A Time Consistent Dynamic Bargaining Procedure in Continuous Time∗

A time consistent dynamic bargaining procedure in continuous time∗

Anna Casta˜ner,† Jes´usMar´ın-Solano‡and Carmen Ribas§

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 nonrenewable 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; common 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 †Dept. Matemàticaeconòmica, financera i actuarial, Universitat de Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain. ORCID: 0000-0003-2265-8607 ‡Corresponding author. Dept. Matemàticaeconòmica,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 §Dept. Matemàticaeconòmica, 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 discounting 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. utility 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 important 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, in a continuous time setting, the so-called dynamic Nash t-bargaining solutions, for the three diﬀerent threat points quoted above. Special attention is paid to time-consistent dynamic bargaining, which is the main contribution of the paper. A common property renewable resource model with log-utilities is analyzed in Section 4. Section 5 studies the joint management in a nonrenewable resource model with general isoelastic utilities. Section 6 concludes the paper.

2 Preliminaries

Let us consider an intertemporal decision problem with several agents in which players can coordinate their strategies in order to optimize their collective payoff. Alternatively, a social planner can look for the solution maximizing the aggregate preferences of all economic agents. In this section, utilities are assumed to be cardinal. If there is a unique and constant discount rate of time preference for all agents, Pareto efficient solutions can be obtained by solving a standard optimal control problem. However, in the case of different discount rates, joint preferences become time inconsistent. As in hyperbolic discounting, when agents lack commitment power but they can cooperate at every instant of time, they act at different times t as sequences of independent coalitions (the t-coalitions). Following Mar´ın-Solano(2015), we call the time-consistent solution for this problem a

2Variable weights are also considered in Yeung and Petrosyan (2015). 3We use the same definition as in Mar´ın-Solano(2015) for this solution concept, which is not strictly cooperative nor Pareto efficient. In de-Paz et al. (2013), it was defined as a time-consistent equilibrium under partial cooperation.

3 t-cooperative equilibrium. If we identify a t-coalition as the “t-agent” in a problem with hyperbolic (or general time-distance) discounting, the t-cooperative equilibrium rule is the decision followed by the “sophisticated agent”. Hence, this solution assumes or allows for cooperation among players at every time t, but is a non-cooperative (Markov Perfect) equilibrium for the non-cooperative sequential game deﬁned by these inﬁnitely many t-coalitions.

Let N be the number of players, and x ∈ X ⊂ R the state variable. For each player i ∈

{1,...,N}, let ci ∈ Ui ⊂ R be her control (decision) variable, c = (c1, . . . , cN ) the corresponding

vector of decision rules, ui(ci) the (cardinal) instantaneous utility function, and ρi the discount 4 rate. We will assume that ui(ci) is continuously diﬀerentiable, increasing and strictly concave . The intertemporal utility function for player i at time t is Z ∞ −ρi(s−t) Ji(x; c1, . . . , cN ; t) = e ui(ci(s)) ds , with (1) t

x˙(s) = g(x(s), c1(s), . . . , cN (s)) , x(t) = xt , (2) PN where g(x(s), c(s)) = f(x(s)) − i=1 ci(s), with f(x) a continuously diﬀerentiable and concave utility (possibly linear) production function, so (2) becomes

N X x˙(s) = f(x(s)) − ci(s) , x(t) = xt . (3) i=1

These conditions facilitate the fulﬁllment of the conditions in Benveniste and Sheinkman (1979) for the concavity and diﬀerentiability of the value functions appearing in our problem, properties that are assumed in our results.

In a cooperative setting, we can aggregate preferences as

N N Z ∞ c X X −ρi(s−t) J (x, c, t) = Ji(x, c, t) = λi e ui(ci(s)) ds , (4) i=1 i=1 t

where λi ≥ 0 are constant numbers. If players have equal weights, we can take λ1 = ··· = λN = 1. Since the discount functions are different, the joint time preferences in (4) are time inconsistent. In order to find time-consistent solutions, the problem can be solved as a noncooperative sequential game with a continuum of “players” (each “player” is each coalition at time t). Hence, we can follow the ideas in Karp (2007), Ekeland and Lazrak (2010) or Yong (2011), who suggested three different procedures to find subgame perfect equilibria for this sequential game. As in Ekeland and Lazrak (2010), we restrict our attention to stationary convergent Markovian strategies, i.e.

strategies ci = φi(x) such that there exists x∞ < ∞ and a neighbourhood U of x∞ such that,

for every x0 ∈ U, the solution to (3) along c = φ(x) = (φ1(x), . . . , φN (x)) converges to x∞. For convergent strategies, the integral in (1) converges.

If c∗(s) = φ(x(s)) is a continuously diﬀerentiable equilibrium rule for Problem (4) subject to

(3), by denoting xt = x, the corresponding value function is

N Z ∞ X −ρi(s−t) V (x, t) = λiVi(x, t) , where Vi(x, t) = e ui(φi(x(s))) ds . (5) i=1 t

4The extension of our results to more general utility functions and multidimensional problems representing, for example, stocks of diﬀerent resources, is straightforward.

4 Since we are looking for stationary equilibrium strategies, the value functions Vi (and hence V ) are time-independent.

Next, following Ekeland and Lazrak (2010) and Ekeland et al (2013), for > 0 andc ¯ =

(¯c1,..., c¯N ),c ¯i ∈ Ui, let ( c¯ if s ∈ [t, t + ] , c(s) = (6) φ(x(s)) if s > t + . If the t-coalition has the ability to precommit its behavior during the period [t, t + ], the

valuation along the perturbed control path c is given by

N Z t+ Z ∞ X −ρi(s−t) −ρi(s−t) Vc (x, t) = λi e ui(¯ci) ds + e ui(φi(x(s))) ds . i=1 t t+

∗ If we expand Vc (x, t) in , we obtain Vc (x, t) = V (x, t)+P (x, φ, c,¯ t)+o(). A decision rule c (s) = φ(x(s)) is called a t-cooperative equilibrium (t-CE) if function P (x, φ, c¯) attains its maximum for

c¯ = φ(x). This is equivalent to saying that, for any admissible c given by (6),

V (x, t) − V (x, t) lim c = P (x, φ, c,¯ t) ≥ 0 →0+ ifc ¯ = φ(x) (in fact, P (x, φ, c,¯ t) = 0 along the t-cooperative equilibrium rule), and strictly negative otherwise. Following de-Paz et al. (2013) (see also Ekeland et al (2013)) it can be proved that,

if a decision rule c = φ(x) is such that the functions Vi(x) in (5) (from now on we will omit the temporal argument) are of class C1 and

 N N  N  X X 0 X  (φ1, . . . , φN ) = argmax{c1,...,cN } λiui(ci) + λiVi (x) · f(x) − cj , (7) i=1 i=1 j=1 

then c = φ(x) is a t-CE rule. By diﬀerentiating (5), from (7) we can easily derive the dynamic programming equation (DPE)

N  N N  N  X X X X  ρiλiVi(x) = max λiui(ci) + λi∇xVi(x) · f(x) − cj . (8) {c ,...,c } i=1 1 N i=1 i=1 j=1 

The corresponding time-consistent decision rule can be group inefficient in the sense that joint payoffs can be lower in the t-cooperative equilibrium than in the feedback noncooperative Nash equilibrium (Mar´ın-Solano(2015)). From a different viewpoint, the t-cooperative equilibria solving (8) can be seen as constrained (to the future behavior of the agents) Pareto efficient.

The previous results can be easily extended to the problem of “maximizing”

N Z ∞ X −ρi(s−t) J(xt, t) = λi(xt, t) e ui (ci(s)) ds (9) i=1 t

subject to (3), where λi(xt, t) ≥ 0, for every i = 1,...,N. Coeﬃcients λi(xt, t) represent the weight 5 6 of agent i at state xt and time t in the t-coalition . Assume that, along the equilibrium rule: (i) ∞ PN R −ρi(s−t) function V (x, t) = i=1 λi(x, t)Vi(x, t), where Vi(x, t) = t e ui(φi(x(s), s)) ds is ﬁnite (i.e.

5 Problems with one decision maker where instantaneous utilities of each t-agent depend also on the current state xt were studied in Yong (2011). 6 PN We do not impose condition i=1 λi = 1 since the solution depends just on the relative weights.

5 the integral converges); (ii) functions λi(x, t), i = 1,...,N, are continuously diﬀerentiable in all

their arguments; (iii) equilibrium rules are (at least) continuous; (iv) value functions Vi(x, t) are of class C1 and

(c1, . . . , cN ) = (φ1(x, t), . . . , φN (x, t)) (10)

 N   N  X X  = argmax{c1,...,cN } λi(x, t) ui(ci) + ∇xVi(x, t) · f(x) − cj i=1 j=1  are continuous functions; (v) and there exists a unique absolutely continuous curve x: [0, ∞] → X PN solution tox ˙(t) = f(x(t))− i=1 φi(x(t), t) with x(0) = x0. Then (c1, . . . , cN ) = (φ1(x, t), . . . , φN (x, t)) is a t-cooperative equilibrium with nonconstant weights. The result follows by observing (see the proof of Theorem 1 in Mar´ın-Solano(2014)) that

N V (x, t) − Vc (x, t) X lim = λi(x, t) [(ui(φi(x, t)) →0+ i=1

N N X X +∇xVi(x, t) · (f(x) − φi(x, t))) − (ui(ci) + ∇xVi · (f(x) − cj))] . j=1 j=1 The equilibrium rule is completely characterized (provided that the corresponding value functions of the diﬀerent players are continuously diﬀerentiable) by condition (10).

3 A dynamic Nash bargaining solution

As an alternative to the t-cooperative equilibrium rule, in this paper we propose to use a dynamic version of Nash bargaining theory. In Nash bargaining theory, payments are obtained as the maximizers of a Nash bargaining function (Nash (1953)). These payments implicitly characterize the weights of players in the whole coalition. Munro (1979) proposed to use Nash bargaining theory with constant weights in a transboundary resource model with asymmetric players. In his model, the threat (status quo or disagreement) point is the payoff associated to a noncooperative Nash equilibrium, and weights are bargained at the beginning of the game. In order to avoid the problems of dynamic inconsistency (implicit in any differential game, see Haurie (1976)) and time inconsistency (due to the different discount rates) of the solution, he assumed full commitment of economic agents, which is unrealistic in most of cases. As a way to overcome these problems, we propose to work within a Markovian formulation, where strategies are derived as the result of repeated negotiations that take place at every moment t.

