How has organized labor responded to the increased bargaining power of multinational corporations

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132.Which of the following is an impediment to cooperation between national unions?

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This section develops a bargaining game model between plant level unions and a multi-unit firm with production sites in different countries of an integrated market.Footnote 4 There are two symmetric countries, denoted A and B. Every economy has two sectors, a perfectly competitive and an imperfectly competitive sector. The perfectly competitive sector acts as a numéraire, with the wage and price levels equal to one. A MNE which has a plant in each country operates in the imperfectly competitive sector. The MNE produces goods for the entire market with no direct substitutes: that is, no interactions occur in the product market. There are some exogenous fixed costs \(G\), large enough so that neither the MNE sets up a new production facility, nor a potential entrant enters the industry. The MNE faces a linear product demand schedule, and products may eventually be exported between countries without extra transportation costs. The two plants have identical technology, and labor is the sole factor of production, with decreasing return to scale. Labor supply is large enough to avoid corner solutions. Any labor required by, or freed up from the MNE is supplied or absorbed by the numéraire sector. However, the MNE hires workers at each plant from a rent-maximizing labor union: in this sector, workers are fully unionized.

The model considers different types of games to derive the patterns of bargaining in the MNE. Firstly, it studies a game where, in the first-stage, the unions and MNE simultaneously and independently decide whether to coordinate wage bargaining among the different plants. Then, the analysis investigates two different games where, in the first two stages, the players move sequentially. In the first game specification, unions are first-movers and decide whether to coordinate activities across plants; then, in the second stage, the MNE chooses whether to conduct negotiations by subsidiary management or general management. In the second sequential game, the order of moves is inverted. First, the MNE selects whether to coordinate wage bargaining among subsidiaries, while in the second stage the unions decide whether to coordinate their activities across plants. The last two stages are common, independent of the simultaneous or sequential choice of coordination of the players in the first-stage(s) of the game. After the bargaining parties’ decisions, wage negotiations take place. These are modelled by the generalized Nash bargaining solution. The wage setting occurs before the employment decisions. Thus, the MNE hires workers according to its necessities (right-to-manage approach). Finally, given the bargained wages, the MNE allocates production among its plants. Notice that, when wage rates are determined at the plant level, it may theoretically occur that unions set different wages. Without transaction costs in the product market, the MNE can shift production to the less expensive plant, until wage rates are equalized. Nevertheless, since the production function presents decreasing return to scale, the reallocation of productive activities from the less to the more economical plant is a finite process before the non-negativity constraint is binding. Thus, wage competition is less intensive with respect to the setting characterized by a constant return to scale technology Borghijs and Du Caju (1999). The majority of the received literature has investigated the patterns of bargaining in the presence of constant return to scale. However, in this model, with the assumptions of an integrated market, linear demand, monopoly unions, and absence of transportation costs, an infinitesimal difference in wage rates is sufficient to shift the entire production towards the less expensive plant; thus, a decreasing return to scale technology seems to be a more realistic assumption. Summarizing, the model investigates a three-stage game in the case of simultaneous decisions and a four-stage game in the case of sequential decisions about coordination activities. The model is solved by backward induction, and the solution concept adopted is that of the sub-game perfect equilibria.

In each plant, the production function (\(q_{i})\) presents decreasing return to scale in the single input, labor (\(l_{i})\),

$$\begin{aligned} q_i =\sqrt{l_i } ,\quad i=A,B, \end{aligned}$$

(1)

while the (inverse) linear product demand function is

$$\begin{aligned} p=a-Q, \end{aligned}$$

(2)

where \(p\) is the common price for the integrated market, and \(Q=\sum _i {q_i } \) is total output. The following Stone-Geary function describes the union utility

$$\begin{aligned} \Omega _i =(w_i -w_0 )l_i ,\quad i=A,B. \end{aligned}$$

(3)

Each union assigns equivalent weight to the wage and employment in its preferences (neutrally oriented union). Positive utility derives from the fact that wages \(w_{i}\) lie above the reservation wage \(w_{0}\), or what workers receive if they are not employed by the MNE. In the present model,\( w_{0}\) is the wage in the numéraire sector which, by definition, is equal to one.Footnote 5

