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A Time Consistent Dynamic Bargaining Procedure in Continuous Time∗

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A Time Consistent Dynamic Bargaining Procedure in Continuous Time∗ A time consistent dynamic bargaining procedure in continuous time∗ Anna Casta~ner,y Jes´usMar´ın-Solanozand Carmen Ribasx Abstract A dynamic Nash bargaining procedure is introduced in a continuous time setting. Decision rules of players are derived as the maximizers of a Nash welfare function. Solutions are obtained for different choices of threat points in case of disagreement. Special attention is paid to a dynamic bargaining procedure giving rise to subgame perfect solutions. This solution concept, which is defined as the dynamic Nash t-bargaining solution with no commitment (or simply time-consistent dynamic bargaining solution), extends the recursive Nash bargaining equilibrium (Sorger, 2006) to a continuous time setting. The underlying idea is that, in case of disagreement, the threat is that players will play a noncooperative Markov Perfect Nash equilibrium just during an instant of time, and new negotiations will take place immediately later. Potential applications include common property resource models with asymmetric players. In particular, two common property resource games are analyzed in detail: a renewable natural resource model with log-utilities, and a nonre- newable resource model with general isoelastic utilities. Different cooperative and noncooperative solutions are compared for both models. Keywords: Dynamic bargaining; time-consistency; differential games; asymmetric players; com- mon property resource games JEL codes: C73, C71, C78, Q20, Q30 1 Introduction In the study of intertemporal choices, if there are several players and utilities are cardinal, a social planner could simply add the intertemporal utility functions of all agents. In the absence of a social planner, if cooperation is permitted, economic agents could decide to coordinate their strategies in order to optimize their collective payoff. Alternatively, Pareto efficient solutions can be found by maximizing a weighted sum (with constant, nonnegative weights) of the intertemporal utility functions. However, although it is typically assumed that all economic agents have the same rate of time preference, there is no reason to believe that consumers, firms or countries have identical ∗This work has been partially supported by MEC (Spain) Grants ECO2013-48248-P and ECO2017-82227-P. Earlier versions of this paper benefited from comments by Larry Karp and Jorge Navas yDept. Matem`aticaecon`omica, financera i actuarial, Universitat de Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain. ORCID: 0000-0003-2265-8607 zCorresponding author. Dept. Matem`aticaecon`omica,financera i actuarial and BEAT, Universitat de Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain. Tf: ++34 934029028. Email: [email protected]. ORCID: 0000-0003-2853-4410 xDept. Matem`aticaecon`omica, financera i actuarial, Universitat de Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain. ORCID: 0000-0002-4347-5300 1 time preferences for utility streams. In that case, in the computation of optimal decision rules, as in hyperbolic discounting, a problem of time-consistency arises: what is optimal for the coalition or the society at time t will be no longer optimal at time s, for s > t. If utilities are cardinal, the most natural way to address the problem of finding time-consistent policies in case agents with heterogeneous discount functions can cooperate - or if there is a social planner aggregating their preferences - is to add the individuals intertemporal utility functions and - as in hyperbolic discounting - look for time-consistent policies. In this approach there are implicit two fundamental assumptions: all players can cooperate at every instant of time t, and the different t-coalitions (coalitions at time t) lack precommitment power for future decision rules. As a result, the solution becomes partially cooperative (agents at the same time cooperate to achieve higher joint payoffs) but partially noncooperative (coalitions at different moments have different time preferences and do not cooperate among them). Somehow, in the setting of intergenerational models, if we interpret an instant of time as a generation, this approach assumes intragenerational but not intergenerational cooperation. As in nonconstant discounting - that can be seen as a problem of aggregating time preferences if agents share the same instantaneous utility function but discount the future at different rates -, time-consistent equilibria can be computed by finding subgame perfect equilibria in a noncooperative sequential game where agents are the different t-coalitions (representing, for instance, different generations). There are many papers that have addressed the issue of aggregating preferences, by adding the intertemporal