We assume that, at time t, players know both the state of the system and their future decision rule c = φ(x(s), s), s > t, as a reaction to their current decisions7. Then, as in the classical Nash bargaining solution, they compare what they get cooperating at time t (or during the time interval [t, t + ], with arbitrarily small) with what they receive otherwise (the status quo, threat point or

reference point). Let us denote the threat point by (W1(x, t),...,WN (x, t)). When players at time t try to reach an agreement and to derive their corresponding actions, they choose their policy in the time interval [t, t + ] as the maximizer of some “distance” between what they obtain in case

7We will not assume, in advance, that there exist stationary strategies. Its existence will depend on each particular problem and the threat point considered in case of disagreement.

6 of agreement and in case of disagreement. We assume that this distance is measured according to the generalized Nash welfare function with strictly positive bargaining powers η1, . . . , ηN

N Y ηi [Vi(x, c1, . . . , cN , t) − Wi(x, t)] . i=1 This solution concept has been characterized (for static games) with a set of three axioms: strong individual rationality, independence of utility calibration and independence of irrelevant alternatives

(Roth (1979)). For η1 = ··· = ηN = 1 we recover the classical Nash bargaining solution satisfying symmetry. However, as Sorger (2006) arguments, in a discrete time 2-player game, the Nash bargaining solution corresponding to the relative bargaining power η = η2/η1 = ln β1/ ln β2 (where

βi is the discount factor of player i) is the limit (as time delay between offers approaches to 0) of the unique perfect equilibrium outcome of a Rubinstein-type alternating-offers model (see, e.g., Binmore (1987)). Then it seems natural, in our framework with asymmetric players, to introduce different bargaining powers ηi related to the discount rates.

For a decision rule (c1, . . . , cN ) = (φ1(x, t), . . . , φN (x, t)) and a set of threat value functions

W (x, t) = (W1(x, t),...,WN (x, t)) verifying Vi(x, φ1(x, t), . . . , φN (x, t), t) − Wi(x, t) ≥ 0, for all i = 1,...,N, let

N Y ηi Π(x, t) = [Vi(x, φ1(x, t), . . . , φN (x, t), t) − Wi(x, t)] . i=1 As in the previous section, take ( c¯ if s ∈ [t, t + ] , c(s) = φ(x(s), s) if s > t + , so that Z t+ Z ∞ −ρi(s−t) −ρi(s−t) Vi,c (x, c, t) = e ui(¯ci) ds + e ui(φi(x(s), s)) ds (11) t t+ and N Y ηi Πc (x, t) = [Vi,c (x, c, t) − Wi(x, t)] . (12) i=1 Then

Πc (x, t) = Π(x, t) + Π1(x, φ, c,¯ t) + o() .

Deﬁnition 1 A decision rule (c1, . . . , cN ) = (φ1(x, t), . . . , φN (x, t)) is called a dynamic Nash t- bargaining solution corresponding to the threat point W (x, t) if

φ(x, t) = arg max Π1(x, φ, c,¯ t) . {c¯}

Alternatively, since

Πc (x, t) − Π(x, t) lim = Π1(x, φ, c,¯ t) →0+ and, for a dynamic t-bargaining solution, max{c¯} Π1(x, φ, c,¯ t) = 0, we can characterize these solutions as the values of c (i.e.c ¯) such that Π (x, t) − Π(x, t) lim c = 0 (13) →0+ It remains to deﬁne the threat value functions. We consider the following three alternatives:

7 1. Full commitment. In case of disagreement at time t, payments are those associated to the Markovian Perfect Nash equilibria (MPNE), for all s ≥ t. 2. Partial commitment. In case of non cooperation at time t, the threat is that agents will not cooperate for a ﬁxed and known time period of length τ. Payments are those associated to the MPNE in the noncooperative game played in the time interval s ∈ [t, t + τ). 3. No commitment. In case of non cooperation at time t, agents can bargain again at time t + , for arbitrarily small.

3.1 Case 1. Dynamic Nash t-bargaining solution with full commitment (BFC)

As a ﬁrst possibility, as threat points in case of disagreement we take the payments associated to n the MPNE, so Wi(x, t) = Wi (x, t), for i = 1,...,N. Hence we have:

Definition 2 A dynamic Nash t-bargaining solution with full commitment (BFC), φfc(x, t) = fc fc fc fc fc fc fc (φ1 (x, t), . . . , φN (x, t)), with payments V (x, t) = (V1 (x, φ (x, t), t),...,VN (x, φN (x, t), t)), is the dynamic Nash t-bargaining solution defined as in Definition 1 where the threat point in case of disagreement at time t, with state x(t) = x, is given as follows: n n n For s ∈ [t, ∞), players apply strategies φ = (φ1 , . . . , φN ) such that

n n n Wi (x, t) = Ji(x, φ , t) ≥ Ji(x, φ−i, σi, t) , ∀i, σi , i = 1,...,N, where σi : X × [t, ∞) → Ui ⊂ R is any possible admissible feedback law for Player i. Hence, the f f threat point is given by (W1 (x, t),...,WN (x, t)).

The BFC solution can be seen as the natural extension, to a setting with dynamic (continuous) bargaining, of the “classical” approach (as in Haurie (1976) or Munro (1979)). This solution concept has two advantages. First, it is easy to implement, as we show below. In addition, if there are bargaining powers η1, . . . , ηN such that the corresponding dynamic Nash t-bargaining solutions exist, the related payoffs are individually rational in the classical sense (and they are not Pareto dominated by MPNE). However, this choice of threat point has two main drawbacks. First, it is unclear what to do if there is multiplicity of MPNE, at least if they are not Pareto ranked. And second, and more importantly, we are assuming some kind of commitment: in case of disagreement at time t, players will not cooperate forever. This rather visceral reaction can make sense in some contexts (for example, most of divorces are irreversible), but becomes unrealistic in many cases, when we assume full rationality of decision makers. In fact, although, by construction, the problem of finding a dynamic bargaining solution has a Markovian nature, the solution itself is not, for this threat point, fully time-consistent (or subgame perfect). For the computation of the decision rule note that, if a dynamic Nash t-bargaining solution fc n with full commitment exists verifying the constraints Vi − Wi > 0, for all i = 1,...,N, from Definition 1 and (13) we have

dΠ N dV fc c X ¯fc i,c Π1(x, φ , c,¯ t) = = λi (x, t) = 0 , d + d + =0 i=1 =0 where

¯fc fc fc n ηi−1 Y fc fc n ηj λi (x, t) = ηi(Vi (x, φ (x, t), t) − Wi (x, t)) · (Vj (x, φ (x, t), t) − Wj (x, t)) . j6=i

8 Hence we have PN ¯fc h fc i i=1 λi (x, t) Vi,c (x, t) − Vi (x, t) lim = 0 →0+

and the problem of maximizing Π1(x, t) transforms into a problem of looking for t-cooperative ¯fc ¯fc ¯fc equilibria if players have nonconstant weights λi (x, t). Since just relative weights λi /λj are relevant, we can divide the expression above by Π(x, t) in order to obtain the equivalent relative weights η λfc(x, t) = i . (14) i fc fc n Vi (x, φ (x, t), t) − Wi (x, t) Finally, we can use (10) with weights given by (14).

3.2 Case 2. Dynamic Nash t-bargaining solution with partial commitment (BPC)

In this case, if players do not cooperate at time t, they will not cooperate during a ﬁxed and known time period of length τ.

Definition 3 A dynamic Nash t-bargaining solution with partial commitment (BPC), φpc(x, t) = pc pc pc pc pc pc pc (φ1 (x, t), . . . , φN (x, t)), with payments V (x, t) = (V1 (x, φ (x, t), t),...,VN (x, φN (x, t), t)), is the dynamic Nash t bargaining solution defined as in Definition 1 where the threat point in case of disagreement at time t, with state x(t) = x, is given as follows:

τ τ τ Wi (x, t) = Ji(x, φ , t) ≥ Ji(x, φ−i, σi, t) , ∀i, σi , i = 1,...,N,

where

• σi : X × [t, ∞) → Ui ⊂ R, i = 1,...,N, is given by ( σ¯i(x(s), s) if t ≤ s ≤ t + τ , σi(x(s), s) = pc φi (x(s), s) if s > t + τ ;

• σ¯i : X × [t, t + τ] → Ui ⊂ R, i = 1,...,N, is any possible admissible feedback law for Player i for the problem with planning horizon [t, t + τ]. pc • φi : X × (t + τ, ∞) → Ui ⊂ R, i = 1,...,N, is the dynamic Nash t-bargaining solution with partial commitment given by Definition 1 and is taken as given; τ τ pc • φj : X × [t, ∞) → Uj ⊂ R, j = 1,...,N, is such that φj (x(s), s) = φj (x(s), s), for s > t + τ. In the definition above, agents take as given the future decision rule, derived through a dynamic bargaining procedure, after time τ. The threat is not to cooperate during the time period [t, t + τ] and possibly to cooperate for s > t + τ 8, by computing the MPNE for the noncooperative game with finite planning horizon [t, t + τ] with final functions Z ∞ −ρi(s−t) pc Fi(x(t + τ), t + τ) = e ui(φi (x(s)), s) ds . t+τ Such limited punishment in case of non cooperation represents, perhaps, the most common situation in real life applications. Note that, if τ goes to infinity, we recover Case 1, whereas in the limit when τ goes to 0 we obtain Case 3. Once the threat point has been computed, for the computation of the dynamic Nash t-bargaining solution with partial commitment, we can proceed as in the case with full commitment and apply (14).

