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We investigate the welfare effect of union activity in a relatively new oligopoly model, the Cournot-Bertrand model, where one firm competes in output (

Union membership in the United States has steadily declined since its peak in the 1950s. A growing number of states have adopted laws that restrict union power, and recent years have seen intense debates over policies such as right-to-work laws, which allow employees at unionized firms to opt out of paying union dues. In order for policymakers to weigh the costs of inefficiencies that can be created by unions against concerns for equity and redistribution, it is important to understand the differential impact of union power in various market structures.

In this paper, we investigate the effect of union activity in duopoly markets and the conditions under which union power has the greatest impact on allocative efficiency. We consider both Cournot and Bertrand models, but our main emphasis is on a relatively new model, the Cournot-Bertrand model. In this case, one firm competes in output (as in Cournot) and the other firm competes in price (as in Bertrand) within the same industry. The main reason for considering this model is that it produces an outcome where firms are heterogeneous, a common feature in the real world. In addition, Tremblay, ^{1}

We consider the three principle bargaining rules, or models of union behavior: the monopoly union (MU), the right-to-manage (RTM), and the efficient bargaining (EB) models. The MU model assumes that firms and unions compete in a two stage game. In the first stage, the union chooses the wage rate that maximizes its objective function, which is generally defined to be a function of wages and employment. In the second stage, each firm observes the union wage and maximizes profit with respect to employment. In the subgame-perfect Nash equilibrium to this game, the union correctly anticipates the firm’s best reply to its wage offer, which is the firm’s labor demand, and then chooses the optimal wage. The firm then maximizes profit, subject to the wage that is set by the union.

The RTM model is similar to the MU model, except that the firm and union collectively bargain over the wage rate in the first stage. This is an attractive feature of the model, as the wage is normally determined through collective bargaining between firms and unions. This model allows us to introduce a parameter that captures the relative bargaining power of firms

In response, the EB model of union behavior was developed. In this case, the firm and union simultaneously bargain over employment and the wage in order to maximize the joint returns of the union and firms. This produces a Pareto efficient wage-employment outcome. The traditional criticism of this model is that the structure of the game is not consistent with reality: firms and unions generally bargain over wages and not employment. A more fundamental problem that has not been discussed in the literature is that when the elasticity of substitution between labor and other inputs is low, an EB agreement forces firms to compete in a Cournot type game. With the extreme Leontief (fixed-proportion) technology, for example, setting the level of employment effectively sets the level of output. With output determined, firms cannot compete in price as Bertrand competitors.

This raises the fundamental question: why should firms compete in output or in price? Both types of actions are observed in the real world. Kreps and Scheinkman [^{2}

Since Cournot competition allows any of the three types of bargaining, a Cournot-type firm may prefer efficient bargaining, which maximizes joint returns. On the other hand, a Bertrand-type firm facing Leontief technology would not be able to take advantage of efficient bargaining without incurring a cost. Such a firm would have to switch to Cournot-type behavior, which could be too expensive. Thus, Bertrand-type firms may prefer to deal with a monopoly union or within a right-to-manage framework.

This may explain why the empirical evidence does not provide unanimous support for any one model of union bargaining. Many empirical studies pool data from a variety of industries, but industries vary considerably in their technologies, demand conditions, and the strategies chosen by firms. In addition, each union model has different strengths and weaknesses. For example, the transaction cost of bargaining would be considerably higher with efficient bargaining compared to a monopoly union framework. Thus, the optimal type of firm-union bargaining is likely to differ by industry. It is also likely to vary across states and countries with different regulations and attitudes about union activity.

These arguments motivate our work on firm-union behavior. We begin by analyzing the familiar MU, RTM, and EB models in a Cournot duopoly market and the MU and RTM models in a Bertrand duopoly market. Then we analyze firm-union bargaining in a Cournot-Bertrand market. We investigate the welfare and policy implications of each of these models. We show that the results are sensitive to the structure of labor and output markets, and that policies that decrease union bargaining power will have more of an effect on allocative efficiency under certain market structures.

