The co-determination decision game with consumption externalities

The present article aims to contribute to the debate on the regulatory policies and institutions concerning the labour market and industrial relations. The main historical labour market institution, which is pervasive in several central European countries is the so-called co-determination, which mixes some characteristics of the competitive labour market and the unionised labour market (e.g., the right to manage and efficient bargaining models). Co-determination—often regulated by law—may also be considered a regulatory tool for harmonising labour relationships and for improving social welfare. However, despite its social harmonising role, from a strict microeconomic approach, the mandatory introduction of co-determination in markets with standard non-network goods cannot represent a Pareto improvement for society. This is because co-determination contributes to enhancing the welfare of workers (trade unions) and consumers, so that both groups are better off, but it also reduces firms’ profitability so that firms are worse off.

By assuming network industries, the present article shows, in a non-cooperative game-theoretic setting, that co-determination can (1) endogenously emerge in the market, i.e., without the need for ad hoc law regulation and enforcement, like previous studies on this issue already showed (e.g., Fanti et al., 2018; Gori & Fanti, 2022; Kraft, 1998, 2001; Kraft et al., 2011), and (2) become a Pareto-superior institution for society. This is because consumption externalities work out by expanding the market demand, in turn, increasing production, consumer surplus, profits and the welfare of workers representatives. The policy implications are clear and far-reaching: mandatory co-determination in network industries is Pareto improving. Indeed, unlike in the previous literature, codetermination can emerge as a Pareto efficient SPNE in a network industry, and as social welfare corresponding to the SPNE when both firms bargain under co-determination, (B,B), is higher than social welfare corresponding to the SPNE when firms are profit maximisers, (PM,PM), the result is twofold: (1) voluntary codetermination is Pareto superior to profit maximisation, (2) if the government chooses to mandatorily introduce codetermination the result for society would be as if it emerged voluntarily and thus society would still get a Pareto superior situation to profit maximisation.

This work then combines two different strands of research belonging to the labour economics and industrial economics literature—co-determination and network externalities—by building on a tractable model describing a strategic competitive framework with quantity-setting firms.

On the one hand, there is increasing attention towards co-determination in several countries of Western Europe. This institution implies, broadly speaking, that employees’ representatives sit on the supervisory board (or similar structures) in large companies. Co-determination is known to be a relevant feature of the German industry (at least since the 1950s), but it is also widespread in other European countries.¹ Its main feature is that employment (production) is jointly determined by owners (or their managers) and workers (or their representatives) in the supervisory board, but the wage is left to be bargained by the federation of firms and the trade unions at the industry level.² More in general, active workers involvement in the firm’s decisions are a crucial element of the European social model.³ Despite this, there is little economic analysis on the effects of co-determination resulting in a gap on the side of policy recipes at the European level: “The practice of board participation and its impact are very hard to gauge, given a general lack of research and evidence.” (Schulten & Zagelmeyer, 1998). However, the number of theoretical and empirical works on the subject is growing.

On theoretical grounds, McCain (1980) represents a pioneering work opening the route to causes for reflections about co-determination, working conditions and labour productivity. Some years later, Kraft (1998) shows—considering a Cournot duopoly with homogeneous products—that profit-maximising firms have the incentive to become bargainers over employment. This in turn implies that co-determination becomes the dominant strategy of the game. However, firms would prefer to have the full power to choose employment (profit maximisation). Then, co-determination emerges as the unique Pareto-inefficient sub-game perfect Nash equilibrium (SPNE) of a non-cooperative co-determination decision game, but firms are entrapped in a prisoner’s dilemma. Subsequently, Kraft (2001) extends his previous work by accounting for a general oligopolistic market and discussing the effects of employment bargaining from both theoretical and empirical perspectives, confirming the existence of a prisoner’s dilemma for a sizable range of the union’s bargaining power. Three other contributions study the effects of R&D activities in codetermined firms. First, Granero (2006)⁴ shows that co-determination could help a firm to increase market share, employment and innovation. Second, Kraft et. al. (2011)⁵ built on a work mixing theoretical and empirical analyses and concluded, by taking the number of patents as a benchmark, that results “do not support the view that co-determination slows down technological progress and reduces innovativity” (Kraft et al., 2011, p. 145). Third, Fanti et. al. (2018) find that the main conclusions of Kraft (1998) and Kraft et al. (2011) may not be robust to a more general setting with horizontal product differentiation. Finally, Fanti & Gori (2019) extend the study of co-determination to a price competition setting with network effects and Gori and Fanti (2022) introduce endogenous co-determination in a Cournot duopoly.