8Alternatively, we can assume that players do not to cooperate in [t, t + τ) and possibly cooperate for s ≥ t + τ

9 3.3 Case 3. Dynamic Nash t-bargaining solution with no commitment: Time-consistent dynamic bargaining (TCB)

If we assume full rationality of economic agents, in case of non cooperation at time t, players can bargain again at time t + , for arbitrarily small. Hence, the threat point (noncooperative behavior) lasts after a time period of length . This is the natural extension to a continuous time setting of the recursive Nash bargaining solution introduced in Sorger (2006). By construction, the corresponding solution is dynamically consistent but typically non Pareto eﬃcient. Equation (14) + cannot be applied in the computation since, in the limit when → 0 , Vi = Wi. However, we can proceed by maximizing directly Π1(x, φ, c,¯ t) inc ¯. We can divide the process of ﬁnding the dynamic Nash t-bargaining solution with no commitment into two steps.

Step 1: Compute the threat point. This can be easily done by taking, in Deﬁnition 3, τ = . Then, for s ∈ [t, t + ], the decision rule in case of no cooperation is given by φ(x(s), s), and for s > t + , φpc(x(s), s). In the limit → 0+ we can obtain the threat point, that we denote as 0 0 0 (W1 ,...,WN ), corresponding to the threat decision rule φ (x, t). Step 2: Compute the equilibrium strategy c = φnc(x, t). If φ(x, s) is the equilibrium (bargaining) decision rule for s > t + , we can write the threat point for player i ∈ {1,...,N} as

Z t+ Z ∞ −ρi(s−t) −ρi(s−t) Wi (x, t) = e ui(φi (x (s), s)) ds + e ui(φi(x (s), s)) ds , t t+ where x(s) is the solution tox ˙ (s) = g(x(s), φ(x(s), s)) with x(t) = x (recall that, from Deﬁnition 3, φ(x(s), s) = φ(x(s), s), for s > t + ). On the other hand, from (11),

Z t+ Z ∞ −ρi(s−t) −ρi(s−t) Vi,c (x, t) = e ui(¯c) ds + e ui(φi(x(s), s)) ds , t t+ where x(s) is the solution tox ˙ (s) = f(x(s), c¯) with x(t) = x, for s ∈ [t, t + ]; andx ˙ (s) = f(x(s), φ(x(s), s)) for s > t + , with the initial condition x(t + ) derived by continuity.

The objective is to maximize Π, as given in (12), at order ; i.e., we have to solve Π1 and φ(x, s) in Deﬁnition 1. Note that Z t+ −ρi(s−t) Vi,c (x, t) − Wi (x, t) = e [ui(c ¯i) − ui(φi (x(s), s))] ds t Z ∞ −ρi(s−t) + e [ui(φi(x(s), s)) − ui(φi(x (s), s))] ds . t+ 0 0 Since, at time t, lim→0+ x(t + ) = x (t) = x and lim→0+ φ (x, t) = φ (x, t), then Z t+ −ρi(s−t) 0 e [ui(¯ci) − ui(φi (x(s), s))] ds = ui(¯ci) − ui(φi (x, t)) + o() t and Z ∞ −ρi(s−t) e [ui(φi(x(s))) − ui(φi(x (s)))] ds = t+ Z ∞ −ρi −ρi(s−t−) e e [ui(φi(x(s))) − ui(φi(x (s)))] ds = t+

10 ∂V nc(x, t) e−ρi [V nc(x (t + ), t + ) − V nc(x(t + ), t + )] = i g(x, c¯) − g(x, φ0(x, t)) + o() . i i ∂x Therefore, ∂V nc(x, t) V nc(x, t) − W 0(x, t) = u(¯c ) − u (φ0(x, t)) + i g(x, c(t)) − g(x, φ0(x, t)) + o() . i i i i i ∂x The Nash welfare function becomes

N Y nc 0 ηi Vi (x, t) − Wi (x, t) = i=1

N η Y ∂V nc(x, t) i u(¯c ) − u (φ0(x, t)) + i g(x, c¯) − g(x, φ0(x, t)) η1+···+ηN + o(η1+···+ηN ) . i i i ∂x i=1

Without loss of generality, we can take η1 + ··· + ηN = 1 (just relative weights are relevant), so

N Y nc 0 ηi Vi (x, t) − Wi (x, t) = i=1

N η Y ∂V nc(x, t) i u(¯c ) − u (φ0(x, t)) + i g(x, c¯) − g(x, φ0(x, t)) + o() . i i i ∂x i=1 According to Deﬁnition 1, we maximize the ﬁrst order term in the Nash welfare function, so we compute

( N η ) ∂ Y ∂V nc(x, t) i u(¯c ) − u (φ0(x, t)) + i g(x, c¯) − g(x, φ0(x, t)) = 0 , ∂c¯ i i i ∂x i i=1 whose solution is given by

N nc X ηi ∂Vi ∂g ∂V nc + 0 i 0 ∂x ∂c¯j i=1 ui(¯ci) − ui(φi (x, t)) + ∂x [g(x, c¯) − g(x, φ (x, t))]

ηj 0 ∂V nc uj(¯cj) = 0 , 0 j 0 uj(¯cj) − uj(φj (x, t)) + ∂x [g(x, c¯) − g(x, φ (x, t))] PN + for j = 1,...,N. Finally note that, in our problem, g(x, c) = f(x)− k1 ck and, in the limit → 0 , strategies become stationary. Hence, the time-consistent dynamic bargaining (TCB) solutions are the decision rules cj = φj(x) solving

ηj 0 ∂V nc uj(cj) = (15) 0 j PN 0 uj(cj) − uj(φj (x)) + ∂x k=1 [φk(x) − ck]

N nc X ηi ∂Vi ∂V nc . 0 i PN 0 ∂x i=1 ui(ci) − ui(φi (x)) + ∂x k=1 [φk(x) − ck] The TCB solution has two clear advantages in comparison with the dynamic Nash t-bargaining solution with partial commitment. First, as we have already commented, it is fully time-consistent (i.e. subgame perfect). And second, it is much easier to compute, in general. The reason is that the dynamic Nash t-bargaining solution with partial commitment gives rise to nonstationary policies and, as a result, the derivation of these strategies can become extremely cumbersome. But this is not the case in the calculation of time-consistent dynamic bargaining solutions if the original economic problem is autonomous. Sections 4 and 5 illustrate these properties for two common property resource games.

11 4 A common property renewable natural resource model with log-utilities

In this section we apply the results in the previous section to the search of cooperative equilibria in a common property resource game with heterogeneous agents coming from the literature of resource economics. The model considers the problem of joint exploitation of a renewable natural resource in case players have logarithmic utilities. For this model, linear strategies exist as the solutions to the dynamic Nash t-bargaining problem, for diﬀerent threat points.

The intertemporal utility function of Player i is given by Z ∞ −ρi(s−t) µi Ji = e ln [(ci(s)) ] ds , (16) t

with µi > 0, for i = 1,...,N. The stock of the resource evolves according to

N X x˙(s) = x(s)(a − b ln x(s)) − cj(s) , x(t) = xt , (17) j=1

with a, b > 0 (see Clemhout and Wan (1985) for extensions including multiple species).

4.1 Noncooperative MPNE and the t-cooperative equilibrium

For general time-distance discount functions with µi = 1, noncooperative MPNE and t-cooperative equilibria were already derived in Mar´ın-Solano(2014). The extension is straightforward and we summarize the results.

4.1.1 Noncooperative MPNE

In players do not cooperate, stationary linear strategies exist and are given by

n n ci (x) = φi (x) = (ρi + b)x (18)

n n n for i = 1,...,N. The corresponding value functions are of logarithmic type, Wi (x) = αi ln x + βi , where

"a − PN (ρ + b) # n n n µi µi j=1 j Wi (x) = αi ln x + βi = ln x + + ln (ρi + b) . (19) ρi + b ρi ρi + b

4.1.2 The t-cooperative equilibrium rule

Again, we restrict our attention to stationary linear strategies. The following proposition summa- rizes our results.

Proposition 1 In Problem (16-17), if linear t-cooperative equilibria (with possibly nonconstant

weights) exist, weights must be of the form λi(x) = νih(x). Within this class of weight functions, 9 tc tc tc by normalizing h(x) = 1, equilibrium rules are given by ci (x) = φi (x) = Ai x, for i = 1,...,N, where tc λiµi Ai = (20) PN λj µj j=1 ρj +b

9Note that just relative weights are relevant

12 tc tc tc and the corresponding value functions are Vi (x) = αi ln x + βi , where

tc µi αi = , ρi + b      PN λ µ N tc µi j=1 j j µi X λjµj βi = a − N λ µ  + ln(λiµi) − ln   . (21) ρi(ρi + b) P j j ρi ρj + b j=1 ρj +b j=1

Proof: From (10) we obtain µ λ (x) φtc(x) = i i . i PN 0 j=1 λj(x)Vj (x) tc tc tc tc tc If linear strategies ci (x) = Ai x exist, then the value functions are given by Vi (x) = αi ln x + βi tc tc tc and necessarily λi(x) = νih(x). The values of Ai , αi and βi follow from Mar´ın-Solano(2014).

Remark 1 If λi(x) = νih(x), where νi are constant numbers, then stationary linear strategies exist

with associated value functions Vi(x) = αi ln x + βi.

