Dynamic Voluntary Contributions to Public Goods with Stock Accumulation J. Cristobal Ruiz-Tagle Department of Agricultural and Resource Economics, University of Maryland, College Park jruiz-tagle@arec.umd.edu Selected Paper prepared for presentation at the Agricultural & Applied Economics Association s 2012 AAEA Annual Meeting, Seattle, Washington, August 12-14, 2012 Copyright 2012 by J. Cristobal Ruiz-Tagle. All rights reserved. Readers may make verbatim copies of this document for non-commercial purposes by any means, provided that this copyright notice appears on all such copies. 1
Abstract This paper aims to test experimentally the fundamental economic incentives for countries to pledge reductions in emissions of green-house gases GHG) at each negotiation of an International Environmental Agreement for Climate Change IEACC). I. Introduction During the past round of negotiations of the UN Climate Change Conference in Durban all parties committed to a legally binding IEACC that will limit the emissions of GHG. The concentration of GHG is a perfect case of a pure global public good, whereby parties pledges for GHG emission reductions when negotiating the details of an IEACC can be thought as voluntary contributions to provide such a public good. Furthermore, concentration of GHGs accumulates over time and depreciates slowly. Thus, fulfilled pledges to reduce GHGs emissions, reduce the flow of GHG into the atmosphere and this, in turn, reduces the stock of GHG. This paper tests experimentally these pledges by looking at them as voluntary contributions to provide a global public good. We focus on two dynamic components: i) the cumulative effect of the emissions of GHG into a stock and ii) the repeated process of pledges for reduction of GHG emissions at each IEACC. The existing literature on voluntary contribution to public goods has largely focused on static public goods only. In this paper, we replicate the main experimental findings for standard static public goods and we contrast them with its dynamic counterpart. Therefore, as in the existing experimental literature, we expect to find contributions above what is predicted by the standard static theory on voluntary provision to public goods. Similarly, in the dynamic setting, we also expect to find contributions above what is predicted by dynamic theory. Preliminary tests with experimental software Z-tree proves that the experiment can be readily implemented with undergraduate students at the Experimental Lab of the AREC Department. Funding to carry out the experiment has been secured and we will soon start running the first rounds of the experiment. 2
II. Background on Voluntary Provision to Public Goods and International Environmental Agreements IEA) When designing and running this experiment, we will pay close attention to all the theoretical and experimental issues studied in the static voluntary literature as these issues will likely be present as well in a dynamic setting. To name a few, the existing experimental literature on voluntary provision of public goods has focus on the following issues: Over-contribution with respect to Nash theoretical predictions. That is, contrary to the standard static theory on voluntary provision to public goods, experimental studies show a tendency towards cooperation. As the reasons for this widely known experimental result, some studies point to altruism warm-glow and other-regarding preferences); error; reciprocity; conditional cooperation contribution conditional on expectations of other s contributions, Fischbacher and Gachter AER 2010)); and punishment of decisions that lead to unfair outcomes Gacther et. al. JPubE 2010). Contributions typically begin at around 50 percent and decline over time as more rounds are played. This can be attributed to a process of learning the free-rider strategies as well as learning about what to expect from others. However, contributions at later rounds of long time-horizon experiments are not lower than contributions at later rounds of short time-horizon experiments. Some authors point out that over-contribution can also be interpreted as signaling some degree of cooperation in early rounds that will decrease throughout the next rounds of the game. Laury and Holt 2008) review literature on interior Nash equilibrium and show evidence that over-contribution decreases when the theory model yields an interior Nash solution. 3
