Bargaining with Habit Formation

Transcription

1 Bargaining with Habit Formation Bahar Leventoğlu Duke University June 2012 Abstract Habit formation is a well-documented behavioral regularity in psychology and economics, however its implications on bargaining outcomes have so far been overlooked. I study an otherwise standard Rubinstein bargaining model with habit-forming players. In equilibrium, a player can strategically exploit his opponent s habit forming behavior via unilateral transfers off the equilibrium path to generate endogenous costs and gain bargaining leverage at no cost to himself on the equilibrium path. Uncertainty about habit formation may lead to a delay in bargaining. 1 Introduction Habit formation is a well documented behavioral regularity in psychology and behavioral economics (Camerer and Loewenstein 2004). Accordingly, human beings form habits for consumption and their current satisfaction level tends to be highly correlated with their past consumption level. Habit formation has been incorporated extensively to a number of research programs in economics, finance, international conflict, management science and social psychology among others. 1 However, the formal bargaining literature Thanks go to Alexandre Debs, Robert Powell and Huseyin Yildirim for helpful comments and suggestions. 1 For example, habit formation has been used to account for the consumption data in the US as well as other countries (Ferson and Constantinides 1991, Braun et al. 1993), 1

2 has so far overlooked its implications in bargaining. In this paper, I examine the role of habit formation in bargaining, and in particular, I explore how players can exploit habit formation to generate bargaining leverage in negotiations. I introduce habit formation into an otherwise standard bargaining model (Rubinstein 1982). 2 Two players collectively receive a flow of v units of consumption good over an infinite time-horizon. The players can consume the good only if they mutually agree to share it. The players discount their future payoffs. They make offers in an alternating fashion in the beginning of every period. If a player s proposal is accepted, the two players share v accordingly forever. If a proposal is rejected, v of the current period perishes. I extend this standard model as follows: When an offer is rejected, player 1 can make a unilateral transfer from his other resources to player 2, which player 2 may accept or reject. In the benchmark case, I model player 2 as an individual that forms habits for consumption and her current satisfaction levels tend to be highly correlated with her past consumption levels (Camerer and Loewenstein 2004). In particular, player 2 uses her consumption from the previous period as a reference point, and pays a cost if her current consumption falls below her past consumption level (Rozen 2010). Her cost is a linear function of the difference between current and previous consumption levels. I refer to the coeffi cient of this linear function as the cost coeffi cient. In the subgame perfect equilibrium of the game, player 1 gains bargaining leverage by exploiting player 2 s habit forming behavior to generate endogenous costs for player 2. Specifically, as long as offers are rejected, which happens only off the equilibrium path, player 1 alternates between making a unilateral transfer one period, which player 2 accepts, and no transfer the following period. Since player 2 pays a cost only when she consumes less than what she has consumed in the previous period, the alternating scheme of unilateral transfers lowers player 2 s continuation payoff off the equilibrium path when an offer is rejected. In turn, she accepts lower offers on the equilibrium path. This bargaining leverage comes to player 1 at no cost since he exploits player 2 s habit formation by unilateral transfers only off and some notable asset pricing anomalies such as the equity premium puzzle (Abel 1990, Constantinides 1990, Campbell and Cochrane 1999). Scholars in growth economics (e.g. Carroll, Overland, and Weil 1997) and monetary economics (e.g. Fuhrer 2000) have also utilized rational models of habit formation for their explanatory and predictive power. 2 See Osborne and Rubinstein (1990) for an extensive review of the literature on bargaining models. 2

3 the equilibrium path without actually making any transfer on the equilibrium path. In addition, a higher cost coeffi cient benefits player 1. Player 2 cannot commit to not accepting transfers by player 1, because momentary benefit of the transfer exceeds the gains in bargaining leverage that player 2 would gain by refusing the transfer. These qualitative features still hold when both players exhibit habit forming behavior or when the proposer is chosen randomly every period. The model predictions survive in the presence of asymmetric information, however, an equilibrium delay may emerge. When player 1 is uncertain about whether player 2 exhibits habit forming behavior or not, if player 2 makes the first offer, a low cost type player 2 can credibly signal her type to player 1 in a separating equilibrium by facing a risk of delay in negotiations. Such a separating equilibrium is possible only if the higher cost coeffi cient is not too high. Otherwise, the low cost and the high cost types pool by making the offer that the high cost type would make in the complete information game. A higher cost coeffi cient benefits player 1 by increasing his expected payoff. It also increases the likelihood of a delay, which only harms player 2 of low cost type. The potential delay disappears when player 1 makes the first offer. More importantly, player 1 continues to exploit player 2 s habit formation without actually making any transfers in the equilibrium of the incomplete information game. That is, the threat of the unilateral transfer is suffi cient to generate bargaining leverage for player 1. Closest to my work is the literature on bargaining games in which players can endogenously determine disagreement payoffs (Haller and Holden 1990, Fernandez and Glazer 1991, Avery and Zemsky 1994, Busch and Wen 1995). In the standard Rubinstein model, disagreement payoffs are fixed and there is a unique perfect equilibrium which is effi cient and has stationary equilibrium offers. In contrast, when disagreement payoffs are endogenous, there may exist multiple equilibria and some of these equilibria may be ineffi cient, explaining wasteful phenomena such as strikes, delay in bargaining and wars. The driving force behind these results is that actions that are taken in the stage game that determines the disagreement payoffs may determine equilibria that are going to be played in future. This allows for multiple equilibria, some of which are ineffi cient. My model is similar to these earlier work in the sense of disagreement payoffs being endogenous. However, in contrast, a player s past consumption level is a payoff relevant state of the game in my model. This has nontrivial implications on model predictions. For example, Haller and Holden (1990) 3