Next we brieﬂy compare these t-cooperative equilibria with the MPNE. • In both cases, a unique steady state exists, so there is no multiplicity of equilibria. We will n tc denote it as x∞ in the MPNE, and x∞ in the t-cooperative equilibria. • As in the standard case with equal and constant discount rates, harvest coeﬃcients are higher in the pure noncooperative solution provided by the MPNE than in the problem with t- 10 n tc cooperation . Note that, from (18) and (20), ci (x) − ci (x) > 0 if, and only if,

N X ρi + b λ µ − λ µ > 0 , j j ρ + b i i j=1 j

and this condition is obviously satisﬁed.

c n • As a result, x∞ > x∞, i.e., the resource is overexploited in the MPNE in comparison with the t-cooperative equilibria. • However, the t-cooperative equilibria can be less eﬃcient. For instance, in Mar´ın-Solano (2015) an example is provided in which, if utilities are cardinal, the joint utility is lower under t-cooperation. The reason is that, unlike the MPNE and the dynamic Nash t-bargaining solution concepts proposed in this paper, t-cooperative decision rules depend in an important tc tc way on weights µi. Note that ci (x)/cj (x) = (λiµi)/(λjµj) and, unlike the case of equal PN tc discount rates, this dependence can reduce in an important way the value of i=1 βi in PN n n tc comparison with i=1 βi (note that αi = αi ). • If, in case of disagreement, agents follow the MPNE, in the search for individually rational

t-cooperative equilibria admitting stationary linear strategies, we should look for weights λi c n verifying βi ≥ βi , for every i = 1,...,N. The dynamic Nash t-bargaining solution with full commitment provides a way (if this solution exists) to ﬁnd weights with this property. 10It is straightforward to check that this property is preserved for general nonconstant time-distance discount functions

13 4.2 Dynamic Nash t-bargaining solution with full commitment (BFC)

The threat point for this solution concept is the noncooperative Markov Perfect Nash Equilibrium. Then we can use (14) for the calculation of the dynamic Nash t-bargaining solution with full tc n tc n commitment. Since Vi (x) − Wi (x) = βi − βi , for every λi ∈ R, the corresponding weights are η λfc = i , (22) i fc n βi − βi n fc fc where βi and βi are given from (19) and (21) with λi = λi . Then, in order to ﬁnd the BFC fc fc n solution, we substitute (22) into Equations (21), and solve the value of βi , provided that βi > βi . It can be checked that the corresponding decision rule is independent on weights µi, as expected. fc n If the constraint βi > βi is not met, then no dynamic Nash t-bargaining solution with full commitment exists.

Remark 2 If a = b = 0, N = 2, ρ1 = 0.01, ρ2 = 0.2s, µ1 = 0.01 and µ2 = 10, the t-cooperative equilibrium (t-CE) rule with equal weights is group ineﬃcient, in the sense that joint payments are higher in the noncooperative MPNE than in the t-CE (Mar´ın-Solano,2014). For bargaining n n powers of the form η1 = 1, η2 = ρ1/ρ2 (Sorger, 2006), we obtain β1 = −25.6052, β2 = −132.972, fc fc fc fc β1 = −23.1472 and β2 = −132.713, with weights λ1 = 0.406842, λ2 = 0.0397278. Note that by assigning the more patient Player 1 a bargaining power much higher than that to the more fc n impatient Player 2, an individually rational BFC solution exists, with V1 (x) − W1 (x) = 2.45796, fc n fc fc n n V2 (x) − W2 (x) = 1.25856, and V1 (x) + V2 (x) > W1 (x) − W2 (x).

4.3 Dynamic Nash t-bargaining solution with no commitment (TCB)

The ﬁrst step consists in computing the threat point in case of disagreement (see Section 3.3). In order to do so, we compute the threat point in case of partial commitment, and take later on the limit when the period of commitment τ goes to zero.

4.3.1 Dynamic Nash t-bargaining solution with partial commitment (BPC): The threat point

The threat point in case of disagreement is given as in Deﬁnition 3. Let us compute it. pc pc pc pc At time t players take as given the decision rule for s > t + τ. Let (c1 , . . . , cN ) = (φ1 , . . . , φN ) be the corresponding future decision rule. Hence Z t+τ Z ∞ −ρi(s−t) −ρi(s−t) pc Ji = µi e ln ci(s) ds + µi e ln φi (x(s), s) ds , (23) t t+τ with ) x˙(s) = x(s)(a − b ln x(s)) − PN c (s) i=1 i for s ∈ [t, t + τ] , (24) x(t) = xt ) x˙(s) = x(s)(a − b ln x(s)) − PN φpc(x(s), s) i=1 i for s > t + τ . (25) x(t + τ) = xt+τ pc Since we are interested in the existence of linear strategies, we will assume that φi (x(s), s) = pc Ai (s)x(s), for s > t + τ. By integrating (25) we obtain

" Z s N ! # −b(s−t−τ) −b(s−z) X pc x(s) = exp e ln xt+τ + e a − Ai (z) dz . (26) t+τ i=1

14 Proposition 2 In Problem (16-17), if, according to Deﬁnition 3, Players follow, for s > t + τ, pc pc strategies φi (x(s), s) = Ai (s)x(s), then the threat points in case of disagreement under partial τ µi τ commitment are given by Wi (x, t) = ln x + βi (t), for i = 1,...,N, where ρi + b " PN ! PN 1 a − (ρj + b) a − (ρj + b) τ −ρiτ j=1 j=1 −ρiτ −(ρi+b)τ βi (t) = µi 1 − e 1 − + 1 + e − 2e ρi b b(ρi + b)    Z ∞ Z s N −ρi(s−t) pc −b(s−z) X pc + e ln Ai (s) + e (a − Aj (z)) dz ds . t+τ t+τ j=1

τ The corresponding decision rules are φi (x(s), s) = (ρi + b)x(s), for s ∈ [t, t + τ].

Proof: See the Appendix. As in the case with full commitment, equation (14) can be used in determining the corresponding dynamic Nash t-bargaining solution with partial commitment. Note that, for s ∈ [t, t + τ], players apply the same threat strategies as in the Nash t-bargaining solution with full commitment (given by the MPNE).

4.3.2 Dynamic Nash t-bargaining solution with no commitment (TCB): The threat point

Next we can use Proposition 2 as the departing point for studying time-consistent bargaining solutions, where threat points can be computed as the limit, when τ goes to zero, of the threat points with partial commitment.

nc nc Proposition 3 In Problem (16-17), if Players follow, for s > t, strategies φi (x(s)) = Ai x(s), the threat point in case of disagreement under no commitment is given by

nc a − PN Anc 0 µi ln Ai j=1 j Wi (x, t) = ln x + + , ρi + b ρi ρi(ρi + b)

0 i = 1,...,N, and the corresponding decision rules are φi (x(s)) = (ρi + b)x(s), for s ∈ [t, t + τ].

Proof: It follows by taking the limit τ → 0+ in Proposition 2. Note that in this limit we can restrict our attention to stationary strategies.

4.3.3 Time consistent dynamic bargaining: equilibrium strategy

For the derivation of the dynamic Nash t-bargaining solution with no commitment, which is fully time-consistent, we substitute the expression of the threat point W 0 and threat strategy φ0 (see Proposition 3) into (15).

Proposition 4 Within the class of linear strategies, time consistent Nash t-bargaining equilibria nc nc (or Nash t-bargaining equilibria with no commitment) φi (x) = Ai x, i = 1,...,N, are the solutions to (ρ + b)η i i = (27) nc h nc PN nc i Ai (ρi + b)(ln Ai − ln(ρi + b)) + k=1(ρk + b − Ak )

N X ηj . nc PN nc j=1 (ρj + b)(ln Aj − ln(ρj + b)) + k=1(ρk + b − Ak )

15 Proof: See the Appendix.

The equation system (27) is highly nonlinear. In the numerical illustrations we will restrict our attention to the case of just two players, N = 2 and values of ηi (bargaining powers) simplifying the equations. In particular, note that, since

(ρi + b)ηi (ρl + b)ηl h nc i = h nc i , nc ln Ai PN nc nc ln Al PN nc A (ρi + b) + (ρk + b − A ) A (ρl + b) + (ρk + b − A ) i ln(ρi+b) k=1 k l ln(ρl+b) k=1 k for all i, l = 1,...,N, if ηi(ρi + b) = ηl(ρl + b), then

" nc N # " nc N # nc ln Ai X nc nc ln Al X nc Ai (ρi + b) + (ρk + b − Ak ) = Al (ρl + b) + (ρk + b − Ak ) ln(ρi + b) ln(ρl + b) k=1 k=1 and Equation (27) becomes

PN nc (ρi + b)ηi j=1 ηjAj h nc i = h nc i . nc ln Ai PN nc nc ln Ai PN nc A (ρi + b) + (ρk + b − A ) A (ρi + b) + (ρk + b − A ) i ln(ρi+b) k=1 k i ln(ρi+b) k=1 k

Therefore,

N N nc N nc X X Aj X Aj η (ρ + b) = η Anc = η (ρ + b) = η (ρ + b) . i i j j j j ρ + b i i ρ + b j=1 j=1 j j=1 j

As a result, for ηi/ηl = (ρl + b)/(ρi + b) (this speciﬁcation of bargaining powers coincides, for b = 0, PN nc with that in Sorger (2006)), then j=1 Aj /(ρj + b) = 1. In particular, for N = 2, if η1 = 1 and η2 = (ρ1 + b)/(ρ2 + b) (just relative powers are relevant), then we have to solve the equation system

nc 2 nc nc nc (A1 ) − (ρ1 + b)A1 ln A1 + A1 ((ρ1 + b) ln(ρ1 + b) − ρ1 − ρ2 − 2b) =

nc 2 nc nc nc (A2 ) − (ρ2 + b)A2 ln A2 + A2 ((ρ2 + b) ln(ρ2 + b) − ρ1 − ρ2 − 2b) , nc nc A1 A2 = (ρ2 + b) 1 − . ρ1 + b

4.4 Comparison of the results: numerical illustrations

We ﬁrst note that, if players are symmetric, µ1 = ··· = µN and ρ1 = ··· = ρN , the symmetric equilibrium is the same for the diﬀerent cooperative solution concepts.

Proposition 5 In Problem (16-17), if players are symmetric, within the class of linear decision rules, the standard symmetric cooperative decision rule is also a dynamic Nash t-bargaining solution with full and with no commitment.

