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


Financial Analysis and Hedging Strategy Analysis of Associated British Foods

. A multinational food company may have to face various risks from its supply chain or other aspects. Hedging strategies by using financial derivatives such as futures contracts are useful for such companies to reduce their risks. This paper aims to make proposals for Associated British Foods (ABF) to reduce its risks by using the future contracts and options of its raw materials. Based on the analysis of the background and financial situation of ABF, it is suggested that future contracts of wheat and other raw materials and their options can be hedged. Long call, short call, bull spread, and bear spread are analyzed respectively based on the latest market data. The paper only takes one example of CHICAGO SRW WHEAT (ZWU2), but it is suggested that a number of related commodities can be hedged in similar methods.

Introduction
With the breakout of the Covid-19 pandemic and the Ukraine war, prices in the food industry are fluctuating dramatically. Fluctuations among the commodities and raw materials would cause a great degree of risk for a company since they affect the revenue and costs of the company significantly. Financial derivatives, as one of the most rapidly developing areas of empirical finance, are crucial to risk management for a company [1]. Future contracts and options, as the main derivatives, are useful to hedge the risks during fluctuations. In addition, it is studied that the bigger size of the company is, the more turnover of hedging contracts the company would like to purchase [2]. Since the end of February 2022, Associated British Foods, one of the biggest multinational companies around the world, has seen a huge decrease in its stock price until today. It is assumed that the outbreak of the Ukraine war and the long-lasting Covid issue has affected badly the company as Mbah and Wasum have pointed out that there might be a worldwide food crisis because of the significant status in food exporting of Russia and Ukraine and it is difficult to recover from the damage of the food supply chain from the Covid pandemic [3][4][5].
In this article, factors of the bad performance of ABF are analyzed from three perspectives including the background information of the company, its business area, and its financial situation. Afterward the risks of the company are analyzed, and they will be reduced by several hedging strategies such as futures contracts and options. Finally, the hedging strategies are suggested in this article and the results will be displayed respectively.
The remainder of the article is structured as follows. In Section 2, the paper shows the company's profile of Associated British Foods. Section 3 is the hedging strategies analysis. In the last, the conclusion is summarized.

Background information
Associated British Foods (AFB) is a huge public limited multinational food processing and retailing company, currently owned by Willington Investments, which has 54.5% of its share value. It was initially founded in 1935 and adapted in 1960 as its current name. It is listed on the London Stock Exchange and is a constituent of the FTSE 100 Index.
Associated British Foods is headquartered in London. As mentioned in its 2021 annual report, the total asset of ABF has reached 16,895 million pounds. There are 53 countries that ABF is operating in, across 5 continents with 123,000 employees. 9 out of 10 UK households are using its grocery brands and it has the largest animal feed business in the UK.

Business area
Associated British Foods has five divisions, including grocery division, sugar division (AB sugar), agriculture division (AB Agri), ingredients division, and retailing division, with large amounts of revenue 3,593m, 1,650m, 1,537m, 1,508m, and 5,593m respectively in 2020. Each division had an optimistic performance in 2021, nearly every sector has a better adjusted operating profit margin than in 2020. Grocery, as its biggest division, has several brands including Twinings Ovaltine, and Acetum in more than 100 countries. AB sugar and AB Agri are the largest sugar producer and cutting-edge food producers for people and animals, while their ingredient businesses are leaders in yeast and bakery ingredients and supply specialty ingredients to the food, nutrition, and feed industries. Primark, as an international retailer with 398 stores in 14 countries, is the flagship of the retailing sector of AFB, which offers a wide selection of products from essentials to the latest trends. All these successful divisions have bonded AFB and provided it with a mature, wide range, healthy business chain.

financial situation
Although the financial performance in the 2021 annual report of Associated British Foods is inspiring and it is announced, in its interim results announcement ended on 5 March 2022 of the past 24 weeks, that the sales and operating profit of ABF have achieved the situation as pre-covid levels, from the end of February 2022 the stock price of ABF has dramatically decreased until today, from 1959.50 GPB to nearly 1526 GPB.