On the side of the empirical evidence, the results of the existing literature on the subject are controversial.⁶ Amongst a few, we recall here that Cable and FitzRoy (1980) find that co-determination positively affects labour productivity, FitzRoy & Kraft (1993) obtain no statistically significant evidence of productivity gains due to co-determination laws, Baums & Frick (1998) study how the behaviour of the German courts during the period 1974–1995 on co-determination issues affects the stock price developments, finding no statistically significant stock market response to court verdicts. More recently, Gorton & Schmid (2004) and FitzRoy & Kraft (2005) respectively pinpoint (1) the negative effect caused by co-determination on the market value of firms, and (2) the positive labour productivity effect of near-parity co-determination, whereas Kraft (2018) considers a model to study empirically the effects of extending co-determination rights on both productivity and bargaining power. He pinpoints no productivity disadvantages of codetermined firms. More generally, increasing co-determination rights appears to be neutral on the side of efficiency. However, it positively affects the bargaining power of labour and modifies the distribution of rents. To sum up, notwithstanding the empirics on co-determination is still a small field and there exist results of opposite signs, co-determination often worsens the performance of firms (productivity, market value, etc.). This is indeed coherent with the theoretical result of the Pareto-inefficiency of the SPNE following the co-determination strategy.

However, a recent authoritative study by Jäger et al. (2021) provides quasi-experimental evidence showing that (mandatory) co-determination has no effects on the wage structure, the labour share, revenues, employment and firm profitability. Then, given the main features of co-determination and the results of Jäger et al. (2021), it seems that co-determination rules lose their effectiveness as devices affecting the behaviour of the firms in the market, possibly being harmful by reducing the labour demand and driving up involuntary unemployment. We can therefore ask whether the theoretical model developed in the present article can capture the key features of co-determination and/or whether there exist significant objective differences between profit maximisation and co-determination. Indeed, one of the goals of this work is to continue giving theoretical support (since Kraft, 1998) to the topic of co-determination by considering firms with market power in a strategic context. In this sense, our contribution should be interpreted to provide a narrative in which co-determination rules are not mandatory, but voluntarily emerge as a device working exactly as the managerial delegation contracts in the models à la Vickers (1985), Fershtman & Judd (1987) and Sklivas (1987), VFJS, in turn, (1) stressing the conditions under which must be applied because it is convenient to firms becoming bargainers under co-determination instead of profit maximisers, and (2) solving the prisoner’s dilemma raised in Kraft (1998), in turn achieving mutually beneficial outcomes. The main difference between a profit-maximising firm and a codetermined firm in the strategic framework adopted here stands about the objective function in the market stage: in the former case, the firm alone chooses production (employment) by maximising profits; in the latter case, the firm chooses production (employment) together the employees’ representatives by maximising a weighted product between profits and the utility function of the firm’s union bargaining unit. From a strategic point of view, this represents a relevant difference, giving rise to an incentive to increase production and reduce profits by opening the route for the emergence of a prisoner’s dilemma. This dilemma can be solved because of the network consumption externalities bringing together Katz & Shapiro (1985) and Kraft (1998), KSK, in the same setting. The implicit assumption behind our model is that each firm can credibly commit to its corporate governance structure in the first stage by bargaining with workers’ representatives, and the co-determination rules are applied as a strategic device (exactly as in the case of managerial firms considering the VFJS modelling setting according to which a firm can credibly commit to its corporate governance structure by strategically hiring a manager in the first stage of a managerial decision game). Each firm, therefore, moves along its reaction function and every choice out of it (including sharing the monopoly output equally, with or without co-determination) does not represent an optimal response to the rival’s choice. Therefore, the commitment power of playing a strategy in a non-cooperative game should be credible (the best reply functions represent the loci of optimal responses), but any other choices out of best replies cannot credibly be implemented with commitment. Unlike the model à la VFJS, co-determination, which can be viewed being part of the agenda of a strategic firm, in a KSK setting can credibly be committed to achieving mutually beneficial outcomes. Some policy implications directly follow. Co-determination can emerge voluntarily in the market and consumers and firms can be better off under co-determination than under profit maximisation; however, the empirical evidence of voluntary co-determination, i.e., in countries in which co-determination is not imposed by law (US for example), is scant so that the fraction of firms with co-determination is negligible. Therefore, our results can be interpreted also in the direction that mandatory co-determination can become a Pareto superior policy from a societal perspective if markets are characterised by network externalities. The present article wants to give the theoretical base to apply this institution efficiently.

Choosing to play the co-determination strategy or the profit maximisation strategy in our non-cooperative game is the same as, for instance, choosing whether to hire a manager as in the VFJS literature, at the early stage of the game. Consequently, the game developed in this article resembles the managerial decision game in which each player credibly chooses to hire or not to hire a manager, and this choice is common knowledge and undeniable. This is a standard assumption of most multi-stage games in the industrial economics literature, e.g., the R&D investment, the managerial decision game, the mode of competition game (Cournot versus Bertrand) and the co-determination decision game. A player’s choice therefore (to play X or to play NX) is always reliable with full commitment. Thus, the choice to collude is not part of this non-cooperative game that indeed follows the structure of any game without binding contracts. To sum up, suppose that a firm can make only two types of binding contracts with their trade union bargaining units: the co-determination contract and no contract (i.e., the profit maximisation contract). Choosing the former implies that the firm will maximise an objective function including a weighted product between profits and its (decentralised) trade union utility, whatever action the competitor takes. Choosing the latter implies that the firm is committing to maximising profits irrespective of the action of the rival. Then, players in two-stage game first simultaneously commit themselves to a type of contract (stage 1) and compete on quantities in the market stage (stage 2), contingent on the chosen types of contracts. The co-determination contract strategy and the no contract strategy do not represent binding contracts for the players but are undeniable due to, e.g., the prohibitive costs of changing the type of contract.