Proof: See the Appendix. Next, we illustrate numerically the results obtained in the noncooperative case with those in a cooperative setting given by the symmetric t-cooperative equilibrium rule and the dynamic bargaining solutions in case of no commitment and full commitment. Since, in the expression of the value function, αi = µi/(ρi + b) for all the solution concepts, in the analysis of eﬃciency levels (i.e. comparison of payments) it suﬃces to compute the values of βi. Value functions and decision rules are given by (18)-(19) (in the case of noncooperation), Proposition 1 with λi = 1 (t-cooperative decision

16 rule with equal weights), Proposition 1 with λi given by (22) (dynamic t-bargaining solution with full commitment) and Proposition 4 (dynamic t-bargaining solution with no commitment or time PN consistent dynamic bargaining). From (17), the steady state is given by x∞ = exp[(a− i=1 Ai)/b]. For the numerical illustrations, we take N = 2 (two players), a = 10, b = 2.18, µ1 = µ2 = 1,

x0 = 100 and ρi ∈ [0.01, 0.19]. For simplicity’s sake, in the dynamic bargaining solutions we take

η1 = 1 and η2 = (ρ1 + b)/(ρ2 + b). Note that, in this case, η2 is very near to η1.

Table 1 presents the results for harvest coefficients Ai and steady states. Harvesting is clearly lower under cooperation, as expected. Joint harvesting, and hence the corresponding steady states, are quite similar in the three cooperation settings considered, with some qualitative differences among players depending on the discount rates. In the two bargaining solutions, harvesting is higher if players are more impatient, being the difference slightly higher in the case of full commitment.

Table 1: Harvest coeﬃcients and steady states MPNE t-CE Bargaining, full commitment Time consistent bargaining n n n tc tc tc fc fc fc nc nc nc ρ1 ρ2 A1 A2 x∞ A1 A2 x∞ A1 A2 x∞ A1 A1 x∞ 0.01 0.01 2.19 2.19 13.1705 1.09500 1.09500 35.9658 1.09500 1.09500 35.9658 1.09500 1.09500 35.9658 0.01 0.07 2.19 2.25 12.8130 1.10980 1.10980 35.4809 1.08715 1.13306 35.4708 1.09063 1.12949 35.4723 0.01 0.13 2.19 2.31 12.4651 1.12420 1.12420 35.0151 1.07951 1.17134 34.9758 1.08637 1.16411 34.9818 0.01 0.19 2.19 2.37 12.1267 1.13822 1.13822 34.5675 1.07207 1.20982 34.4814 1.08221 1.19884 34.4946 0.07 0.07 2.25 2.25 12.4651 1.12500 1.12500 34.9894 1.12500 1.12500 34.9894 1.12500 1.12500 34.9894 0.07 0.13 2.25 2.31 12.1267 1.13980 1.13980 34.5175 1.11715 1.16306 34.5079 1.12062 1.15949 34.5094 0.07 0.19 2.25 2.37 11.7975 1.15422 1.15422 34.0639 1.10950 1.20133 34.0266 1.11636 1.19410 34.0324 0.13 0.13 2.31 2.31 11.7975 1.15500 1.15500 34.0395 1.15500 1.15500 34.0395 1.15500 1.15500 34.0395 0.13 0.19 2.31 2.37 11.4773 1.16981 1.16981 33.5802 1.14715 1.19306 33.5712 1.15062 1.18949 33.5726 0.19 0.19 2.37 2.37 11.1657 1.18500 1,18500 33.1155 1,18500 1,18500 33.1155 1,18500 1,18500 33.1155

Table 2 illustrates eﬃciency by showing gains under cooperation over the noncooperative MPNE. Although gains are relatively low for players with higher discount rates, all the cooperative solutions Pareto dominate payments given by the MPNE.

Table 2: Eﬃciency analysis t-CE Bargaining, full commitment Time consistent bargaining tc n tc n fc n fc n nc n nc n ρ1 ρ2 V1 − V1 V1 − V2 V1 − V1 V1 − V2 V1 − V1 V1 − V2 0.01 0.01 30.685 30.685 30.685 30.685 30.685 30.685 0.01 0.07 33.4160 4.0013 31.3260 4.2938 31.6530 4.2493 0.01 0.13 36.1300 1.9580 31.9610 2.2658 32.6110 2.2194 0.01 0.19 38.8290 1.2110 32.5920 1.5200 33.5720 1.4739 0.07 0.07 4.3836 4.3836 4.3836 4.3836 4.3836 4.3836 0.07 0.13 4.7633 2.1599 4.4727 2.3133 4.5177 2.2900 0.07 0.19 5.1408 1.3467 4.5612 1.5520 4.6515 1.5210 0.13 0.13 2.3604 2.3604 2.3604 2.3604 2.3604 2.3604 0.13 0.19 2.5596 1.4813 2.4071 1.5836 2.4307 1.5680 0.19 0.19 1.6150 1.6150 1.6150 1.6150 1.6150 1.6150

17 5 An exhaustible resource model with general isoelastic utilities

As a second example, we solve a model describing the joint management of an exhaustible natural resource in case players have general isoelastic utilities with diﬀerent marginal elasticities. The intertemporal utility function of Player i is given by Z ∞ −ρi(s−t) Ji = e ui(ci(s)) ds (28) t with  1−σi ci − 1  if σi > 0, σi 6= 1, for i ∈ {1,...,J} ui(ci) = 1 − σi  ln ci if σi = 1, for i ∈ {J + 1,...,N} The stock of the resource evolves according to

N X x˙(s) = − ci(s) , x(t) = xt . (29) i=1 In this model, the stock of the resource converges in all the solution concepts to the steady state x∞ = 0.

5.1 Noncooperative MPNE and the t-cooperative equilibrium

The problem in which players do not cooperate and have constant discount rates has been studied in detail in the literature. For the sake of completeness, and to ﬁx notation, we present next a sketch of the derivation of noncooperative MPNE in (stationary) linear strategies. For the time-consistent cooperative equilibrium rule, we center our attention in weights for which linear strategies can exist.

5.1.1 Noncooperative MPNE

n First, for j ∈ {1,...,J}, σj 6= 1 and, if φl (x) denotes an equilibrium strategy for Player l 6= j, then Player j aims to solve    c1−σj − 1  j n0 X n  max − Wj (x) cj + φl (x) , {cj } 1 − σj  l6=j 

n where Wj (x) denotes the value function of Player j. From the maximization problem we obtain −1/σj n h n0 i n n n n cj = Wj (x) . Since we are looking for strategies of the form ci = φi (x) = Ai x, Ai ≥ 0, −σ n n 1−σj n n n j for i = 1,...,N, then Wj (x) = αj x + βj , where αj = Aj /(1 − σj). By construction, n the solution veriﬁes the conditions φi (0) = 0 and limt→∞ x(t) ≥ 0, as expected. PN n PN n Next, note that the solution tox ˙(s) = − i=1 ci (s) = − i=1 Ai x(s) with the initial condition PN n n n PN n x(t) = xt is given by x(s) = xt exp[− i=1 Ai (s − t)]. Then, cj (s) = Aj xt exp[− i=1 Ai (s − t)]. By substituting in the expression of the value function and assuming that the integrals converge we obtain

1−σj n − PN An(s−t) A x e i=1 i − 1 Z ∞ (cn(s))1−σj − 1 Z ∞ j t n −ρj (s−t) j −ρj (s−t) Wj (xt) = e ds = e ds t 1 − σj t 1 − σj

18 n 1−σj Z ∞ Z ∞ (Aj xt) −ρ −(1−σ ) PN An τ 1 −ρ τ = e j j ( i=1 i ) dτ − e j dτ . 1 − σj 0 1 − σj 0

n 1−σj n σj n By identifying xt = x, since Wj (x) = x / (Aj ) (1 − σj) + βj we obtain

N X 1 An = ρ + (1 − σ ) An , βn = − . (30) j j j i j ρ (1 − σ ) i=1 j j

n Next, for k ∈ {J +1,...,N}, uk(ck) = ln ck and, proceeding in a similar way, we obtain Ak = ρk. n n n n Hence, ck = ρkx and Wk (x) = αk ln x + βk , with

PN n n 1 n ln ρk i=1 Ai αk = , βk = − 2 . (31) ρk ρk ρk From (30) and (31) we can easily derive:

 PN ρ  ρ + (1 − σ ) l=1 l for i ∈ {1,...,J} n i i PJ Ai = 1 − j=1(1 − σj) (32)   ρi for i ∈ {J + 1,...,N}

 !−σi  1 PN ρ  ρ + (1 − σ ) l=1 l for i ∈ {1,...,J} αn = 1 − σ i i PJ (33) i i 1 − j=1(1 − σj)   1/ρi for i ∈ {J + 1,...,N}

 −1  − (ρi(1 − σi)) for i ∈ {1,...,J} n  N β = X (34) i ln ρ /ρ − ρ /ρ2 for i ∈ {J + 1,...,N}  i i l i l=1 Note that, if

PN J (1 − σi) ρl X ρ > l=1 , with (1 − σ ) − 1 6= 0 , for i =,...,J, i PJ j j=1(1 − σj) − 1 j=1

n the extraction rates Ai of all agents are strictly positive, the integrals in (30) converge, so that the value functions are well deﬁned, and limt→∞ x(t) = x∞ = 0.

5.1.2 Time-consistent cooperative equilibria with nonconstant weights

In general, t-cooperative equilibria with constant weights seem to be extremely cumbersome to compute for this simple problem. In fact, no linear decision rules exist unless all marginal elasticities coincide. In order to ﬁnd analytical solutions, we introduce nonconstant weights. The following proposition characterizes the functional form of weight functions necessary for the existence of linear decision rules in the computation of the t-cooperative equilibria with nonconstant weights.

Proposition 6 In Problem (28-29), if linear t-cooperative equilibria with nonconstant weights ex-

σi σi σl σi−σl ist, then weight functions are of the form λi(x) = νix h(x), i.e. λi(x)/λl(x) = (Ai /Al )x , for i, l = 1,...,N, with h(x) an arbitrary continuously diﬀerentiable function.

Proof: See the Appendix.

19 A natural question arises in this context. Do linear strategies always exist for arbitrary weight

σi functions of the form λi(x) = νix h(x)? For this family of weight functions, from (48) we have

−1/σi  N  1/σi X σj 0 ci = ν  νjx Vj (x) x . j=1

It is necessary to prove that there are value functions V1(x),...,VN (x) such that the quantity PN σi 0 i=1 νix Vi (x) is a constant number. Since we are interested in the existence of linear strategies, 1−σj we take, for j ∈ {1,...,J}, Vj(x) = αjx + βj, and for k ∈ {J + 1,...,N}, Vk(x) = αk ln x + βk.

