Dynamic Voluntary Contribution to a Public Project under Time-Inconsistency

Transcription

1 Dynamic Voluntary Contribution to a Public Project under Time-Inconsistency Ahmet Altınok and Murat Yılmaz September 10, 2014 Abstract We study a general voluntary public good provision model and introduce timeinconsistent agents who have β δ preferences. There is a public project and finitely many agents where each agent is allowed to contribute any amount she likes, in any period she likes before the project is completed. The agents have discontinuous preferences over the total contribution with a jump when there is provision. There is complete information about the environment but imperfect information about others individual actions: each period, each agent observes only the total contribution made, not the other agents individual contributions. Assuming the agents are time-inconsistent and sophisticated, we characterize the set of equilibria. We compare the set of equilibria under sophisticated time-inconsistent agents to that under time-consistent agents. We show that any equilibrium outcome for the timeconsistent agents is also an equilibrium outcome for the time-inconsistent agents. Also, there is an equilibrium outcome for time-inconsistent agents, which is not an equilibrium outcome for the time-consistent agent. Moreover, we show that for any given project, the sophisticated time-inconsistent agents complete the project weakly earlier than the time-consistent agents. Keywords : time-inconsistency, dynamic voluntary contribution, β δ preferences. JEL Classification Numbers: Arizona State University, Department of Economics. aaltinok@asu.edu Phone: Boğaziçi University, Department of Economics, Bebek, Istanbul, muraty@boun.edu. tr. Phone: Fax: Web site: 1

2 1 Introduction Public projects such as building a library collection, a public recreational facility, a railroad or subway take a lot of time; generally for weeks, months and even years. Such a period of time is necessary for many voluntary contribution plans which finance the public project. This paper addresses such a joint project, which is socially desirable in the sense that its total benefit is more than its cost, where the classic free rider problem and a general discounting scheme are both present. A widely studied set of questions, when forming a contract over the contributions in advance is not possible, consists of whether the project will be completed and if it will be, how long will it take to complete it and what will be the contribution scheme for each agent. When the nature of the problem has a dynamic component, the agents time-preferences become important. There is evidence showing that agents tend to postpone costly actions, like finishing a report or filing taxes, but they rarely postpone benefits. Thus, agents may have put less weight on the future benefits and more on the immediate costs, yielding to time-inconsistent preferences. In particular, agents may have a higher discounting between the current and the next period than the discounting between any other two successive periods, that is, agents time-preferences may change over time. 1 the questions posed above are worth exploring under time-inconsistent preferences. Therefore, We consider the dynamic voluntary contribution model introduced in Marx and Matthews 2000), MM henceforth, and introduce time-inconsistency into the model. 2 analyze the set of equilibria under time-inconsistency and compare it to that of the timeconsistent agents. More importantly, we provide a sensible comparison of the two environments in terms of their number of periods to complete a given project and show that time-inconsistent agents finish a given project earlier than the time-consistent agents. In the baseline model, each agent can contribute any amount she likes, at any period, and the contributions are non-refundable. At the end of each period, each agent learns the total amount of contribution at that period, but they do not observe the individual contributions of the other agents. The cost of the public good is common knowledge. Each agent has a discontinuous benefit function, in the sense that there is a benefit jump whenever the public project is completed. Also, each period, each agent has a marginal 1 Frederick, Loewenstein, O Donoghue 2002) provide an extensive overview of the literature on timeinconsistency. Also see Loewenstein and Prelec 1992) for an extensive survey on anomalies in intertemporal choice. 2 Admati and Perry 1991) also provide a similar model, however, Marx and Matthews 2000) use a more general model. We 2

3 benefit from the total contribution made in that period even when the public project is not completed. 3 preferences, through β δ preferences. 4 Within this framework, we introduce time-inconsistency on agents An agent with β δ preferences has a discount factor βδ between the current and the next period, and a discount factor δ between any two future successive periods. If an agent is fully aware of her time-inconsistency, she is a sophisticated time-inconsistent agent, otherwise she is a naïve agent. In this paper, we focus on the case of sophisticated agents and investigate the effect of agents timeinconsistency on the set of equilibria and also on the equilibrium number of periods to complete a given project, compared to those for the time-consistent agents. The Nash equilibria and the nearly efficient perfect Bayesian equilibria of the public good provision with the time-consistent agents are characterized by MM. One of their main results is that by allowing agents making contributions slowly over time, efficient outcomes can be constructed and the agents are willing to complete the project. Here, we first focus on the set of equilibria under both agent types and compare them. We show that for any given project, any equilibrium outcome for the time-consistent agents is also an equilibrium outcome for the time-inconsistent agents. We also show that for any project, there is an equilibrium outcome for the time-inconsistent agents, which is not an equilibrium outcome for the time-consistent agents. More importantly, we show that, for any given project, the time-inconsistent agents finish the project weakly) earlier than the time-consistent agents. 5 Intuitively, since the sophisticated agents are aware of their inconsistency, they tend to guard themselves against the future selves by contributing higher amounts in the earlier periods. Thus, they reach the total contribution needed for completion of the project earlier than the time-consistent agents. We also study the efficiency induced by how fast the project is completed. We show that with time-inconsistent agents, under certain conditions, for a given project, the equilibrium 3 This marginal benefit can be zero or positive. Both cases are allowed in the model. 4 Phelps and Pollak 1968) first developed β δ preferences and then these preferences were used in various settings. Among these settings, see for instance, Laibson 1997) and O Donoghue and Rabin 1999a,1999b,2001, 2008). Also see O Donoghue and Rabin 2000) for for a number of applications based on β δ preferences, capturing time-inconsistency. 5 Here, there is a number of different time-consistent agents we can compare time-inconsistent agents to. Following Chade et.al 2008), for a given number of periods t, we consider a time-consistent agent with a discount factor ˆδ t, such that 1 + ˆδ t + ˆδ t ˆδt t = 1 + βδ + βδ βδ t Thus, for a given δ and β, there is a discrete set of corresponding time-consistent agents with discount factor depending on the number of periods, {ˆδ t } t=2. We provide a general comparison by considering any possible ˆδ t with t = 2,...,. 3