Another pillar of modern economies is represented by network consumption externalities.⁷ These kinds of externalities are of increasing importance and there exists a burgeoning Industrial Organization literature pioneered by Katz & Shapiro (1985) which is growing rapidly. For instance, in an oligopoly context, Hoernig (2012), Bhattacharjee and Pal (2014), and Chirco & Scrimitore (2013) show that the standard results of the managerial delegation literature may change when markets deal with network goods, whereas Fanti and Buccella (2017, 2018) investigate whether and how the network effects may modify the common wisdom regarding the bargaining agenda (between unions and firms) and corporate social responsibility. More recently, Buccella et al. (2022) and Choi & Lim (2022) respectively extend the R&D investment decision game and the commitment decision game to a network industry with linear demand pinpointing the key role of the network strength in determining the SPNE.

Empirical evidence about network effects for industries located in countries with the institution of co-determination (e.g., Germany) also exists. For example, by focusing on the specific case of telecommunications,⁸ Doganoglu and Grzybowski (2007) account for network effects in the German mobile telecommunication market by estimating a system of demand functions for mobile subscribers in Germany (data on mobile subscriptions was collected from the Internet site run by the German regulator—RegTP) from January 1998 to June 2003. They found that network effects played a significant role in the diffusion of mobile services in Germany. More specifically, they conclude that, as a proxy that measures the intensity of the network effect, if the previous period’s total installed base increased by 1%, current period sales would surge on average by 0.69% (which is considered a strong network effect).⁹ Another example is represented by the work of Baraldi (2008) considering 30 OECD countries from 1989 to 2006 to specify and estimate a model of consumer demand for mobile telephone calls aimed at identifying the extent of network externalities. This work shows that also for countries such as Austria and Germany the network effect is significantly large (though smaller than that found by Doganoglu & Grzybowski, 2007), thus confirming that the competition analysis under co-determination in network industries (such as the telecommunications industry) should consider the extent and intensity of network effects.

Given this empirical background, different labour market institutions may affect the consumers’ expectations about the total sales of the (network) goods. Despite the possible theoretical and empirical relevance of positive consumption externalities on market outcomes, the issues related to network goods have been ignored in the literature on co-determined industries. This article aims to fill this gap by providing a theoretical analysis based on a strategic competitive framework with quantity-setting duopoly firms.

The article differentiates between exogenous and endogenous co-determination in a network Cournot duopoly. In the former case, results show that (exogenous) co-determination emerges as a Pareto-efficient SPNE. This, in turn, implies that network goods solve the prisoner’s dilemma raised in Kraft (1998) by letting the co-determination decision game move from a prisoner’s dilemma (there is a conflict between self-interest and mutual benefit of co-determination) to an anti-prisoner’s dilemma or deadlock (no conflict exists between self-interest and mutual benefit of co-determination). This holds in the cases of homogeneous and heterogeneous trade unions or bargaining strength. In the latter case (endogenous co-determination), results extend Gori & Fanti (2022) to a network industry by enriching the spectrum of Nash equilibria that can correspondingly emerge, including the anti-prisoner’s dilemma that cannot be observed in the non-network case.

The rest of the article proceeds as follows. Section 2 builds on a non-cooperative co-determination decision game (exogenous co-determination with homogeneous trade unions or bargaining strength) played in a quantity-setting network duopoly with homogeneous goods. Section 3 extends the model to horizontal product differentiation. Section 4 introduces endogenous co-determination in a network industry by extending Gori and Fanti (2022). Section 5 studies the co-determination decision game (exogenous co-determination) by assuming heterogeneous trade unions. Section 6 concentrates on welfare analysis. Section 7 outlines the main conclusions. The Appendix provides some analytical details.

This section aims at studying a Cournot duopoly considering a non-cooperative two-stage co-determination decision game in which in the first, decision-making stage each owner must choose to be either a codetermined or profit maximisation firm (exogenous co-determination with homogeneous bargaining strength), whereas in the second stage, they compete à la Cournot in the product market.

The economy consists of two types of agents, firms and consumers. It is bi-sectorial with a competitive industry producing the numeraire goods \(m\) and a duopolistic industry in which firm \(i\) and firm \(j\) produce horizontally differentiated products of variety \(i\) and variety \(j\), respectively \(\left( {i,j = \{ 1,2\} ,\;i \ne j} \right)\).

Different from the traditional industrial organisation literature (that assumes that a demand for a good is independent of one another), we assume that there are network externalities in consumption. This implies that one person’s demand also depends on the demand of other consumers. The mechanism of network effects considered in the present article work follows the tradition initiated by Katz & Shapiro (1985) so that the surplus that a firm’s client obtains increases (resp. reduces) with the number of other clients of this firm if the network consumption externality is positive (resp. negative).

