steps 1 to 4 can be neutralized by offering a Nash solution transfer in step 5. The player with the right to manage takes the preferences of his opponent into account, since by eliminating inefficiency the proposing player can increase his payoff in balance with that of the opponent and eliminate the need for a conflict resolution transfer, which would otherwise have to be paid. It follows that the identity of the player who actually sets the variable values is irrelevant (in Worker Power, Surplus Sharing and the Wage-Employment Outcome 19 a sense to be qualified later).10 Cooperation therefore does not require the union to have an explicit involvement in the actual management of the firm. A lack of empirical evidence on the union's direct involvement in the management of the firm cannot therefore be used to deny the existence of cooperation between the union and management. For the division of a fixed size surplus in the cooperative framework, Binmore, Rubinstein and Wolinsky (1986) have shown that the nonsymmetric Nash bargaining solution may be seen as a shortcut description of a noncooperative bargaining procedure with two- sided offers and the alternative opportunities of the union and management taken into account in case of disagreement. Such a procedure could be used as an alternative justification for our cooperative approach.11 However, unlike in Binmore et. al. (1986) and other bargaining models, our model has the advantage that the size of the surplus is not fixed but depends on the players' future strategies, and, more importantly, on the past policies as reflected in the state variables. From Lemma 2, the disagreement payoffs depend crucially on whether 3 > aL or 1 < aL. In the first case all members are employed and the management's profit is positive in the transition to the stationary state of the MPE. In the second case only the core members are employed and the management is in danger of being pushed to a zero profit position if wage differentials are low during the transition. In the open shop the management can prevent zero profit (0 < aL) from happening by keeping the wage differential sufficiently high (0 = cL). However, in the closed shop the union may set sufficienly low wage differentials (by moving from 0 > aL to 1 = aL) in order to improve its bargaining position vis a vis the management. Of course, for both 1 > aL and 1 < aL, the imposition of a binding constraint (13 = aL) involves an efficiency loss. It might therefore be preferable for the management to accept a weaker bargaining position in the closed shop if this is compensated for by the gain from improved efficiency. Our strategy is as follows. We first derive the BMPE if every member remains employed in disagreement, which might be achieved only if e aL is imposed in the open 10 The disagreement payoffs of course do depend on the identity of the player who controls each instrument. 11 If this were to be done, the Nash cooperative transfer solution of step 5 might be replaced by the following noncooperative game. At step 5 both players either agree with the plan or either one may refer the conflict to an arbitrator. The arbitrator gives the right to make a take-it-or-leave-it offer to the union with probability -yu and management with probability -ym. The selected player makes a take-it-or-leave-it-offer requiring the opponent to pay T. If T is paid, the game returns to the cooperative framework. If T is not paid, the conflict continues with the parties operating in the (inefficient) MPE of the previous section. By subgame perfection, the threat of sticking to this equilibrium after the opponent refuses to pay T is credible. Of course, in this setting, the offerer will set T such that the receiver of the offer gets the same net-payoff by accepting to pay as he would get by refusing to pay and the offerer acquires the whole surplus S. Ex- ante, the expected payoff of disagreeing is the RHS of Equation (19). 