4 that completes the project fastest is actually the social welfare maximizing equilibrium. Then, since the equilibria we construct for our main result that compares the number of completion periods for both type of agents) is the fastest one, under the conditions we provide, it is sound to look at the fastest equilibria for comparison purposes. The voluntary public good provision problem and its efficiency properties have been widely studied. In a sequential contribution game under complete information where the amount of the public good to be provided is continuous, Varian 1994) show that the ability of the first mover to credibly commit to a certain level of contribution aggravates the free rider problem. Bag and Roy 2011), however, show that, under incomplete information, a sequential contribution mechanism may perform better than the simultaneous contribution game, in terms of total expected contributions. Admati and Perry 1991) consider an alternating contribution game under both full-refunds and no refunds cases. They investigate whether efficient equilibria still exist if the contributions are divided into small sums over time. The answer is negative; each of the two players has an incentive to let the other player contribute in the future. Fresthman and Nitzan 1991) also consider a dynamic public good provision problem with flow benefits. They also have negative results which hinge on the fact that a player can sometimes increase the level of future contribution by lowering the current contribution and this is a potential incentive for her to free ride on the future contributions of other players. On the contrary, if there is imperfect information about individual actions aggregate contribution is observed but not the individual contributions) and if players can contribute more than once and at any time, then efficiency is achieved under certain conditions, as Marx and Matthews 2000) show. 6 Compte and Jehiel 2003) introduce agent asymmetries to the model of Admati and Perry 1991) and show that their main result is not robust. In a model of private provision of public goods, Bagnoli and Lipman 1992) show that the set of undominated perfect equilibrium outcomes is identical to the core. The private provision mechanism for a discrete public good through both contribution game and subscription game is studied by Barbieri and Malueg 2008a, 2008b) and they provide some interim inefficiency results. Barbieri and Malueg 2010) restrict attention to piecewise-linear equilibrium, and show that the subscription game achieves the outcome of the optimal mechanism for a profit maximizing seller. 7 6 The horizon must be long, the players must have similar preferences and they must be patient enough. 7 Also see Bliss and Nalebuff 1984), Laussel and Palfrey 2003), Ledyard and Palfrey 1999, 2002, 2007), Lu and Quah 2009), Morelli and Vesterlund 2000) for more on efficiency properties of the outcomes of the private provision of public goods. 4

5 Time-inconsistent behavior, captured through β δ preferences, is also widely studied in various contexts, those focusing on individual decision making and those including strategic interaction. O Donoghue and Rabin 1999a, 2000, 2001, 2008) focus on individual decision making under β δ preferences, with no strategic consideration. On the other hand, O Donoghue and Rabin 1999b), Dellavigna and Malmendier 2004), Gilpatric 2008) and Yılmaz 2013) focus on contractual relationships under β δ preferences. O Donoghue and Rabin 1999b) introduces a moral hazard problem in the form of unobservable task-cost realizations and assumes that the agent is risk-neutral. Gilpatric 2008) focuses on a contracting problem with time-inconsistent agents assuming that profit is fully determined by the effort. DellaVigna and Malmendier 2004) consider a monopolistic firm who faces time-inconsistent agents and designs an optimal two-part tariff. Yılmaz 2013) considers a repeated moral hazard problem with the standard trade-off between risk and incentives, and characterizes the optimal contract. Chade et.al. 2008) study infinitely repeated games where players have β δ preferences. They characterize the equilibrium payoffs and show that the equilibrium payoff set is not monotonic in β or δ. 8 To our best of knowledge time-inconsistent preferences have not been studied in a dynamic voluntary contribution problem. Our paper contributes to the literature in this aspect. Section 2 provides the details of the model. In Section 3, equilibrium analysis is carried out. Section 4 provides the comparison of the set of equilibria of the two agent types, as well as the comparison on the equilibrium number of periods to complete a given project. Section 5 concludes and discusses a number of extensions. Some of the proofs are given in the Appendix. 2 The Model There are n 2 players and a public good with a cost X > 0. In each period t 0, each player decides how much to contribute for the public good. Player i s contribution in period t is z i t). Let zt) z 1 t),..., z n t)) denote the contribution vector in period t. Let z zt) t=0 be the entire contribution sequence. The aggregate contribution in period t is given by Zt) n j=1 z jt). Each player can observe her own past contributions and the aggregate contribution of the other players past contributions in each period. Hence, the personal history in the beginning of the period t for the player i is h t 1 i 8 There are other related papers that study time-inconsistency in other contexts. These include Akın 2009, 2012), Sarafidis 2006) and Strotz 1956). 5