The issue of network externalities on the side of consumers has become relevant especially due to the tremendous growth of internet-related activities (e.g., online games, telephone and so on). We pinpoint that in what follows we will use the terms “network externality”, “consumption externality” or “network consumption externality” interchangeably. In the words of Katz and Shapiro (1985, p. 424): “… the utility that a given user derives from the good depends upon the number of other users who are in the same “network” as is he or she. The scope of the network that gives rise to the consumption externalities will vary across markets…”.

The model used in the present article directly departs from Katz and Shapiro (1985), which has been the base for more recent contributions on the subject, such as Hoernig (2012), Chirco & Scrimitore (2013), Song & Wang (2017), Buccella et al. (2022) and Choi & Lim (2022), who consider the representative consumer approach with quadratic utility and linear demand in markets with network goods. Following the model proposed by Katz and Shapiro (1985) in the main text of their article (in which firms do not commit to an announced output level), the utility function of consumers (and then their marginal willingness to pay) depends on expected network sizes (network \(i\) and network \(j\), \(i,j = \{ 1,2\}\), \(i \ne j\)). Consumers choose to buy the product of network \(i\) or product of network \(j\) before the actual network sizes are known to them. This implies that consumers first form the expectations about the size of networks and then the duopolistic firm \(i\) and firm \(j\) play a non-cooperative co-determination decision game based on Cournot rivalry. This is done by considering consumers’ expectations as given. Consumers are rational and their expectations are realised in equilibrium.

This section extends the results of Sect. 2 to the case of horizontal product differentiation \(\left( { - 1 < d < 1} \right)\). Therefore, the inverse market demand for product of variety \(i\) is expressed by Eqs. (2a, 2b). Tables 6 and 7 summarise the equilibrium values of quantity and profits in this case.

Let \(n_{a} (\beta ,d)\), \(n_{b} (\beta ,d)\) and \(n_{c} (\beta ,d): = 1 - \frac{\sqrt \beta }{{1 + d}}\) be three threshold values of \(n\) such that \(\Delta_{a} = \Pi_{i}^{B/PM} - \Pi_{i}^{PM/PM} = 0\), \(\Delta_{b} = \Pi_{i}^{PM/B} - \Pi_{i}^{B/B} = 0\) and \(\Delta_{c} = \Pi_{i}^{PM/PM} - \Pi_{i}^{B/B} = 0\) for any \(i,j = \{ 1,2\} ,\;i \ne j\), respectively. Then, Lemma 3 and Propositions 2 and 3 clarify the outcomes of the network-co-determination game at stage 1 in the case of heterogeneous products.

The proof of Lemma 3 and Propositions 2 and 3 follows by applying the same line of reasoning used to show Lemma 2 and Proposition 1. Specifically, Proposition 2 shows that product differentiation allows (PM,PM) to become the unique SPNE, i.e., product differentiation works out against co-determination. This is in line with the results obtained by Fanti et al. (2018). However, the interaction between network externalities and product differentiation can bring to light an interesting (and counterintuitive) outcome. As an increase in both the degree of product differentiation and the strength of the network effect allows firms to increase their profits. Indeed, a reduction in \(d\) (increase in the market power) in a network industry can make the profitability of a codetermined firm larger than that of a profit-maximising firm due to the outward shift in the market demand that B promotes compared to PM. In other words, product differentiation strengthens the working of the network effects as a device increasing profits under co-determination. This allows to let (B,B) become the Pareto-efficient outcome of the game for a wider range of values of \(\beta\) and \(n\).

Proposition 4 shows this result, which is strictly related to Propositions 2 and 3 and also illustrated in Fig. 2 by contrasting Panels (A) and (C), related to product substitutability, plotted for \(d = 0.8\) and \(d = 0.5\), respectively, and Panels (B) and (D), related to product complementarity, plotted for \(d = - 0.8\) and \(d = - 0.5\), respectively.

The results obtained in the previous section allow having some policy recipes (mandatory co-determination versus voluntary co-determination) depending on the values of the main parameters of the problem. However, one of the drawbacks of the proposed approach (following the original idea of Kraft, 1998) is an exogenous degree of co-determination (i.e., the strength with which trade unions negotiate with firms). Unlike this, firms might decide to bargain not with any trade union, but with the trade union exerting a bargaining effort just allowing them to maximise profits. Indeed, in actual economies, there may be distinct types of union bargaining units that should not necessarily be appreciated by the firm as part of the bargaining process. To consider this heterogeneity, this section speculates in this direction and extends the model of exogenous co-determination by assuming that each firm is aware of the union’s attitude at the time of bargaining and then chooses to bargain with a union bargaining unit under co-determination only whether the firm’s bargaining power is the profit-maximising one. In doing this, we assume that the firm has the right to choose the composition of the board of representatives (including or not workers’ representatives) to make production decisions. This amounts to say that firms may choose the optimal union’s bargaining effort by choosing the optimal corresponding number of workers’ representatives to be co-opted within the supervisory board.

A rationale for this approach is that even focusing only on Europe, we can observe significantly different degrees of co-determination in different countries. Consequently, a natural motivation for endogenous co-determination is that, in an international context, firms can choose the country where they produce their brands and therefore hire workers, which in practice allows firms to choose the version of co-determination that arises from local labour relationships.