Laury and Holt 2008) also review the effect of differences in the endowment of subjects within the group and point out that there could be a feeling of fairness in which wealthier individuals under-contribute whereas poorer individuals over-contribute Chan et.al 1996a). Furthermore, contrary to theoretical predictions, experimental evidence shows that group size seems to have a lesser impact on contributions to public goods as compared to marginal percapita returns to contributions MPRC). On the other hand, largely motivated by the provision of public goods in negotiations of international environmental agreements IEA), a new literature has emerged on the formation of coalition to provide public goods. Most of the studies in this area basically break down the process of voluntary contributions into a two-step process: in the first step individuals decide whether to join a coalition that will provide the public good with benefits to all, even those not joining the coalition); whereas in the second step, once they know how many individuals joined the coalition, individuals decide how much to contribute to the public good. Yet, these models may violate the voluntary principle of contributions to public goods. That is, when joining a coalition, these models assume that some individuals will contribute to the public good even beyond what they would voluntarily contribute in the absence of the coalition. For instance, these models assume that the minimum amount that each member of the coalition will be determined by majority rule. That is, those in the coalition that would want to contribute less than what the majority rule determines as a minimum contribution, would find themselves forced to contribute what is determined by the majority, not what they would voluntarily like to contribute. Other models on coalition formation even test treatments in which, once individuals decide to join a coalition, they would have to contribute a large amount of their endowment to the public good so that to maximize the payoffs of those in the coalition. That is, they would be forced to a contribution that is Pareto efficient despite their individual incentives to contribute a lesser amount. Not surprisingly, experimental results show that these coalitions would be joined by very few members. 4
Nevertheless, as mentioned before, since all parties have committed to a legally binding IEACC at the UN negotiations in Durban, we do not find appropriate a two-step model of formation of coalitions when analyzing the dynamics of pledges to reduce GHG. Instead, we will focus on a simple model of voluntary contribution to public goods paying particular attention to its dynamic features. In a recent study, Battaglini, Nunari and Palfrey 2010) propose a theoretical model of provision of dynamic public goods under alternative institutional mechanisms: autarky and legislature. They test the performance of the legislature mechanism as compared to the autarky mechanism. 1 The authors test experimentally the performance of these two mechanisms in their performance to provide a dynamic public good. 2 In the jargon of Battaglini et. al., the concentration of GHG in the atmosphere, can be best characterized as an autarky mechanism. However, unlike Battaglini et. al., we allow for decay in the stock of the public good and we contrast the existing experimental literature on static voluntarily contribution to public goods with its dynamic counterpart. A parallel literature is that of common pool resources CPR) with a few studies that test experimentally the optimal extraction path of a renewable natural resource such as underground water Gardner et al 1997, Sute et al 2012). This literature looks at the experimental performance of different legal rules for extraction of a natural resource. Furthermore, for ease of mathematical analysis and experimental tractability, this literature has focused on extraction models in a finite time horizon. However, as we will explain below, we will focus on an infinite time horizon model. 1 One of the salient assumptions of their model is that, in order to attain uniqueness in the theoretical dynamic equilibrium, the authors assume that the stock of the public good presents no decay over time. We believe that this assumption plays a crucial role on driving their experimental results. Furthermore, this assumption allows for an infinite accumulation of the public good and does not bring much realism in the context of provision of public goods as pledges to reduce emissions of GHG into the atmosphere. 2 Autarky refers to the case in which individual agents, such as sovereign nations, decide on the provision of public goods, whereas the legislature mechanism involves the election of a representative authority that chooses the level of provision of the public good. However, in the provision of a global public good, such as the reduction in GHG emissions, there is no representative authority, or global government, that can actually implement a centralized mechanism such as a legislature. 5