6 z i τ), Z i τ)) t 1 τ=0 where Z i τ) Zτ) z i τ). There is a contributing horizon T such that contributions are not allowed after T, and any z i t) 0 is feasible for t T and only z i t) = 0 is feasible for t > T. A strategy for player i maps each h t 1 i into a contribution, z i t), that is feasible in the following period. Agents have infinite amount of private good to contribute, that is, the budget constraints are not binding. Also, the contributions are non-refundable. Given a feasible contribution sequence, z, the completion period T z) is the smallest t with t τ=0 Zτ) X. The cumulative contribution for player i at the end of period t is x i t) t τ=0 z iτ). The aggregate cumulative contribution; cumulation, is Xt) n j=1 x jt). The total benefit player i receives from the project at any period t depends on the cumulative contribution, through f i Xt)), where f i is the benefit function. f i Xt)) > 0, the project generates a benefit in period t, even if t < T. 9 If Alternatively, if f i Xt)) = 0 for all t < T, benefits are not generated until the project is completed. This is the case of a binary project such as the building of a bridge. We consider the following benefit function. f i X) = { λ i X V i X < X X X where λ i is the player i s marginal benefit from a non-completing cumulative contribution, V i is the benefit for player i from the completed project, and b i is the benefit jump received upon completion and every period following the completion period, that is b i V i λ i X as shown in the figure below. We assume 1 > λ i 0, and b i 0, which implies 0 λ i V i / X for all i N. Note that λ i = 0 yields the binary benefit function and b i = 0 yields the continuous benefit function. The cost function enters quasi-linearly and the cost of a contribution is borne at the period when it is made. We focus on the case where no agent is willing to complete the project alone, but it s efficient to provide the public good. That is, we have V i < X < n j=1 V j 9 We write T instead of T z), to ease notation. 6

7 benefit V i f i X) b i λ i X cumulation for any player i. X Players have β-δ preferences, that is, each player, at any period, discounts benefits and costs according to the discounting scheme 1, δβ, δ 2 β, δ 3 β,..., δ t β,...). βδ is the discount factor between the current period and the next period, however the discount factor between two adjacent periods in the future is δ. The agent is time-inconsistent when β < 1. A time-inconsistent agent can be fully aware, partially aware or fully unaware of his time inconsistency. Denote the agent s belief about his true β by β. As in the literature, a time-inconsistent agent is sophisticated when he is fully aware of his inconsistency, that is, when β = β < 1. The agent is partially naive when β < β < 1 and fully naive when β < β = 1. The agent is naive when β < β 1. When β = 1, the agent is time-consistent and the model becomes identical to the one in MM, which, thus, becomes a special case of our model. For a given feasible contribution sequence, z, player i s present discounted overall net payoff from the initial period to the end of the completion period of the project, T z), is U i z) 1 δ1 β))[f i X0)) z i 0)] + T z) t=1 δ t β[f i Xt)) z i t)] Now, we write the payoffs in a tractable way for dynamic programming purposes, 7

8 including the discontinuous, linear benefit function specified above. U i x i, X i )) 1 δ1 β))[λ i X0) x i 0)] + T z) t=1 δ t β[λ i Xt) x i t)] + δ T z) βb i where the first term is the net benefit received in period t = 0, the second term is the net benefit received within the periods 1 and T z), and the last term is the benefit jump received in every period t > T z), normalized by 1 δ1 β). This displays the payoffs as discounted sum of benefits and costs that are each borne in just one period, where the completion period depends on the contribution sequence z. 3 Equilibrium Analysis We first consider the static version of the game which we use to construct the set of equilibria in the dynamic version. A strategy profile in the static game, z 1, z 2,..., z n ), yields an aggregate contribution Z = n i=1 z i, and player i receives payoff f i Z) z i. Denote the total contribution of other players by Z i = Z z i. Player i s best response to Z i < X, is either to finish the project by contributing the rest of the amount required, that is, X Z i ; or to contribute nothing. 10 The marginal benefit from completing the project is f i X) f i Z i ) = V i λ i Z i, and the marginal cost of completing the project is X Z i. Marginal benefit is larger than the marginal cost if and only if the amount needed to complete the project is less than the critical contribution, which is the maximum amount that player i is willing to contribute. To get this critical contribution, we solve the equality X Z i = V i λ i Z i for X Z i. That is, X Z i = V i λ i Z i + λ i X λi X = V i + λ i X Z i ) λ i X, which implies 1 λi )X Z i = V i λ i X. Thus, we get c i X Z i = V i λ i X 1 λ i = b i 1 λ i The reaction function of player i, for Z i < X, is given by z BR i Z i ) = { 0 X Z i > c i X Z i X Z i c i 10 Intermediate amounts are dominated, because the marginal benefit, λ i, is less than 1. 8

9 Whenever b i = 0, that is, whenever there is no benefit jump, the critical contribution level is zero, c i = 0. So, contributing nothing is a dominant strategy for each player. In the dynamic version of the game, let g = g 1 t),..., g n t)) t=0 be a feasible sequence of nonnegative contributions. Then, the corresponding aggregate contribution in period t is Gt) n i=1 g it) and the aggregate of all players contributions other than player i s contribution in period t is G i t) Gt) g i t). As in MM, for g to be a candidate equilibrium outcome, it has to be feasible and wasteless. Recall, g is feasible if g i t) 0 is for t T and z i t) = 0 for t > T, where T is the contributing horizon. g is wasteless if Gt) = 0 for all t > T g) where T g) is the completion period under the sequence g. We consider the strategies with maximal punishments, that is, the grim g strategy profile in which g is played in each period as long as Gt) is observed, otherwise no player ever contributes again. Then, g is a Nash equilibrium outcome if and only if the grim g profile is a Nash equilibrium. MM provide a necessary and sufficient condition, under contributing constraint, for players not to deviate to contributing zero as long as no deviation has occurred yet, when all the players are time-consistent: 11 λ i G i t) δ T t b i + [λ i Gt) g i t)] + τ=t+1 δ τ t [λ i Gτ) g i τ)] for all i N and t T. The generalization of this constraint to the environment with sophisticated time-inconsistent players is 1 δ1 β))λ i G i t) δ T t βb i +1 δ1 β))λ i Gt) g i t))+ for all i N and t T. τ=t+1 δ τ t β[λ i Gτ) g i τ)] 1) The left side is the current value of the player s payoff if she deviates to zero in period t and thereafter, dropping the terms due to previous contributions. The right side is her payoff if she does not deviate, dropping the same terms from previous periods and inconsistently discounting to period t. Given that the other players do not deviate play grim g strategies), player i prefers to contribute according to g in period t and thereafter, rather than to deviate to zero, if and only if 1) holds. Clearly, the gap between right hand side and left hand side increases if the players shift some of their current contributions to the future. Hence, by shifting, player 11 See Marx and Matthews 2000) for the details on how to get this condition. 9