Let us first assume the existence of a continuum of firm-specific unions differentiated amongst them based on their relative attitude to bargaining \(\left( {0 < 1 - \beta_{i} \le 1} \right)\). The research question, which is novel and follows Gori and Fanti (2022), arising in this context is the following. Do firms always prefer to bargain with a trade union with little bargaining power? The answer is not so obvious, and this section aims to show that the strategic interacting effects between the degree of product differentiation and the strength of the network effect may lead a quantity-setting duopoly firm to bargain with a union unit with a sizeable bargaining power, as this choice allows a firm to maximise profits.

The stages of the game change and become the following. At stage 1 (the contract stage), each owner must choose to be either a codetermined or profit-maximising firm. At stage 2 (the union-strength stage) the owner of each firm chooses to bargain with a union bargaining unit only whether its bargaining attitude is exactly the profit-maximising one. At stage 3 (the market stage), firms either choose the quantity in the output market in the case of profit maximisation or bargain it together with unions in the case of co-determination. The game follows the backward induction logic.

We now briefly discuss the key features of a network-co-determination non-cooperative (three-stage) game with quantity competing firms, complete information and endogenous co-determination. Of course, equilibrium outcomes are still those reported in Tables 6 and 7 (Sect. 3) if both firms are profit maximising (PM) so that \(\beta_{1} = \beta_{2} = 1\). When both firms are codetermined (B), the Nash bargaining function \({\rm N}_{i} = \Pi_{i}^{\beta } Z_{i}^{1 - \beta }\) modifies to become \({\rm N}_{i} = \Pi_{i}^{{\beta_{i} }} Z_{i}^{{1 - \beta_{i} }}\). Then, firm 1 aims at bargaining with a type-1 union bargaining unit with an effort or bargaining strength \(\beta_{1}\) to choose the quantity of product of variety 1. Correspondingly, firm 2 bargains with a type-2 union bargaining unit with an effort or bargaining strength \(\beta_{2}\) to produce the quantity of product of variety 2. Then, there will be reaction functions depending on \(\beta_{1}\) and \(\beta_{2}\) that should be used to compute quantities of firm1 and firm 2, in turn, allowing to compute profits of firm \(i\) \(\left( {i,j = \{ 1,2\} ,\;i \ne j} \right)\) as follows:

As each firm chooses to bargain with its union bargaining unit if and only if there exists a profit-maximising bargaining power, we get the following reaction-bargaining-function of firm \(i\), that is:

By using the corresponding counterpart of (20) for firm \(j\), one can get the optimal value of firm \(i\)’s bargaining strength (outcomes are symmetric), that is

The expression in (21) gives all the couples \((n,d)\) such that the owner maximises profits by choosing to be a bargainer under co-determination and augments the result of Gori and Fanti (2022). The expression in (21) is meaningful if and only if \(\beta_{i}^{*(B/B)} \le 1\). This condition implies thatshould hold, otherwise, there would be no economically meaningful profit-maximising value of \(\beta_{i}\). Equation (22) tells us that each firm would decide to be codetermined by choosing a profit-maximising bargaining effort if and only if the network externality is strong enough, otherwise it would prefer to be a profit-maximiser. In the case of positive network externalities, the condition in (21) is meaningful for any \(0 < n < 1\) and \(- 1 \le d < 1\) so that (22) is always fulfilled. In the case of negative network externalities, the condition in (21) is meaningful only whether (22) holds, that is for any \(- n_{\beta } (d) < n < 0\) and \(- \sqrt 3 /2 < d < \sqrt 3 /2\).

By substituting (21) into (19) for \(\beta_{i}\) one gets profits of firm \(i\) under optimal co-determination and network externalities, that is

When firm 1 is codetermined (B) and firm 2 is profit maximiser (PM), firm 1 bargains with type-1 union bargaining unit with an effort \(\beta_{1}\) and firm 2 does not bargain at all \(\left( {\beta_{2} = 1} \right)\). Then, by considering quantities and prices as a function of \(\beta_{1}\) profits of firm 1 and firm 2 are the following:and

The profit-maximising bargaining power \(\beta_{1}\) is the following:

In the case of positive network externalities, the condition in (26) implies that \(\beta_{1}^{*(B/PM)} \le 1\) for any \(0 < n < 1\) and \(- 1 < d < 1\). In the case of negative network externalities, the condition in (26) is meaningful only whether \(- n_{\beta } (d) < n < 0\) and \(- \sqrt 3 /2 < d < \sqrt 3 /2\). By substituting (26) into (24) and (25) for \(\beta_{1}\) one getsand

To sum up, Table 8 summarises the equilibrium outcomes of the optimal bargaining strength in the cases of both symmetric and asymmetric behaviours and Table 9 refers to the corresponding values of firms’ profits (payoff matrix).

