Present Bias in Renewable Resources Management Reduces Agent’s Welfare

: This paper analyses the effects of myopic and present - biased preferences on the welfare of a naive agent when she is engaged in an intertemporal harvesting activity from a stock of renewable resources. The analysis is conducted by taking into account also the nature of present - biased behaviors as phenomena that is derived from a dual system of discounting and of response to short and long - term stimuli. In the task of harvesting from a stock of renewable resources, the present biased preferences of a naive agent create a conflict between the long run benefit of the agent and the short run desire. Thus, this paper demonstrates and argues that in the decision - making, which involves intertemporal choices in renewable resources management, the prevalence of naive behavior, strongly influenced by the emotional - affective system, can lead to a reduction in the overall utility enjoyed by the individual due to the present bias. two different harvesting plans that both arise from the intertemporal preferences of the agent: two mutually excludable harvesting plans — the optimal harvesting path and the biased plan. The paper has shown that the first plan can guarantee the greater welfare for the agent.


Introduction
Intertemporal resources management is frequently subjected to risks of inefficiency and mistakes.
Often people encounter difficulties in defining intertemporal choices and consistently allocating consumption over time. Economic theory generally assumes conventional exponential discounting, where future benefits are discounted at a constant rate. A discount rate that differs from the exponential one generates time-inconsistent plans and myopic behaviors (Strotz 1956).
Unfortunately, people often behave contradictory to the time-consistency assumption. Several studies underline the existence of non-compliant behaviors to the precepts of time consistencyfor a review see Loewenstein and Pralec (1992) and Frederick et al. (2002). Controlled experiments in the laboratory have shown that people exhibit a systematic tendency to discount the near future more than the distant one (Loewenstein and Pralec 1992). This depends on the impulsive behaviors of people in following the short run benefit despite its effects in the long run. Furthermore, intertemporal choices seem to be better represented by hyperbolic discounting rather than by the exponential one (Laibson 1997), implying that people make short-sighted decisions where costs and benefits are involved. These kinds of behaviors are interpreted as a lack of self-control or presentbiased preferences Rabin 1999, Laibson 1997).
In the last years, some studies have started to explore the application of non-constant discount rate in resource management (Settle and Shogren 2004) and the environment (Brekke andJhoansson-Stenman 2008, Karp 2005), discussing issues related to the present-biased preferences in these contexts -in particular, the dichotomy between biased agents and rational ones (Hepburn, Duncan and Papachristodoulou 2010). However, the effect of present bias on agent welfare in the field of resources management has not yet been investigated. For these reasons, this paper conducts an analysis of the effect of present biased preferences in the welfare of the agent, when she is involved in renewable resources harvesting. The analysis is conducted taking into account also the nature of present-biased behaviors as phenomena that are derived by a dual system of discounting with the agent's cognitive foundations.
The investigation proceeds as follows: preliminary a retrospective in the relation between time inconsistency and present biased preferences is presented. In section 3 the origin of present biased behaviors are described taking care to expound the complexity of this phenomena in an interdisciplinary dimension. In section 4 the harvesting model that concerns the exploitation of a stock of renewable resources is presented and the analysis on the effect of the adoption of a non-constant discount rate in this framework is conducted. Finally, the results obtained, showing that the present biased preference of a naive agent in the harvesting activity generate a lower welfare level for the agent, are discussed.

A retrospective on time inconsistency and present bias
Standard economic models usually assume the exponential discounting, such that the agent discounts the future with a constant discount rate. This assumption implies time consistency that means that the future choices defined in the present, by the maximization of the present value, will still be optimal choices in the future. Time consistency is guaranteed when the discount rate is independent from the time. However, theoretical and experimental studies have widely shown a higher discount rate over the short time and a lower discount rate in the distant one (Frederick et al. 2002, Laibson 1997. In presence of time dependence, a violation of the stationary postulate of Koopmans (1960) occurs. This violation generates time inconsistency because an optimal choice at time t may no longer be so when the task is verified at a time that follows t (Strotz 1956). This condition could generate preference reversal, which implies that the preference ordering defined at a given time can be reversed in the future.