10 i s incentive to contribute; to play according to grim g strategy profile is increased. Rearrangement of 1) yields the constraint, the under contributing constraint, which guarantees that players do not deviate downward, that is, they do not free ride: 1 δ1 β))1 λ i )g i t) δ T t βb i + τ=t+1 δ τ t β[λ i Gτ) g i τ)] 2) for all i N and t T. The left-side of 2) is player i s net cost of contributing g i t) and the right side is her continuation payoff if she does not deviate; given the grim-g strategies. The interpretation of the right side is the payoff she gives up by not contributing g i t). There is also another incentive for players to deviate, which is to complete the project prematurely. This upward deviation incentive is killed by another constraint, the over contributing constraint. In the MM environment with time-consistent agents, this is given by λ i 1) X ) t Gτ) + b i δ T t b i + τ=0 τ=t+1 δ τ t [λ i Gτ) g i τ)] for all i N and t T. 12 The generalization of this constraint to the environment with time-inconsistent agents is then given by 1 δ1 β)) [ λ i 1) for all i N and t T. X ) ] t Gτ) + b i δ T t βb i + τ=0 τ=t+1 δ τ t β[λ i Gτ) g i τ)] This constraint guarantees that player i does not find it profitable to contribute the extra amount to complete the project before it is supposed to be finished. The left hand side is the extra payoff from completing the project by contributing g i t) + X t Gτ), on top of what she gets when she contributes according τ=0 to g. The right hand side is the continuation payoff she loses if she deviates. Now, we show that these two constraints, over and under contributing constraints, are necessary and sufficient for a grim-g outcome g to be a Nash equilibrium outcome. That is, these two conditions are enough to not just deter two kinds of deviations but also deter any other kinds of deviations. This is shown in MM for time-consistent agents, in Theorem 1. We modify the proof of this result in MM, to accommodate time-inconsistent agents. 12 See Marx and Matthews 2000) for the details on how to get this condition. 3) 10

11 Theorem 1 A grim-g outcome g is a Nash equilibrium outcome if and only if it satisfies over-contributing and under contributing constraints. Proof. The proof is very similar to the one in MM. We modify it to account for the δβ discounting. See Appendix for details. MM also show that the over-contributing constraint is implied by the under-contributing constraint under certain conditions. We show that this result still holds in our model with time-inconsistent agents. Corollary 1 For a candidate outcome g, let T be the the number of periods the project is completed under the grim-g profile. If g satisfies under-contributing constraint, then it also satisfies over-contributing constraint if either i) T = 0 or ii) c i = 0 for all i or iii) T < and g i T 1) + GT ) c i for all i. Proof. The proof is very similar to the one in MM. We modify it to account for the δβ discounting. See Appendix for details. 4 Comparing the Equilibria We are interested in the comparison of equilibrium outcomes under time-consistency and under time-inconsistency, in terms how fast they finish a given project. To do this, we restrict attention to the projects completed in finite time by both type of agents. For such an equilibrium to exist when players are time-consistent, MM provide sufficient conditions: some player must have a discontinuous benefit function, the contributing horizon, T, must be long enough and the discount factor must be large enough. We will not focus on this existence result in our environment, instead we take it as given. Thus, we assume that these conditions hold, and then compare the number of periods the time-inconsistent agents finish a given project with that for the time-consistent agents. We construct the equilibrium profile recursively; as in MM, starting with the completion period T, in which the under-contributing constraint is g i T ) bi 1 λ i = c i for player i. Define c i 0) c i. Given that c 1 0),..., c n 0)) is contributed in period T, then binding under-contributing constraints give us a sequence c i k) k=0 for each i where c i 1) = b i c i 0) = 1 λ i δβ λ i c j 0) 1 δ1 β) 1 λ i j i 11

12 for all k δβ λ i c i k) = c j k 1) + 1 δ1 β) 1 λ i j i 1 δ)1 β) 1 δ1 β) δc ik 1) We define an equilibrium outcome g as follows, where in the very first period, t=0, each player i contributes a fraction of c i and in every other period t > 0, each player i contributes the amount c i T t). 14 Thus, X RT 1) c RT ) RT 1) it ) for t = 0 gi t) c i T t) for 0 < t T 0 for t > T Note that, g t) satisfies the under-contributing constraint by construction. If T is finite, then since g i T 1) + G T ) c i = c i 0) for all i, Corollary 1 implies that g t) also satisfies the over-contributing constraint. Thus, it is a Nash equilibrium outcome. Now to proceed to the comparison of the outcome with the time-inconsistent agents and the one with the time-consistent agents, we consider plausible discount factors for the time-consistent agents. We follow the analysis in Chade et. al. 2008) to find the time-consistent agent who has a discount factor that corresponds to the average discount factor of the sophisticated time-inconsistent agent. That is, for a given number of periods t, we consider a time-consistent agent with a discount factor ˆδ t, such that 1 + ˆδ t + ˆδ t ˆδt t = 1 + βδ + βδ βδ t Thus, for a given δ and β, there is a discrete set of corresponding time-consistent agents with discount factor depending on the number of periods, {ˆδ t } t=1. Note that when t =, we have ˆδ = δβ. Even in an equilibrium which completes the project in finitely many 1 δ+δβ periods, V i is received every period following the completion period. Thus, the relevant corresponding time-consistent agent seems to be the one with ˆδ. However, we provide a more general comparison by considering any possible ˆδ t with t = 2,...,. Now, we show that any equilibrium outcome for the time-consistent agents with any 13 Note that, if we insert β = 1 in c i k), then we get the same sequence in MM. 14 Player i is willing contribute at most c i k) in period T k, provided that each player j contributes exactly c j T τ) in period τ > t. In the equilibrium we construct, player i contributes these critical amounts, c i k), for every period T k, except in the very first period. In the first period, period 0, players may need to contribute less. When each player contributes c i τ) in period T τ, the remaining total contribution to be made from T k on is given by Rk) = τ = 0 k i N c iτ). Thus, at t = 0, the remaining amount of contributions, RT ), must be at least as big as X. That is, T satisfies RT 1) < X RT ). 12