Last stage: optimal allocation of production among MNE’s plants

The MNE maximizes profits by choosing the total quantity for the integrated market. The two plants’ respective costs determine the optimal allocation between them. From (1), given wages, total and marginal costs at each plant are \(TC_i =w_i q_i^2 \) and \(MC_i =2w_i q_i\), respectively. It follows that the global marginal cost for the MNE is

$$\begin{aligned} MC=\frac{2w_i w_j}{w_i +w_j }Q. \end{aligned}$$

Total and marginal revenue are \(TR=({a-Q})Q\) and \(MR=a-2Q\). Standard optimization techniques (see Borghijs and Du Caju 1999) yield the following productive allocation at each plant

$$\begin{aligned} q_i (w_i ,w_j )=\frac{aw_j }{2(w_i w_j +w_i +w_j )},\quad i,j=A,B;\quad i\ne j. \end{aligned}$$

(4)

Thus, the labor demands are

$$\begin{aligned} l_i (w_i ,w_j )=\left[ {\frac{aw_j }{2(w_i w_j +w_i +w_j )}} \right] ^2\quad i,j=A,B ;\quad i\ne j, \end{aligned}$$

(5)

with \(\partial q_{i}/ \partial w_{i} <0,\,{\partial l_i } / {\partial w_i <0},\,{\partial q_i } / {\partial w_j >0}\) and \({\partial l_i } /{\partial w_j >0}\), that is, output and employment in each plant are negatively dependent on the plant’s wage rate and positively related to the wage rate in the other plant. This means that plant workers are in competition with each other in the labor market. Notice also that \({q_i } / {q_j }={w_j } / {w_i }\): the necessary condition of equalization of the marginal costs of production across plants is satisfied. Given global production, total cost is minimized and, hence, the MNE maximizes profits. Finally, subsidiary \(i\)’s profits can be written as \(\pi _i =pq_i -w_i l_i \). Making use of (2), (4) and (5), it can be shown that profits are equal to

$$\begin{aligned} \pi _i =\left[ {\frac{a^2w_j }{4(w_i +w_j +w_i w_j )}} \right] , \end{aligned}$$

with \({\partial \pi _i } /{\partial w_i }<0\) and \({\partial \pi _i } / {\partial w_j }>0\), as expected: an increase in subsidiary \(i\)’s bargained wage decreases its profits, while an increase in subsidiary \(j\)’s wage increases subsidiary \(i\)’s profits.

Second-to-last stage: bargaining outcomes

This subsection reports on all possible bargaining outcomes, according to the strategic choice of the negotiating parties (unions and MNE subsidiaries) regarding the coordination/lack of coordination of their activities. In the following, this notation is adopted: \(N\) stands for no coordination, \(C\) for coordination; the first letter refers to unions, the second to the MNE. Therefore, four different bargaining regimes may occur: both parties do not coordinate their activities, that is, full decentralization (NN regime, i.e. plant level negotiations); MNE general management coordination (NC regime, i.e. headquarter agents of the MNE bargain separately and simultaneously with the two unions); union coordination (CN regime, i.e. the delegates of the “European” union coordinate their activities and bargain separately and simultaneously with the management of the two subsidiaries of the MNE); full centralization (CCregime, i.e. two-union/general management level negotiations). With the obtained results, the bargaining patterns in equilibrium are then derived.

Full decentralization

Suppose that neither the MNE nor the unions choose to coordinate their activities. That is, negotiations at the MNE take place between subsidiaries’ management and the respective plant level unions. Under the NN regime, maximization of the following Nash Product determines the wage rate at each subsidiary

$$\begin{aligned} w_i =\mathop {\arg \max }\limits _{w_i } \left\{ {NP_i =(\Omega _i )^\alpha (\pi _i )^{1-\alpha }} \right\} \quad i=A,B \end{aligned}$$

(6)

where \(\alpha \in (0;1)\) is the exogenous relative bargaining power of the unions, assumed to be symmetric across plants. In case of breakdown of negotiations, the outside option of both parties equals zero. Similarly, to Horn and Wolinsky (1988a, b), each MNE’s subsidiary is in a bilateral monopoly relation with the local labor union. Thus, the wage rate at subsidiary \(j\) affects union \(i\)’s objective function only due to its indirect effect on \(l_{i}\). The FOC for wage maximization is

$$\begin{aligned} \alpha \pi _i \left[ {l_i +\frac{\partial l_i }{\partial w_i }(w_i -1)} \right] =-(1-\alpha )\left[ {(w_i -1)l_i } \right] \left( {\frac{\partial \pi _i }{\partial w_i }}\right) \quad i=A,B. \end{aligned}$$

(7)

Given symmetry, the equilibrium wage under NN is

$$\begin{aligned} w_{NN} =1+\left[ {\sqrt{\alpha ^2+\alpha +1} -(1-\alpha )} \right] . \end{aligned}$$

(8)

The term in brackets represents the rent over the reservation wage. As expected, \({\partial w_{NN} } / {\partial \alpha >0}\): higher bargaining power of the union increases the equilibrium wage: unions capture a higher share of the MNE’s rents. Substituting (8) into (5), the labor demand at each subsidiary in equilibrium is

$$\begin{aligned} l_{i,NN} =\left[ {\frac{a}{2(2+\alpha +\sqrt{\alpha ^2+\alpha +1} )}} \right] ^2, \end{aligned}$$

(9)

with \({\partial l_{i,NN}} /{\partial \alpha <0}\). Further substitutions into the relevant expressions allow computation of the MNE’s profits and global union utility, reported in Table 1.