4 and Fernandez and Glazer (1991) study wage bargaining between a firm and a union. The union may strike in case of disagreement, which is costly for both the firm and the union workers. In this model, not striking is a Nash equilibrium of the stage game and therefore the unique equilibrium of the standard Rubinstein game without a strike option is an equilibrium of the game with the strike option (Busch and Wen 1995, Corollary 1). In contrast, for a region of parameters, there is no equilibrium in my model in which player 1 does not make any unilateral transfer on and offthe equilibrium path. Therefore the unique equilibrium of the standard Rubinstein game without a transfer option is not an equilibrium in my model even though no transfer is the unique Nash equilibrium of the stage game. This is because a transfer today changes preferences over consumption in future via habit formation and player 1 finds it optimal to manipulate player 2 s future preferences. This is not solely an equilibrium phenomenon as in the earlier work, in which actions today do not change fundamentals in future in particular, in these models, players continue to have the same preferences regardless of what actions have been played in the past, however, players past actions may determine equilibria that are going to be played in future. Finally, the prospect theory of international conflict (Levy 1996, 1997a, 1997b; also see Berejikian 2004) hypothesizes that political leaders of adversary states behave differently when they are bargaining over gains than when they are bargaining over loses (Levy 1996). Although this is close to my work in terms of exploring non-standard preference patterns in bargaining, my work does not rest on prospect theory: First, players payoff functions are weakly concave everywhere in my model, 3 and second, the insights that are derived from my model are fundamentally different than that come from prospect theory. I introduce the complete information bargaining model with habit formation in the next section and discuss the equilibrium in Section 3. I consider two extensions of the benchmark model in Section 4. I first consider the case when both players exhibit habit forming behavior, and then the case when the proposer is selected randomly every period. In Section 5, I study the case when player 1 is uncertain whether player 2 exhibits habit forming behavior or not. I defer all the technical analysis to the appendix. 3 Prospect theory (Kahneman and Tversky 1979) postulates that an individual evaluates alternatives with respect to a reference point and assigns value to gains and losses with respect to the reference rather than to final assets. The value function is generally concave for gains, convex for losses and steeper for losses than for gains. 4

5 2 Bargaining with habit forming players The benchmark model is an extension of Rubinstein (1982) bargaining model: There are two players 1 (he) and 2 (she). The players bargain over a flow of v units of a perishable consumption good that they can share and consume only if they mutually agree to do so. The players make offers in an alternating and deterministic order. The player that is selected to make an offer makes a proposal (v x, x) where x is player 2 s share and v x is player 1 s share. If an agreement is reached, the players consume their shares of the flow thereafter. When a player rejects a proposal, he makes the next proposal. Before the next proposal, player 1 offers to make a unilateral transfer of y 0 to player 2, which costs player 1 ψy, ψ 0. Then player 2 decides whether to accept or reject the transfer. She consumes y if she accepts the transfer. The game proceeds to the next period and continues until one player accepts the other s proposal. Player i discounts future payoffs by δ i [0, 1). Let z it denote player i s consumption in period t and y t be player 1 s unilateral transfer in period t. Player 1 s per-period payoff is z 1t ψy t, which is his consumption minus the cost of unilateral transfer in the given period. Then his lifetime utility is given by δ1(z t 1t ψy t ) where Player 2 s lifetime utility is given by δ2 t [z 2t φ[z 2,t 1 z 2t + t t { z2,t 1 z [z 2,t 1 z 2t + = 2t if z 2,t 1 > z 2t 0 otherwise φ[z 2,t 1 z 2t + captures player 2 s cost from her habit for past consumption. If player 2 s current consumption is at least as much as her consumption in previous period, i.e. z 2t z 2,t 1, then player 2 does not face any additional cost, and her per-period payoff is her current consumption. If her current consumption is less than her consumption in the previous period, i.e. z 2t < z 2,t 1, then player 2 pays a cost of φ(z 2,t 1 z 2t ) and her per-period payoff is her current consumption minus the cost. This cost increases linearly with the difference between consumption levels in current and previous 5

6 periods. The marginal cost coeffi cient φ 0 measures how costly it is for the player when there is a gap between the current and past consumption levels. The higher φ is, the greater costs player 2 pays. The only part of the cost that is exogenous is φ, and later I discuss how this parameter can vary, for example by consumption type, regime type of states or sanction type when the model is applied to international bargaining and sanctions. The benchmark model captures scenarios in which player 1 s income from other resources is large enough in comparison to unilateral transfers he could make so that he does not suffer any cost from habit forming behavior. I discuss the equilibrium of the benchmark model in the next section. Then I show that the qualitative features of this complete information game is robust to various changes in modelling assumptions on habit forming behavior of player 1, the bargaining protocol and the information structure. 3 Equilibrium A strategy profile of offers, unilateral transfers and acceptance/rejection decisions is a subgame perfect equilibrium if it is a Nash equilibrium in every subgame. Formally, x it [0, v is player i s offer for player 2 s share in period t; y it 0 is player 1 s unilateral transfer when player i s offer of x it is rejected; a it (x) {accept, reject} is player i s decision to accept or reject an offer of x made by the other player in period t; T t (y) {accept, reject} is player 2 s decision to accept or reject a transfer of y made by player 1 in period t I use the following convention: t represents a period that player 1 makes an offer. So I refer to 1 s offer periods as...t 2, t, t + 2,... and 2 s offer periods as..., t 1, t + 1, t Player 1 s strategy is denoted by σ 1 = (x 1t, y 1t, y 2t, a 1t ) t=1,2,..., player 2 s strategy is denoted by σ 2 = (x 2t, a 2t, T t ) t=1,2,.... A player makes an offer or an acceptance/rejection decision every other period. One can set x it and a it arbitrarily in periods that are not relevant for player i. This will not cause any confusion in the analysis. A strategy profile (σ 1, σ 2 ) is a subgame perfect equilibrium if the continuation of (σ 1, σ 2 ) forms a Nash equilibrium at every subgame. An equilibrium is Markov perfect if strategies depend only on payoff relevant state of the game, which is player 2 s previous period 4 In general, let 1 make offers at t = τ + 2n, n = 0, 1, 2,... and 2 make offers at t + 1 or t 1, where τ = 1 if 1 makes the first offer and τ = 0 if 2 makes the first offer. 6

7 consumption. I focus on the time-invariant Markov perfect equilibrium of the game in which strategies do not depend on time. When there is no habit formation, the model reduces to the standard Rubinstein bargaining game, which has a unique and effi cient equilibrium with stationary offers of Let x 1 = δ 2(1 δ 1 ) 1 δ 1 δ 2 v and x 2 = 1 δ 1 1 δ 1 δ 2 v φ = (1 δ 1)ψ δ 1 (1 δ 2 ). The following proposition characterizes optimal unilateral transfers when player 2 accepts all unilateral transfers. I defer all the proofs to the Appendix. Proposition 1 Suppose that player 2 commits to accepting all unilateral transfers. In a subgame perfect equilibrium, player 1 s unilateral transfer when 1 rejects 2 s offer is given by { 0 if φ φ y 2,t+1 = x 1,t+2 if φ > φ Since no transfer is always an option for player 1 and y 2,t+1 > 0 when φ > φ, it must be the case that player 1 benefits from this transfer. In other words, player 1 gains bargaining leverage by exploiting player 2 s habit forming behavior, which in turn hurts player 2. However player 2 has the option of rejecting a transfer. The next proposition states that player 2 will accept y 2,t+1 = x 1,t+2 in any subgame perfect equilibrium when φ > φ. Proposition 2 Suppose that φ > φ. In any subgame perfect equilibrium, player 2 accepts y 2,t+1 = x 1,t+2. That y 2,t+1 = x h 1 > 0 and player 2 accepts the transfer is a striking result. This result differentiates my model from earlier work that studies bargaining with endogenously determined disagreement payoffs (Haller and Holden 1990, Fernandez and Glazer 1991, Avery and Zemsky 1994, Busch and Wen 1995). My model is similar to these earlier work in the sense of disagreement payoffs being endogenous. However, in contrast, a player s past consumption level 7