13 discount factor, ˆδ t ) is also an equilibrium outcome for the sophisticated time-inconsistent agents. Moreover, there are strategy profiles which are Nash equilibria under the setting with time-inconsistent agents while they are not under the time-consistent setting with a discount factor equal to the average discount factor of the sophisticated time-inconsistent agent. Proposition 1 For all X,δ,β and ˆδk, any equilibrium outcome g {g i t)} t=0 for the time-consistent agents with discount factor ˆδ k, is also an equilibrium outcome for the time-inconsistent agents. Proof. Let g {g i t)} t=0 be an arbitrary equilibrium outcome for a given project size X and ˆδ k, for time-consistent agents with discount factor ˆδ k. Suppose g completes the project in T periods. Note that, a contribution scheme g is a Nash equilibrium outcome if and only if the grim-g is a Nash equilibrium strategy profile [Proof]. Thus, the grim-g is a Nash equilibrium strategy profile, thus, the under contributing constraints must be satisfied. That is, 1 λ i )g i t)) ˆδ T t k b i + τ=t+1 ˆδ τ t k λ i Gτ) g i τ)) is satisfied for all t {0, 1, 2,.., T 1}, and for period T, we have g i T ) b i 1 λ i Now, we check whether this sequence g satisfies the under contributing constraints of the time-inconsistent agents. For period T, it is the same constraint, g i T ) thus, it is trivially satisfied. For any period t {0, 1, 2,.., T 1}, we need to check 1 λ i )g i t)) δt t β 1 δ + δβ b i + τ=t+1 δ τ t β 1 δ + δβ λ igτ) g i τ)) b i 1 λ i, Recall 1 + ˆδ t + ˆδ t ˆδt t = 1 + βδ + βδ βδ t. First, we show that ˆδ t is strictly increasing in t. To see this, first note that 1 + ˆδ t + ˆδ t ˆδt t = 1 + βδ + βδ βδ t implies ˆδ t 1 + ˆδ t ˆδ t t 1 ) = βδ1 + δ δ t 1 ). Since, δ > ˆδ t, we have ˆδ t > δβ. Also, βδ t > ˆδ t t t. To see this, suppose otherwise, that is, assume ˆδt βδ t. Then, βδt < ˆδ ṱ t since δ δ δ > ˆδ t 1 t. Thus, ˆδ t > βδ t 1. Repeating this argument we get ˆδ t s > βδ s for all s = 1,..., t, which implies 1 + ˆδ t + ˆδ 2 t t ˆδt > 1 + βδ + βδ βδ t, which is a contradiction. 13

14 Thus, we get ˆδ t t < βδ t. Now, to see ˆδ t+1 t is strictly increasing in t, add ˆδ t to the left hand side, and βδ t+1 to the right hand side of and get 1 + ˆδ t + ˆδ 2 t ˆδ t t = 1 + βδ + βδ βδ t 1 + ˆδ t + ˆδ 2 t ˆδ t t + ˆδ t+1 t < 1 + βδ + βδ βδ t + βδ t+1 since ˆδ t+1 t < βδ t+1, which is because ˆδ t < δ and ˆδ t t < βδ t. Thus, to get the equality 1 + ˆδ t+1 + ˆδ 2 t ˆδ t t+1 + ˆδ t+1 t+1 = 1 + βδ + βδ βδ t + βδ t+1 we must have ˆδ t+1 > ˆδ δβ t. Also, lim t ˆδt =, which follows from the equality 1 δ+δβ ˆδ t=1 t = t=1 βδt. Since ˆδ t is strictly increasing in t and lim t ˆδt = δβ, we get 1 δ+δβ ˆδ t δβ. Since, ˆδ 1 δ+δβ t < δ for any t, we get ˆδ t s δs β for any s 1. 1 δ+δβ Thus, we get 1 λ i )g i t)) ˆδ T t k b i + τ=t+1 δ T t β 1 δ + δβ b i + ˆδ τ t k λ i Gτ) g i τ)) τ=t+1 δ τ t β 1 δ + δβ λ igτ) g i τ)) for all t {0, 1, 2,.., T 1}. That is, grim-g is also a Nash equilibrium strategy profile for the time-inconsistent agents, thus, g is a Nash equilibrium outcome for the timeinconsistent agents. Proposition 2 For all X,δ,β, there exists a sequence of nonnegative contributions, g {g i t)} t=0, such that g is a Nash equilibrium outcome for the time-inconsistent agents while it is not a Nash equilibrium outcome for the corresponding time-consistent agents. Proof. Let g {g i t)} t=0 be an arbitrary equilibrium outcome for a given project size X, for the inconsistent agents, where g completes the project in T periods, such that the under contributing constraint binds for some t {0, 1, 2,..., T 2}. Since grim-g strategy profile is a Nash equilibrium strategy profile, the under contributing constraint 14