Table 1 Coordinated/not coordinated bargaining outcomes

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MNE subsidiaries’ coordination in bargaining

Assume now that the MNE negotiates with the headquarters representatives. One may imagine a situation where the MNE sends one agent representing the total interests of the firm to each subsidiary Zhao (1995). Since the cost of sending the headquarters agents may be realistically supposed negligible with respect to the total profit amount, the assumption that the MNE does not sustain any costs to coordinate bargaining activities applies. Negotiations occur separately but simultaneously at each plant. In such a regime, maximization of the following expression leads to the wage rate for the \(i\)th subsidiary

$$\begin{aligned} w_i =\mathop {\arg \max }\limits _{w_i } \left\{ {NP_i =(\Omega _i )^\alpha (\pi _i +\pi _j -\pi _j^*)^{1-\alpha }} \right\} \quad i,j=A,B;\quad i\ne j, \end{aligned}$$

(10)

where \(\pi _j^*\) is the MNE’s outside option in case of failure of the negotiations. Union \(i\)’s outside option is equal to zero. In the present context, the MNE’s disagreement profits (alternatively seen as lock-out funds) may be defined as \(\pi _j^*=\left[ {\frac{a^2w_i^*}{4(w_i^*w_j +w_i^*+w_j)}} \right] \), where \(w_i^*\) is the equilibrium wages of this regime charged to subsidiaries \(i\). That is, the headquarters agent at the subsidiary \(i \)assumes that, during the negotiations at the plant \(j\), the agent in \(j\) takes for granted that an agreement is reached at the plant \(i\) on the equilibrium wage. The first-order condition for wage maximization is

$$\begin{aligned}&\alpha \pi _i \left[ {l_i +\frac{\partial l_i }{\partial w_i }(w_i -1)} \right] \nonumber \\&\quad =-(1-\alpha )\left[ {(w_i -1)l_i } \right] \left( {\frac{\partial \pi _i }{\partial w_i }+\frac{\partial \pi _j }{\partial w_i }}\right) \quad i,j=A,B ;\quad i\ne j, \end{aligned}$$

(11)

because, in equilibrium, \(w_i^*=w_j^*\) and, therefore, \(\pi _j =\left[ {\frac{a^2w_i^*}{4(w_i^*w_j^*+w_i^*+w_j^*)}} \right] =\pi _j^*\). Given the symmetry assumption, equilibrium wages are

$$\begin{aligned} w_{NC} =1+\left[ {\frac{\sqrt{\alpha ^2+10\alpha +1} -(1-\alpha )}{2}} \right] , \end{aligned}$$

(12)

where the term in brackets is the rent over the reservation wage, with \({\partial w_{NC} } / {\partial \alpha >0}\). Nevertheless, from comparison of Eqs. (8) and (12), it occurs that \(w_{NC} >w_{NN} \quad \forall \alpha \in (0;1)\). The rationale for this result may be found by inspection of the first-order conditions in Eqs. (7) and (11). In the NN regime, each subsidiary management takes into account only the negative effect of the negotiated wage during bargaining on its subsidiary profit (the term \({\partial \pi _i } /{\partial w_i })\). Additionally, in the NC bargaining regime the MNE’s headquarters agents also internalize the positive effect of the wage increase on the other subsidiary (the term \({\partial \pi _j }/ {\partial w_i })\). In other words, the general management of the MNE considers the aggregate profits of both plants when bargaining with the union of plant \(i\), whereas decentralized bargaining at plant level implies that each subsidiary takes into account only its own profits. This implies that subsidiary \(i\)’s position is weaker during negotiations while, through the recognition of this internalization effect by MNE headquarters agents, the bargaining position of each union at the respective plant improves. As a consequence, the negotiated wages are higher than in the NN case: the MNE will accept payment of higher wages than in the case of decentralization.