In this paper we borrow from the theoretical developments by Fershtman and Nitzan 1991) who analyze a continuous-time model of voluntary provision of public goods. They show that, as compared to standard non-cooperative games of provision of public goods, when the optimal contribution depends on the current accumulated level of the stock the incentives to free-ride become further aggravated. In the next section, we depart from Fershtman and Nitzan s model by constructing the discrete-time counterpart of dynamic voluntary contributions to public goods. Unlike Fershtman and Nitzan s model, we assume a payoff function that has a linear-quadratic term in both the private and the public good. 3 III. A Model of Dynamic Public Goods with Stock Accumulation In our model individuals benefit both from the consumption of private and public goods. We assume an additively separable payoff function that is quadratic in both the consumption of private ) and public good ). Thereby, individual i's payoff is given by 1) We will further assume that individuals receive an equal endowment that they can decide to allocate either to private consumption or to contribute in the provision of the public good, so that. A. Static setting The previous literature has largely analyzed the static setting. In this setting, the provision of public good is given by the aggregate contribution across all N individuals, that is,. Summing 1) across 3 This assumption, when analyzing the static game, allows us to obtain unique dominant strategies in the Nash contributions to the public good. 6
individuals and maximizing the aggregate payoff function ) with respect to individual consumption of the private good ) ) Yields that, at the optimum, marginal benefit from consumption of one unit of private good equal to the aggregate marginal cost in terms of forgone consumption of public good ). must be 2) Solving for the Pareto efficient consumption level of private good ), it yields ) and provision of public good 3) On the other hand, maximization of the individual payoff function 1) Yields the classic Nash result. For every individual, marginal benefit from consumption of one unit of private good must equals the marginal cost that individual faces in terms of forgone consumption of public good. 4) Equation 4) shows that when deciding optimal consumption, individual does not fully account for the social costs of his consumption decision. Furthermore, comparison of equations 2) and 4) show that the marginal payoff of individual s consumption of private good equates to a much lower marginal cost as compare to what is Pareto efficient as in equation 2)). Therefore, individuals s consumption of the private good is higher than the Pareto efficient consumption, and hence, the Nash provision of public 7
good is lower than under the Pareto efficient scenario. The actual static Nash results for consumption of private good, and provision of the public good are given by 4 5) Furthermore, it can easily be shown that for small values of the Pareto efficient level of the public good is larger than the Nash equilibrium. In fact as goes to zero the difference goes to B. Dynamic setting In a dynamic setting, the aggregate voluntary contribution to a public good at time t, constitutes a flow whereas the accumulated level of the public good at time t constitutes a stock the dynamic setting we assume that the stock accumulates over time according to:. In 6) Unlike the static setting, in the dynamic setting the current available stock does not depend solely on the contemporaneous sum of individual s contribution in the current period t, but it also depends on contributions during all periods before t as well as on the decay factor. Furthermore, in the dynamic setting, the payoff function in 1) can be rewritten as 7) As in the static case,. For the analysis that follows we define 8) 4 The quadratic term in the consumption of the private good in 1) allows for the best-response Nash strategies to be unique, regardless of the aggregate level of the public good. The importance of this will become apparent in the experimental section. Furthermore, the quadratic term in the public good allows for a bounded accumulation of the stock in the dynamic steady state equilibrium. 8
so that 6) can be expressed as [ ] 9) Dynamic Pareto Efficient solution In this section, we will analyze the dynamic equilibrium and optimal trajectories that will yield a maximum payoff for all individuals throughout time. Similar to the static case, we maximize the aggregate payoff function by summing 1 ) across individuals ) subject to the dynamic transition equation 9). [ )] [ ] The current value discrete-time Hamiltonian for this problem is [ ] [ ] F.O.C. : ) 10) : ) 11) [ ] Defining [ ] as the net discount on the part of the stock of that carries over to the next period, equation 10) can be rearranged as 9