Define \(n_{b}^{1} (d) = n_{c}^{1} (d) = 1 - \frac{1}{{\sqrt {1 - d^{2} } }} = n_{\beta } (d)\), and let \(n_{b}^{2} (d)\) and \(n_{c}^{2} (d)\) be two threshold values of the strength of the network effect such that the corresponding profit differentials \(\Delta_{b} = \Pi_{i}^{PM/B} - \Pi_{i}^{B/B} = 0\) and \(\Delta_{c} = \Pi_{i}^{PM/PM} - \Pi_{i}^{B/B} = 0\) \(\left( {i,j = \{ 1,2\} ,\;i \ne j} \right)\). The shape of \(n_{\beta } (d)\), \(n_{b}^{2} (d)\) (red line) and \(n_{c}^{2} (d)\) (black line) is depicted in Fig. 3 in the parameter space \((n,d)\).¹⁷ The red region in the figure refers to the couples \((n,d)\) corresponding to which every firm does not find it convenient to bargain with its trade union under co-determination. Result 4 classifies the outcomes of the network-co-determination game at stage 1, where each owner must choose to be either a codetermined or profit-maximising firm under endogenous co-determination, for the case of positive consumption externality. The case of negative consumption externality is more difficult to disentangle analytically and economically. To this purpose, the red area in Fig. 3 highlights the unfeasible parameter space of optimal co-determination.

The main findings emerging from Result 4 and Fig. 3 (positive and negative externalities) under endogenous co-determination are in line with those obtained under exogenous co-determination. The figure shows that the larger the degree of product substitutability and the larger the network effect, the lower the optimal bargaining effort of the firm needed to maximise profits. The red area represents the unfeasible parameter space of optimal co-determination, where firms behave as profit maximisers and co-determination can be applied only through legislation. In all other cases, co-determination can emerge through voluntary agreements (irrespective of the number of employees). When products are substitutes, a voluntary co-determination agreement is efficient when the strength of the network effect is sufficiently large. However, it is possible to have also multiple mixed Nash equilibria corresponding to which only one firm voluntarily chooses to be codetermined. In this case, no one has a dominant strategy and both equilibria are Pareto-efficient. The solution to the game may emerge from the credible disclosure of a player’s will to not play B. Then, the rival will be forced to (be the first to) play B to avoid obtaining a lower pay-off unilaterally.

To complete the analysis of the network-co-determination game, this section considers the case of exogenous heterogeneous co-determination presented until Sect. 3. The main assumptions that hold here partly follow those presented in Sect. 4. Therefore, under exogenous co-determination with heterogeneous bargaining strength, we assume, unlike Sect. 4, that the firm does not have the right to choose the composition of the board of representatives to make production decisions. This amounts to saying that firms cannot choose the optimal union’s bargaining effort by choosing the optimal corresponding number of workers’ representatives to be co-opted within the supervisory board. Then, like Sects. 2 and 3, each firm takes the bargaining effort as given and then considers the case of exogenous co-determination in a context in which there exists a continuum of firm-specific unions differentiated amongst them based on their relative attitude to bargaining \(\left( {0 < 1 - \beta_{i} \le 1} \right)\). The stages of the game, therefore, turn out to be the same as those used to solve the games presented in Sects. 2 and 3.

The payoff matrix is determined by the following equations:and

Given Eqs. (29)–(31), the outcomes of the network-co-determination game with exogenous co-determination and heterogeneous bargaining strength at stage 1 are summarised in Figs. 4, 5, 6. The figures are depicted in the space \((\beta_{1} ,\beta_{2} )\) for different values of \(n\) and \(d\), showing the loci of points such that (1) the profit differential of firm 1 \(\Delta_{1,a} = \Pi_{1}^{B/PM} - \Pi_{1}^{PM/PM} = 0\), i.e., \(\beta_{1,a} (n,d)\), and the profit differential of firm 2 \(\Delta_{2,a} = \Pi_{2}^{PM/B} - \Pi_{2}^{PM/PM} = 0\), i.e., \(\beta_{2,a} (n,d)\), where \(\beta_{1,a} (n,d) = \beta_{2,a} (n,d) = \beta_{a} (n,d) = \frac{{(1 - n)^{2} [1 + (1 - n)(1 - d^{2} )]^{2} }}{{(2 - n)^{2} }}\), and (2) the profit differential of firm 1 \(\Delta_{1,b} = \Pi_{1}^{PM/B} - \Pi_{1}^{B/B} = 0\), i.e., \(\beta_{1,b} (\beta_{2} ,n,d)\), and the profit differential of firm 2 \(\Delta_{2,b} = \Pi_{2}^{B/PM} - \Pi_{2}^{B/B} = 0\), i.e., \(\beta_{2,b} (\beta_{1} ,n,d)\), where \(\beta_{1,b} (\beta_{2} ,n,d) = \frac{{(1 - n)^{2} [\beta_{2} + (1 - n)(1 - d^{2} )]^{2} }}{{(1 - n + \beta_{2} )^{2} }}\) and \(\beta_{2,b} (\beta_{1} ,n,d) = \frac{{(1 - n)^{2} [\beta_{1} + (1 - n)(1 - d^{2} )]^{2} }}{{(1 - n + \beta_{1} )^{2} }}\).