8 is a payoff relevant state of the game in my model. This has nontrivial and novel implications on model predictions. For example, Haller and Holden (1990) and Fernandez and Glazer (1991) study wage bargaining between a firm and a union. The union may strike in case of disagreement, which is costly for both the firm and the union workers. In that model, not striking is a Nash equilibrium of the stage game and therefore the unique equilibrium of the standard Rubinstein game without a strike option is an equilibrium of the game with the strike option (Busch and Wen 1995, Corollary 1). In contrast, if φ > φ, there is no equilibrium in my model in which player 1 does not make any unilateral transfer on and off the equilibrium path. Therefore the unique equilibrium of the standard Rubinstein game without a transfer option is not an equilibrium when φ > φ in my model even though no transfer is the unique Nash equilibrium of the stage game. I discuss the time-invariant Markov perfect equilibrium next. Define the following: x h 1 = x h 2 = δ 2 1 δ 1 δ 2 + (1 δ 2 )φ δ 2 (1 δ 1 )ψ (1 δ 1)v, 1 + (1 δ 2 )φ 1 δ 1 δ 2 + (1 δ 2 )φ δ 2 (1 δ 1 )ψ (1 δ 1)v, y h 1 = 0 and y h 2 = x h 1 The following proposition summarizes the Markov perfect equilibrium of the game with habit formation. Proposition 3 (i) (Low Cost) If φ φ, in the time-invariant Markov perfect equilibrium of the game, player 1 does not make any unilateral transfer when an offer is rejected; if player 1 made a unilateral transfer of y after rejecting 2 s offer in the previous period, then player 1 offers ˆx 1 (y), where ˆx 1 (y) = { (1 δ2 )φy + δ 2 x 2 if y x 1 1+(1 δ 2 )φ otherwise, x 1 1+(1 δ 2 )φ he accepts any offer smaller than or equal to x 2 and rejects any other offer; player 2 offers x 2, accepts any offer greater than or equal to ˆx 1 (y) and rejects any other offer. Player 2 accepts any transfer by player 1. ˆx 1 (0) = x 1 so that the habit formation has no effect on offers on the equilibrium path. (ii) (High Cost) If φ > φ, in the time-invariant Markov perfect equilibrium of the game, player 1 does not make any unilateral transfer when 2 8

9 rejects his offer and he makes a unilateral transfer of x h 1 when he rejects 2 s offer; if player 1 made a unilateral transfer of y after rejecting 2 s offer in the previous period, then player 1 offers ˆx h 1(y), where { (1 ˆx h δ2 )φy + δ 1(y) = 2 x h 2 if y x h 1 x h 1 otherwise, he accepts any offer smaller than or equal to x h 2 and rejects any other offer; player 2 offers x h 2, accepts any offer greater than or equal to ˆx h 1(y) and rejects any other offer. Player 2 accepts any transfer by player 1. When φ φ, there is no unilateral transfer on the equilibrium path and ˆx 1 (0) = x 1. That is, the players offer x 1 and x 2 in equilibrium, so habit formation does not have any effect on equilibrium behavior. When φ > φ, player 2 s habit formation changes the equilibrium behavior of both players. On the equilibrium path, y 2 = x h 1 and ˆx h 1(x h 1) = x h 1 so that the players offer x h 1 and x h 2 in equilibrium. The first interesting observation is that player 1 takes advantage of player 2 s habit formation to decrease the equilibrium offers for player 2: x h 1 < x 1 and x h 2 < x 2 This bargaining leverage comes to player 1 at no additional cost since the offers are accepted immediately in equilibrium and player 1 never makes a unilateral transfer on the equilibrium path. The timing of 1 s unilateral transfer off the equilibrium path reveals how player 1 takes advantage of player 2 s habit forming behavior. First, player 1 alternates between making a unilateral transfer one period and no transfer the following period. This creates a cost for player 2 in periods with no unilateral transfer and decreases her continuation payoff when an offer is rejected off the equilibrium path. This outcome is also predicted when the proposer is selected randomly every period (see Section 4.2). Second, the timing of the unilateral transfer off the equilibrium path matters. Player 1 can make a unilateral transfer either after rejecting player 2 s offer or after player 2 rejects his offer. It is optimal for Player 1 to make the unilateral transfer after he rejects player 2 s offer. Intuitively, player 2 collects a larger share of surplus when her offer is accepted. Player 1 can recover some of this surplus by threatening player 2 with a rejection. After rejecting player 2 s offer, he can unilaterally transfer the amount that he will 9

10 propose next period. This transfer will create costs for player 2 in the next period if she rejects it, therefore she will accept smaller offers next period. Then player 1 s continuation payoff will be higher, since he can induce player 2 to accept smaller offers next period. Anticipating player 1 s actions off the equilibrium path and his continuation payoff, player 2 rationally makes and accepts smaller offers in comparison to the case without habit formation. Player 1 s bargaining leverage gain can be demonstrated more clearly at the threshold φ = φ. Habit formation has no impact on the equilibrium behavior when φ φ. Substituting φ = φ in x h i, the limit of x h i as φ approaches to φ from above become ψ > 0 implies x h,lim 1 = δ 1 δ 2 (1 δ 1 δ 2 )(δ 1 + (1 δ 1 )ψ) (1 δ 1)v and x h,lim 2 = 1 δ 1 1 δ 1 δ 2 v x h,lim 1 < x 1 and x h,lim 2 = x 2 The jump from x h,lim 1 to x 1 is surprising. Although player 1 cannot utilize habit formation when φ φ, he can utilize it when φ is just slightly above φ. This discontinuity shows the strategic opportunity player 1 gains from player 2 s habit formation off the equilibrium path. The closed form solution yields intuitive comparative statics. dx i dφ < 0 and dx i > 0 for both i = 1, 2 when φ > dψ φ. 5 In other words, both players decrease their equilibrium offers for player 2 s share when the cost from habits, φ, increases or the cost of unilateral transfer, ψ, decreases. Also dφ x h i 2 ψ φ dψ > 0 and > 0 for both i = 1, 2 when φ > φ. 6 In words, if player 1 s cost of unilateral transfers gets larger, it becomes more diffi cult for him to exploit player 2 s habit forming behavior. This completes the discussion of the benchmark model. Next, I discuss the extensions of the model. 5 x h 1 φ = (1 δ 2)δ 2(1 δ 1)v [1 δ 1δ 2+(1 δ 2)φ δ 2(1 δ 1)ψ < 0 and xh 2 2 φ = δ2(1 δ2)(δ1+(1 δ1)ψ)(1 δ1)v [1 δ 1δ 2+(1 δ 2)φ δ 2(1 δ 1)ψ < dφ dψ = (1 δ1)ψ x δ 1(1 δ 2) > 0, h2 1 ψ φ = 2δ 2(1 δ 1) x h 1 1 δ 1δ 2+(1 δ 2)φ δ 2(1 δ 1)ψ φ > 0 and xh2 2 [1+δ 1δ 2+(1 δ 2)φ+δ 2(1 δ 1)ψ(1 δ 2)δ 2(1 δ 1) 2 v [1 δ 1δ 2+(1 δ 2)φ δ 2(1 δ 1)ψ > 0. 3 ψ φ = 10