utility functions, of heterogeneous agents in a dynamic setting. Li and Lofgren (2000) and Gollier and Zeckhauser (2005) have restricted their attention to efficient but time- inconsistent policies (although Hara (2013) shows, for a continuous time equilibrium model, that the more heterogeneous the consumers are in their impatience, the more dynamically consistent the representative consumer is). Recent papers (Ekeland and Lazrak (2010), Ekeland et al. (2015), Karp (2017)) have justified the introduction of weighted sums of exponentials in nonconstant dis- counting models with one decision maker as a way to represent the time preferences in overlapping generation models. Breton and Keoula (2014), Karp (2017) and Millner and Heal (2017) search for time-consistent policies in case players are symmetric in their preferences on consumption (i.e. util- ity functions) but heterogeneous in their time preferences. Mar´ın-Solanoand Shevkoplyas (2011), de-Paz et al. (2013), Ekeland et al. (2013) and Mar´ın-Solano (2015) study time-consistent equilibria in case players are asymmetric, not just in their discount rates, but also in their utility functions. However, the time-consistent \cooperative" solutions obtained in this way can have some im- portant drawbacks. First of all, such time-consistent solutions are not Pareto efficient, in general1. In addition, time-consistent cooperative equilibria seem to be extremely cumbersome to compute (see e.g. de-Paz et al. (2013)). In this paper we propose, instead, to work within the framework of a dynamic Nash bargaining theory in a continuous time setting, where negotiations are done at every instant of time. Since Haurie (1976), it is well-known that the classical Nash bargaining theory is problematic when applied to dynamic/differential games, because problems of dynamic inconsistency are likely to 1Jackson and Yariv (2015) show that every Pareto efficient and non-dictatorial method of aggregating utility functions is time-inconsistent if time preferences of the individuals are heterogeneous. Previously, Zuber (2011) proved that Paretian social preferences are consistent and stationary if, and only if, all agents have the same discount rate of time preference. Mar´ın-Solano(2015) showed with a simple example that, assuming that utilities are cardinal, time-consistent cooperative solutions can be inefficient for the group: joint payments can be higher if players act in a fully noncooperative way. 2 arise. Our proposal departs from the recursive Nash bargaining solution introduced in Sorger (2006). This solution concept was proposed for dynamic games with heterogeneous players in a discrete time setting. According to this solution, knowing the decision rule at future periods s = t + 1; t + 2;::: , agents look for a weighted Nash bargaining solution in which the status quo or threat point is given by the payoffs of the players if they do not cooperate just at period t. As a result, weights of players become time-varying and the corresponding solution is time-consistent2. Our contributions in this paper are the following. We introduce, in a continuous time setting, a dynamic Nash bargaining procedure by maximizing a Nash welfare function under three different threat (status quo, disagreement or reference) points. First we consider the noncooperative feedback (Markov Perfect) Nash equilibria. As a result, disagreement leads to the noncooperative outcome at perpetuity. In case solutions exist for this problem, they are, by construction, individually rational. However, such visceral reaction (not collaborating at perpetuity) is unrealistic, in general, and, by construction, the corresponding bargaining solution is not fully time-consistent. As a second option, we take as threat value functions the payments obtained by agents if, in case of disagreement, they do not cooperate just for a finite time period of length τ. The corresponding dynamic bargaining solution becomes more realistic, but it is still not fully time-consistent. In addition, it seems to be very difficult to compute, moreover existence and uniqueness is not guaranteed. However, the threat point can be used in the derivation of a third solution concept, which is fully time-consistent, by taking as threat point the one obtained from the previous solution in the limit when τ ! 0+. The corresponding time-consistent dynamic bargaining solution is the natural extension of the Sorger's approach (Sorger (2006)) to a continuous time setting. Finally, we illustrate the theoretical findings by computing the dynamic bargaining solutions for two common property resource games. We compare the results obtained within the class of linear strategies. The paper is structured as follows. Section 2 presents the general model. It also includes some results on the t-cooperative equilibrium rule3 that will be needed later. Section 3 introduces and characterizes,
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