20 Worker Power, Surplus Sharing and the Wage-Employment Outcome shop. The full union is the opponent of the management12. We then derive the BMPE if only the union core remains employed in disagreement, which in some cases will be achieved only if 03! aL is imposed in the closed shop. In this game, the management plays against the core of the union. If one of the equilibria is an interior solution, we look whether the player who controls hiring can at any time do better by exercizing his control and improving his threat position. In what follows we use subscript n to denote variables in the stationary state of the noncooperative MPE and subscript d to denote the lifetime disagreement payoffs of switching from the cooperative to the non-cooperative mode. The two have to be distinguished because a switch to a noncooperative mode may not immediately result in a stationary state as seniority rule precludes the hiring of outsiders as long as union members are laid off and attrition takes time for L > Ln. Lemma 5: Assume that in the closed shop the management accepts the constraint P UL and in the open shop the union accepts the constraint P ! &oaL. Assume also that the management has access to perfect capital markets. It follows that by augmenting the sequence of decisions by step 5, the identity of the player controlling each instrument in a BMPE becomes irrelevant. Furthermore, after the balancing constraint is imposed at the start of a play, it is no longer binding. Proof: See Appendix A. We next examine the cases where the management bargains with the entire union and the core members, respectively. D. Management v. the Entire Union As we pointed out in the introduction, in accepting an efficient outcome with a lower wage and higher employment, the union in the static case runs the risk that the management will want to increase its profit by returning to the marginal product curve and cause unemployment of union members. In the dynamic context, the analogous risk is that after the union has accepted a cooperative outcome with large membership and wage differentials, the management may be tempted to switch to the noncooperative game and increase its short term profit by moving to the marginal product curve, temporarily forcing a lower average wage at the high employment level on the union. High employment will be temporary, however, because by Lemma 2 retiring insiders will not be replaced until membership falls to 12 Note that the ratio of the wage bill of the union core and the full union in agreement is equal to the one in disagreement and depends on the wage differentials agreed upon in the past. Since the Nash bargaining solution is invariant for linear transformations of the utility function, the same solution is obtained in the game against the full union as in the game against the core of the union. Worker Power, Surplus Sharing and the Wage-Employment Outcome 21 Ln at t =t+1n(L(t)/Ln)/m. Without admissions, the share of the surviving insiders grows at the rate m offsetting the effect of the mortality rate in discounting. The payoffs of the management and the union after switching to the non-cooperative mode are given by t* Pd r (- + 1I[(1-p)L e'(*-) U1 Yd = (e-r(r-r)Le-m(r-'tjpd- + 1 PL0e't*-)} m+r-g (28) which simplifies to +1 mp Lnr'P H Pd = (1 - ) L[1 + - _- r+mp r L r (29) d P 1 + g --) L. -r r+m0 m+r-g, L Proposition 3: In BMPE where the management bargains with the entire union under P > ozL, employment is set at the level of the nonunionized sector paying the competitive wage $ and employing the same technology. The stationary state with L > ImM is characterized by p Pm r+m-g (30) L m+r m-g g = m [1 - (L)] with 5(L) = (-y!! + LP r+m r r+mp x { 1 + Y-M + [(1-Y) g-M(1-0) - y(-P)m _n m r m+r-gn r L (31) L = 1 if 0(L) !g P (32) L satisfies f(L) = p otherwise Proof: See Appendix A. 22 Worker Power, Surplus Sharing and the Wage-Employment Outcome Equations (26) and (27) are reformulations of the participation constraint for the new entrants and the balancing constraint on the gains from cooperation, respectively.13 If the seniority induced growth rate of wages is sufficiently large, then by equation (23) employment is set at the same level as in an otherwise identical non-unionized firm. The result holds if the union is not too powerful (u < 1) and the firm opearates a closed shop (gn > m(1-)). It