Fig1. stock price changes of ABF from Feburary to July
As reported by Simply Wall St (2022), in the past 12 months, the stock price has shrunk by 34% and 11% down in the past quarter. Compared with the loss of the broader market (5.2%), Associated British Foods has a worse performance (32%) even with dividends included. The shareholder returns of ABF are moving downwards, by -5.3% in the past quarter and -27.3% in the past year, as the entire UK food market and UK food industry are also dropping, no matter in the past quarter or in the past 12 months. The reasons for the recession in the stock price of ABF and the entire industry are complicated and diversified. The first factor is the COVID-19 pandemic, which affects the cost of transportation of goods and the demand for the products of ABF. It is reported that at the beginning of the pandemic, air freight was significantly damaged by the decrease in traveling passengers. Global air cargo capacity was 26% lower than normal and nearly 80% lower in Europe and Latin America. Container and truck transport are also negatively impacted and have fallen somewhere in-between. The pandemic has influenced consumer demand and led to a dramatic downward shift in sales of food away from home, which are using ingredients from food suppliers such as ABF. In some areas the demand even reached about zero due to the lockdown in the past two years, causing huge damage to the entire food supply market [4]. On the other hand, the Ukraine war has caused a drastic shift in the global supply of wheat, corn, sunflower oil, and fertilizer, since exports from Russia and Ukraine account for 28%, 15%, 66%, and 16% of the global exports from 2016 to 2020 and have shown an increasing trend before the war. The shortage in exports has led to a significant rise in the food price index around the world [5]. As a multinational company that has business around the world, ABF has to bear the higher cost of producing food.

potential risks of the company
The first risk is commodities price risk, this may come from the Ukraine war, and as exports of wheat, corn, and other raw materials that ABF utilizes to produce its products are decreasing, the prices of such commodities may fluctuate considerably in months [6][7][8]. Since the prices of the raw materials, such as wheat and corn, have already risen sharply, whether their prices will return to a pre-war situation or keep soaring is still unknown [9].
In addition, foreign currency risk may affect the company's net profit. As a group that conducts businesses worldwide, the transactions that it makes are in all kinds of currencies [10]. Consequently, the group is exposed in the risks of transaction costs due to the movement in exchange rates of different foreign currencies.

hedging strategies to reduce risks
To start with, commodities price risk can be reduced by trading futures contracts. The simplest way of hedging risk would be long call options and long put options. Taking CHICAGO SRW WHEAT (ZWU2) expiring in December 2022 as an example, there are several option contracts that ABF can utilize.
Firstly, if ABF is trading long call options, for example, at a strike price of $850/ contract, it will have the right to buy 10 contracts. If the prices of wheat are still rising (current price is $851.6/ contract) and higher than $850/ contract, the company will exercise it, and the profit it can earn by the option is (future price-$850-option premium) per contract. But if the prices drop or remain under $850/ contract, the company will not exercise it, its loss by this option will be the option premium.
Oppositely, if ABF is trading short-call options, for example, at a strike price of $860/ contract, it will have the right to sell 20 contracts. If the price is lower than $860/ contract, the counterparty will not exercise the option and the profit for selling the option would be equal to the option premium. While the prices keep rising and higher than $860/ contract, the loss for selling the option would be (future price-$860+option premium) per contract.
There are both benefits and flaws for each strategy, the profit of the long call option could be unlimited while it is difficult to have a positive profit since the option premium is relatively high. It would be easier for the short-call selling to earn a profit if the price is not increasing significantly, but the seller is bearing a risk of unlimited loss. Similar ideas would apply for long put and short put options. As a result, the company can combine long call and short call options to create a bull spread or a bear spread to reduce its risk.
A bull spread would be helpful when the prices are expected to rise and even if not, the loss of the company would not be huge. It consists of a long call option with a lower strike price and a short call option with a higher strike price. By using the examples mentioned above, the new model would use a long call option at a strike price of $850/ contract and a short call option at a strike price of $860/ contract. It shows that if the future price is under $850/ contract, the loss of the company would be (long call premium-short call premium)/ contract, which is lower than the loss if simply using a long call or short call. When the future price is between $850 and $860/ contract, the profit would be (future price-$850-long call premium+ short call premium), which is also higher than a single long call or short call, and the profit can be more easily to be positive. Although the profit can not be unlimited positive, the risk of taking a great loss is hedged by this strategy.  Similarly, by using a long call with a higher strike price (K2) and a short call with a lower strike price (K1), the company can create a bear spread. The mechanism of the two spread models is basically the same with the only difference that a bull spread is expecting a higher future price and is hedging the risk of a lower future price while a bear spread is expecting the future prices in an opposite way.
As for ABF, the two spreads can effectively control the risks from the fluctuations of the future prices of its raw materials. The commodity contracts that ABF can make may not only be restricted to wheat, soybeans, rice or oats are also appropriate commodities that the firm can hedge.
To hedge the risk of foreign currencies, the idea would be the same as commodity futures contracts. By using different currency futures contracts and the two spreads mentioned, ABF can also hedge the dramatic changes in exchange rates of different currencies.

