Bargaining solutions at work: Qualitative differences in policy implications

Abstract

Traditional bargaining theory characterizes solutions to bargaining problems by their properties in the utility space. In applications, however, one is usually interested in the implications of the conflict resolution within the economic environment, where the properties of axiomatic bargaining solutions are less well understood. By means of a standard bargaining model for the labor market we demonstrate that economic policy implications may be very sensitive to the choice of the bargaining solution. More specifically, the induced employment effects of a change in the reservation wage under the Nash-solution may differ substantially from those under the Kalai–Smorodinsky, the egalitarian, or the equal-loss solution. Hence, the choice of the bargaining solution is not innocuous, even if one is only interested in qualitative policy conclusions.

Introduction

In many bargaining situations we observe that parties negotiate on items such as prices and quantities rather than on utility levels directly. However, classical bargaining theory, originating in Nash's (1950) seminal paper, takes a welfarist point of view, according to which the solution to a bargaining problem should only depend on the set of feasible utility allocations and the status quo utilities of the parties involved. Only few authors have challenged this view and have studied solutions to bargaining problems on economic environments. Kihlstrom et al. (1981) analyze how bargaining solutions respond to changes in the degree of risk aversion of one player. Chun and Thomson (1988) and Chen and Maskin (1999) study monotonicity properties of bargaining solutions in resource allocation problems. Roemer (1988) considers the domain of pure exchange economies. He points out that relevant information is lost by presenting a bargaining problem in the utility space rather than in the underlying economic environment. Hence, the class of possible solutions on the utility domain is smaller than the class of allocation mechanisms on the domain of economic environments. Moreover, axioms like monotonicity that can be well motivated in a resource allocation problem may become unnecessarily restrictive when formulated directly in utility space. If instead, one tries to characterize allocation mechanisms on the economic environment, then the counterparts of the traditional axioms may have no bite any more and several mechanisms may satisfy, for example, the counterparts of the Nash axioms (Nash, 1950).¹

In this paper we address an additional issue that arises when bargaining solutions are applied in economic environments: the problem that the behavior of different solutions on the economic domain is not well understood. It is, however, this behavior of the economic variables which is at the center of interest in economic applications. Yet, up to now the restrictions implied by the choice of a particular bargaining concept on economic variables are unclear. In labor markets, for example, policy makers are typically concerned about how certain policy variables influence wages and employment, and not so much about how they influence utilities. Nevertheless, when modeling labor market conflicts between unions and employers' federations the applied literature appeals to the classic bargaining solutions defined and characterized in the utility space. The choice of the bargaining solution is hardly ever discussed. It may in fact be innocuous as long as one is only interested in policy conclusions and these are robust against changes in the solution concept. It is the purpose of this paper to demonstrate that this is not the case in general and that, to the contrary, economic policy implications may be very sensitive to the choice of the bargaining solution. Hence, unless one has good reasons to prefer one bargaining solution over another from a descriptive point of view, one should be aware of the fact that policy recommendations may be severely biased.

We study the sensitivity of policy conclusions with respect to the choice in the bargaining solution by means of a classic bargaining problem between a labor union and an employers' federation as it was first studied by McDonald and Solow (1981). For this bargaining problem we investigate the responsiveness of the negotiated wage and employment on changes in the reservation wage, which can be manipulated by the government through different channels, for example through the social security or the tax system. Since the Nash bargaining solution is by far the most frequently used solution in the applied literature, we take it as a benchmark and compare its behavior on the economic domain with the behavior of another prominent bargaining solution, namely the Kalai–Smorodinsky solution (Kalai and Smorodinsky, 1975), henceforth KS-solution. Our direct comparison between these bargaining solutions in a labor market shows that while a lower reservation wage unambiguously leads to higher employment and to a lower wage rate for the Nash bargaining solution, the same policy has an ambiguous employment effect (together with a lower wage rate) for the KS-solution. And indeed there are specifications of the utility and production functions where a higher reservation wage does lead to a higher employment level. Alternative solution concepts, notably the egalitarian solution (Kalai, 1977), and the equal-loss solution (Chun, 1988), display the same ambiguity of the employment effect. Hence, among the classic bargaining solutions we have analyzed, the Nash-solution appears to be the only one which rules out seemingly counterintuitive effects in the underlying economic variables. From this example of labor market negotiations we conclude that the choice of a bargaining solution is indeed far from being innocuous and should not be disregarded.

In their seminal paper on labor market negotiations McDonald and Solow (1981) have also studied the behavior of both the Nash- and the KS-solution on the economic domain. Their analysis focuses, however, on changes in (real) product demand. In this way they provide a possible explanation for small, unsystematic real wage movements which are seemingly independent of the business cycle. McDonald and Solow (1981) find that both, the Nash as well as the KS-solution may give rise to perverse countercyclical wage flexibility (i.e. wages decrease if revenues increase), but that this phenomenon is more likely to arise for the KS-solution. They do not thoroughly analyze, however, changes in the bargaining outcome with respect to movements in the reservation wage. Referring to the KS-solution they only state (p. 906) that “the effects of changes in w¯ [the reservation wage] are complicated.” By contrast, our paper demonstrates that there are situations where changes in the reservation wage bring about perverse employment effects under the KS-solution, which never occur under the Nash bargaining solution.

The outline of the paper is as follows. In Section 2 we derive the Nash bargaining and the Kalai–Smorodinsky solution in the framework of labor-market negotiations. Section 3 contains the comparative static results for the KS-solution, which we compare with the corresponding effects for the Nash bargaining solution. Finally, Section 4 concludes. All proofs are relegated to Appendix A and a numerical example is provided in Appendix B.