15 holds: 1 δ + δβ)1 λ i )g i t)) δ T t βb i + τ=t+1 for all t {0, 1, 2,.., T }. Moreover for period t we have, δ τ t βλ i Gτ) g i τ)) 1 δ+δβ)1 λ i )g i t )) = δ T t βb i +δβλ i Gt +1) g i t +1))+...+δ T t βλ i GT ) g i T )) Now, we consider the corresponding time-consistent agents with ˆδ k and check the under contributing constraint at period t for these agents. 1 λ i )g i t )) = δ T t β 1 δ + δβ) b δβ i + 1 δ + δβ) λ igt + 1) g i t + 1)) δt t β 1 δ + δβ) λ igt ) g i T )) > ˆδ T t k b i + ˆδ k λ i Gt + 1) g i t + 1)) ˆδ T t k λ i GT ) g i T ) where the first equality follows from the binding under contributing constraint for the time-inconsistent agent at period t. The strict inequality follows from the fact that ˆδ s < δs β for all finite s > 1. Note that T t > 1 since t < T 1. Thus, at period 1 δ+δβ t, the under contributing constraint for the time-consistent agents does not hold. Thus, grim-g is not a Nash equilibrium strategy profile, hence not an equilibrium outcome, for the time-consistent agents. Now, we show our main result. When it comes to compare the time-inconsistent agents to the time-consistent agents with a discount factor ˆδ t, in the sense of completion period of any given project, surprisingly; sophisticated time-inconsistent agents finish the project no later than the time-consistent agents do. Let T SO X) be the minimum number of periods that the sophisticated time-inconsistent agents finish the project of size X. Let T T Cˆδ k ) X) be the minimum number of periods that the corresponding time-consistent agents, with the discount factor ˆδ k, finish the project of size X. Theorem 2 For any given project size X and for any δ,β, ˆδ k, λ i, b i > 0 and n 2, we have T SO X) T T Cˆδ k ) X). Proof. First, we show that a grim-g strategy profile with binding critical values is the fastest, that is, T i X), is the number of periods induced by the grim-g with binding 15

16 critical values, where i {SO, T Cˆδ k )}. To see this, suppose otherwise, that is, there is another equilibrium which finishes the project faster than the equilibrium profile with the binding critical values. Thus, at least one player at some period must be contributing more than her critical value for that period, violating the under contributing constraint. Thus, by Theorem 1, this profile is not a Nash equilibrium outcome. Thus, the grim-g equilibrium outcome with binding critical values is the fastest. Now, we show that for any δ, β, ˆδ k and for any given values of λ and n, we have t=0 c δβ i t) t=0 cˆδ k i t) 4) for any T 1, where c δβ i t) is the critical value of the time-inconsistent agent i in period t, and k cˆδ i t) is the critical value of the corresponding time-consistent agent i, with the discount factor ˆδ k, in period t. To see this, we look at the critical values for the timeinconsistent agent. c δβ i 1) = c δβ δβ i s) = 1 δ1 β) for all s 2. ) λi c δβ i 0) = b i 1 λ i δβ 1 δ1 β) 1 λ i j i c δβ j ) λi 1 λ i j i c δβ j 0) s 1) + cδβ i s 1)δ [ ] 1 δ)1 β) 1 δ1 β) The critical values of the corresponding time-consistent agent with a discount factor ˆδ k cˆδk i 0) = b i 1 λ i i s) = ˆδ λ i k 1 λ i cˆδ k j i cˆδ k j s 1) for all s 1. Note that as k, ˆδ k δβ. 1 δ1 β) Now, we compare the critical values. Note that c δβ i 0) = b 1 λ = cˆδ k i 0) 16

17 and c δβ δβ i 1) = 1 δ1 β) ) λi 1 λ i j i c δβ j 0) ˆδ k λ i 1 λ i j i cˆδ k j 0) = cˆδ k i 1) which follows from the fact that Proposition 2. c δβ δβ i s) = 1 δ1 β) ) λi δβ 1 δ1 β) ˆδ k for any k, which was shown in the proof of 1 λ i j i δβ > 1 δ1 β) j i c δβ j s 1) + cδβ i s 1)δ ) λi 1 λ i j i c δβ j s 1) ˆδ λ i k k cˆδ 1 λ j s 1) = k cˆδ i s) i t=0 [ ] 1 δ)1 β) 1 δ1 β) for all s 1, where the first inequality follows from the fact that c δβ i s 1)δ 1 δ)1 β) > 0. 1 δ1 β) δβ The second inequality follows from the fact that ˆδ 1 δ1 β) k for any k and c δβ j s 1) k cˆδ j s 1) for all j and for all s 1. Thus, we get c δβ i t) k cˆδ i t) for all i and for all t. Thus, c δβ i t) k cˆδ i t) Thus, for the cumulative contributions we get t=0 c δβ i t) t=0 i i t=0 cˆδ k i t) This implies that the time-inconsistent agents always achieve a weakly higher cumulative contribution at any period. Thus, they will always achieve a given X weakly earlier than any corresponding time-consistent agents, that is, TSO X) T X). T Cˆδ k ) The intuition behind this result is that the time-inconsistent agents are sophisticated, thus they know that their future selves may tend to postpone contributions and cause the project be finished later. Thus, the current selves of time-inconsistent agents contribute more, relative to the time-consistent agents, in early periods in order to guard themselves against the future selves. Thus, with achieving higher cumulative contribution levels in the early periods, time-inconsistent agents manage to finish the project earlier than the time-consistent agents. 17