Figure 4 represents the non-network industry \(\left( {n = 0} \right)\) and contrasts the cases of perfect substitutability \(\left( {d = 1} \right)\), Panel A, with the cases of imperfect substitutability \(\left( {d = 0.7} \right)\) and complementarity \(\left( {d = - 0.7} \right)\), Panel B, which are symmetric in the emergence of Nash equilibria. Figure 5 represents a network industry with positive consumption externalities \(\left( {n = 0.5} \right)\) and contrasts the cases of perfect substitutability \(d = 1\), Panel A, with the cases of imperfect substitutability \(\left( {d = 0.7} \right)\) and complementarity \(\left( {d = - 0.7} \right)\), Panel B, which are symmetric in the emergence of Nash equilibria. Figure 6 represents a network industry with negative consumption externalities \(\left( {n = - 0.5} \right)\) and contrasts the cases of perfect substitutability \(d = 1\), Panel A, with the cases of imperfect substitutability \(d = 0.5\) and complementarity \(\left( {d = - 0.5} \right)\), Panel B, which are symmetric in the emergence of Nash equilibria.

The figures clearly show, as expected from the results of previous sections, that (1) for a given value of the network strength (including the case of non-network industry) product heterogeneity (by increasing the firm’s market power) favours the emergence of (PM,PM) as the unique Nash equilibrium of the game, (2) for a given value of the extent of product differentiation a positive consumption externality (by shifting outward the market demand) favours the emergence of (B,B) as the unique Nash equilibrium of the game, and (3) for a given value of the extent of product differentiation a negative consumption externality (by shifting inward the market demand) favours the emergence of (PM,PM) as the unique Nash equilibrium of the game. In all the cases, a high degree of heterogeneity between the bargaining strength of firm 1 (union 1) and firm 2 (union 2) favours the emergence of asymmetric Nash equilibria in which only one firm chooses to be a codetermined entity. The codetermined firm will be the one in which the union is bargaining with the lower bargaining power. In this case, the firm is incentivised to follow the union’s request to increase employment and production.

The welfare analysis is conducted by considering the most analytically tractable model, i.e., exogenous homogeneous co-determination. We pinpoint, however, that the results and the rankings presented in this section hold also for the case of exogenous heterogeneous co-determination and endogenous co-determination.

The classical notion of social welfare \(\left( W \right)\), which is also the one employed in the pioneering work of Kraft (1998), is \(W = CS + PS\), where \(CS = \frac{1 - n}{2}\left( {q_{1}^{2} + q_{2}^{2} + 2dq_{1} q_{2} } \right)\) and \(PS = \Pi_{1} + \Pi_{2}\). However, as also pinpoint in Kraft (2006), an alternative measure of social welfare, including the utility functions of each trade union, can safely be used (see also Buccella et al., 2023). In this case, therefore, social welfare would be given by \(W = CS + PS + Z\), where \(Z = Z_{1} + Z_{2}\). For analytical tractability, we employ the simplest definition, but results hold a fortiori by including the utility of the trade unions.

The consumers’ surplus, producers’ surplus and social welfare under (PM,PM) are respectively given by:and

The consumers’ surplus, producers’ surplus and social welfare under (B,B) are respectively given by:and

The results emerging from a direct comparison between (36) and (39), (37) and (40), and (38) and (41) are the following: \(CS^{B/B} > CS^{PM/PM}\), \(PS^{B/B} > PS^{PM/PM}\) if and only if \(n > n_{c} (\beta ,d)\), as was already pinpointed in the article, and \(W^{B/B} > W^{PM/PM}\).

By also considering the asymmetric case (B,PM) (or (PM,B)) one can show that the ranking \(W^{B/B} > W^{B/PM} > W^{PM/PM}\) always holds in all the models of the article.

This definitively implies that mandatory co-determination results in a Pareto superior allocative scenario compared to neoclassical profit maximisation when (B,B) emerges as the unique Pareto-efficient Nash equilibrium of the network-co-determination game played by quantity-setting firms. In this regard, the role of the network effect is relevant as this scenario holds when the (positive) network externality is sufficiently high.

The present article extends the strand of research dealing with the institution of co-determination by considering network externality in consumption by considering Katz & Shapiro (1985) and Kraft (1998) in the same setting. Co-determination is an institution that plays a significant role in the protection of workers’ rights and the improvement of working conditions. Amongst other North European countries, it has become relevant in the German industry since at least 1976 (The German Co-determination Act), extending co-determination rules to all industries and firms with more than 2000 employees, by also affecting the designing of German industrial policy.

In modern economies, industries producing goods that generate positive consumption externalities has become relevant (e.g., software, mobile phone) and work in the direction of increasing the quantity available in the market, in turn, also expanding the market size (or reducing it in the case of negative network effects).

The empirical and historical evidence has shown that production decisions/incentives are different between profit maximisation and co-determination. The empirical literature on co-determination has shown that quantity, prices, profits and wages under co-determination are different than under profit maximisation. Specifically, productivity, profits, stock returns, wages and employment adjustment, labour share, bargaining power and other relevant variables on the firm side have been widely analysed by many studies such as Svejnar (1982), FitzRoy & Kraft (1993, 2005), Kraft & Ugarkovic (2006), Gorton & Schmid (2004), Fauver & Fuerst (2006) and Kraft (2018). For instance, Gorton & Schmid (2004, p. 895) pinpoint that co-determination appears “to succeed in altering the objective function of the firm”, Fauver and Fuerst (2006, p. 677) state that “… our analysis suggests that the judicious use of labor representation can increase firm value”, whereas Kraft (2018) finds that co-determination leads to a significant increase in workers’ bargaining power and the distribution of rents. Unlike these studies, Jäger et al. (2021) provide quasi-experimental evidence to show no effects of co-determination on the wage structure, the labour share, revenue, employment or profitability.