The preference reversal is coherent with the observed behavior of agents that show diminishing impatience such that the future is discounted with a declining discount rate (Hepburn, Duncan and Papachristodoulou 2010). Evidence of this kind of behavior is widely reported and several observations clarify that time affects choices (Thaler 1981, Della Vigna 2009, Frederick, Loewenstein and O'Donoghue 2002. Impulsivity and misevaluations of immediate rewards are included between the behavioral and cognitive origins of the preference reversal (Ainslie 1992, Shefrin and Thaler 1988, Benabou e Pycia 2002. Therefore, preference reversal and time inconsistency generate a conflict between long run preferences and immediate choices, which consequently creates a conflict between the initial intentions of the agent and the realized choices. Preference reversal, impulsive choices, and the impatience to obtain immediate rewards can be explicated by the presence of a hyperbolic discount (Ainslie 2005). 1 It is also usual to define as "present bias" the baseline behavior that is derived from hyperbolic or quasi-hyperbolic discounting: greater impatience in the short run with a declining discount rate for a more distant future.
1 Or quasi-hyperbolic discount. 5 Present biased preferences imply that immediate benefits drive the choices despite the long run interest; and thus, they can induce the agent to myopic decisions. Present biased preferences are widely observed in several frameworks: low saving rate (Ashraf, Karlan and Yin 2006, Harris e Laibon 2001, Laibson 1997, Laibson, Ripetto and Tobacman 1998; health contexts (Pol and Cairns 2002); drug, smoking or buying addictions (Gruber and Koszegi 2001, Thaler and Shefrin 1981, Wertenbroch 1998, Frederick, Loewenstein and O'Donoghue 2002; and procrastinating behaviors Rabin 1999, Bernabou andTirole 2003). Furthermore, Cropper and Laibson (1998) have analyzed the non-Pareto efficiency in the context of project evaluation when agents have time inconsistent plans.
There are some contributions to the literature that show how the non-constant discount interacts with resource management and climate change policy. Settle and Shogren (2004) explored the application of the hyperbolic discount rather than the usual constant one, in the context of natural resource management. Karp (2005) analyzed the role of the hyperbolic discount in a model of global warming, and Brekke and Johansson-Stenman (2008) analyzed the contribution of behavioral economics in the field of climate change. The present bias has consequences in the intergenerational framework. In fact, Winkler (2006) showed that in the presence of hyperbolic discounting, there is a potential conflict between economic efficiency and intergenerational equity in public good investments. Furthermore, in the framework of intergenerational renewable resource harvesting, the present bias generates negative externalities on the welfare of future generations, reducing the resource stock even if the current generation has other-regarding preferences. This happens when the naive agent's behavior has no commitment (Persichina 2019 b). Moreover, the present bias also affects the agent's decisions in the exploitation of resources in terms of disruption of cooperative behaviors. Indeed, the present-biased preferences can trigger a strategy that directs the community to excessively increase the harvesting level even in the presence of cooperative intentions because the behavior of naive agents can activate a dynamic of cascading defections from the cooperative strategy (Persichina 2019 a). Besides, under the hyperbolic discount, the undesired collapse of the natural resources can occur when the agent is naive (Hepburn, Duncan and Papachristodoulou 2010).

Roots of present bias and the dual system of discounting
An evolutionary origin seems involved in the existence of the present bias. Some authors assign the 6 existence of myopic behaviors and present biased preferences to evolutionary pressures (Godwy, Barkley Rosser jr. and Roy 2013); for example, Dasgupta and Maskin (2005) argue that uncertainty and waiting costs have contributed to the emerging of present-biased behaviors. Furthermore, there are evidences that the evolutionary components of these behaviors are widely rooted in human and non-human animals (Ainslie 1974, Green andMyerson 1996). 2 For example, the ability to ordinate the numbers in a correct cardinal order is not an innate ability of humans, this fact confirms the ancestral roots of present bias (Godwy, Barkley Rosser jr. and Roy 2013). In fact, studies conducted on indigenous populations of Amazonia show that these populations do not have an exact numeric ordering, although they have a non-verbal numerical sense. Therefore, when they have to define a spatial ordering for increasing quantities, the space interval between the numbers becomes smaller and smaller (Pica, et al. 2004). Conversely, American adults define a spatial ordering that shows an equidistant space between the numbers; the logarithmic spatial ordering of the Amazonian populations is similar to the ordering of kindergarten pupils who only in the second year of school arrive at spacing the numbers equidistantly (Stiegler and Booth 2004).