11 4 Extensions of the benchmark model The benchmark model predicts how and when player 1 can exploit player 2 s habit forming behavior in bargaining. The qualitative features of these predictions are robust to several extensions. I will discuss below the case where player 1 also exhibits habit forming behavior, as well as the case where the proposer is selected randomly every period. In Section 5, I study the case where player 1 is uncertain about whether player 2 exhibits habit forming behavior or not. In order to simplify the analysis, I will assume that player 2 accepts player 1 s transfers, so I will not study her decision to accept or reject a transfer in the appendix. 4.1 Two-sided Habit Formation Set ψ = 1. That is, player 1 s consumption falls by the amount of his unilateral transfer to 2. In addition, assume that player 1 also pays a cost due to habit formation whenever the consumption level falls below that in the previous period. If he consumes z 1t, his lifetime utility is given by δ1 t [z 1t γ[z 1,t 1 z 1t + where t { z1,t 1 z [z 1,t 1 z 1t + = 1t if z 1,t 1 > z 1t 0 otherwise γ 0, and γ[z 1,t 1 z 1t + captures player 1 s cost from his habit for past consumption. Define φ = (1 δ 1)(1 + γ), δ 1 (1 δ 2 ) x hh δ 2 (1 δ 1 ) 1 = (1 δ 2 )(1 + φ) γδ 2 (1 δ 1 ) v, x hh 2 = (1 δ 1 ) (1 + (1 δ 2 )φ) (1 δ 2 )(1 + φ) γδ 2 (1 δ 1 ) v, y hh 1 = 0 and y hh 2 = { x hh 1 if φ > φ 0 if φ φ 11

12 The following summarizes the equilibrium: Proposition 4 (i) (Low Cost) If φ φ, in the time-invariant Markov perfect equilibrium of the game, player 1 does not make any unilateral transfer when an offer is rejected; if player 1 made a unilateral transfer of y after rejecting 2 s offer in the previous period, then player 1 offers ˆx 1 (y), where ˆx 1 (y) = { (1 δ2 )φy + δ 2 x 2 if y x 1 1+(1 δ 2 )φ otherwise, x 1 1+(1 δ 2 )φ he accepts any offer smaller than or equal to x 2 and rejects any other offer; player 2 offers x 2, accepts any offer greater than or equal to ˆx 1 (y) and rejects any other offer. ˆx 1 (0) = x 1 so that habit formation has no effect on offers on the equilibrium path. (ii) (High Cost) If φ > φ, in the time-invariant Markov perfect equilibrium of the game, player 1 does not make any unilateral transfer when 2 rejects his offer and he makes a unilateral transfer of x hh 1 when he rejects 2 s offer; if player 1 made a unilateral transfer of y after rejecting 2 s offer in the previous period, then player 1 offers ˆx hh 1 (y), where { (1 ˆx hh δ2 )φy + δ 1 (y) = 2 x hh 2 if y x hh 1 x hh 1 otherwise, he accepts any offer smaller than or equal to x hh 2 and rejects any other offer; player 2 offers x hh 2, accepts any offer greater than or equal to ˆx hh 1 (y) and rejects any other offer. The expression of φ is very similar to the expression of φ. When player 1 makes a unilateral transfer of y in the original model, his cost is ψy; in this new setting, the cost has two components: the direct cost y is the decrease in his consumption, and γy is the cost thanks to habit formation. φ becomes equal to φ when ψ = 1 and γ = 0. ˆx hh 1 (x hh 1 ) = x hh 1 so that the players offer x hh 1 and x hh path. An increase in player 1 s cost coeffi cient γ increases x hh i player 2 and harming player 1. 2 on the equilibrium, benefiting 12

13 4.2 Random offers Suppose that, at the beginning of each period, player i is selected to make an offer with probability π i, where π 1 + π 2 = 1. π i may represent player i s relative bargaining power. When there is no habit formation, i.e. φ = 0, there will be no unilateral transfer in equilibrium and equilibrium offers are given by x r δ 2 π 2 1 = (1 δ 1 )v 1 δ 2 π 1 δ 1 π 2 x r 1 δ 2 π 1 2 = (1 δ 1 )v 1 δ 2 π 1 δ 1 π 2 Consider φ > 0. In the time-invariant Markov perfect equilibrium, when offers are rejected off the equilibrium path, player 1 makes a positive unilateral transfer every other period. Player 1 s optimal transfers do not depend on past actions, but equilibrium offers depend on player 1 s unilateral transfer in the previous period. Let s {0, 1} denote the state of the game at the beginning of a period. Then s = 0 if player 1 was prescribed to make no unilateral transfer in the previous period, and s = 1 if he was prescribed to make a transfer in the previous period. Denote by ˆx i0 (y) the offer that player i makes if i is chosen to make an offer, s = 0, and he made a transfer of y in the previous period. Similarly, denote by ˆx i1 (y) the offer that player i makes if i is chosen to make an offer and s = 1, and he made a transfer of y in the previous period. Define φ 1 δ 1 = δ 1 π 1 (1 δ 2 ) and let (x 11, x 21, x 10, x 20 ) solve the following linear equation system, δ 2 x 11 = 1 + (1 δ 2 )φ (π 1x 10 + π 2 x 20 ) x 21 = (1 δ 1 )v + δ 1 (π 1 x 10 + π 2 x 20 ) x 10 = (1 δ 2 )x 11 + δ 2 (π 1 x 11 + π 2 x 21 ) x 20 = (1 δ 1 )(v + x 11 ) + δ 1 (π 1 x 11 + π 2 x 21 ) Also { (1 δ2 )φy + δ ˆx 11 (y) = 2 (π 1 x 10 + π 2 x 20 ) if y x 11 x 11 otherwise 13