12) At the dynamic optimum, marginal benefit from consumption of one unit of private good equal to its contemporaneous marginal cost in terms of forgone consumption of public good must be ) plus the discounted value of an additional unit of the stock of public good for future periods. We now solve for the steady-state Pareto Efficient level of public good and consumption of private good yields. At the steady state. Combining equations 10) and 11), and assuming, yields the dynamic Pareto efficient consumption of private good and provision of public good. [ ] [ ] 13) Notice that in a situation of full depreciation of the stock,, then the dynamic Pareto Efficient result above reduces to its static counterpart in 3). Furthermore, it can be shown that both, Pareto efficient level of private consumption and stock of Pareto efficient public good are higher in the dynamic setting than in the static setting. 5 Furthermore, from the FOCs of the optimization problem above equations 10) and 11)), we derive optimal paths towards steady-state equilibrium in 13) for both consumption of the private good and the stock of public good. { [ )]} [ ] ) 14) and initial conditions ;. 5 Solving for the consumption of private good ) conditional on the stock yields ) The derivative of the expression above with respect to efficient consumption of private good is lower than it s dynamic counterpart, is greater than zero. Therefore, we conclude that the static Pareto 10
Along the optimal path, consumption of the private good increases over time,. Furthermore, as long as, consumption of the public good also increases over time, that is,. 6 De-centralized equilibrium I: Open-loop solution Unlike the previous section, in this section we analyze the decentralized Nash equilibrium when individuals decide on an optimal path of consumption of private good and contribution to the public good throughout time. In this section we analyze the open-loop equilibrium so that we will assume that individuals do not re-optimize their contributions once the provision of public goods is realized. In the next section, we will relax this assumption and will allow for dynamic re-optimization. Maximizing the individual payoff function 7) subject to the dynamic transition equation 9) ) [ ] The current value discrete-time Hamiltonian for this problem is [ ] F.O.C. : ) 15) : ) 16) [ ] 6 Notice that the largest value that can take is 0.25. Therefore, for and, this constraint is unlikely to be binding. 11
[ ] Rearranging equation 15), we have that at the decentralized dynamic optimum, marginal benefit equals its contemporaneous marginal cost plus the discounted value of the stock of public good that carries over for future periods. 17) In the decentralized analysis, consumption of private good carries a lower marginal cost than the one under the dynamic Pareto efficient setting in equation 12). In the decentralized solution, individuals do not carry the full cost of their consumption decisions. Therefore, as individual consumption of private good is higher than social optimum, the stock of public good provision in the open-loop equilibrium is lower at every point in time as compared to the Pareto efficient solution. Furthermore, the open-loop optimal consumption of private good carries a higher marginal cost ) that its static counterpart in equation 4). In fact both private consumption and stock of public good are higher in the dynamic open-loop setting than in the static setting. 7 At the steady-state, the dynamic open-loop consumption of private good and stock of public good are [ ] [ ] 18) Solving for the consumption of private good ) conditional on the stock yields ) The derivative of the expression above with respect to good is lower than it s dynamic counterpart, is greater than zero. Therefore, the static Nash consumption of private 12
Similar to the analysis in the previous section, from the FOCs above equations 15) and 16)), we derive optimal paths towards the open-loop steady-state equilibrium. { [ )]} [ ] ) 19) and initial conditions ;. Comparing equations in 14) with 19) above, we observe that under the decentralized open-loop optimal path, consumption of both private good and public good grow at a lower rate than under the Pareto efficient optimal path. 20) De-centralized equilibrium II: Feed-back Nash solution In a feed-back Nash dynamic setting, at each period of time individuals can make their strategies conditional on the observable accumulated) contributions of others up to that time, the current stock of Fershtman and Nitzan, 1991). In other words, equilibrium strategies must be Sub-game Perfect Equilibrium SPE) they must be an equilibrium in each and every stage of the game so that equilibrium strategies must incorporate the information available up to the current period, the current level of stock, and make contributions conditional on the observable. 8 8 In this section we focus on Markov Perfect Equilibrium MPE) strategies. 13
Bellman s principle of optimality requires that equilibrium strategies should be SPE. That is, equilibrium strategies should be optimal for the current period as well as for all future periods after. In order to obtain the optimal SPE equilibrium strategies, we solve the following Bellman equation. { } 21) [ ] FOC : [ ] 22) Envelope Condition [ ] 23) Furthermore, by firs-order Taylor series expansion on the last term of the value function in 21) we obtain the following discrete-time Hamilton-Jacobi-Bellman HJB) equation { [ ]} 24) [ ] Maximization of the discrete-time HJB equation yields [ ] 25) Combining equations 22), 23)and 25) above we arrive to the following feed-back strategy [ ] 26) [ ] 14