18 Example 1. Here, we provide an example, in which T SO X) < T T Cˆδ k ) X). For the set of parameters below, the time-inconsistent agents finish the project at TSO = 4. However, the corresponding time-consistent agents, with a discount factor ˆδ defined above, finish the same project at T T Cˆδ ) = 5.15 The parameters are specified as follows. δ 0.8 β 0.7 b 10 λ 1 8 n 4 X 68 Given these values of the parameters, the critical contribution levels are k=0 k=1 k=2 k=3 k=4 Total c δβ 80 k) cˆδk) The required amount X = 68 will be collected by the time-inconsistent agents within T = 4 periods. However, the maximum amount that the corresponding time-consistent agents are willing the contribute in this environment is the summation of the second row which is Hence, the project, which requires the total of amount of 68 to be provided, will be finished by the time-consistent agents within T = 5 periods. Now, we turn to the question of whether the fastest equilibrium is efficient or not, in terms of social welfare. We show that under certain conditions, the fastest equilibrium maximises the social welfare. Proposition 3 When λ i = λ and b i = b for all i = 1,..., n, the grim-g equilibrium which completes the project fastest is efficient if 1. λn 1 or 2. λn < 1 and 1 λ 1 λn β 1 δ+δβ) ) 1 β + T β1 δ) 15 If we would get the corresponding time-consistent agent as the one with a discount factor ˆδ 5 for instance, we would satisfy the condition even easier, since the critical contributions would be smaller in that case. Thus, they finish the project no earlier than T = 5. 18

19 Proof. The proof is in the Appendix. When λn 1, in the equilibrium the marginal benefit of one unit of cumulative contribution, λn, is larger than the marginal cost of contributing one unit, which is 1, in every period. Thus, the net marginal benefit, which is positive, is delayed if players postpone their contributions. Thus, in an equilibrium with critical contributions binding, postponing the contributions is not efficient. When λn < 1, the net marginal benefit is negative, however, postponing the contributions also delays the benefit jump, b. This implies that there is a tradeoff and under a condition on the model parameters, the fastest equilibrium is efficient. 1 λ Now, we interpret this condition. When n increases, the left hand side, increases, 1 λn thus it s easier to satisfy the condition. 16 That is, it is more likely for the fastest equilibrium to be efficient. To see this, suppose with n players postponing the contributions is not efficient. Now, if there are n + 1 players, the overall contribution per player will be smaller, thus, the gain from postponing is also smaller. But the loss of postponing the benefit jump is the same as in the case with n players. Thus, postponing is also not efficient with n + 1 players, that is, the fastest equilibrium is still going to be efficient. When the number of periods to finish the project is T, the loss from postponing the benefit jump is δ T β1 δ)b. But when the number of periods to finish the project is T +1, this loss, which is δ T +1 β1 δ)b, is smaller. Thus, as T increases there is more incentive to postpone the contributions and the fastest equilibrium is less likely to be efficient. When λ is larger, the marginal benefit from cumulative contributions is larger in every period. Thus, if postponing is not increasing the payoff with a small λ, then it cannot be payoff increasing with a larger λ. The effect of β on this condition is not trivial. The right hand side of the condition, β 1 β ), + T strictly increases with β if T > 1/1 δ) 2, where T is the number 1 δ+δβ) β1 δ) of periods to finish the project for the time-inconsistent agents. 17 If T is large enough, the derivative of the right hand side of the condition will be positive and if β is smaller, the right hand side is smaller, thus it is easier to satisfy the condition. Thus, for large enough T, or large enough X, as the degree of time-inconsistency increases through a lower β) it is more likely for the fastest equilibrium to be also the efficient one. 16 Note that as n increases, the number of periods to finish the project weakly decreases, which makes the right hand side of the condition smaller. Thus, it becomes even easier to satisfy the condition. 17 Note that as β changes T may also change. 19

20 5 Conclusion The model of voluntary contribution, that we studied, lets the players contribute any amount in any period as they wish, by observing the aggregate contribution at each period. We introduced sophisticated time-inconsistent agents with linear discontinuous preferences to the model and characterized the Nash equilibria as well as the shortest time period in which the project is finished. Our main result is regarding the comparison of completion period of specific projects. We find that any project is completed by the time-inconsistent agents weakly earlier than the corresponding time-consistent agents. This is surprising because time inconsistent agents tend to be postpone costly actions. Intuitively, a time-inconsistent agent commits to higher contributions in the earlier periods in order to guard herself against the potentially low contribution levels of the future selves. For the future research, we plan to look at the naive agent case, where the timeinconsistent agent is not aware of her time-inconsistency. We also plan to generalize our results to the heterogeneous agents case, where some portion of the population is time-consistent and rest is time-inconsistent. 20

21 6 Appendix Proof of Theorem 1. Let T be the number of periods the project is completed under the grim-g profile. Then, the payoff of player i under the grim g profile is Ui g) = 1 δ1 β))λ i G0) g i 0)) + δ τ βλ i Gτ) g i τ)) + δ T βb i Suppose player i contributes a non-completing z i g i t) in period t and then never contributes again. Then, her payoff from such deviation is t 1 U iz i, t) = 1 δ+δβ)λ i G0) g i 0))+ δ τ βλ i Gτ) g i τ))+δ t βλ i G i t) 1 λ i )z i t)) τ=1 τ=1 Since λ i < 1 and z i t) 0, we have U iz i, t) U i0, t). Now, suppose player i contributes an amount ẑ i t) = g i t) + X t < T, hence, finish the project at period t < T immediately. Then, her payoff is t 1 U i t) = 1 δ + δβ)λ i G0) g i 0)) + δ τ βλ i Gτ) g i τ)) τ=1 + δ t βλ i G i t) + δ t βb i 1 λ i )ẑ i t)) = U i0, t) + δ t βb i 1 λ i )ẑ i t)) t τ=0 Gτ) in period Finally, consider a deviation in which player i contributes an amount z i g i t) in period t < T, which does not complete the project in period t, and then contributes a positive amount at some t > t as well. However, this type of deviation is dominated by contributing zero in every period following period t, whenever b i 1 λ i )g i t) + X t τ=0 Gτ). When b i > 1 λ i )g i t) + X t τ=0 Gτ), then it is again dominated, this time by contributing exactly g i t) + X t τ=0 Gτ) in period t. Note that the three type of deviations considered above are exhaustive, that is, there is no other deviation we need to check. Thus, for player i, any kind of deviation becomes non-profitable if and only if U i0, t) U i g) for each t T and U i t) U i g) for each t < T. These two conditions for each i are equivalent to the under-contributing and overcontributing constraints. Thus, under-contributing and over-contributing constraints are necessary and sufficient for the grim-g outcome to be a Nash equilibrium outcome. Proof of Corollary 1. If T = 0, over-contributing constraint is vacuously satisfied. 21