Putting the expression in (12) into (5), subsidiary \(i\)’s labor demand in equilibrium is

$$\begin{aligned} l_{i,NC} = \left( {\frac{a}{5+\alpha +\sqrt{\alpha ^2+10\alpha +1} }}\right) ^2, \end{aligned}$$

(13)

with \({\partial l_{i,NC}} /{\partial \alpha <0}\). By comparison with Eqs. (9) and (13), it results that \(l_{i,NN} >l_{i,NC} \quad \forall \alpha \in (0;1)\): higher bargained wages in the NC regime reduce the labor demand at each site for the MNE. Finally, after subsequent substitutions, the MNE profits and global union utility are obtained, reported in Table 1.

Union coordination in bargaining

Assume now that the MNE decides to participate in negotiations with subsidiary management while unions coordinate their bargaining activities, i.e. through negotiations conducted by delegates of the European Industry Federations unions. Bargaining takes place independently but simultaneously at each subsidiary. By assumption, unions sustain an exogenous per member (symmetric) transaction cost \(\tau \ge 0\) for the coordination of activities. The total amount of this cost, \(\tau l_i \), has to be deducted from the union’s rent in the case of separate bargaining (Borghijs and Du Caju 1999). A reduction in unions’ transaction costs is the measure of the degree of labor market integration. In the CN bargaining regime, the following Nash product’s maximization determines the wage for the \(i\)th subsidiary

$$\begin{aligned} w_i&= \mathop {\arg \max }\limits _{w_i } \left\{ {NP_i =\left[ {(w_i -\tau -1)l_i +(w_j -\tau -1)l_j -D_j^*} \right] ^\alpha (\pi _i )^{1-\alpha }} \right\} \nonumber \\&\quad i,j=A,B; \quad i\ne j \end{aligned}$$

(14)

where \(D_j^*\) is the outside option of the industry unions’ federation. MNE subsidiary \(i\)’s outside option in the absence of coordination equals zero. The disagreement utility of the unions’ federation may have different specifications (Horn and Wolinsky 1988a, b; Dobson 1994; Santoni 2009; Mukherjee 2010). In the present context, the unions’ federation disagreement utility might be defined as \(D_j^*=(w_j -\tau -1)\left[ {\frac{aw_i^*}{2(w_i^*w_j +w_i^*+w_j )}} \right] ^2\)(alternatively seen as the utility flows during temporary negotiations breakdown), where \(w_i^*\) is the equilibrium wages of this regime charged to subsidiaries \(i\). Similarly to the previous case, the delegate of the unions’ federation in \(i\) assumes that, during negotiations at the plant \(j\), the delegate in \(j\) takes it for granted that an agreement is reached at the plant \(i\) at the equilibrium wage.

The first-order condition for wage maximization is, for \(i,j=A,B; \quad i\ne j\),

$$\begin{aligned}&\alpha \pi _i \left[ {l_i +\frac{\partial l_i }{\partial w_i }(w_i -\tau -1)+\frac{\partial l_j }{\partial w_i }(w_j -\tau -1)} \right] \nonumber \\&\quad =-(1-\alpha )\left[ {(w_i -\tau -1)l_i } \right] \left( {\frac{\partial \pi _i }{\partial w_i }}\right) \end{aligned}$$

(15)

because, in equilibrium, \(w_i^*=w_j^*\); therefore, \((w_j -\tau -1)l_j \!=\!(w_j^*\!-\!\tau -1) \left[ {\frac{aw_i^*}{2(w_i^*w_j^*+w_i^*+w_j^*)}} \right] ^2=D_j^*\). Comparison of (7) and (15) shows that the difference between uncoordinated/coordinated unions lies in the fact that, in the latter case, they consider both the effect of \(w_{i}\) on \(l_{i}\) and the impact of their own wage on the labor demand at the other plant. In other words, unions internalize the cross effects of an increase in their wage rates on overall employment levels (Davidson 1988; Horn and Wolinsky 1988a, b). Since in the present context workers at each plant are perfect substitutes in production activity, an increase in subsidiary \(i\)’s wage rate raises the labor demand at subsidiary \(j \)(and vice versa): that is, \({\partial l_j }/{\partial w_i >0}\). Hence, the bargaining coordination across plants for labor unions should in principle be more profitable than uncoordinated bargaining. Nevertheless, the presence of transaction costs may offset the unions’ gains. Whether coordination is beneficial for unions depends crucially on the size of these costs.