Hence, as underlined by Godwy et al. (2013), these results effectively suggest that the non-constant discount has deep origins in the human behavior. Furthermore, some research in the field of cognitive neuroscience support a non-constant discount rate and find two different systems designed to process discounting: one for immediate rewards and another for the delayed ones. In particular, two distinct brain areas related to the definition of intertemporal choices are identified (S. McClure, et al. 2004). The first area, namely the limbic and paralimbic, is an area of the brain that is heavily innervated by the dopaminergic system and is connected to short-term rewards (Breiter and Rosen 1999, Knutson, et al. 2001, McClure, Berns and Montague 2003, while the other area belongs to the frontoparietal region, an area that supports higher cognitive functions (Loewenstein, Rick and Cohen 2008). Moreover, in the field of cognitive neuroscience, some experiments show the activation of the limbic circuit just before choices that provide an immediate reward (S. McClure, et al. 2004); similar conclusions have been reached by Hariri et al. (2006) and McClure et al. (2007).
In this discussion, it is worth mentioning that the limbic system is the seat of reaction processes that are impulsive and emotional (Hariri, Bookheimer andMazziotta 2000, Pattij andVanderschuren 2008). The limbic system -which is the most ancient part of the human brain -also includes the amygdala (Isaacson 1974) whose functions are significantly correlated with emotional activities 7 (Cardinala, et al. 2002, Hariri, Tessitore, et al. 2002. Conversely, in the presence of choices that reflect deeper consideration for future gains areas afferent to the neocortex are relevantly activated, whereas there is no prevalent activation of the limbic system (S. McClure, et al. 2004). The neocortex, exclusive to mammals, is the most recently formed brain area from an evolutionary perspective. The neocortex's areas are markedly developed in humans (Rachlin 1989) and play a role in appropriate, deliberative cognitive activities (Miller andCohen 2001, Smith andJonides 1999). It is, therefore, possible to assume that consumer choices in an intertemporal context define a dualism between the limbic system -whose responses are characterized by rapid impulsivity and emotion -with a prevalent activation of this system in response to short-term choices, and the deliberative-cognitive system, afferent to areas of the neocortex, which is slower and more balanced.
The joint involvement of the two systems in the decision-making process is further supported by Bechara (2005), Bechara et al. (1999), Damasio (1994), andLeDoux (1996). A distinction, between the two systems of response to short and long-term stimuli, can be defined: the information about immediate rewards is subjected to the substantial involvement of the impulsive system, while a more appropriate reflective system refers to decisions about long-run rewards. Therefore, it is congruous to assert that the intertemporal decision-making process and the time inconsistency that arises out of this process is driven by the interaction of these two coexistent systems, coherently with the complexity of human nature (Shefrin and Thaler 1988, Loewenstein 1996, Metcalfe and Mischel 1999.
The wide variety of fields and contexts in which the present bias emerges, the evolutionary hypothesis, the psychological foundations, the systematic manifestations of the phenomena of procrastination and over-consumption, as well as the presence of impatience, temptation, and lack of self-control, clearly outline a profile of an economic behavior that resides outside the barriers of the pure rational behavior which assumes time consistency. Hence, the present bias is a specific peculiarity of decisional heuristics about intertemporal choices, in particular in contexts where the long run plans can be object of revision over the short run and where the long run outcomes depend on a continuum of instantaneous or short run choices. Frequently, resource dilemmas have the characteristics of the context just described. In fact, resource dilemmas describe a situation in which long run and short run choices can come into conflict, exposing the agent to the risks related to the present bias; particularly, in the context of the exploitation of renewable resources.

Decrease in agent's welfare due to the present bias
In this section the analysis of the effect of the present bias on the welfare on a naive agent is conducted. The harvesting model adopted in the analysis concerns the exploitation of a stock of renewable resources, R(t). The dynamic of the growth of resources is given by the following equation: where ( , ( )) ≥ 0, with the constant g > 0, 3 is the growth rate, and h(t) is the harvested amount at time t such that the stock of resources is reduced over time, dR/dt < 0, when the exploitation rate exceeds the natural growth rate, ℎ( )/ ( ) > ( , ( )). 4 The interval from 0 to T is the lifetime of the agent. In this model, the resources are materials; consequently, a negative stock of resources is impossible: where R0 is the initial stock at time 0. The strictly positive initial stock and the growth rate are known by the agent, the amount harvested is not restorable in the stock of resources, such that: Moreover, the agent is subjected to a capacity constraint and a resources constraint.