Solutions obtained from other choices of functions Vi(x), i = 1,...,N, will be nonlinear. Then 0 −σj 0 −1 PN σi 0 PJ PN Vj (x) = (1−σj)αjx , Vk(x) = αkx and i=1 νix Vi (x) = j=1 νj(1−σj)αj + k=J+1 νkαk so !1/σi ν c = i x . (35) i PJ PN j=1 νj(1 − σj)αj + k=J+1 νkαk From (35) it becomes clear that, for the existence of linear strategies, it is necessary that coeﬃcients PJ PN αi, i = 1,...,N, exist such that j=1 νj(1 − σj)αj + k=J+1 νkαk > 0. Finally, for the above mentioned family of weight functions, the corresponding t-cooperative decision rules are presented.

σi Corollary 1 In Problem (28-29), for λi(x) = νix h(x), i = 1,...,N, if t-cooperative equilibria tc tc tc with nonconstant weights exist, they are of the form ci (x) = Ai x, for Ai > 0, with corresponding tc tc 1−σi tc tc tc tc value functions Vi (x) = αi x + βi for i = 1,...,J, and Vi (x) = αi ln x + βi if i = tc tc tc J + 1,...,N, where coeﬃcients Ai , αi and βi are the solution to the equation system

!1/σi ν Atc = i i PJ tc PN tc j=1 νj(1 − σj)αj + k=J+1 νkαk  ! 1 (Atc)1−σi  i for i ∈ {1,...,J} αtc(x) = 1 − σ PN tc i i ρi + (1 − σi) l=1 Al   1/ρi for i ∈ {J + 1,...,N}  −1  − ((1 − σi)ρi) for i ∈ {1,...,J} tc  N β = X i (ln Atc)/ρ − Atc/ρ2 for i ∈ {J + 1,...,N}  i i l i l=1

Proof: See the Appendix.

5.2 Dynamic Nash t-bargaining solution with full commitment (BFC)

Once we have computed both the MPNE (which is the threat point in case of disagreement for this n solution concept, Wi (x), for i = 1,...,N) and the time-consistent equilibria, we can use (14) for the calculation of the BFC. Since just relative weights are relevant, we can take h(x) = 1/x. From the results in Section 5.1.1 and Section 5.1.2 we have:

fc fc Proposition 7 If BFC solutions exist, they are given by Corollary 1 with weight functions λi (x)/λj (x) = fc fc σi−σj fc fc σi−1 fc (νi /νj )x (e.g., λi (x) = νi x for h(x) = 1/x) and coeﬃcients νi given by

( fc n fc ηi/(αi − αi ) for i ∈ {1,...,J} νi = fc n (36) ηi/(βi − βi ) for i ∈ {J + 1,...,N}

20 n n fc fc tc tc with αi and βi given by (33)-(34), and αi and βi the values of αi and βi in Corollary 1 for PJ fc PN fc νi given by (36). BFC solutions exist if j=1 νj(1 − σj)αj + k=J+1 νkαk > 0 (where σj 6= 1 fc fc n fc n and σk = 1), so that Ai > 0, and αi > cαi for i ∈ {1,...,J}, βi > βi for i ∈ {J + 1,...,N}.

Proof: It follows from (14). The existence of BFC solutions is not guaranteed for general utility functions, and conditions fc n fc n for the positivity of Ai and αi > αi for i ∈ {1,...,J}, βi > βi for i ∈ {J + 1,...,N}, are too cumbersome to be expressed analytically. In the case of logarithmic utility functions (J = 0), the corresponding weights are given by η λfc = νfc = i . (37) i i fc n βi − βi

5.3 Dynamic Nash t-bargaining solution with no commitment (TCB)

As in the example studied in Section 4, we compute ﬁrst the threat point in case of partial commitment, we take later the limit when the period of commitment goes to zero, and we compute ﬁnally the TCP solution.

5.3.1 Dynamic Nash t-bargaining solution with partial commitment (BPC): The threat point

For the computation of the threat point, from Deﬁnition 3, at time t players take as given the pc pc pc pc decision rule for s > t + τ. Let (c1 , . . . , cN ) = (φ1 , . . . , φN ) be the corresponding future decision rule. Hence Z t+τ Z ∞ −ρi(s−t) −ρi(s−t) pc Ji = e ui(ci(s)) ds + e ui(φi (x(s), s)) ds , (38) t t+τ with ) x˙(s) = − PN c (s) i=1 i for s ∈ [t, t + τ] , (39) x(t) = xt ) x˙(s) = − PN φpc(x(s), s) i=1 i for s > t + τ . (40) x(t + τ) = xt+τ pc Since we are interested in the existence of linear strategies, we will assume that φi (x(s), s) = pc Ai (s)x(s), for s > t + τ. By integrating (40) we obtain

pc pc − PN R s Apc(z) dz x(s) = Λ (t + τ, s)xt+τ , with Λ (t + τ, s) = e i=1 t+τ i . (41)

Proposition 8 In Problem (28-29) if, according to Deﬁnition 3, Players follow, for s > t + τ, pc pc strategies φi (x(s), s) = Ai (s)x(s), then the candidates to threat points in case of disagreement under partial commitment are given by ( τ 1−σi τ τ αi (t)x + βi (t) for i ∈ {1,...,J} , Wi (x, t) = τ τ αi (t) ln x + βi (t) for i ∈ {J + 1,...,N} ,

τ τ and the corresponding decision rules are φi (x(s), s) = Ai (s)x(s), for s ∈ [t, t + τ]. For every τ τ τ t ∈ [0, ∞), functions αi (t), βi (t) and Ai (t), i = 1,...,N, are the solutions to the integral equation system ( τ −1/σi τ [(1 − σi)αi (t)] for i ∈ {1,...,J} , Ai (t) = τ −1 [αi (t)] for i ∈ {J + 1,...,N} ,

21  h t+τ −1 R −ρi(s−t) τ τ 1−σi  (1 − σi) e [Ai (s)Λ (t, s)] ds+  t for i ∈ {1,...,J} , τ (1−σ ) ∞ pc 1−σ i α (t) = τ i R −ρi(s−t) pc i i [Λ (t, t + τ)] t+τ e (Ai (s)Λ (t + τ, s)) ds   1/ρi for i ∈ {J + 1,...,N} ,   −1/((1 − σi)ρi) for i ∈ {1,...,J} ,  t+τ −ρiτ βτ (t) = R e−ρi(s−t) ln [Aτ (s)Λτ (t, s)] + e Λτ (t, t + τ) i t i ρi ∞ pc for i ∈ {J + 1,...,N} ,  R −ρi(s−t) pc + t+τ e ln [Ai (s)Λ (t + τ, s)] ds τ − PN R s Aτ (z) dz where Λ (t, s) = e i=1 t i .

Proof: See the Appendix. As in the case with full commitment, if the threat (or reference) point obtained from Proposition 8 can be computed, equation (14) can be used in determining the corresponding dynamic Nash t- bargaining solution with partial commitment. It is interesting to realize that players with log-utilities (i ∈ {J+1,...,N}) apply the same threat strategies as in the Nash t-bargaining solution with full commitment (given by the MPNE), although the corresponding payments and, therefore, threat points are diﬀerent, in general. On the contrary, this is not the case for players with marginal elasticities diﬀerent to one. For i ∈ {1,...,J}, threat points are computed as the solutions to a highly nonlinear system of integral equations. Existence (and uniqueness) of solutions is not guaranteed for this very complicated equation system. However, we can use Proposition 8 as the departing point for studying time-consistent bargaining solutions, where threat points can be computed as the limit when τ goes to zero.

5.4 Dynamic Nash t-bargaining solution with no commitment (TCB): the threat point

As in the other t-bargaining solutions, the ﬁrst step consists in computing the threat point. We can proceed by taking the limit, when τ goes to zero, of the strategies derived in Proposition 8.

nc nc Proposition 9 In Problem (28-29), if Players follow, for s > t, strategies φi (x(s)) = Ai x(s), then the candidates to threat points in case of disagreement under no commitment (time-consistent bargaining) are given by ( 0 1−σi 0 0 αi x + βi for i ∈ {1,...,J} , Wi (x) = 0 0 αi ln x + βi for i ∈ {J + 1,...,N} ,

0 0 0 0 0 and the corresponding decision rules are φi (x(s)) = Ai x(s). Functions αi , βi and Ai , i = 1,...,N, are given by ( −1/σ (1 − σ )α0 i for i ∈ {1,...,J} , A0(x, t) = i i i 0−1 αi for i ∈ {J + 1,...,N} ,

 0 1−σi (Ai )  PN 0 for i ∈ {1,...,J} , 0 (1−σi)[ρi+(1−σi) A ] αi = i=1 i  1/ρi for i ∈ {J + 1,...,N} ,

( − 1 for i ∈ {1,...,J} , 0 (1−σi)ρi β = 0 P 0 i ln Ai i=1N Ai − 2 for i ∈ {J + 1,...,N} . ρi ρi Proof: It follows by taking the limit τ → 0+ in Proposition 9. Note that in this limit we can restrict our attention to stationary strategies.

22 5.4.1 Time consistent dynamic bargaining: equilibrium strategy

Once we have computed the threat point we use (15) for the derivation of the dynamic Nash t-bargaining solution with no commitment.