Using equation 26) above and the transition equation for 9), we derive optimal paths towards the feed-back steady-state equilibrium. 27) { [ ) ]} Assuming a linear quadratic value function of the form ), the following equilibrium strategies constitute a SPE for the provision of the public good K 28) And hence, the resulting steady-state level of the public good is ) 29) where,and ) ) [ ]. Furthermore, given the linearity of the equilibrium strategies, it can be shown that these strategies are dynamically stable globally asymptotically stable, GAS) Basar and Olsder, 1999). 15
Still pending is to contrast 28) and 29) with dynamic open-loop equilibrium in equation 18) Extensions: uncertainty in the dynamic setting still work in progress) In future research we would like to expand the theoretical model outlined above to allow for a more realistic feature of dynamic voluntary contribution to public goods: uncertainty about the welfare effect of a variable stock of a public good in future periods. Let us assume for now that the stock of the public good is not fully determined by a deterministic series of voluntary contributions but, instead, is also determined by a random shock with mean zero and variance. Also, assume that this random shock is non-stationary such that its variance is larger as we move further away from the initial period. That is,. Furthermore, we assume that the process of accumulation of the stock of public good will have a negative effect on the net effect of the random shock. Thereby, we add a term to equation 4) such that the net effect of the random shock will be smaller the larger the accumulated stock of the public good. The purpose of this term is to allow for a larger degree of uncertainty about the events in the distant future. This uncertainty will be diminished, however, for large values of the stock of public good. Testing experimentally this sort of uncertainty will be the focus of attention of future research. IV. Recruitment and Experimental Sessions 16
A. Recruitment The experimental session will be advertised via email in the list serve of the University of Maryland FYI digest system), which reaches all faculty and students of the University of Maryland College Park. It will be explained in this email that the experiment aims to reach undergraduate students interested in participating this experiment for a show up payment of $8, plus an extra amount based on performance, to average a total payment of around $20. Furthermore, there will be an URL link to the recruitment website for these experiments ORSEE 2.09) where participants can register by entering some basic data such as name, email address and main field of studies see Figure 1 below). No further information is given at this point regarding the time and location of experimental sessions. Instead, once they have register to participate in the experiments, participants will receive an automated email to confirm they email address and in which there will be another URL to register for a specific experimental session, listing the time and location of these sessions See figure 3 below). A total of 24 participants will be accepted to register at each session and there will be an automated reminder email. Fig. 1: Participant Registration Form 9 Organization Recruitment System for Experiments in Economics. 17
Fig. 2: Experiment registration page B. Experimental Sessions The experiment will be carried out at the experimental lab of the Department of Agricultural and Resource Economics at the University of Maryland. The lab sits a maximum of 24 individuals each paired to a networked computer station. As participants show up at the pre-specified time of the experimental session, they will be assigned a computer station and hanged a consent form 10.This experiment requires any amount of participants that is multiple of 4, so that participants will only be accepted on the basis of keeping numbers to meet this requirement. 11 10 In order to meet the requirements set by the university s Institutional Review Board. 11 The remaining participants will be dismissed after payment of a $X show-up fee and will be invited to sign up for a new session in the ORSEE website. 18
The experiment begins with a series of introductory slides on the computer screen that explain the rules of the experiment, how participants can make money depending on their performance and the details of possible actions and payoffs of the experiment. This section will be intended to be as brief and thorough as possible so that the rules and incentives of the experiment can be fully understand in the minimum amount of time possible. Furthermore, in order to ensure full understanding, there will be a comprehension quiz and a short series of practice rounds that will yield no earnings. SCREEN SHOTS OF QUIZ? B.1. Experimental Treatments Table 1: Experimental Treatments Treatment Beliefs 1 round 1 Static PG 10 rounds of 1 period each. 2 Dynamic PG 2 periods 3a More than 10 periods 3b Once there is full understanding of the rules, actions and payoffs of the experiment we will precede to the first computerized treatment. In the static game each individual is randomly assigned to a specific group of size N=4. 