The capacity constraint implies that in each period, the agent cannot harvest an amount of resources greater than hmax, a value that is strictly positive and finite, such that, considering the non-restorable condition: The resource constraint implies that the agent cannot harvest at time t more than the amount of 3 R(0) > 0 implies f(g, R(0)) > 0, and R(t) = 0 implies f(g, R(t)) = 0. 4 When ∂f(g,R(t))/∂R(t) = 0, the growth rate is a constant exponential one. 9 resources available: There are no exchange markets in the model, so the agent's welfare depends only on the amount harvested and enjoyed in each time. The utility function of the agent is defined in the usual manner: where (ℎ( )) is monotonic and strictly concave on ℎ( ) in the interval [0, hmax]: The discount factor ( ) represents the degree of impatience of the agent, 5 such that: Continuity for the harvesting amount on the interval [0, hmax] is assumed. Finally, the system defined assumes that it is impossible for the agent to avoid the total exploitation of the resources before the end of her lifetime, if she continuously harvests the amount hmax in all the periods. So defining with = {ℎ (0), . . . , ℎ ( ), . . . , ℎ ( )} a generic harvesting profile inside the set of all the feasible harvesting plans, ∈ { }, given 0 , , ( , ( )), this last assumption can be expressed as: and 5 The assumptions exclude the case of pleasure in procrastination, ′ ( ) > 0, and neutrality in the harvesting time, which implies ′ ( ) = 0 with ( ) = 1 ∀ ∈ (0, ).
The assumptions (9) and (10) imply that in at least one period ℎ( ) < ℎ . Considering that the agent tends to distribute her consumption over time, avoiding to finish the resources before time T, it is assumed that the agent's intertemporal preferences are given such that: This means that at time 0, the agent formulates the harvesting plan, avoiding to harvest amounts equal to hmax in all the periods until time if this implies the depletion of the resources before the time T. This is consistent with the dependency of welfare on the harvested amount at each time, generating utility only in the period in which the amount is harvested.
The optimal harvesting plan evaluated in absence of present bias guarantees the time consistency of the future decisions and corresponds to the long run harvesting plan evaluated at time 0. In fact, in the standard rational model, the agent can accurately define her exact optimal path of harvesting, keeping her bond with the initial optimal plan formulated at the beginning, and she will do this throughout her life. As discussed in the previous sections, this implies that the discount factor must be expressed in an exponential manner that guarantees time consistency; but, present bias makes an exponential discount factor impossible.
When the agent's preferences incorporate the properties of the non-constant discount factor just enounced, the process of maximization can lead the agent to a harvesting plan that differs from the plan defined at time zero. In this case, the harvesting plan of the agent is defined with the amounts that derive time after time by the instantaneous maximization of the utility function under the same condition of but with a non-constant discount rate. The resulting plan is labelled as a biased harvesting plan, , and defined as: A discount factor like that one expressed in (13)  with < 0, then: So, the amount effectively harvested at time , ℎ( ), is greater than the amount predicted for the same period when the optimal harvesting plan was evaluated at time t, t < .
The implications for the harvesting plan in this model can be expressed in the following proposition: . If under the assumption of present bias defined in (13) and given the conditions (9) and (11), the agent develops an expected harvesting amount formulated at time t, with t < tb, 0 < ℎ ( ) < ℎ , then in the time interval [0,T], there exists at least one period, tb, such that: Thus, the present bias induces the agent to harvest an amount greater than the optimal one evaluated without the bias, leading the agent outside of the optimal harvesting path. So, by inducing the reevaluation of the amount harvested at time , the present bias generates a differentiation between the two possible harvesting plans of the agent. Now, the question is: does a different harvesting profile determined by the present bias imply a reduction of the agent's welfare, and if so, does it happen because of the present-bias?
The agent faces two different harvesting plans that respond to two different systems of discounting: -1 -The plan that responds to the short run, expressed by Hbias, where the amount harvested at each period is affected by the present bias, re-evaluating the harvesting plan time after time; and -2 -the long run plan, Hopt, where the plan of harvesting formulated at time zero excludes the effect of the present bias and is confirmed each time.