14 and x 10 if y x 11 ˆx 10 (y) = x 10 φ(y x 11 ) if x 11 < y x x otherwise where x = x 10+φ(1 δ 2 )x 11 1+(1 δ 2 )φ < x 10. Notice that ˆx 10 (0) = x 10 and ˆx 11 (y) = x 11, which are player 1 s offers on the equilibrium path when he did not deviate from his equilibrium transfer in the previous period. Player 2 s equilibrium offers are x 20 and x 21 independently of past actions. The following summarizes the equilibrium: Proposition 5 (i) (Low Cost) If φ φ, in the time-invariant Markov perfect equilibrium of the game, player 1 does not make any unilateral transfer when an offer is rejected; if player 1 made a unilateral transfer of y in the previous period, then player 1 offers ˆx r 1(y), where ˆx r 1(y) = (1 δ 2 )φy + δ 2 (π 1 x r 1 + π 2 x r 2) if y δ 2(π 1 x r 1 +π 2x r 2) 1+(1 δ 2 )φ δ 2(π 1 x r 1 +π 2x r 2) 1+(1 δ 2 )φ otherwise, he accepts any offer smaller than or equal to x r 2 and rejects any other offer; player 2 offers x r 2, accepts any offer greater than or equal to ˆx r 1(y) and rejects any other offer. ˆx r 1(0) = x r 1 so that habit formation has no effect on offers on the equilibrium path. (ii) (High Cost) If φ > φ, in the time-invariant Markov perfect equilibrium of the game, (0) if s = 0, i.e. player 1 s equilibrium strategy in the previous period was not to make a unilateral transfer, and he made a transfer of y 0, player 1 offers ˆx 10 (y), accepts any offer smaller than or equal to x 20 and rejects any other offer; makes a unilateral transfer of x 11 if an offer is rejected; player 2 offers x 20, accepts any offer greater than or equal to ˆx 10 (y) and rejects any other offer. (y) if s = 1, i.e. player 1 s equilibrium strategy in the previous period was to make a unilateral transfer of x 11, and he made a transfer of y 0, player 1 offers ˆx 11 (y), accepts any offer smaller than or equal to x 21 and rejects any other offer; makes no unilateral transfer when an offer is rejected; player 2 offers x 21, accepts any offer greater than or equal to ˆx 11 (y) and rejects any other offer. 14

15 As I mentioned before, these equilibrium predictions have important implications for international bargaining. In particular, the alternating pattern of unilateral transfers in equilibrium implies that sanctions will be effective if the target state has not been subject to sanctions yet and a temporary ease of sanctions will be more effective than constant sanctioning if the target state has been sanctioned for some time and has adapted, Player 1 s equilibrium offers are smaller than player 2 s offers in both states, i.e. x 1s < x 2s for s {0, 1}, and x 11 is the smallest offer made in equilibrium. Comparative statics analysis yields intuitive results. An increase in π 1, φ or δ 1 and a decrease in δ 2 decreases every x is, increasing player 1 s payoff in equilibrium. 5 Incomplete Information Consider the benchmark model with ψ = 1. Suppose that player 1 does not know if player 2 exhibits habit formation or not. Let φ {φ l = 0, φ h }, φ < φ h. If φ = φ l = 0, then player 2 does not exhibit habit formation. If φ = φ h, then player 2 exhibits habit formation. Player 2 s type is her private information. It is common knowledge that she exhibits habit formation with probability θ, i.e. P r(φ = φ h ) = θ. The outcome is predicted by Bayesian Nash equilibrium. Under complete information, if φ = φ l, then y i = 0 for i = 1, 2 and the equilibrium offers are given by x 1 = δ 2(1 δ 1 ) 1 δ 1 δ 2 v and x 2 = 1 δ 1 1 δ 1 δ 2 v and if φ = φ h, the equilibrium offers are given by x h 1 = δ 2 (1 δ 1 ) (1 δ 2 )(1 + φ h ) v and xh 2 = 1 + (1 δ 2)φ h (1 δ 2 )(1 + φ h ) (1 δ 1)v, in that case, y 1 = 0 and y 2 = x h 1. Assume that player 2 makes the first offer. When different types of player 2 make different offers in a separating equilibrium, the continuation game reduces to a complete information game. Define φ = δ 2 (1 δ 1 δ 2 ) (1 δ 2 )(δ δ 1 δ 2 ) 1 > φ and α = x 2 x h 2 x 2 δ 2 x 1 15

16 The following proposition summarizes the Bayesian equilibrium when player 2 makes the first offer. Proposition 6 Assume that player 2 makes the first offer. (i) If δ 1 < 1 δ2 2 δ 2 or φ h [φ, φ, the following is a separating equilibrium: Type φ l offers x 2, which player 1 accepts with probability 1 α. Type φ h offers x h 2 < x 2, which player 1 accepts with probability 1. Player 2 rejects any other offer bigger than x h 2, accepts any other offer smaller than x h 2. If player 2 offers x 2, then player 1 updates his belief to φ = φ l, otherwise he updates his belief to φ = φ h and the players play the equilibrium of the associated complete information games. The equilibrium payoff of both types of player 2 is x h 2. (ii) If δ 1 > 1 δ2 2 δ 2 and φ h > φ, there does not exist any separating equilibrium. The following is a pooling equilibrium: Both types of player 2 offer x h 2. Player 1 rejects offers x > x h 2 and accept offers x x h 2. If player 2 offers x x h 2, then player 1 updates his belief to φ = φ h and plays according to the equilibrium of the associated complete information game. The equilibrium payoff of both types of player 2 is x h 2. Since x h 2 is a decreasing function of φ h, α increases with φ h. That is, the likelihood of a delay when player 1 receives an offer of x 2 increases. However, since player 1 is indifferent between accepting and rejecting x 2, this increase in the likelihood of delay does not decrease his expected payoff. In contrast, since x h 2 decreases with φ h, his payoff of v x h 2 increases. Type φ l s expected payoff decreases because of the delay. Consider the game in which player 1 makes the first offer. There is no separating equilibrium of this game in which different types of player 2 separate themselves by their acceptance/rejection decision. This is because player 2 of type φ h can achieve a higher payoff by imitating φ l, after which the game turns into a complete information game with φ = φ l and φ h collects the highest payoff she can. Therefore, the continuation game after a potential rejection is an incomplete information game in which player 2 makes the offer. By Proposition (6), the equilibrium payoff of both types of player 2 is x h 2 whether they play a separating or pooling equilibrium in the continuation incomplete information game. Then the payoff from rejecting player 1 s offer is δ 2 x h 2 for both types of player 2, so that player 1 offers δ 2 x h 2 and both types accept in equilibrium. The following proposition summarizes the Bayesian equilibrium when player 1 makes the first offer. 16