Given the symmetry, from (15) equilibrium wages are

$$\begin{aligned} w_{CN}&= 1+\left\{ \left[ {\frac{\sqrt{(4+\tau )\alpha ^2+2(\tau ^2+2\tau -2)\alpha +(\tau +2)^2} }{2}} \right] \right. \nonumber \\&\left. +\ [(2\alpha -1)+\frac{\tau }{2}(\alpha +1)] \right\} . \end{aligned}$$

(16)

The term in brackets represents the union rent. Analytical inspection reveals that \({\partial w_{CN} } /{\partial \alpha >0}\), as expected, and \({\partial w_{CN} } / {\partial \tau >0}\): an increase in transaction costs for coordinating activities is shifted on higher wage demands by unions. Placing the wage rate in (16) into (5), the labor demand in equilibrium at subsidiary \(i\) is

$$\begin{aligned} l_{i,CN} =\left[ {\frac{a}{(4+\tau )(\alpha +1)+\sqrt{(4+\tau )\alpha ^2+2(\tau ^2+2\tau -2)\alpha +(\tau +2)^2} }} \right] ^2\quad \end{aligned}$$

(17)

with \({\partial l_{i,CN} }/ {\partial \alpha <0}\) and \({\partial l_{i,CN} } / {\partial \tau <0}\). This is so because both higher unions’ relative bargaining power and transaction costs increase the bargained wage, and this, in turn, implies a reduction in the labor demand at each site for the MNE (the wage/employment trade-off). Subsequent substitutions lead to the expressions for the MNE profits and global union utility reported in Table 1.

Full coordination

Finally, suppose that both the MNE and the labor unions choose to conduct negotiations by coordinating their bargaining activities. Negotiations take place independently and simultaneously at the two MNE subsidiaries (the CC case in Fig. 1). As before, labor unions incur exogenous per member coordination costs while the MNE does not. In the CC bargaining regime, the maximization of the following Nash product determines the wage rate for the \(i\)th subsidiary

$$\begin{aligned} w_i&= \mathop {\arg \max }\limits _{w_i } \left\{ {NP_i =\left[ {(w_i -\tau -1)l_i +(w_j -\tau -1)l_j } \right] ^\alpha (\pi _i +\pi _j )^{1-\alpha }} \right\} \nonumber \\&\quad i,j=A,B;\quad i\ne j. \end{aligned}$$

(18)

In the case of a breakdown in negotiations, the two parties’ outside option now equals zero.Footnote 6 For \(i,j=A,B;\, i\ne j\), the first-order condition for wage maximization is

$$\begin{aligned}&\alpha (\pi _i +\pi _j )\left[ {l_i +\frac{\partial l_i }{\partial w_i }(w_i -\tau -1)+\frac{\partial l_j }{\partial w_i }(w_j -\tau -1)} \right] \nonumber \\&\quad =-(1-\alpha )\left[ {(w_i -\tau -1)l_i +(w_j -\tau -1)l_j } \right] \left( {\frac{\partial \pi _i }{\partial w_i }+\frac{\partial \pi _j }{\partial w_i}}\right) \end{aligned}$$

(19)

Given the symmetry, in equilibrium the wage rates are

$$\begin{aligned} w_{CC} =1+\left[ {(3\alpha +\tau (1+\alpha )} \right] \end{aligned}$$

(20)

where the term in brackets is the union rent over reservation wage, with \({\partial w_{CC} } / {\partial \alpha >0}\) and \({\partial w_{CC} } / {\partial \tau >0}\). Substituting the wage rate in (20) back into (5), the labor demand in equilibrium at subsidiary \(i\) is

$$\begin{aligned} l_{i,CC} =\left[ {\frac{a}{2[(3+\tau )(1+\alpha )]}} \right] ^2 \end{aligned}$$

(21)

with \({\partial l_{i,CC} } /{\partial \alpha <0}\) and \({\partial l_{i,CC} } /{\partial \tau <0}\). By comparison to Eqs. (16) and (20), and Eqs. (17) and (21), it results that \(w_{CN} \ge w_{CC} \) and \(l_{i,CC} \ge l_{i,CN} \) if and only if transaction costs are above a certain threshold level, namely \(\tau \ge \tau ^*\), with \(\tau ^*=\frac{\sqrt{4\alpha ^2+5\alpha +1} }{\alpha +1}-1\). In such a case, the MNE, through implemented coordination, recovers its bargaining position when it opposes coordinated unions. This is so because the MNE reduces unions’ outside option from \(D_j^*\) to zero. This coordination effect overcomes both the cross effects of employment and the internalization of profit externalities (see Eqs. (19), (15) and (11)), which leads to lower negotiated wages and higher employment levels. On the contrary, if \(\tau <\tau ^*\), it follows that \(w_{CC} >w_{CN} \) and \(l_{i,CN} >l_{i,CC} \): despite unions’ outside option is falling to zero, the coordination advantages for the MNE are not sufficient to dominate the internalization of employment and profit externalities. Thus, full coordination leads to negotiated wages higher than in the presence of union coordination alone. This finding partially reverses the MNE’s position with respect to the case of uncoordinated unions. In that situation, the MNE has no incentives in conducing wage negotiations with the headquarters agents, since the internalization of subsidiary profit externalities puts the unions in a stronger position. Finally, further substitutions allow for the evaluation of the expressions for the MNE profits and global union utility, as reported in Table 1.Footnote 7