Consider a market with two firms (1 and 2) that bargain over a wage contract with a single (industry wide) union. Product differentiation exists and is driven by consumer taste for variety (Beath and Katsoulacos, [^{3}
_{1}(_{1},_{2}) = 1 − _{1} − _{2}
_{2}(_{1},_{2}) = 1 − _{2} − _{1}
_{i}_{i}

Firms have no monopsony power and face the same Leontief technology. Capital and other inputs are assumed to be in sufficient supply so that firm _{i}_{i}_{i}_{i}_{i}_{i}_{i}q_{i}_{i}

The objective of unions remains largely unknown (Lawson, [_{o})_{1} + _{2} (or _{1} + _{2}), _{o} is the competitive wage or the opportunity cost of labor. If the union has no bargaining power, then the equilibrium wage will equal _{o}. For simplicity, _{o} is set to 0.^{4}

We use this framework to investigate the MU, RTM, and EB models when firms compete in a duopoly output market. We begin by considering the traditional Cournot and Bertrand models. These models are well known and provide results that are used to compare with those of the Cournot-Bertrand model. For the remainder of this section, we assume that each firm’s choice of strategic variable (output or price) and the type of firm-union bargaining rule (MU, RTM, or EB) are exogenously given. The effect of endogenizing these choices will be discussed in

In the first two firm-union bargaining models, players compete in a two-stage game. The wage is determined in the first stage (I), using either a MU or RTM bargaining rule. In the second stage (II), firms simultaneously compete in output. In the EB model, the wage and firm levels of employment (and output) are chosen in the first stage of negotiation, effectively eliminating Stage II.

We first consider the MU model. We calculate the subgame-perfect Nash equilibrium (SPNE) to this game using backwards induction. In Stage II of the game, firms simultaneously choose output. The Nash equilibrium (NE) in this stage game is:^{5}
_{i}_{i}

In terms of notation, superscript

Next, we consider the RTM model where the union and firms collectively bargain over the wage rate in Stage I. The NE in Stage II is the same as in (3). In Stage I, the union and firms correctly anticipate the best-replies in Stage II and use this information to bargain over ^{6}
^{a}_{1} + _{2})^{(1 − a)}
^{7} When

Finally, we consider the EB model, where all negotiations take place in Stage I. That is, players maximize Equation (5) with respect to _{1} = _{1,} and _{2} = _{2}. The fixed-proportion technology forces firms to compete in output (Cournot) in Stage I and eliminates Stage II of the game. The NE to this game is:

In this section, we assume the same game structure as above, except that firms compete in price instead of output in Stage II. This eliminates the possibility of efficient bargaining, however, because such bargaining would dictate output levels and make Bertrand (price) competition impossible. As a result, only the MU and RTM models are relevant in the Bertrand framework.

In the Bertrand model, the inverse demand Equations (1) and (2) must be transformed so that the choice variables are on the right-hand side of each equation. This produces the following demand function for firm ^{8}

In Stage I, the wage rate is determined. With the MU model, the union correctly anticipates the firms’ NE behavior in the future period and maximizes U(

In the RTM model, the union and firms collectively bargain over the wage rate in Stage I. The NE in Stage II is unchanged, and the Nash bargaining equation is the same as Equation (5). In Stage I, the SPNE is solved by maximizing (5) with respect to

The model of interest that has not been investigated in the union context is the Cournot-Bertrand model. Here, Firm 1 competes in output, as in Cournot, while Firm 2 competes in price, as in Bertrand. The other characteristics of the game are the same as in the Cournot and Bertrand cases. Because one firm competes in price, efficient bargaining is not possible with the Cournot-Bertrand model.^{9}

Operationally, the inverse demand Equations (1) and (2) are transformed so that the choice variables are on the right-hand side of each equation. This produces the following demand system:
_{1}(_{1}, _{2}) = (1 − ^{2})_{1} + _{2}
_{2}(_{1}, _{2}) = 1 − _{2} − _{1}

In each model, the SPNE is calculated using backwards induction. In Stage II of the game, Firm 1 chooses output and Firm 2 chooses price, decisions that are made simultaneously. The NE of this stage game is:^{10}

Consistent with the analysis of the Cournot-Bertrand model without a union (Tremblay and Tremblay [

In Stage I, the players correctly anticipate future NE behavior. With the MU model, the union maximizes U(

In the RTM model, there is collective bargaining over the wage rate in Stage I, based on the Nash bargaining Equation (5) and the anticipated best replies in Stage II. The SPNE is:

Because this is the first application of the Cournot-Bertrand model in the presence of a union, we discuss the results in some detail. First, the degree of product differentiation has a dramatic effect on the outcome. When ^{11}

Second, union bargaining power affects equilibrium values. As in the Cournot and Bertrand models, union welfare increases and firm profits decrease with union bargaining power (i.e., as ^{12} This implies that the union is better able to extract rents from more profitable firms.