17 Proposition 7 Assume that player 1 makes the first offer. Player 1 offers δ 2 x h 2. Both types of player 2 accept offers x δ 2 x h 2 and reject offers x < δ 2 x h 2. If an offer is rejected, player 1 updates his belief to Pr(φ = φ h ) = 1, and the players play the equilibrium strategies of the complete information game with φ = φ h. If player 1 knew that φ = φ h, he would have offered x h 1 < δ 2 x h 2. Thus, player 2 of type φ h collects an information rent of R(φ h ) = δ 2 x h 2 x h 1. In the complete information game, player 2 of type φ h is hurt by an increase in φ h, since x h 1 and x h 2 are decreasing functions of φ h. She is also hurt by an increase in φ h in the incomplete information game for the same reason. However, both R(φ h ) and R(φ h )/x h 1 are increasing functions of φ h. That is, player 1 pays her a larger information rent in both absolute and relative values as φ h increases. 6 Conclusion In this paper, I examine the role of habit formation in bargaining, and show how players can exploit their opponent s habit formation to generate endogenous costs for the opponent. Introduction of habit formation brings in new strategic tools and incentives. In the subgame perfect equilibrium of the complete information game, a player can exploit the habit forming behavior of his opponent off the equilibrium path. This increases his equilibrium payoff in comparison to the case with no habit formation. Since unilateral transfers are made only off the equilibrium path, this increase in payoff comes for free to the player. The qualitative features of the models are robust to several extensions. Introduction of informational asymmetry may cause delay in negotiations without changing the qualitative predictions. These findings have direct implications in international bargaining. In particular, the carrot-stick approach is a preferred policy in international negotiations. The conventional wisdom is that offering a combination of rewards and punishments may be effective in getting one s opponent to make concessions in bargaining. My findings suggest that, even a free carrot may sometimes improve the hand of a negotiating country if the citizens of the opponent country exhibit habit forming behavior. The international relations literature mostly studies such problems in isolation that is, independent of other possible coercive tools, for example, the actual use of force (e.g. see Jonge Oudraat 2000 for a criticism of the 17

18 sanctions literature). It is quite possible that such policies can work more effectively when they are a part of a comprehensive strategy. That includes the use of force as an outside option. I leave this problem for future research. 18

19 References [1 Abel, Andrew B "Asset Prices under Habit Formation and Catching up with the Joneses." American Economic Review. 80(2): [2 Avery, Christopher and Peter B. Zemsky (1994), Money Burning and Multiple Equilibria in Bargaining, Games and Economic Behavior, 7, [3 Busch, Lutz-Alexander and Quan Wen Perfect Equilibria in a Negotiation Model. Econometrica 63(3): [4 Berejikian, Jeffrey D International Relations Under Risk: Framing State Choice. SUNY Press, Albany, NY. [5 Braun, Phillip A., Constantinides, George M. and Wayne E. Ferson "Time Nonseparability of Aggregate Consumption: International Evidence." European Economic Review. 37(5): [6 Campbell, John Y. and J. H. Cochrane "By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior." Journal of Political Economy. 107(2): [7 Carroll, Christopher D., Overland, Jody, and David N. Weil "Comparison Utility in a Growth Model." Journal of Economic Growth. 2(4): [8 Camerer, C., and G. Loewenstein "Behavioral Economics: Past, Present, and Future" in Advances in Behavioral Economics, ed. by C. Camerer, G. Loewenstein, and M. Rabin. Princeton University Press. [9 Constantinides, George M "Habit Formation: A Resolution of the Equity Premium Puzzle." Journal of Political Economy. 98(3): [10 Fernandez, Rachel and Jacob Glazer "Striking for a Bargain Between Two Completely Informed Agents," American Economic Review 81: [11 Ferson, Wayne E. and George M. Constantinides "Habit persistence and durability in aggregate consumption: Empirical tests." Journal of Financial Economics. 29(2):

20 [12 Fuhrer, Jeffrey C "Habit Formation in Consumption and Its Implications for Monetary-Policy Models." American Economic Review. 90(3): [13 Haller, Hans Hermann and Steinar Holden "A Letter to the Editor on Wage Bargaining." Journal of Economic Theory 52: [14 Jonge Oudraat, Chantal de "Making Economic Sanctions Work." Survival 42(3): [15 Kahneman, Daniel and Amos Tversky "Prospect theory: An analysis of decisions under risk." Econometrica. 47: [16 Levy, Jack S "Loss Aversion, Framing and Bargaining: The Implications of Prospect Theory for International Conflict." International Political Science Review 17(2): [17 Levy, Jack S. 1997a. "Prospect Theory and the Cognitive-Rational Debate." In Nehemia Geva and Alex Mintz, eds., Decisionmaking on War and Peace: The Cognitive-Rational Debate. Boulder, CO: Lynne Rienner, pp [18 Levy, Jack S. 1997b. "Prospect Theory, Rational Choice, and International Relations." International Studies Quarterly. 41(1): [19 Osborne, Martin J. and Ariel Rubinstein Bargaining and Markets. Academic Press. [20 Pape, Robert "Why Economic Sanctions do not Work." International Security 2: [21 Rozen, Kareen "Foundations of Intrinsic Habit Formation." Econometrica. 78(4): [22 Rubinstein Ariel "Perfect Equilibrium in a Bargaining Model." Econometrica. 50(1):

21 A Equilibrium Analysis In equilibrium, each player chooses an optimal action at every subgame. I will assume that a player accepts an offer when he is indifferent between accepting and rejecting it given the continuation equilibrium. If player j is better off by accepting player i s offer, then player i can increase his/her payoff by slightly decreasing player j s share, which player j will continue to accept. Therefore player i s optimal offer makes player j indifferent between accepting and rejecting i s offer. Consider an equilibrium at which offers and transfers are accepted. Given 1 s strategy for transfers, y it, the conditions for the equilibrium offers that are accepted are given by x 1t 1 δ 2 φ [y 2,t 1 x 1t + = (y 1t φ [y 2,t 1 y 1t + ) [ x2,t+1 + δ 2 φ [y 1t x 2,t+1 1 δ + 2 (1) v x 2,t+1 = ψy 2,t+1 + δ 1 (v x 1,t+2 ) 1 δ 1 1 δ 1 (2) In the first equation, the left hand side is player 2 s payoff if she accepts x 1t. x 1t 1 δ 2 is her lifetime utility from consuming x 1t forever and φ [y 2,t 1 x 1t + is the audience cost that she pays in period t. Since there is no change in the consumption level thereafter, there is no further cost after period t. The right hand side is her payoff if she rejects the offer x 1t. In that case, she consumes y 1t, the unilateral transfer by player 1 and pays the audience cost of φ [y 2,t 1 y 1t + in period t, and then her offer of x 2,t+1 will be accepted in period t+ 1, which she will consume thereafter, and she will pay a one-period cost of φ [y 1t x 2,t+1 + in period t + 1. In the second equation, the left hand side is player 1 s payoff if he accepts 2 s offer x 2,t+1 in period t+1 and the right hand side is his payoff if he rejects it. In that case, player 1 s offer of x 1,t+2 will be accepted in period t + 2, and he will pay a one-period unilateral transfer cost, ψy 2,t+1. To simplify the expressions, multiply both sides of (1) by (1 δ 2 ) and multiply both sides of (2) by (1 δ 1 ). I will use the following equations in the analysis. 21