Conclusion
The global food industry has reached a low ebb as there is a significant reduction in exports of wheat, corn, sunflowers, and fertilizer from Ukraine and Russia while the industry is still recovering from the damage caused by the outbreak of Covid-19. Associated British Foods, a multinational food company, has suffered from the two disasters and its stock price has decreased by 34% in the last quarter. The shareholder returns have also declined by 5.3% in the last quarter, as the whole industry has been in a similar situation. By analyzing the background information and the financial situation of ABF, this article has found several factors that caused the bad performance of the company and the risks that the company may be facing.
In this article, commodity risk is mainly discussed and the potential solutions are based on this type of risk. Hedging strategies by using financial derivatives such as futures contracts and options are analyzed to reduce the risks of the company where CHICAGO SRW WHEAT (ZWU2) is taken as an example for a more straightforward version of the effect of hedging. The hedging strategies suggested in the article consist of the long call, short call options, and bull spread and bear spread.
Although specific data have been analyzed in this article, it still has some deficiencies. The first flaw is that the detailed performance of ABF is not clear, including the changes in its costs and revenue in the last quarter because the latest report of ABF has not come out. If enough data has been released then it is easier to find the accurate reason for the bad performance thus a better solution for reducing the risks may be come up with. In addition, as the market of futures and options is fluctuating from time to time, the real result of hedging may be different from the assumption in this article. However, the idea and basic directions of the hedging strategies still hold.


Does the shape of economic recovery matter? An alternative unit root test with new smooth transition model

The subject of economic recovery after the Coronavirus pandemic has received much attention in the media and by academics in recent years. Pandemic experience creates a new transition between the pre-pandemic era trend and the post-pandemic era trend related to the major economic indicators’ time series path. This paper offers a new smooth transition model and a unit root testing procedure to test null of non-stationary against the alternatives of stationary that allow for a pit shape smooth transition from the pre-pandemic trend to post-pandemic trend. The properties of the test statistics are investigated with several simulation studies. Also, the new model and unit root testing procedure are applied to industrial production index, consumer price index and the unemployment rates of Global 8 countries and results state the usefulness of these new tests.

for the new unit root test statistics. An empirical study, which investigates the Coronavirus' impact on time series trends of selected Global 8 (G8) countries' several macroeconomic indicators, is reported in Chapter 4. Finally, the fifth chapter summarizes the study.

A new approach for smooth transition model and unit root tests
The LNV method introduces three models that allow one to estimate two linear trends around one specific time point of structural change. This structural change is represented with a logistic smooth transition function and the purpose of this function is to provide a proper estimation of the "S" shaped transition between two linear trends. On the other hand, the LNV approach also represents a very useful tool to develop various type of non-linear unit root testing processes. The smooth transition is a difficult concept to perceive and is confused by many researchers because there are many studies on it. Nonlinear dynamics in a time series can be considered in two frameworks. The first of these is the structural change (break) dynamic, which determines the nonlinear trend over time. The other is the regime change dynamic, which considers the change in the trend path according to the past values of the time series of interest. Considering this basic distinction in nonlinear time series, unit root tests on the concept of smooth transition can be categorised as in Table 1.
The LNV approach offers the following three exponential smooth transition regression models, which could be also considered as alternatives against the null of difference stationary, are employed: where ε t is a zero mean stationary error, t is a trend variable, and S t (γ, t b ) is the time varying exponential function that could be written as: In equation (4), the t b parameter indicates midpoint time of smooth transition and the γ parameter controls the speed of transition. The process starts with Equations (1)-(4) defining a smooth structural change in the trend of a time series. Moreover, it is important to remember that smooth transition is controlled by the variable t and the parameter of t b in (4), not by the lags of the time series y t . This is why it is a time varying structural change process, and it could not be categorised as a regime switching process like Smooth Transition Autoregressive Model (STAR), Logistic STAR (LSTAR) and Exponential STAR (ESTAR) models. After the LNV's path breaking introduction of the time varying smooth structural change modelling approach, Sollis et al. (1999) offer more advanced time driven exponential function instead of (4) and could be shown as: The roles of γ and t b are the same as with (4). The key parameter of (5) is θ. θ controls the asymmetry of the smooth transition and when it is equal to 1 (θ = 1) G t function turns into a symmetric logistic function as introduced by the LNV approach.
The logistic functions of (4) and (5) are proposed to fit the "S" shape nonlinear trend of a time series. Therefore, these functions are not capable of estimating the impact of the coronavirus pandemic on economic time series. As mentioned before, the pandemic mostly created "V" or "U" shaped nonlinear trends in economic indicators because of the recovery measures that economic authorities applied. It is clear that some modifications are needed for them to fit within the LNV models to the pandemic related nonlinear trend of time series. Firstly, the models of (1), (2) and (3) are expanded with a binary variable that could be defined as: where T is the size of the sample. The exponential smooth transition regression models could be written as:  Kapetanios et al. (2003) Ö zcan and Yurdakul (2022) Kruse (2011) Hepsag (2021) a These studies considers both structural change and regime switching dynamics together.
where ε t is a zero mean stationary error and t is a trend variable. In the same perspective, the unit root testing procedure is introduced with the three exponential smooth transition regression models above, which could be also considered as alternatives against the null of difference stationary. The second modification is the new function. P t (γ, t b ) is the time varying exponential function that could be written as: where γ controls width of pit shape or in other words, the speed of reaching the bottom point of the transition and should be greater than zero (γ > 0). t b indicates the mid-point of the transition (0 < t b < 1), α 2 and α 1 determines asymmetry. If α 1 + α 2 > α 1 , this means that t b is greater than the mean before t b . Conversely, when α 1 + α 2 < α 1 , this means that t b is smaller than the mean before t b . Additionally, α 2 = 0 indicates symmetry. Before defining unit root test statistics that consider stationary around a special form of smooth transition between two linear trends, the null hypothesis of non-stationary could be stated as follows: where, as in common practice, ω t is a zero mean and unit variance stationary process. Moreover, the LNV method proposes the usage of proper test statistics to test null of non-stationary in a two-step procedure. Following the same process, firstly, the smooth transition models (8), (9) and (10) are estimated with the nonlinear least square (NLS) estimation method. By doing this, the NLS residuals can be estimated. In this study, sequential quadratic programming algorithm is used to get NLS residuals from the smooth transition models.
Secondly, test statistics are estimated for residuals (ε t ) from (13), (14) and (15) with a Dickey and Fuller (1981) (DF) type autoregressive model where φ t is stationary error with the zero mean for an optimal lag length of k. Three test statistics p α , p α(β) and p αβ , which are connected with models (13), (14), and (15) respectively, could be calculated from the DF t ratio associated with ρ in (16):