In this section, we discuss the effects of different union bargaining rules (MU, RTM, and EB) and different output models (Cournot, Bertrand, and Cournot-Bertrand) on firm, union, and social welfare. Our results help explain why more than a single bargaining rule may be used in the real world. We also consider the conditions under which union power has a greater or lesser effect on allocative efficiency.

Two comparative static results hold in all of the models discussed above. First, both firms and the union benefit from product differentiation. Product differentiation increases the market power of firms, creating greater exploitable profit opportunities for all players. Second, an increase in the bargaining power of the union (an increase in

Regarding the output market, consider the case where there is no union and firms have the choice of competing in output or price. In models with the demand and cost structures described above, the dominant strategy is to compete in output (Singh and Vives [^{13} To illustrate, when ^{14} Firms that compete in the Bertrand game and the Bertrand-type firm (Firm 2) in the Cournot-Bertrand game earn the lowest profit.

Nevertheless, there are several reasons why it may be more profitable for one or both firms to compete in price instead of output. As discussed in the introduction, the Kreps and Scheinkman [

In the models discussed above, the union has a different preference ordering over bargaining rules than do the firms. In the RTM model, for example, firm welfare is always highest with Cournot behavior, while union welfare is always highest for Bertrand behavior and lowest for Cournot behavior. In addition, the union is not always better off with efficient bargaining (assuming no additional transfers from firms). To illustrate, when ^{B–RTM}^{CB–RTM}^{C–RTM}^{C–EB}^{15}

These firm and union welfare results help explain why the empirical evidence does not clearly support any one model of union bargaining. Clearly, players benefit collectively from using efficient bargaining, but this requires Cournot behavior when technology is close to being fixed-proportion. Thus, we may not observe efficient bargaining in all unionized markets, especially where there is Bertrand competition. First, Bertrand-type firms will prefer MU or RTM over EB if the cost of switching from Bertrand to Cournot behavior is sufficiently high. In addition, our model demonstrates that there are cases where unions prefer Bertrand-type markets.

Another factor that would affect union activity is the degree of heterogeneity within a given industry, as in the Cournot-Bertrand model. With imperfect and incomplete information, greater heterogeneity between firms within an industry would raise the transaction cost of reaching a collective bargaining agreement. In an industry such as this, MU and RTM bargaining, which have only one bargaining instrument (_{1}, and _{2}). In this case, MU or RTM bargaining may be preferred over EB by firms and the union.

The final welfare issue involves the effect of union activity on allocative efficiency. Two conditions guarantee allocative efficiency in output and input markets. First, the union has no bargaining power. Second, firms have no market power (as buyers of labor services or sellers of output). In the input market, this causes w to equal the competitive wage, which is 0 in our model. In the output market, this causes price to equal marginal cost. With the demand and cost functions used above, the market is allocatively efficient when industry production is ^{AE}^{2}). As expected, Bertrand competition is more socially efficient than Cournot-Bertrand and Cournot competition.^{16} In the RTM model, when ^{AE}^{B–RTM}^{CB–RTM}^{C–RTM}^{17}

Industry production falls as union bargaining power increases, except in the efficient bargaining model, where industry output is unaffected by union power. Thus, the presence of a union cannot improve efficiency in this model.^{18} The degree of inefficiency generated by union power differs, however, depending on market structure. Consider the total quantity, ^{AE}

These derivatives are plotted on

The impact of union power on output, according to market structure and product differentiation.

In this paper, we analyze market outcomes in oligopoly markets when firms face an industry wide union. Three popular models of union behavior are considered: the monopoly union, right-to-manage, and efficient bargaining models. We investigate three output market models: the Cournot, Bertrand, and Cournot-Bertrand models. This is the first paper to investigate the effect of union activity in a Cournot-Bertrand setting.