22 x 1t (1 δ 2 )φ [y 2,t 1 x 1t + = (1 δ 2 ) [ y 1t φ [y 2,t 1 y 1t + + δ 2 [ x2,t+1 (1 δ 2 )φ [y 1t x 2,t+1 + (3) v x 2,t+1 = (1 δ 1 )ψy 2,t+1 + δ 1 (v x 1,t+2 ) (4) I will solve for a time-invariant Markov perfect equilibrium with x it = x i and y it = y i for all t A.1 Equilibrium with no habit formation Suppose that φ = 0. In that case subgame perfection requires y 1 = y 2 = 0 and conditions 3 and 4 for the equilibrium offers reduce to x 1 = δ 2 x 2 v x 2 = δ 1 (v x 1) which yields x 1 = δ 2(1 δ 1 ) 1 δ 1 δ 2 v and x 2 = 1 δ 1 1 δ 1 δ 2 v A.2 Equilibrium with habit formation When player 2 exhibits habit forming behavior, player 1 may find it optimal to take a unilateral action to gain bargaining leverage in the negotiations. Next, I will solve for 1 s optimal unilateral transfers under the assumption that player 2 commits to accepting all transfers. A.2.1 Optimal choice for unilateral transfer y i when player 2 accepts Assume that player 2 commits to accepting all transfers. First consider player 1 s decision for a unilateral transfer when 1 rejects 2 s offer. Proposition 8 In a subgame perfect equilibrium, player 1 does not make any unilateral transfer when his offer is rejected. That is, y 1t = 0 for all t in every subgame perfect equilibrium. 22

23 Proof. Consider the node at which 2 rejects 1 s offer x 1t in period t. At this node of the game, 1 s continuation payoff as a function of y 1t is given by (after multiplying by (1 δ 1 )) where x 2,t+1 is determined by u r1 = (1 δ 1 )ψy 1t + δ 1 (v x 2,t+1 ) v x 2,t+1 = (1 δ 1 )ψy 2,t+1 + δ 1 (v x 1,t+2 ) Then x 2,t+1 y 1t = 0 so that which implies that du r1 dy 1t = (1 δ 1 )ψ < 0 y 1t = 0 in equilibrium. That is, if 2 rejects 1 s offer, it is optimal for 1 not to make any unilateral transfer in that period. Next consider the unilateral transfer by 1 after 1 rejects 2 s offer. Proposition 9 Assume that player 2 commits to accepting all transfers. In a subgame perfect equilibrium, player 1 s unilateral transfer when 1 rejects 2 s offer is given by { 0 if φ φ y 2,t+1 = x 1,t+2 if φ > φ where φ = (1 δ 1)ψ δ 1 (1 δ 2 ) Proof. Consider the node at which 1 rejects 2 s offer x 2,t+1 in period t + 1. At that node of the game, 1 s continuation payoff as a function of y 2,t+1 is given by (after multiplying by (1 δ 1 )) where x 1,t+2 is determined by u r2 = (1 δ 1 )ψy 2,t+1 + δ 1 (v x 1,t+2 ) x 1,t+2 (1 δ 2 )φ [y 2,t+1 x 1,t+2 + = (1 δ 2 ) [ y 1,t+2 φ [y 2,t+1 y 1,t δ 2 [ x2,t+3 (1 δ 2 )φ [y 1,t+2 x 2,t

24 Substituting y 1t = 0 for all t from Proposition 8, I obtain x 1,t+2 (1 δ 2 )φ [y 2,t+1 x 1,t+2 + = (1 δ 2 )φy 2,t+1 + δ 2 x 2,t+3 (5) If y 2,t+1 > x 1,t+2 then this equation reduces to so that and (1 + (1 δ 2 )φ)x 1,t+2 = δ 2 x 2,t+3 dx 1,t+2 dy 2,t+1 = 0 du r2 dy 2,t+1 = (1 δ 1 )ψ < 0 This implies that 1 s utility increases by an decrease in y 2,t+1 when y 2,t+1 > x 1,t+2 so that y 2,t+1 x 1,t+2 in equilibrium. Then equation (5) reduces to so that which implies Then where x 1,t+2 = (1 δ 2 )φy 2,t+1 + δ 2 x 2,t+3 dx 1,t+2 dy 2,t+1 = (1 δ 2 )φ du r2 dy 2,t+1 = (1 δ 1 )ψ + δ 1 (1 δ 2 )φ du r 12 dy 2,t+1 < 0 if φ < φ = 0 if φ = φ > 0 if φ > φ φ = (1 δ 1)ψ δ 1 (1 δ 2 ) When y 2,t+1 x 1,t+2, this implies that 1 s utility increases with y 2,t+1 only if φ > φ, that is, if the cost coeffi cient is suffi ciently large. Then in equilibrium, { 0 if φ φ y 2,t+1 = x 1,t+2 if φ > φ Next I will prove that it is optimal for player 2 to accept player 1 s transfer. 24

25 A.2.2 Player 2 s optimal action to accept or reject a transfer Consider a continuation game after player 2 rejects player 1 s offer of x 1,t. Suppose that player 1 makes a transfer of y 1t 0. Since player 2 will make the next offer and x 2,t+1 is independent of y 1t, it is optimal for player 2 to accept y 1t. Next consider player 2 s decision to accept or reject y 2,t+1 = x 1,t+2 after player 1 rejects her offer of x 2,t+1. Then [y 2,t+1 x 1,t+2 + = 0. If she accepts y 2,t+1, then x 1,t+2 is given by x 1,t+2 = (1 δ 2 ) [ y 1,t+2 φ [y 2,t+1 y 1,t+2 + +δ2 [ x2,t+3 (1 δ 2 )φ [y 1,t+2 x 2,t+3 + Since y 1,t+2 = 0 in any subgame perfect equilibrium, this yields x 1,t+2 = (1 δ 2 )φy 2,t+1 + δ 2 [ x2,t+3 (1 δ 2 )φ [y 1,t+2 x 2,t+3 + Then player 2 s payoff from accepting y 2,t+1 is (1 δ 2 ) [ y 2,t+1 φ [y 1t y 2,t δ2 x 1,t+2 (6) If she rejects the transfer, then player 1 s offer the next period is given by x 1,t+2 = (1 δ 2 )y 1,t+2 + δ 2 [ x2,t+3 (1 δ 2 )φ [y 1,t+2 x 2,t+3 + = δ 2 [ x2,t+3 (1 δ 2 )φ [y 1,t+2 x 2,t+3 + and her payoff from rejecting y 2,t+1 is (1 δ 2 )φy 1t + δ 2 x 1,t+2 Player 2 accept the transfer if (6) is greater than or equal to (??). equivalently, [ y2,t+1 φ [y 1t y 2,t+1 + δ2 φy 2,t+1 + φy 1t 0 (7) Player 1 might have deviated in the past, so y 1t is not necessarily equal to zero. If y 1t > y 2,t+1, (7) is equivalent to 1 + (1 δ 2 )φ 0 which holds, so player 2 accepts y 2,t+1. If y 1t y 2,t+1, (7) becomes (1 δ 2 φ)y 2,t+1 + φy 1t 0 25