Proposition 10 Within the class of linear strategies, (time consistent) dynamic Nash t-bargaining nc nc equilibria φi (x) = Ai x, i = 1,...,N, are the solutions to

PN nc nc −σi (1 − σi)ηi(ρi + (1 − σi) k=1 Ak )(Ai ) nc N N = F (A ) , P nc nc 1−σi 0 1−σi nc 1−σi P 0 nc (ρi + (1 − σi) k=1 Ak )[(Ai ) − (Ai ) ] + (Ai ) (1 − σi) k=1(Ak − Ak ) (42) for i = 1,...,J, and η i = F (Anc) , (43) nc PN nc ρi(ln Ai − ln ρi) + k=1(ρk + b − Ak )

nc nc nc for i = J + 1,...,N, with A = (A1 ,...,AN ),

J nc −σj nc X (1 − σj)ηj(Aj ) F (A ) = N N P nc nc 1−σj 0 1−σj nc 1−σj P 0 nc j=1 (ρj + (1 − σj) k=1 Ak )[(Aj ) − (Aj ) ] + (Aj ) (1 − σj) k=1(Ak − Ak )

N X ηj + , nc PN nc j=J+1 ρj(ln Aj − ln ρj) + k=1(ρk + b − Ak ) 0 and Ai given from Proposition 9.

Proof: It is similar to the proof of Proposition 4.

5.5 Comparison of the results

For the calculation of the dynamic Nash t-bargaining solution with no commitment, it is clear that the equation system (42-43) is highly nonlinear. In the case of logarithmic utilities, if ηi/ηj = ρj/ρi, for all i, j = 1,...,N (see Sorger, 2006), by reproducing the calculations in Section 4.3.3 we obtain PN nc j=1 Aj /ρj = 1. In this case, for N = 2, we have to solve the equation

nc 2 nc nc nc nc 2 nc nc (A1 ) − ρ1A1 ln A1 + A1 (ρ1 ln ρ1 − ρ1 − ρ2) = (A2 ) − ρ2A2nc ln A2 + A2 (ρ2 ln ρ2 − ρ1 − ρ2) ,

nc nc with A2 = ρ2(1 − A1 /ρ1). In order to establish comparisons, we ﬁrst note that, in case of equal marginal elasticities and discount rates, as in the previous example with log-utilities, the symmetric equilibrium is the same for the diﬀerent cooperative solution concepts.

Proposition 11 In Problem (28-29), if σi = σ and ρi = ρ, for i = 1,...,N, the (standard) symmetric cooperative decision rule is also a dynamic Nash t-bargaining solution with full and with no commitment.

Proof: See the Appendix. Next, we illustrate numerically some of the previous results. Note that there are some diﬀer- ences in comparison with the model in Section 4. First, η1 can be very diﬀerent to η2 and, since b = 0, relative extraction rates Ai/Aj will be much more unequal, so comparisons of the results become clearer. In addition, all solution concepts converge to the same steady state x∞ = 0. We

23 center our attention in comparing the two bargaining solutions (with full commitment and with no

commitment). We consider N = 2 (two players), ui(ci) = ln ci (logarithmic utilities), η1 = 1 and 11 η2 = ρ1/ρ2 . Discount rates ρ1, ρ2 belong to the rank [0.01, 0.19]. Table 3 shows the extraction rates corresponding to the dynamic bargaining solutions with full commitment and with no commitment for some values of the discount rates.

Table 3: Harvest coeﬃcients. Dynamic bargaining with full/no commitment fc fc nc nc fc fc nc nc ρ1 ρ2 A1 A2 A1 A2 A1 + A2 A1 + A2 0.01 0.01 0.005000 0.005000 0.005000 0.005000 0.010000 0.010000 0.01 0.07 0.002628 0.051602 0.003378 0.046354 0.054230 0.049732 0.01 0.13 0.002045 0.103416 0.002796 0.093654 0.105461 0.096450 0.01 0.19 0.001738 0.156970 0.002452 0.143404 0.158708 0.145856 0.07 0.07 0.035000 0.035000 0.035000 0.035000 0.070000 0.070000 0.07 0.13 0.029306 0.075575 0.031737 0.071061 0.104881 0.102797 0.07 0.19 0.025944 0.119580 0.029596 0.109669 0.145524 0.139265 0.13 0.13 0.065000 0.065000 0.065000 0.065000 0.130000 0.130000 0.13 0.19 0.058481 0.104528 0.061328 0.100368 0.163009 0.161696 0.19 0.19 0.095000 0.095000 0.095000 0.095000 0.190000 0.190000

In both solution concepts, global extraction A1 +A2 increases with an increment of the discount rates, as expected. The increase is lower in the case of no commitment (time-consistent bargaining). Note also that a higher divergence of discount rates induces also an increase in the joint extraction

rates. For instance, global extractions rates are higher with ρ1 = 0.01 and ρ2 = 0.13, with an

average discount rateρ ¯ = 0.07, compared with the sum of extraction rates if ρ1 = ρ2 = 0.07. The same tendency is observed for other values of discount rates not included in Table 3. At an individual level, for a fixed value of the discount rate of the first player, an increase in the discount rate of the second player induces a reduction of the extraction rate of the first player and an increase of the extraction rate of the second player. Table 4 compares the payments for both players between the two dynamic bargaining solutions considered.

Table 4: Two players with η1 = 1 and η2 = ρ1/ρ2 (logarithmic utility) fc nc fc nc P2 fc nc ρ1 ρ2 V1 − V1 V2 − V2 i=1(Vi − Vi ) 0.01 0.07 -70.0700 0.6141 -69.4559 0.01 0.13 -121.3900 0.2295 -121.1605 0.01 0.19 -162.9300 0.1197 -162.8103 0.07 0.13 -1.5637 0.3506 -1.2131 0.07 0.19 -3.1585 0.2819 -2.8766 0.13 0.19 -0.4434 0.1773 -0.2661

It is interesting to observe that, if ρ1 6= ρ2, joint payments are higher under the time consistent

11Due to the high nonlinearity of the equation systems involved, other speciﬁcations of the parameters increase the computational tractability of the problem in an important way

24 dynamic bargaining solution with no commitment compared with those under the dynamic bargaining solution with full commitment. However, although the latter Pareto dominates non cooperative payments under the MPNE, this property is no guaranteed in the former (time-consistent) solution.

6 Concluding remarks

In this paper we have introduced a dynamic Nash bargaining procedure for different threat points. The solution provided in case of full commitment, if it exists, Pareto dominates payments provided by the noncooperative Markov Perfect equilibria, but it is not fully time-consistent. As an alternative, we propose using a different approach, in which threat of noncooperative behavior is maintained just during an instant of time. This second solution concept, that we call dynamic Nash t-bargaining solution with no commitment (or time-consistent bargaining), is the natural extension (or, to be more precise, adaptation) to a continuous time setting of the recursive Nash bargaining solution introduced in Sorger (2006) in discrete time. These ideas have been applied to the study of two common property resource games: a model on the joint management of a renewable resource with log-utilities, and a model of an exhaustible natural resource with general isoelastic utilities. The latter admits a straightforward extension if a linear production function is introduced. We think that the dynamic bargaining procedure proposed in the paper can be valuable in the search of cooperative solutions, not just for problems in which players exhibit unequal discount rates, but also in case they discount the future at the same rate but have different preferences (i.e. utility functions). Concerning the applications in resource games, the renewable resource model analyzed in the paper has the simplifying property that it has a unique positive steady state. Other models with different production functions admitting two steady states could be studied.

References

[1] Benveniste LM, Scheinkman JA (1979) On the Diﬀerentiability of the Value Function in Dynamic Models in Economics. Econometrica 47: 727–732 [2] Binmore K (1987) Nash bargaining theory II. In: Binmore, K., Dasgupta, P. (Eds.), The Economics of Bargaining. Basil Blackwell Ltd., Oxford, pp. 61–76 [3] Breton M, Keoula MY (2014) A Great Fish War Model with Asymmetric Players. Ecological Economics 97: 209–223 [4] Clemhout S, Wan HY (1985) Dynamic common property resources and environmental problems. Journal of Optimization Theory and Applications 46: 471–481 [5] de-Paz A, Mar´ın-Solano J, Navas J (2013) Time-Consistent Equilibria in Commons Access Re- source Games with Asymmetric Players under Partial Cooperation. Environmental Modeling and Assessment 18: 171–184 [6] Ekeland I, Karp L, Sumaila R (2015) Equilibrium resource management with altruistic overlapping generations. Journal of Environmental Economics and Management 70: 1–16 [7] Ekeland I, Lazrak A (2010) The golden rule when preferences are time inconsistent. Mathe- matics and Financial Economics 4: 29–55 [8] Ekeland I, Zhou Qinlong, Long Yiming Long (2013) A new class of problems in the calculus of variations. Regular and Chaotic Dynamics 18: 553–584

25 [9] Gollier C, Zeckhauser R (2005) Aggregation of Heterogeneous Time Preferences. Journal of Political Economy 113: 878-896 [10] Hara, C (2013) Heterogeneous Impatience and Dynamic Inconsistency. Preprint [11] Haurie A (1976) A Note on Nonzero-Sum Differential Games with Bargaining Solution. Jour- nal of Optimization Theory and Applications 18: 31–39 [12] Jackson MO, Yariv L (2015) Collective Dynamic Choice: The Necessity of Time Inconsistency. American Economic Journal: Microeconomics 7: 150–178 [13] Karp L (2007) Non-constant discounting in continuous time. Journal of Economic Theory 132: 557–568 [14] Karp L (2017) Provision of a public good with multiple dynasties. Economic Journal 127: 2641–2664 [15] Li CZ, Lofgren KG (2000). Renewable Resources and Economic Sustainability: A Dynamic Analysis with Heterogeneous Time Preferences. Journal of Environmental Economics and Management 40: 236-250 [16] Mar´ın-SolanoJ (2014) Time-consistent equilibria in a differential game model with time inconsistent preferences and partial cooperation. Dynamic Games in Economics: 219–238. Springer. Series: Dynamic Modeling and Econometrics in Economics and Finance, Vol. 16 [17] Mar´ın-SolanoJ (2015) Group inefficiency in a common property resource game with asymmetric players. Economics Letters 136: 214–217 [18] Mar´ın-SolanoJ, Shevkoplyas EV (2011) Non-constant discounting and differential games with random horizon. Automatica 47: 2626-2638 [19] Millner A, Heal G (2017). Discounting by committee. Preprint [20] Munro GR (1979) The Optimal Management of Transboundary Renewable Resources. The Canadian Journal of Economics 12: 355–376 [21] Nash JF (1953) Two-person cooperative games. Econometrica 21: 128–140 [22] Roth A (1979) Axiomatic Models of Bargaining. Springer-Verlag, Berlin [23] Sorger G (2006) Recursive bargaining over a productive asset. Journal of Economic Dynamics and Control 30: 2637–2659 [24] Yeung DWK, Petrosyan LA (2015) Subgame consistent cooperative solution for NTU dynamic games via variable weights. Automatica 59: 84–89 [25] Yong J (2011) A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control and Related Fields 1: 83–118 [26] Zuber S (2011) The aggregation of preferences: can we ignore the past? Theory and Decision 70: 367–384