12 Experimental subject receive an endowment of units of experimental currency UEC) of which they can chose to allocate between a private good yielding payoff for the individual only), and a public good, that yields a payoff for every participant in the same group. The first treatment Treatment 1) will consist of a static public goods game in which experimental subjects will be asked to contribute conditional on contributions made by others in their group Fischbacher and Gachter, 2010). That is, subjects will state their conditional contribution for each possible level of the average contribution of others in the group. That is, experimental subjects will be presented a full contribution schedule as in the table below [in the actual experiment we will limit the number of values 12 Participants will never know who is in their same group 19
to 11 as oppose to 21. That is, the possible level of contribution of others in the group ranges from 0 to 10, not from 0 to 20 as in the table below]. There will be only one round or repetition of this treatment. As explained in the following section, Treatment 1 will only be instrumental for the analysis of the experimental results as it is expected that will allow us to categorize subjects into one of the following three categories: always free-riders, conditional cooperators and always cooperators Fischbacher and Gachter, 2010). Next, we will carry on the second treatment, the static unconditional contribution to a public good Treatment 2). Following the approach that is being extensively used in the experimental literature of public goods, and unlike the previous treatment) subjects are asked to state their unconditional contribution to the public good. In this treatment UEC, to allocate either to consumption of a private good, or to a public good. Consistent with most current experimental literature on contributions to public goods, there will be 10 rounds or repetitions) of this treatment. 20
In Treatment 3a we will first introduce individuals to the concept of stock of the public good and we will explain how, unlike the private good, the stock of public good accumulates over time. In order to allow for an easy transition from static to dynamic public good setting, Treatment 3a consists of a 2-period dynamic public good game. In this treatment, there will be only two periods and the stock accumulates from period 1 to period 2. Therefore, in Treatment 3a participants are told that the aggregate contribution to the public good across all participants in the same group in period 1 will be carried over to the next period period 2) with a decay of. Thereby, at the onset of period 2 all participants start off with a stock of public good equal to and then, in period 2, participants have to make the same decision as in the previous period, that is, how much of their endowment to allocate to private consumption and to the public good. This treatment ends after period 2 so there is no further accumulation of the stock of public good beyond that period. In Treatment 3b, we will introduce the concept of dynamic public good game with an unknown ending period. For tractability of the analysis of the results, we will first run 10 rounds of experimental public good with accumulation of stock. Then, after the tenth round we will implement a random stopping rule for the dynamic process. That is, once subjects have played 10 rounds with accumulation of stock of public good, on each subsequent round, there will be a random draw say, the roll of a dice) that will determine whether another period is to be played. If another period is to be played, the stock of the public good that has been accumulated up to that period carries over to the following one. We repeat this process until the random draw calls for the stopping of the dynamic game. 13 Diagrammatic representation of Payoffs In the introductory section of the experiment we will introduce the concept of payoffs and how they are connected to individual and group performance. Instead of using directly the payoff function in equation 1) above, we break it down into diagrammatic representation and, in a way that can be easily understood by individuals with no mathematical background, we show how participant s decisions on provision of public good affects both individual and group payoffs that See Figure 3 below). Thereby, we will present participants with a payoff table that represents the payoff equation in 1) in which each 13 This process of a random stopping rule allows for the possibility of infinite number of periods so that the experiment emulates an infinite time-horizon game. Dal Bo, AER 2010). 21