To compare the two plans in terms of the agent's welfare, referring to the concept of total utility of the agent is necessary. In particular, it is useful to separate the concept of decision utility from hedonistic pleasure derived by the instant utility enjoyed by the agent (Kahneman and Sugden 2005). In this sense, the concept of utility is defined following utilitarian philosophers such as Bentham, where utility is logically separated from what choices are made (Read 2007). The instant utility is the hedonic value of a moment of experience utility (Kahneman and Thaler 2006), such that the total utility is derived by a temporal profile of instant utilities. Following this approach, a time-neutral weighting of the outcomes is considered (Kahneman, Wakker and Sarin 1997). Hence, the total utility of the periods from zero to T is given by the sum of the instant utilities of all periods, allowing the total utility to be expressed as and given by: such that the agent's welfare is evaluated by the comparison of the different profiles of the total instant utilities.
As said before, this paper aims to understand if an overharvesting generated by the present bias, as showed in the preposition 1, can generate a reduction in the total enjoyed instant utility of the agent, and that this possible welfare's reduction is determined by the discounting peculiarity of the present bias. In accord with this aim, preliminary the investigation studies the possibility that the adoption of the biased harvesting plan can imply a lower total enjoyed utility than the optimal harvesting plan. This comparison is done using a 3 periods model (present, near future, distant future) and it is presented in the appendix. This comparison of the level of total utility between the optimal long run plan and the biased short run shows that the agent's utility is greater in the optimal harvesting plan.
In fact, the utility derived by the increase in harvesting at time , determined by the present bias, is smaller than the decreased utility given by the difference between the total amount that will be harvested following the optimal harvesting plan and the amount that will be effectively harvested under the present-bias hypothesis.
We can so assume that in front to the two alternative harvesting plans, it is possible that the increased utility derived by a higher amount in the present is less than the decreased utility derived from the amount enjoyed in the future: The main question, now is if the present bias is the element that generates the reduction of the agent's welfare. As it will be showed soon, it is possible to assert that the peculiarity of the present biased time discounting generates the reduction of the agent's welfare in presence of a lower total enjoyed utility determined by a biased harvesting profile. To show this assertion, it is helpful to use the utility function with present bias preferences that offers the essential peculiarity of the no constant discounting. The present biased preferences are expressed in the following intertemporal utility function: where β, not greater than 1, represents the present bias. 8 When β = 1 the discounting guarantees time consistency (absence of present bias) with an exponential discount factor, consequently the optimal harvesting plan is followed. When β is smaller than 1, (13) holds.
Proceeding to show the involvement of present bias in the welfare reduction: with {H} is defined the set of all possible harvesting profiles, and a generic profile is defined as Hi ={hi(0),…,
Consequently, at time 0: and this implies: 8 This form of present biased preferences was originally used by Phelps and Pollak (On Second-best National Saving and Game-equilibrium Growth 1968) in the intergenerational context. 9 The system admits just one optimal solution and at time zero Hopt is preferred to all other feasible plans. 10 Strictly monotonicity in the utility function is assumed.
Because the agent faces an intertemporal decision-making process in which at each time she defines her harvesting amount, at time , she will reevaluate her harvesting profile, choosing an amount ℎ ( ) > ℎ ( ) because at this time ≻ . This implies that at time , Consequently, Because (22) implies: then (24) can be true only if < 1. This shows that the strategy , which leads to a total utility enjoyed that is lower than , can be implemented only if a non-exponential time discount is adopted.
Hence, in conclusion, the consequence of the present-bias on the agent's welfare when she faces the task of intertemporal harvesting of renewable resources can then be summarized in the following proposition.
Proposition 2: Given the utility function of the agent expressed in (19), with ≤ 1, two possible harvesting plans can be derived by the decision making process of the agent: the first one, = {ℎ (0), . . . , ℎ ( ), . . . , ℎ ( )} , in which = 1 , and the second one, = {ℎ (0), . . . , ℎ ( ), . . . , ℎ ( )}, in which < 1. The adoption of the plan , for effect of the present bias, can lead the agent to obtain a total utility lower than in the plan evaluated at time 0, , such that, Hence, between the short run biased harvesting plan and the long run optimal one, it is the second that can ensure to generate higher welfare for the agent. The short run biased harvesting plan, , can be implemented if and only if the discount factor applied by the agent incorporates the peculiarities of the present bias.