22 Let T > 0 and t < T. Then, condition ii) and iii) independently imply g i t) + T τ=t+1 Gτ) c i for all i. Since, λ i < 1, c i = and X T τ=0 Gτ), we get λ i 1)X b i 1 λ i t Gτ)) + b i 1 λ i )g i t) τ=0 This, together with the under-contributing constraint, implies 1 δ1 β)) 1 δ1 β))1 λ i )g i t) δ T t βb i + [ λ i 1) X τ=t+1 ) ] t Gτ) + b i δ T t βb i + τ=0 δ τ t β[λ i Gτ) g i τ)] τ=t+1 δ τ t β[λ i Gτ) g i τ)] which is the over-contributing constraint. Proof of Proposition 3. Recall the critical contributions c0) = b 1 λ c1) = δac0) c2) = c3) =.. ct 1) = c0) δa + δa +. δa + δa + ) 1 δ)1 β)δ δac0) 1 δ + δβ ) 2 δac0) 1 δ)1 β)δ 1 δ + δβ ) T 2 1 δ)1 β)δ δac0) 1 δ + δβ ) T 1 δac0) 1 δ)1 β)δ 1 δ + δβ where δ = δβ < δ < 1, and A = λ n 1). Also, note that δa + 1 δ)1 β)δ < δ. 1 δ+δβ 1 λ 1 δ+δβ When an agent is willing to do this contribution scheme, it means that the total payoff coming to the agent is greater than zero so that he is willing to participate to the project. 22

23 In more formal way, uc) = δ T βb 1 λn)[δ T βc0)+δ T 1 βc1)+δ T 2 βc2)+...+δβct 1)+g0)] 0 5) Now that the payoff with the critical contributions which finishes the project at the earliest period T is the one above, the question is whether an agent can increase his payoff by extending the finishing time 1 period i.e. whether he gets more payoff or not if the project is finished within T + 1 period instead of T. The best way to do it is to move each contribution 1 period later i.e. gt) that used to be contributed at period t will be done at period t + 1. Interior amount of postponing is dominated by either postponing nothing or postponing the whole amount. Thus, let us denote the new payoff by uc ) which is; uc ) = δ T +1 βb 1 λn)[δ T +1 βc0)+δ T βc1)+δ T 1 βc2)+...++δ 2 βct 1)+δβg0)+0] 6) The question is now that is uc) uc ) positive or negative. We want it to be nonnegative. Hence we need uc) uc ) = δ T βb1 δ) 1 λn)[δ T βc0)1 δ) + δ T 1 βc1)1 δ) + δ T 2 βc2)1 δ) δβct 1)1 δ) + g0)1 δβ)] 0 by dividing both sides 1 δ) we get δ T βb 1 λn)[δ T βc0) δβct 1) + g0) 1 δβ 1 δ ] 0 Note that if λn 1, it holds obviously, and this proves the first claim. We will; now, more focus on the case where 0 < λn < 1. Now, by writing all of the criticals in terms of 23

24 c0), we get uc) uc ) 1 δ = δ T βb 1 λn) [ δ T βc0) + δ T 1 β δac0) ] 1 λn) δ T 2 β 1 λn) δ T 3 β δa + δa + ) 1 δ)1 β)δ δac0) 1 δ + δβ ) 2 1 δ)1 β)δ δac0) 1 δ + δβ. 1 λn) δβ δa + ) T 2 1 δ)1 β)δ δac0) 1 δ + δβ 1 λn) 1 δβ 1 δ g0) note that, since g0) ) T 1 δa + 1 δ)1 β)δ 1 δ+δβ δac0) we have uc) uc ) 1 δ δ T βb 1 λn) [ δ T βc0) + δ T 1 β δac0) ] 1 λn) δ T 2 β 1 λn) δ T 3 β δa + δa + ) 1 δ)1 β)δ δac0) 1 δ + δβ ) 2 1 δ)1 β)δ δac0) 1 δ + δβ. 1 λn) δβ 1 λn) δa + 1 δβ 1 δ ) T 2 1 δ)1 β)δ δac0) 1 δ + δβ δa + ) T 1 1 δ)1 β)δ δac0) 1 δ + δβ 24

25 and also since δa + 1 δ)1 β)δ 1 δ+δβ uc) uc ) 1 δ < δ we have, > δ T βb 1 λn)c0) δ T β + δ T 1 β δa δ T 1 β δa + 1 δβ δt 1 δa }{{} 1 δ T 1 of them Hence we get uc) uc ) 1 δ [ > δ T βb 1 λn)c0) δ T β + T 1)δ T 1 β δa + 1 δβ ] δt 1 δa 1 δ Now we need δ T βb 1 λn)c0) [ δ T β + T 1)δ T 1 β δa + 1 δβ 1 δa ] 1 δ δt 0. To get that, divide both sides by δ T b βb and plug c0) as and δ δβ as to get 1 λ 1 δ+δβ 1 λ 1 λn 1 + βa 1 δβ A T 1) + 1 δ + δβ 1 δ 1 δ + δβ Now plug A into above inequality as λ n 1) to get 1 λ λn 1) 1 λn λ ) βt 1) n 1) 1 λ 1 δ + δβ + 1 δβ 1 δ)1 δ + δβ) Thus we get the condition which is necessary for the second claim to be true. 1 λ 1 λn βt 1) 1 δ + δβ + 1 δβ 1 δ)1 δ + δβ) ) β 1 β = 1 δ + δβ) β1 δ) + T This completes the proof. 25

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