column reads as follows. The first column denotes the individual s contribution to the public good Your Contribution, or YC) whereas the second column denotes its corresponding payoff only from the individual s allocation to private good Payoff from Own-conumption, POC). In order to calculate the payoff from the accumulated contributions across the same group, individuals have to make a guess of what they think the average contribution by others in their same group will be the guess in the example is that the average contribution by the others in the group equals 5). This guess can be change and updated at any time and so will be the payoffs associated with it. Thus, the third column denotes the payoff by the aggregate contributions Payoff Total Stock, PTS) for each value of own-contributions and the fourth column is the Total Payoff TP, sum of second and third columns). Fig. 3:Diagrammatic representation of payoffs 22
V. Analysis and Expected Findings In order to start off from solid ground, in this experiment we will first attempt to replicate the previous experimental findings on static voluntary provision to public goods Treatment 2). Consistent with the current literature, we expect to find contributions above what is predicted by the Nash equilibrium as well as contributions that will decline over subsequent rounds towards the Nash prediction. Next, we depart from the static case to test experimentally the more interesting dynamic setting described in this document. We will test whether individuals adjust their contributions in response to the accumulation process of the stock of public good. The current experimental literature contrasts the static Nash predictions with the actual behavior in the lab. Unlike the current static experimental literature, in the dynamic counterpart we set parameter values in our theory model and derive openloop and feed-back equilibrium paths or trajectories equations 14) and 19)). In the dynamic section we contrast these dynamic trajectories with the experimental trajectories from treatments 3a and 3b. 14 Furthermore, recent experimental literature show that a large proportion of individuals can be thought of as conditional cooperators. That is, individuals that will contribute to a public good as long as others contribute to the public good Fischbacher and Gachter, 2010). If a group of experimental subjects are 14 That is why we force Treatment 3b to have at least 10 periods, so that to yield a contribution trajectory. 23
conditional cooperators, then, in the dynamic part of the experiment Treatments 3a and 3b) they will be more sensitive to the accumulated level of stock at each period of time and so they may condition their contributions to the observed stock. Treatment 1 seeks to identify these individuals as they will be likely to behave in line with the theoretical predictions contribution trajectories). On the other hand, identifying individuals that are always free-riders will allow us to check whether they present a similar behavior in the dynamic setting of the experiment and whether they can be categorized as myopic when it comes to contributions in a dynamic setting. 15 To conclude, in this paper we test whether experimental subjects adjust their contributions in line with the theoretical prediction for dynamic public goods. In addition, in future research we will test the effects of uncertainty on the provision of public goods probably a simplified version of the model above REFERENCES Andreoni, J. 1995. Cooperation in Public-Goods Experiments - Kindness or Confusion. American Economic Review 85 4):891-904. Barrett, Scott. 2007. Why Cooperate?: The Incentive to Supply Global Public Goods, OUP Catalogue: Oxford University Press. Battaglini, Marco, Salvatore Nunnari, and Thomas R. Palfrey. 2010. Political Institutions and the Dynamics of Public Investment. Princeton University. Dal Bo, P. 2005. Cooperation under the shadow of the future: Experimental evidence from infinitely repeated games. American Economic Review 95 5):1591-1604. Fershtman, C., and S. Nitzan. 1991. Dynamic Voluntary Provision of Public-Goods. European Economic Review 35 5):1057-1067. Fischbacher, U., and S. Gachter. 2010. Social Preferences, Beliefs, and the Dynamics of Free Riding in Public Goods Experiments. American Economic Review 100 1):541-556. Gachter, S., D. Nosenzo, E. Renner, and M. Sefton. 2010. Sequential vs. simultaneous contributions to public goods: Experimental evidence. Journal of Public Economics 94 7-8):515-522. Laury, Susan K., and Charles A. Holt. 2008. Chapter 84 Voluntary Provision of Public Goods: Experimental Results with Interior Nash Equilibria. In Handbook of Experimental Economics Results, edited by R. P. Charles and L. S. Vernon: Elsevier. 15 In the common pool resource experimental literature, subjects present a myopic behavior when players behave as if each period of the dynamic game is a single period of a static game. That is they neglect the fact that current contribution increases the future value of the stock of public good. 24
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