Conclusion and final remarks
This paper has defined a discount system that is expressed by the coexistence of two discount forms: an emotional, rapid, and impulsive system for responding to short-term stimuli and a reflective system suitable for the long term. This system of intertemporal discounting is consistent with -and is a part of -the complexity of the decision-making process that characterizes human beings. This complex process is based on the existence of a highly integrated decision-making system composed of two simultaneous main circuits: the affective-emotional, where the emotional component is predominant in the dynamics of decision-making and the cognitive-deliberative, which is delegated to greater mediation in defining what actions to take given the input received. In this system, a conflict between the long run and the short run in the decision output can occur. The reason of the involvement of the present bias in this conflict has been presented and discussed. The discount system in which two potential discount patterns coexist -the long run with the constant discount rate and the short run with the non-constant discount -generates two different harvesting plans that both arise from the intertemporal preferences of the agent: two mutually excludable harvesting plans -the optimal harvesting path and the biased plan. The paper has shown that the first plan can guarantee the greater welfare for the agent.
Before this investigation, at the best of the knowledge of who write, the relationship between the present-bias and the agent's welfare has not been adequately explored in the literature. Studies on specific applications involving the management of renewable resource stocks, when addressing the basic question of behavior and decisions related to harvesting by naive agents, have focused on the effects in terms of resource management efficiency and resource conservation or depletion, implicitly assuming that the agent's choices will always maximize her utility. This implicit assumption, which ignores the impact of the present bias on welfare, arises from not considering the naive biased/not-biased agent dichotomy as an element of an individual agent's system of preferences. In fact, addressing issues on the lifetime welfare of individuals involved in managing renewable resources inevitably involves a contraposition that can be defined as a conflict of choices between those that are biased by current emotions and the rational unbiased. The second kind of choice is defined in the absence of present bias, that it is when the system of intertemporal discounting is oriented toward overall well-being. Conversely, present-biased choices lead individuals to a calculation that is predominantly oriented toward the short term and disregards their long run preferences. This conflict is part of the decision process of the agent with the dichotomy biased/not-biased choices in the process of realization of the agent's preferences.
This paper shows that in the decision-making that involves intertemporal choices in renewable resources management, the prevalence of naive behavior, strongly influenced by the emotionalaffective system, can lead to a reduction on the overall welfare of the agent due to the present bias.
The comparison of the two harvesting plans has shown that the utility derived by the increase in the instantaneous utility determined in the present by the present bias, could not compensate the future decrease in utility determined by the adoption of the biased harvesting plan instead of the optimal one. These conclusions pose a question about the effective intertemporal maximization of the wellbeing of the naive agent when she adopts a present biased harvesting behavior. It should be noted that a harvesting plan derived from present bias could be not sufficient to allow a definition of effective maximization of the individual's overall well-being when she is in a condition in which she cannot cope with the excessive impulsive component in the immediate present.
These results underline that a naive individual involved in the intertemporal management of renewable resources could not adopt an harvesting plan that properly maximize her overall wellbeing according to her long run preferences independently from her ability or possibility to commit her behaviors, or to balance the immediate impulsivity with the long run welfare. Hence, the reduced welfare derived from the implementation of a strategy dominated by the impulsivity inherent in present bias, highlights problems that are relevant to maintaining a given level of resources but also shows the need to identify tools that can ensure effective implementation of strategies that are not so strongly dominated by the present bias during the management of renewable resources. In the context in which the agent faces the risk of making decisions on the spur of the present bias, suitable nudges or instruments could be required to offer to the agent the possibility to commit her harvesting plan to her long run preferences.
For the interval [1,T] the amount defined in at time 0 is an expected amount, so ℎ ( ) can be recall as [h( )] 0 . Where the subscript indicated that it is the expectation evaluated at time 0 about the amount that will be harvested at time tb.
We know from (9) and (10)  It is assumed that tb is the first period in which:  (11), this guarantees also that the condition 1 holds -such that: ∄ < : 0 < [h(t)] 0 < ℎ .
The conditions 1, 2, 3 and 4 make that the postulate 1 holds, consequently the amount effectively