First stage(s): strategic choices and sub-game equilibrium bargaining patterns

This subsection investigates the coordination choice of the parties given the bargaining outcomes. Firstly, the focus is on the simultaneous game; then, the model analyzes the sequential game where the MNE is the first mover.

Simultaneous moves game

Let us at first consider the bargaining patterns arising as equilibrium of the game within the MNE by analysing the simultaneous moves game, where the parties select their coordination strategy at the same time, each unaware of the other party’s strategic choice. In this case, the game as a whole is a three-stage game. The MNE and the unions decide whether to coordinate negotiations by comparing global profits and utility levels attained. The bargaining power of the parties, \(\alpha \), and the unions’ per member fees, \(\tau \), affect the relative outcomes.

Proposition 1

Under the simultaneous move game, not to coordinate bargaining is the dominant strategy for the MNE. If transaction costs are low, unions find it advantageous to coordinate negotiations whatever their relative bargaining power. Nevertheless, there is a threshold value of coordination costs \(\tau ^{**}(\alpha )\) such that, for \(\tau \le \tau ^{**}\) , a CN regime emerges in equilibrium while, for \(\tau \le \tau ^{**}\), a NN regime arises.

By using the results in Table 1, it is directly obtained that \(\pi _{CN} >\pi _{CC} \) and \(\pi _{NN} >\pi _{NC}\quad \forall \alpha \in (0,1)\wedge \tau \in (0,\infty )\): independently of the unions’ strategic choice, not to coordinate bargaining activities is the dominant strategy for the MNE. The rational is that the headquarters agents internalize subsidiaries’ profits and, therefore, take into account the aggregate profits of the company during negotiations, putting unions in a stronger bargaining position. Hence, the general management of the MNE may agree to pay higher wages, whereas decentralized bargaining implies that each subsidiary management considers only plant profits.

Given that the MNE always plays \(N\), the unions’ best response is to play the strategy \(N\) if \(\Omega _{NN} >\Omega _{CN} \). This occurs when \(\tau >\tau ^{**}(\alpha )\) (whose expression is not reported due to analytical complexity), with \(\mathop {\lim }\nolimits _{\alpha \rightarrow 0} \;\tau ^{**}=1.02\). Here, unions do not coordinate: full decentralization arises as the bargaining regime in equilibrium. That is, if unions have sufficiently strong bargaining power, the presence of high transaction costs more than offset the gains from coordination.

On the other hand, when \(\tau \le \tau ^{**}(\alpha )\), it follows that \(\Omega _{CN} \ge \Omega _{NN} \), and the unions’ best response is to coordinate negotiations. Thus, the combination of the parameters is such that unions’ partial centralization arises in equilibrium. The threshold value \(\tau ^{**}\) depends on the parties’ bargaining power, with \({\partial \tau ^{**}} / {\partial \alpha }<0\): high levels of unions’ bargaining power decrease the threshold for coordination. If per member fees are not too high, namely \(\tau ^{**}\left| {_{\alpha =1} =0.232} \right. \), unions find it advantageous to coordinate negotiations whatever their relative bargaining power. However, when unions are weak (low \(\alpha \)), their coordination incentives are high. Hence, unions’ partial centralization is the equilibrium of the game also in the presence of comparatively high transaction costs. These analytical results are graphically summarized in Fig. 1, which depicts the two areas representing the Nash equilibria of the game in the \((\alpha ,\tau )\)-plane.

Sequential move game

Let us focus now on the sequential move game, where one of the bargaining parties has knowledge about the strategic decision of the other party. In this model, in the first stage the MNE decides whether to coordinate wage bargaining between plants. Then, in the second stage, unions decide whether to coordinate negotiations.