The present article shows that network externalities play a relevant role in a strategic competitive quantity-setting duopoly à la Kraft (1998) with codetermined firms. This is because co-determination per se allows firms to produce more than standard profit maximisation and a network externality broadens this effect. Unlike Kraft (1998), who showed that co-determination is a Pareto-inefficient SPNE (prisoner’s dilemma) in a non-network market with homogeneous products, this article identified a solution to the dilemma by letting co-determination become a Pareto-efficient SPNE of a network industry in a quantity-setting duopoly. The result is also extended to the case of horizontal product differentiation. Finally, as co-determination enhances the consumer’s surplus and the workers’ utility, we must emphasise that this institution may represent a Pareto-superior policy in comparison to profit maximisation. Therefore, in network industries co-determination may voluntarily emerge as the endogenous (Pareto-efficient) outcome of a Cournot strategic competitive framework, thus coming not only from legislative rules. Alternatively, mandatory co-determination can become a Pareto superior policy from a societal perspective.

The economic reasons for the main results are clear: as in the previous literature, the key strategic effect of co-determination is that each codetermined firm produces a higher output for any given output of its opponent. The reason is that, due to co-determination, employee representatives are part of each firm’s supervisory board. Formally, this is captured by assuming that co-determination changes the objective function of the firms from conventional profits to a Nash bargaining maximand that is made up of both profits and workers’ labour income. As workers want to increase employment for any given wage, each codetermined firm will have the incentive to increase its output level accordingly. Hence, relative to a situation without co-determination, the codetermined firms will end up producing more output, then the market price will fall, and profits will fall as well. This implies that co-determination will lead to a more competitive outcome. In this context, the novelty of the paper is to explore the impact of network externalities on consumption. The results from the analysis point to these externalities as relevant. The main results suggest that (positive) consumption network externalities can preserve the positive impact of co-determination on consumer surplus but, at the same time, they can also contribute to higher profits. This is because the (positive) externality on the demand side contributes to expanding the market demand and makes consumers willing to pay a higher price.

Results hold for the cases of (1) exogenous homogeneous or heterogeneous co-determination, according to which firms cannot choose the composition of the board of representatives but can bargain with decentralised unions having identical or different bargaining efforts, which is taken as given in the bargaining between the two parties, and (2) endogenous co-determination, according to which firms can choose the composition of the board of representatives (including or not workers’ representatives) by choosing the bargaining effort that maximises its profits and then correspondingly choosing the optimal number of workers’ representative to be co-opted within the supervisory board.

Definitively, in network industries co-determination emerges as an institution that harmonises labour relationships and improves social welfare. To the extent that the network industries are becoming predominant in the economies, “co-determination” may be the “natural” institution of the labour market, which—from a societal point of view—is Pareto-superior to the neoclassical institution. Therefore, the policy implications are obvious and far-reaching. Though the empirical relevance of voluntary co-determination is indeed low, our results do not refer to the case of voluntary co-determination as an institution being observed in the market, but rather that mandatory co-determination results in higher social welfare than profit maximisation. This is because consumers are better due to an increase in output (in turn increasing consumer surplus) and this benefit more than offsets the reduction in the firms’ profits. However, before our work, this institution was considered harmful to firms as it contributed to reducing their profitability. Instead, in markets with positive consumption externalities co-determination can be beneficial also to firms (Pareto-efficient SPNE), in turn, resulting in a win–win solution for society. Consequently, mandatory co-determination may be Pareto superior to profit maximisation.

Thus, co-determination voluntarily emerging in non-network markets from non-cooperative games such as those by Kraft (1998) and Fanti et al. (2018) represents a Pareto-inefficient SPNE for firms. Unlike this, the present article provides the conditions to let it become a Pareto-efficient outcome. Accordingly, our theoretical prediction is that co-determination can be a strategic variable enabling an efficient outcome in network markets. In this sense, our work may offer a future research direction for econometricians to study whether in network markets the presence of voluntary co-determination may be less negligible than in standard non-network markets. The point, therefore, is not that the fraction of firms with voluntary co-determination is negligible but rather that the fraction of firms with voluntary co-determination in network markets is less negligible than in standard non-network markets.

The article complements the analysis of co-determination and price competition in a network industry developed by Fanti & Gori (2019). However, considering the competition (or mode of competition) game played by codetermined firms that choose whether be quantity or price competitors, the outcome that endogenously emerges is always the quantity competition.

The potential future development of this study can follow, e.g., the framework developed by Yang et al. (2015) about manufacturers’ channel structures. In this regard, the theoretical analysis of co-determination in strategic competitive industries is still silent thus representing a promising research agenda.