is easy to check that #(.) is increasing in L and, for L = 1, it shifts up when -y is increased.14 The interest cost of an upwards sloping wage profile increases with the interest rate. If the interest rate is increased for L = 1, -' eventually reaches the upper bound /, g is set at m(1-0) to guarantee full employment after a transition from a cooperative to noncooperative game, and L is decreased below the efficient employment level to satisfy the balancing constraint. In the closed shop, for sufficiently low interest rate, there exist equilibria with efficient employment satisfying the balancing constraint if union's power is sufficiently small (y ; -y(r)). Increasing the union's power decreases the wage differential and, once -y > -y(r), employment is decreased. Since membership cannot fall below Lmin, the BMPE and the MPE may coincide if L < 1 and the fixed cost of production is sufficiently large. Although management's profit in the stationary state decreases with the wage differential and is lower than that in the nonunionized sector, the present value of the profit at the time of the founding of the firm is the same as in the nonunionized sector if the wage differential is unconstrained and employment is then chosen efficiently. To see this, note that workers take into account lifetime rents: they are willing to accept low initial wages in return for high wages later, provided they can maintain the constant consumption level they would achieve with the constant wage in the nonunionized sector. At the founding of the firm, the management thus generates high initial profits by offering the first generations of workers low initial wages. When the first generations have gained enough seniority, however, profits will become lower. The effects will cancel out in the long run, as both the firm and workers receive the same long-term payoffs as in the nonunionized sector. 15 13 We note that the balancing constraint is linear in -y. Equation (27) implies that pl r+m-g) = 'd for 'Y 1 0 - (p-H)/r = Pd for - = 0 and V satisfies (26). 14 Since 1 is linear in y, it is sufficient to show that ! is increasing in L for -y=O and y= 1.Note that &O'/ay is nonnegative for r=O and for r= cc and that this derivative has a unique maximum in r. 15 For the conclusion: this is the equivalent for the union of the Coase conjecture for the monopolist of a durable good. It claims that by subgame perfection, further sales after serving the high valuation buyers in the. market cannot be ruled out. But the high valuations have a reason to wait until the monopolist lowers its price. In a frictionless market all buyers will wait and almost all sales will be at marginal cost. Here also union power results in the same outcome as the competitive market). (Since discounted profits are decreasing over time, balancedness requires also decreasing wage differentials Worker Power, Surplus Sharing and the Wage-Employment Outcome 23 The firm's discounted profit is less than in the non-unionized sector if the wage differential is bounded and employment is inefficient. In order to reduce this inefficiency or to ensure that a solution with L > Lmin exists for a given set of parameters, the wage growth must be reduced below m(1-#). In that case the profit in disagreement falls discontinuously to zero, however, since the bargaining position of the firm is weak as only Lmin members are employed in the transition period. E. Management v. the Core of the Union We now derive the disagreement payoffs for the BMPE with 3 < aL, when in the transition to the stationary state of the MPE the union's core of ILin members drives up the wage and imposes zero profits and employment Lmin on the firm. Only the core members expect lifetime employment in disagreement and they are the decisionmakers in step 5. Their objective is the lifetime income of the marginal core member cmin, with law of motion mL=in m(r-Cd c emn= e (33) The common objective is 01(t) = D(~~@)oc()t ~) tdr(34) where, by [Al], ce(r)= 'T and the wage of cnn(r) is obtained after adjusting the base wage for the cumulated wage differential G. The lifetime income for the newly admitted outsider