Fig. 1

simultaneous moves game, Nash equilibria

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Second stage: unions’ choice of coordinated vs. not coordinated bargaining

In the second stage, unions choose whether to coordinate bargaining. This decision depends on the level of the global utility (that is, the sum of the utilities) the two unions receive under the choice of the MNE as regards the conduct of negotiations. Unions’ global utility is, in turn, affected by the relative bargaining power and per member coordination fees.

Coordination in bargaining among unions is advantageous if \(\Omega _{CN} \ge \Omega _{NN} \) when the MNE does not coordinate, and if \(\Omega _{CC} \ge \Omega _{NC} \) when the MNE negotiates with headquarter agents. Previous direct comparisons of the unions’ payoffs have shown that, if the MNE does not coordinate negotiations, \(\Omega _{NN} \ge \Omega _{CN} \) if \(\tau >\tau ^{**}(\alpha )\), and \(\Omega _{CN} >\Omega _{NN} \) if \(\tau \le \tau ^{**}(\alpha )\). On the other hand, if the MNE coordinates bargaining, unions’ coordination outcome dominates independent plant level negotiations when the transaction costs are lower than (equal to) the threshold value

$$\begin{aligned} \tau ^{***}(\alpha )\le -2\frac{\alpha [(\alpha ^2+2\alpha -3)\sqrt{\alpha ^2+10\alpha +1} -5\alpha +7\alpha ^2+\alpha ^3-15]}{(\alpha +1)^2[(\alpha +3)\sqrt{\alpha ^2+10\alpha +1} +8\alpha +\alpha ^2+3]}, \end{aligned}$$

with \(\tau ^{***}\left| {_{\alpha =0} =0} \right. \) and \(\tau ^{***}\left| {_{\alpha =1} =0.232} \right. \), while \(\Omega _{NC} \ge \Omega _{CC} \) if \(\tau >\tau ^{***}(\alpha )\). Further analytical inspection reveals the following results. Firstly, unions coordinate negotiations to improve their position vis-à-vis the MNE when transaction fees are not too costly, irrespective of their degree of relative bargaining power. Secondly, an inverse U-shaped relation (with the maximum reached at the value of \(\alpha =1 /3\)) between transaction costs and unions’ bargaining power exists: weak and strong unions are better off with independent plant level negotiations. If unions are extremely weak, small amounts of per member fees are sufficient to generate costs high enough to offset the benefits of coordination. On the other hand, if unions are strong, they prefer independent negotiations because the gains from internalization (the cross effects of wage increase on total employment) are not sufficient to counterbalance the cost of coordination.

First stage: the MNE’s decision and sub-game equilibriums

In the first stage of the game, the MNE chooses whether to negotiate with the headquarters agents. To select its bargaining strategy, the MNE evaluates the profits associated to every regime and takes into account the unions’ strategic choices about coordination. The following analysis shows how the unions’ strategic decisions affect the MNE’s choice, and derives the sub-game perfect bargaining regime in equilibrium.

Proposition 2

Under the sequential move game, for low values of coordination costs, namely \(\tau \le \tau ^{***}\) , a CN regime emerges in equilibrium for every unions’ bargaining power. If \(\tau \le \tau ^{***}\) , unions do not coordinate and, for \(\tau \le \tau ^{**}\) , a NC regime emerges in equilibrium while, for \(\tau \le \tau ^{**}\) , a NN regime arises.

If the MNE does not coordinate bargaining activities, unions will select the strategy \(C\) for \(\tau \le \tau ^{**}\) while, for \(\tau > \tau ^{**}\), they will play the strategy \(N\). The rationale has been found in the analysis of the simultaneous game: when \(\tau \le \tau ^{**}\) and unions do not have strong bargaining power, coordination gains more than offset the costs, while the reverse holds true if \(\;\tau >\tau ^{**}\). Instead, if the MNE coordinates, unions will play \(C\) if \(\tau \le \tau ^{***}\). This is because low transaction costs make coordination worthy for unions whatever is their bargaining power. On the other hand, for\(\;\tau >\tau ^{***}\), they negotiate independently at plant level. These outcomes generate three different regions in the\((\alpha ,\tau )\)-plane, as Fig. 2 shows.