26 y 1t y 2,t+1 implies (1 δ 2 φ)y 2,t+1 + φy 1t (1 δ 2 φ)y 1t + φy 1t = (1 + (1 δ 2 )φ)y 1t 0 so that (7) holds and player 2 accepts y 2,t+1. This completes the proof of Proposition 2. Next I will solve for the equilibrium offers. A.2.3 Optimal offers with low cost: φ φ On the equilibrium path of a time-invariant Markov perfect equilibrium, x it = x h i and y it = y h i for all t. Substituting y h 1 = y h 2 = 0 from propositions 8 and 9, conditions 3 and 4 for the equilibrium offers reduce to which yields x h 1 = δ 2 x h 2 v x h 2 = δ 1 (v x h 1) x h 1 = x 1 and x h 2 = x 2 Consider a period following y 2,t 1, which may be different than his equilibrium transfer of zero. By propositions 8 and 9, y 1t = y 2,t +1 = 0 and are independent of past decisions for all t t. Also x 2,t+1 and x 1,t+2 are independent of y 2,t 1, because player 1 will choose y 1t = 0 and the game will revert to its time-invariant Markov perfect equilibrium path after t. So x 2,t+1 = x 2 and x 1,t+2 = x 1. Substitute these in (3) and obtain x 1t (1 δ 2 )φ [y 2,t 1 x 1t + = (1 δ 2 )φy 2,t 1 + δ 2 x 2 If y 2,t 1 x 1t, [y 2,t 1 x 1t + = 0 so that x 1t = (1 δ 2 )φy 2,t 1 + δ 2 x 2 and y 2,t 1 x 1t becomes equivalent to y 2,t 1 (1 δ 2 )φy 2,t 1 + δ 2 x 2 δ 2 x 2 y 2,t 1 x (8) 1 + (1 δ 2 )φ 26

27 If y 2,t 1 > x 1t, (3) becomes x 1t (1 δ 2 )φ(y 2,t 1 x 1t ) = (1 δ 2 )φy 2,t 1 + δ 2 x 2 x 1t = x So player 1 s optimal offer in a time-invariant Markov perfect equilibrium is given by ˆx 1 (y t 1 ) where { (1 δ2 )φy + δ ˆx 1 (y) = 2 x 2 if y x x otherwise Notice that ˆx 1 (0) = δ 2 x 2 = x 1. Consider a period following y 1t, which may be different than his equilibrium transfer of zero. Since y 2,t+1 and x 1,t+2 are independent of y 1t, x 2,t+1 is independent of y 1t in (4) so that x 2,t+1 = x 2. A.2.4 Optimal offers with high cost: φ > φ On the equilibrium path of a time-invariant Markov perfect equilibrium, x it = x h i and y it = y h i for all t. Substituting y h 1 = 0 and y h 2 = x h 1 from propositions 8 and 9, conditions 3 and 4 for the equilibrium offers reduce to which yields x h 1 = x h 2 = x h 1 = (1 δ 2 )φx h 1 + δ 2 x h 2 v x h 2 = (1 δ 1 )ψx h 1 + δ 1 (v x h 1) δ 2 1 δ 1 δ 2 + (1 δ 2 )φ δ 2 (1 δ 1 )ψ (1 δ 1)v and 1 + (1 δ 2 )φ 1 δ 1 δ 2 + (1 δ 2 )φ δ 2 (1 δ 1 )ψ (1 δ 1)v φ > φ implies that (1 δ 2 )φ > δ 2 (1 δ 1 )ψ so x h 1 > 0 and x h 2 > 0. Consider a period following y 2,t 1, which may be different than his equilibrium transfer of x h 1. x 2,t+1 and x 1,t+2 are independent of y 2,t 1, because player 1 will choose y 1t = 0 by propositions 8 and the game will revert to its time-invariant Markov perfect equilibrium path after t. So x 2,t+1 = x h 2 and x 1,t+2 = x h 1. Also by propositions 8 and 9, y 1t = 0 and y 2,t +1 = x 1,t +2 = x h 1 27

28 and are independent of past decisions for all t t. Substitute these in (3) and obtain x 1t (1 δ 2 )φ [y 2,t 1 x 1t + = (1 δ 2 )φy 2,t 1 + δ 2 x h 2 If y 2,t 1 x 1t, [y 2,t 1 x 1t + = 0 so that x 1t = (1 δ 2 )φy 2,t 1 + δ 2 x h 2 and y 2,t 1 x 1t becomes equivalent to y 2,t 1 (1 δ 2 )φy 2,t 1 + δ 2 x h 2 y 2,t 1 If y 2,t 1 > x 1t, (3) becomes δ 2 x h (1 δ 2 )φ = xh 1 x 1t (1 δ 2 )φ(y 2,t 1 x 1t ) = (1 δ 2 )φy 2,t 1 + δ 2 x h 2 x 1t = x h 1 So player 1 s optimal offer in a time-invariant Markov perfect equilibrium is given by ˆx h 1(y t 1 ) where { (1 ˆx h δ2 )φy + δ 1(y) = 2 x h 2 if y x h 1 x h 1 otherwise Notice that ˆx h 1(x h 1) = x h 1. Consider a period following y 1t, which may be different than his equilibrium transfer of zero. Since y 2,t+1 and x 1,t+2 are independent of y 1t, x 2,t+1 is independent of y 1t in (4) so that x 2,t+1 = x h 2. Next, given the offers and players decisions to accept and reject offers, I will solve for player 2 optimal action when player 1 makes an offer. A.3 Player 2 s optimal action to accept or reject a transfer Consider a continuation game after player 2 rejects player 1 s offer of x 1,t. Suppose that player 1 makes a transfer of y 1t 0. Since player 2 will make the next offer and x 2,t+1 is independent of y 1t, it is optimal for player 2 to accept y 1t. I summarize this in the proposition. 28