is obtained after adjusting for the cumulated wage differential G. The participation constraint is therefore given by (35) Let 6 be a dummy variable taking on the value 8 = 0 in the closed shop and 6 = 1 in the open shop. Lemma 6: If a and g are constant, ft oeL and [A],[P] hold. Then Le= L before t* at which the stationary state of the MPE is entered, with 24 Worker Power, Surplus Sharing and the Wage-Employment Outcome L __ t* = t + [In(-L) + In( )]A/m L, x-1+p where xe[(1-P),1] satisfies (36) Ln [x-1+p L 9 Proof: See Appendix A. Note that x depends neither on L nor on p and that 8x _ 6(x-1+p3) 2 0 where 6 x ag lnL m(1-P)-gx (37) The discounted wage bill accruing to the core members is obtained by multiplying the lifetime income of the least senior core member by the seniority weighted number of core members, taking attrition into account. It is given by I= I is characterized by Worker Power, Surplus Sharing and the Wage-Employment Outcome 25 (1-3)L4 L L. x-+3 Pd - [ r LFT L r+m-g where (39) PL K L mg Inx-l1+p A n mm m [___m r+m-g, L L 0 Z +g~ K gn(1-n)+(8n- Zn r+m p 1m rn+m-g L m+r m-g (40) T_ m r+m-g (41) L m+r m-g 1 _ 1 y,(r=m-g) y,r 7 - (p -m (42) L g Ap LO-

O O Proof of Lemma 2 Assume that at t membership is L(t), that for t: 7:5=t* there is no hiring, membership L(7) falls by attrition and L (r) satisfies (14), and that at t the management increases membership from L(t ) to L. By [Pl], the new members must be employed. By Lemma 1 this will be the case if the Uf is at least as large with as without hiring. By the seniority - clause, before hiring the new members, all redundant workers must be recalled. After giving the share a to the new hires so that the participation constraint is satisfied, the fraction [1- a(L - L(t *)] of 9 is available to the surviving members. By attrition, the share a(t) of class t has grown to exp[m(t* -t)]a(t). Adjustifg this share for the wage differential between class ce(t ) and class t and the wagedifferential Gi gives U, with hiring. Since management's profit is increasing in employment, in the open shop the management makes the adjustment as soon as possible. Note that if fewer workers were hired and employed, from Lemma 1 and aL=O, employment can be increased to L. Since management profit is increasing in L, the management switches to the values of (15) as soon as the union employs the new hires at tOE1 Proof of Lemma 3 The proof adapts the usual argument in deriving the current value Hamiltonian (see e.g., Kamien and Schwartz, ..). In the present case, the discount factor depends on the control variables and the value function appears in a constraint. The union's objective is given by (4) and is maximized under the participation constraint with Lagrangean multiplier 4 and the laws of motion of the state variables Dt, L, a, 4, with costates XD, XL, Xce , i' respectively. From (5), for any path, 28 Worker Power, Surplus Sharing and the Wage-Employment Outcome XD(r) = a4'(t)/8D(r) = -1(r) If Dt-Xp > 0 and x x/(D-X4), partial integration making use of the transversality conditions yields J(t) = f Dr[aJi-%,(r+m-g)]+AXa(m-g-at)+,[(r+m-g)4#-ap] t +1X(4mL)+p(4-4')-XDDt-1,a-X1(Q-LL } dr (1) Pointwise maximization w.r.t. 4' results in p = -din(Dt-X(4). Pointwise maximization w.r.t. L and a yields the differential equations for the costates of L and a. Note that for the optimal policy max J(t) = 4'(t), so that -fp(s)ds (t)=fe ' [H*+p(D-A*a-.L a +iLL]dv +X'()a(t)+ Q(t)L(t) t by subtracting X,'(t)4)(t) from both sides and dividing through (1- XD(t)). O Proof of Proposition 2 0 O 00 Let I>0. From (22),d(X a) = (p+at)X a- 0 , and in the stationary state p=r. From (22),dX L = (p+m)X L-ap'. Since aH /a=X L-a4 = 0, in the stationary state (21) follows. Combining (21) with the participation constraint yields (22). O Proof of Lemma 4 The proof proceeds as in Lemma 3. However in the closed shop, the hiring in step 1 is set by the union and t* is determined so as to maximize lifetime income of the least senior member with lifetime employment. Before t no hiring takes place and at t one moves to the values of Proposition 2. O Proof of Lemma 5 Consider the case when the union