Three main results emerge from this study. First, efficient bargaining produces the best outcome for firms and the union combined, suggesting that one should only observe efficient bargaining in the real world. Yet, the empirical evidence does not clearly support the efficient bargaining model over other models. Previous studies and our analysis help explain this result. In an industry with a relatively fixed-proportion technology, efficient bargaining forces firms to compete in output (as in Cournot). From the firm’s perspective, however, there are market conditions under which Bertrand or Cournot-Bertrand competition is preferred to Cournot competition (Kreps and Scheinkman, [

Finally, even though firms bargain with a single union and face the same demand and cost conditions, the equilibrium outcome is asymmetric in the Cournot-Bertrand model. The Cournot-type firm produces greater output and earns greater profits than the Bertrand-type firm. This is in contrast to the Cournot and Bertrand models where firms are symmetric in equilibrium. Because the cost of reaching a firm-union agreement may be higher with efficient bargaining and in markets with more heterogeneous firms, firms and unions alike may prefer a monopoly union or a right-to-manage bargaining rule over efficient bargaining in markets with greater diversity.

From a policy perspective, the main channel through which lawmakers can adjust the equity-efficiency balance of union activity is by changing the relative strength of union bargaining power. Policymakers may wish to increase union power to address inequality. For example, because the evidence shows that market power contributes to income or wealth disparities, the government may wish to promote unions as a way of re-distributing these gains from the wealthy to the poor (see Tremblay and Tremblay [

There are a variety of policies that lawmakers use to influence union bargaining power. As mentioned above, right-to-work laws that allow employees to choose not to join a union or pay union dues, even when they benefit from union negotiations, have been enacted in several states, and were the subject of a heated controversy recently in Michigan. These laws can be interpreted as weakening union bargaining power. Binmore,

If policymakers wish to promote unions for equity reasons, they may prefer to do so in markets in which union power has less of an effect on output. In our models, union power is the least distortionary in Cournot markets, in that a change in union power has the smallest effect on industry output in the Cournot model. Unions decrease output by the greatest amount in Bertrand markets. Thus, industries that typically compete as Cournot, such as heavy manufacturing, are more efficient places from society’s perspective to increase union power. In Bertrand-type industries, such as software and telecommunications, the efficiency loss from increasing union power to address equity issues will be greatest. Industries that compete as Cournot-Bertrand have an intermediate impact on allocative efficiency, but approach the Bertrand result when the degree of product differentiation is extremely low and extremely high. Our work makes it clear that it will be more difficult to identify an optimal policy in Cournot-Bertrand type markets because firms are so diverse, and because the social effect of union power varies so greatly with the degree of product differentiation.

We would like to thank Carol Tremblay and two anonymous referees for their helpful comments on an earlier version of the paper.

The authors declare no conflict of interest.

For example, Petrakis and Vlassis [

As a referee points out, however, in practice it can be difficult to determine if a firm is setting price or quantity.

That is, this demand system derives from the representative consumer’s utility function U(_{1}, _{2}) = (_{1} + _{2}) − (_{1}^{2} + _{2}^{2} + 2_{1}_{2})/2 +

Alternatively, from the union’s perspective one can think of w as the markup of the union wage over the competitive wage.

The comparative static results are as expected. Greater product differentiation (i.e., a decrease in

For a discussion of this bargaining approach, see Roth [

For example, Binmore,

The comparative static results are as expected. Greater product differentiation (i.e., a decrease in

Another possibility is that the union uses a different bargaining strategy with each firm. This would allow for efficient bargaining with the Cournot-type firm and MU or RTM bargaining with the Bertrand-type firm. This would impose high transaction costs on the union, however.

The comparative static results are as expected. Greater product differentiation (i.e., a decrease in

Tremblay and Tremblay [

From the profit equations in (16),

One can prove this by comparing the profit equations in (6), (11), and (16).

In general, the relative performance of the profits obtained from efficient bargaining depends upon the specific values of

In general, the relative performance of the profits obtained from efficient bargaining depends upon the specific values of

This can be proven by inspecting the output equations in (6), (11), and (16).

The relative performance of the efficient bargaining model depends upon the specific values of

Of course, unions can promote efficiency when facing a monopsonist or oligopsonists.