26 APPENDIX

Proof of Proposition 2: From (26) we can write

Z t+τ Z ∞ Z t+τ −ρi(s−t) −ρi(s−t) pc −ρi(s−t) Ji = µi e ln ci(s) ds + µi e ln[Ai (s)x(s)] ds = µi e ln ci(s) ds t t+τ t

 N  e−ρiτ Z ∞ Z s X +µ ln x + µ e−ρi(s−t) ln Apc(s) + e−b(s−z)(a − Apc(z)) dz ds . (44) i ρ + b t+τ i  i j  i t+τ t+τ j=1 Next, according to Deﬁnition 3, Player i aims to solve    τ N  ∂Wi X τ  max µi ln ci + x(a − b ln x) − ci − φj  , ci ∂x  j=1,j6=i 

∂W τ −1 hence cτ = µ i . By assuming linear strategies, cτ = φτ (x(s), s) = Aτ (s)x(s) and i i ∂x i i i

τ µi τ Wi (x, t) = τ ln x + βi (t) . (45) Ai (t) Alternatively, by substituting in (44) we obtain

Z t+τ Z t+τ e−ρiτ τ −ρi(s−t) τ −ρi(s−t) Wi (x, t) = µi e ln Ai (s) ds + µi e ln x(s) ds + µi ln xt+τ t t ρi + b   Z ∞ Z s N −ρi(s−t) pc −b(s−z) X pc +µi e ln Ai (s) + e (a − Aj (z)) dz ds . (46) t+τ t+τ j=1 τ By solving (24) for ci(s) = Ai (s)x(s) we get   Z s N −b(s−t) −b(s−z) X τ ln x(s) = e ln x + e a − Aj (z) dz t j=1 so, by continuity,   Z t+τ N −bτ −b(t+τ−z) X τ ln xt+τ = e ln x + e a − Aj (z) dz . t j=1

By substituting in (46) and identifying the result with (45) we obtain, after several calculations, τ τ Ai = ρi + b and βi as given in Proposition 2. 0 nc µi µi 0 Proof of Proposition 4. Since u (ci ) = nc = nc and φi (x) = (ρi + b)x, then ci Ai x Z ∞ nc −ρi(s−t) nc Vi (x) = e µi ln[Ai x(s)] ds , (47) t where N X nc x˙(s) = x(s)[a − b ln x(s)] − Ak x(s) , x(t) = x , k=1 whose solution is " # a − PN Anc x(s) = exp e−b(s−t) ln x + k=1 k (1 − e−b(s−t)) . b

27 nc 0 By substituting the equation above into Equation (47) and diﬀerentiating, we easily obtain (Vi ) (x) = µi 1 0 nc 0 nc 0 . The result follows by substituting the expressions for u (ci ), φi (x) and (Vi ) (x) into ρi + b x Equation (15). Proof of Proposition 5: The t-cooperative equilibrium rule coincides with the standard coopera- tc tive solution if ρi = ρ, and if players are symmetric it is given by Ai = (ρ+b)/N. From Proposition 4 it is straightforward to check that we obtain the same result, within the class of linear decision rules, for the dynamic t-bargaining solution with no commitment. For the case of dynamic bargaining with full commitment, note that, from Equations (19), (21) and (22), if ρ1 = ··· = ρN = ρ, fc then λiµi = λlµl, for all i, l = 1,...,N. Therefore, from Equation (20), Ai = (ρ + b)/N. Proof of Proposition 6: From (10) we solve   J 1−σj N N ! N ! X cj − 1 X X 0 X  max λj(x) + λk(x) ln ck − λi(x)Vi (x) · cl . {c1,...,cN } 1 − σj j=1 k=J+1 i=1 l=1  By applying the ﬁrst order condition we obtain, for all i = 1,...,N,

!1/σi λi(x) ci(x) = φi(x) = . (48) PN 0 l=1 λl(x)Vl (x)

If linear strategies φi(x) = Aix, Ai ≥ 0, for i = 1,...,N, then, from (48), we obtain

N ! X 0 σi λi(x) = λl(x)Vl (x) (Aix) , l=1

σ PN 0 i.e. weights must be of the form λi(x) = νixi h(x), with h(x) = l=1 λl(x)Vl (x) so λi(x)/λl(x) = σi σl σi−σl (Ai /Al )x . Proof of Corollary 1: PN c The solution to (29) is x(s) = xt exp[− i=1 Ai (s − t)]. By proceeding as in the derivation of the MPNE,

 fc 1−σi ! 1 (Aj ) 1  x1−σi − for i ∈ {1,...,J}  1 − σ PN fc ρ V fc(x) = i ρi + (1 − σi) l=1 Al i . (49) i PN fc  1 1 fc l=1 Al  ln x + ln Ai − 2 for i ∈ {J + 1,...,N} ρi ρi ρi The result follows from (35) and (49). Proof of Proposition 8: By substituting (41) in (38) we can easily derive Z t+τ −ρi(s−t) Ji = e ui(ci(s)) ds + Γi(t, τ, xt+τ ) , (50) t where ( ∞ 1−σi R −ρi(s−t) pc pc 1−σi −ρiτ [xt+τ t+τ e (Ai (s)Λ (t + τ, s)) ds − e /ρi]/(1 − σi) for i ∈ {1,...,J} Γi = ∞ −ρiτ R −ρi(s−t) pc (e /ρi) ln xt+τ + t+τ e ln[Ai(s)Λ (t + τ, s)] ds for i ∈ {J + 1,...,N} (51) We have to compute the Markovian Nash equilibria for Problem (50-51) with state dynamics (39). According do Deﬁnition 3, Player i aims to solve    τ N  ∂Wi X τ  max ui(ci) − ci + φj  , ci ∂x  j=1,j6=i 

28 τ −σi ∂Wi whose solution is ci = . ∂x τ We are restricting our attention to the existence of linear strategies, so we assume that φi (x(s), s) = ∂W τ Aτ (s)x(s), for i = 1,...,N. For these strategies, i = [Aτ (t)]−σi [x]−σi and i ∂x i

( 1−σi [Aτ (t)]−σi x + Bτ (t) for i ∈ {1,...,J} W τ (x, t) = i 1−σi i (52) i τ −1 τ [Ai (t)] ln x + Bi (t) for i ∈ {J + 1,...,N}

τ τ − PN R s Aτ (z) dz By integrating (39) we obtain, for s ∈ [t, t+τ], x(s) = xtΛ (t, s), where Λ (t, s) = e j=1 t j . For i ∈ {1,...,J}, from (50)-(51) we have

Z t+τ (φτ (x(s), s))1−σi − 1 τ −ρi(s−t) i Wi (x, t) = e ds + Γi(t, τ, xt+τ ) = t 1 − σi

1−σ pc 1−σ Z t+τ [Aτ (s)x(s)] i − 1 Z ∞ (A (s)Λpc(t + τ, s)) i e−ρiτ −ρi(s−t) i 1−σi −ρi(s−t) i e ds+xt+τ e ds− = t 1 − σi t+τ 1 − σi (1 − σi)ρi Z t+τ [Aτ (s)Λτ (t, s)]1−σi Z ∞ [Apc(s)Λpc(t + τ, s)]1−σi 1 1−σi −ρi(s−t) i 1−σi −ρi(s−t) i xt e ds+xt+τ e ds− . t 1 − σi t+τ 1 − σi (1 − σi)ρi − PN R t+τ Aτ (z) dz Since, by continuity, xt+τ = xte j=1 t j , by identifying xt = x, " Z t+τ [Aτ (s)Λτ (t, s)]1−σi τ −ρi(s−t) i Wi (x, t) = e ds + (53) t 1 − σi

Z ∞ pc pc 1−σi # (1−σ ) [A (s)Λ (t + τ, s)] 1 [Λτ (t, t + τ)] i e−ρi(s−t) i ds x1−σi − . t+τ 1 − σi (1 − σi)ρi Finally, from (52),

[Aτ (t)]−σi 1 Z t+τ τ i −ρi(s−t) τ τ 1−σi αi (t) = = e [Ai (s)Λ (t, s)] ds 1 − σi 1 − σi t Z ∞ p 1−σi −ρi(s−t) pc pc 1−σi + [Λ (t, t + τ)] e [Ai (s)Λ (t + τ, s)] ds t+τ and τ 1 Bi (t) = − . (1 − σi)ρi

In a similar way, for i ∈ {J + 1,...,N}, from (50)-(51) and (39) we obtain

Z t+τ Z t+τ τ −ρi(s−t) τ −ρi(s−t) τ Wi (x, t) = e ln φi (x(s), s)) ds + Γi(t, τ, xt+τ ) = e ln [Ai (s)x(s)] ds t t

e−ρiτ Z ∞ Z t+τ −ρi(s−t) pc pc −ρi(s−t) τ τ + ln xt+τ + e ln [Ai (s)Λ (t + τ, s)] ds = e ln [Ai (s)Λ (t, s)] ds ρi t+τ t 1 e−ρiτ Z ∞ τ −ρi(s−t) pc pc + ln x + Λ (t, t + τ) + e ln [Ai (s)Λ (t + τ, s)] ds ρi ρi t+τ and, from (52), the result follows. Proof of Proposition 11: The t-cooperative equilibrium rule coincides with the standard coop- ρ erative solution if ρ = ρ, and it is given by A = . It is easy to check that we obtain the same i i Nσ result for the dynamic Nash t-bargaining solution with full (Proposition 7) and no (Proposition 10) commitment.