Fig. 2

Sequential game, MNE first-mover. Bargaining regimes in equilibrium

Full size image

The first region is defined by the set of points \(\tau =0\) and \(\tau \le \tau ^{***} (\alpha )\). In this region, the unions coordinate wage negotiations irrespective of whether the MNE coordinates. Therefore, the MNE compares \(\pi _{CN} \) and \(\pi _{CC}\). As previously seen, for \(\tau <\tau ^{***}, \pi _{CN} >\pi _{CC} \): the MNE does not negotiate with the headquarter agents. Thus, the CN regime arises in equilibrium. The set of points \((\tau \in (\alpha =0)\left| {0\le \tau <1.02}\right. )\cup \tau ^{***}(\alpha )<\tau \le \tau ^{**}(\alpha )\) delineates the second region. In this area, if the MNE coordinates with the headquarters agents, the unions will negotiate separately at each plant, while if the MNE does not coordinate, unions will negotiate wages with industry delegates. Thus, the MNE evaluates \(\pi _{NC} \) and \(\pi _{CN} \). Direct comparison of payoffs shows that \(\pi _{NC} >\pi _{CN} \), and, therefore, the NC regime arises as equilibrium of the game. Finally, the part of the \((\alpha ,\tau )\)-plane delimited by the set of points \((\tau \in (\alpha =0)\left| {1.02<\tau } \right. )\cup \tau ^{**}(\alpha )<\tau \) characterizes the third region. At these levels of transaction fees, unions’ gains from coordination are cancelled out. Thus, unions bargain independently at each plant, no matter what is the MNE’s choice. The MNE selects its strategy by comparing \(\pi _{NC} \) and \(\pi _{NN} \). Since, in this region, the profits for the MNE are such that \(\pi _{NN} >\pi _{NC} \), full decentralization, the NN regime, arises in equilibrium. The results differ from those of Bárcena-Ruiz and Garzón (2002) due to the absence of transaction costs and the equal bargaining power between the bargaining parties in wage negotiations. Moreover, the general framework presented in this work, by taking into account the cross and coordination effects of negotiations, determines a broader set of bargaining structures in equilibrium than those in Borghijs and Du Caju (1999).

To summarize, the analysis in the previous subsections has shown that, in the presence of strong unions and high coordination costs, the bargaining pattern is identical for both structures of the game: full decentralization arises in equilibrium. The difference between the two game structures arises in the presence of relatively low coordination fees. On the one hand, in the simultaneous game, firm partial coordination is the other possible regime in equilibrium. On the other hand, in the sequential game, in the presence of sufficiently low union transaction costs and for every level of their relative bargaining power, also unions partial coordination arises in equilibrium. Therefore, the bargaining regimes in equilibrium are sensitive to the hypothesis regarding the timing of the game.

Extensions of the model

Many of the qualitative results of the model are also valid under different specifications. Nonetheless, some specificities appear.

Firstly, it can be easily checked that the change in the order of moves (unions as first-movers) in the model described above leads to the identical outcome of the simultaneous moves game.

Secondly, different specifications of the bargaining parties’ outside options have been also tested. To evaluate the potential benefits of union coordination, it has been considered the more favorable bargaining position for the MNE during wage negotiations. Therefore, the outside options in the Nash product have been set equal to zero for unions and the monopoly production with a unique plant serving the whole market for the MNE. It has been found that: (1) the bargaining outcomes of the NN and NC regimes are the same; and (2) the CC and CN regimes are equivalent as well. As a result, the MNE chooses to coordinate bargaining, irrespective of the unions’ choice, if \(\pi _C \ge \pi _N \). Similarly, the unions choose to coordinate their bargaining activities, irrespective of the MNE choice, if \(\Omega _C \ge \Omega _N \). This different model design has led to the following results. In the simultaneous move game, the organization of the wage bargaining predominantly occurs with decentralized unions in equilibrium, while the MNE is indifferent whether to coordinate wage negotiations. However, for low levels of the transaction costs, there are two areas where a Nash equilibrium does not exist. On the other hand, in the sequential move games with the MNE as first-mover, the bargaining regimes with unions’ decentralization are still predominant, but for low levels of transaction costs and high unions’ bargaining power, unions’ coordination arises in equilibrium, while the MNE is still indifferent to its organization of wage negotiations.Footnote 8

Finally, Buccella (2013) investigates the effects of fixed transaction costs for unions. The main findings are as follows. If the firm acts as the first-mover, an inverse U-shaped relation between fixed coordination costs and unions’ bargaining power exists. For low and high union bargaining strength, relatively small transaction costs make coordination not advantageous, while coordinated activities turn out to be beneficial for intermediate values. The multi-plant firm generally prefers not to coordinate wage negotiations. Hence, the predominant bargaining regimes are NN and CN. However, if the unions are strong enough and the fixed transaction costs are not very low, there is an area where the NC regime arises in equilibrium. On the other hand, if the simultaneous moves and the sequential game with unions as the first movers are investigated, similar patterns of bargaining arise, with the only difference that the NC regime is no longer an equilibrium of the game.

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