sets the values of the variables and 0 aL. Everybody hence has lifetime employment and the union's objective is the maximization of the lifetime income of the youngest member. Let K be the financial wealth of the firm and k be the savings (retained earnings) of the firm. Note that both P - K and Pd - K refer to the present value of futtire profits earned in the firm and that P - Pd is independent of K. Let qi be the RHS of the state equation of the state variable xi (xi # P,4',K) and let hi 0 be any constraint other than the balancing constraint. Extending the argument of Lemma 3, one Worker Power, Surplus Sharing and the Wage-Employment Outcome 29 obtains the Hamiltonian Ho = a< -(r+m-g)Z +1,(rP+p +k-L )+1rK+k) + -Td P-Pd] +EX q1+EA (3) Y. Y, i J 0 a a0 0,0 o Since dX K and dxp = (p-r)o and H 0 k =XO +K=O , it follows that v=0. From aH /ag =a+X p=0 it follows that -Xp =a>0. . The Hamiltonian of the management is obtained by dividing H through by ce. The optimal choice of the union and management is therefore the same. This result holds if some control variables are chosen by the management and some by the union. The case a aL yields the same results. Proof of Proposition 3 From Lemma 4, if the constraint fl > aL is not binding, the identity of the playerowho proposes a plan is irrelevant for a balanced equilibrium. If the union proposes, Xp = The problem is equivalent with the problem of Proposition 2, and therefore (21) is satisfied for p = La and L = 1 is chosen as in the nonunionized sector. The participation constraint and the balance constraint determine the wage bill and the wage differential. In the constrained case, the constraints determine the solution. l Proof of Lemma 6 Let Z+6gn r +m -(1 -8)g,n (4) Z = r(1-a.L)+(m+r)a.L(t*) In the stationary state of a closed shop, from Lemma 1 and Lemma 3, t* in (23) satisfies a (c,(t*)) em(rc(r9)j3 = a(t)G(ce(t*),t)ZY which for constant g simplifies to x-- e(m-)(-c(t) e-(m-Xt* e(m-9)(t((t)) = e(mfXt*-) L m (6) p iti Lma From Proposition 2 and the definition of Z 30 Worker Power, Surplus Sharing and the Wage-Employment Outcome e0 * - mZ-rg, m(m+r-g) L. 1 Ln (7) e-*- =x " + = P[- ](7 m(r+m-g,) (r+m)(m-gn) L P L and (34) follows. From 0 g < m, the RHS of the relation defining x in (34) is increasing and concave and zero (and therefore less than the LHS) for x = 1 - fl. Since g < m(1 - fl), the RHS exceeds the LHS for x = 1 and Ln > Lmin. Since the LHS is linear, a unique solution for x satisfying (30) exists. In the stationary state of an open shop it follows from (16), Le = Lmin and the definition of Z that (36) is satisfied. From Proposition 1, (37) is satisfied. Lemma 2. O Proof of Lemma 7 The lifetime income 4d for class cmin(t) is given by the RHS of (23). Since d (d4d/a(t,cmin),it follows that d'd =ed-(m-(g(cmin)-a(cmin)(cmin))dcmin)lild- Since in the transition to the MPE, from Lemma 1, Le=Lmin, = -t + (r+m-9(Ci.)e.,i.l)0Zd -aQ d.P! (8) D [(t*,c (t*))G(c (t*),t)a(t*,t). In the closed shop it follows from (23) that ad/1t* =0. In the open shop it follows from (16) that a0d ---d_ =t(t*,c (t*))G(c (t*),t)a(t *,t)Y, x1(r+m-g)[1-a,(L.-L(t)e-MW-to)] a (t ,t) -Z Using a(t*,t)/a(t,cmin)=exp[m(t*-t)]o/a(cmin) and the definitions of Z and x results in (39] Proof of Proposition 4 From Lemma 5, if # < aL does not bind, the balancing constraint is not binding for T > t. The state variables of interest are Dt,4,a,G, P,L,K and cmin. From Lemma 3, the control variabless are optimal if they maximize the current value Hamiltonian Worker Power, Surplus Sharing and the Wage-Employment Outcome 31 Ho = Gasp - [r+m-g(c,)6.]d + a[m-g-af]+lGG[g-g(c )e.,] + 1L[-mLJ10) + X,(rP+q+k-LP) + X1rK+k) + XcL e-m(r-c-/t(cm ) + p[-Go] O The first order condition ofop yields Xp =-aG. Differentiating both sides and using the 0 0 0 definition of p we obtain A =at. It follows that d[I-XG G]=p[R-XG G] and that d[4- xa a]=(p+al)[4-xa a]. Also Xc = [p+m 0i]1c-[ ~-ooG]g(c )ma Note that f and g have an effect on r and on cmin- 1(7). The first order conditions are therefore AL ~ - X:a2 0 0 0 Take XG =4/G and Va =4/a, so that the first order condition for g is satisfied and xc 0=0.G=bGadX -