Simulation studies
This section contains Monte-Carlo simulation studies that are performed to obtain critical values, the finite sample size, and empirical power values of the test statistics p α , p α(β) , and p αβ under different data generation processes (DGP).

Finite sample size
To observe specific size distortions under various conditions, a Monte-Carlo experiment is designed with the following DGP: Δy t = φΔy t− 1 + η t y 0 = 0 and η t ∼ N(0, 1) φ varies among {− 0.5, 0, 0.5} and autoregressive lag k varies among {0, 1, 4}. Simulation results are reported in Table 3. It could be seen that, for φ = 0 choosing k = 0 causes serious size distortions. Also, when k = 0, φ < 0 leads large empirical sizes and φ > 0 creates small empirical sizes. Lastly, increasing autoregressive lag produces empirical sizes that are below the nominal level. These findings are consistent with the reported results found in Leybourne et al. (1998).

Empirical power
Another important simulation study for the test statistics is needed to be done to investigate power values. For simplicity, the powers of p α and DF τ t are calculated. In order to obtain the stationary series under various conditions, a new DGP is offered as: In DGP (19), asymmetry parameter α 2 varies among 0 (symmetry case), 5 and 10 (asymmetry case), the mid-point of transition t b is set 0.5 and 0.8 and the speed of transition γ is chosen as 0.01 and 0.1. The computed power values are reported in Table 4. The findings indicate that over-fitting lag leads to significant power declines and, as expected, p α statistics has more power than τ t with increasing asymmetry (α 2 ). However, changing of the mid-point of transition (t b ) and the speed of transition (γ) do not cause noticeable power differentiations.

Empirical study
As discussed in Islam et al. (2020) and Al-Thaqeb et al. (2020), the macroeconomic indicators that the pandemic most affected are total production, employment and general price level. It is expected that the negative effects of the pandemic will decrease production and increase prices and unemployment. In order to get consistent results from nonlinear unit root tests, high frequency monthly industrial production index (IPI), consumer price index (CPI) and unemployment rates of Global 8 (G8) countries are chosen to apply the new method that is proposed in this paper. The data were observed monthly between September 2013 and December 2021. Also, natural logarithms of the IPI and the CPI are used, and series were collected from OECD Short-Term Economic Indicators database. Although it was suspended from the G8, Russia is not excluded, but the unemployment rates of Russia are not available in the related database. Moreover, the European Union is not considered as a one united economy for empirical application. Estimated test statistics of τ t (DF) and p αβ , autoregressive lags (k) and mid-point of transition (t b , Date) are reported in Table 5. Additionally, the estimated smooth transitions of IPI, CPI, and unemployment rates for each economy and two trend lines that occurred before and after the Coronavirus era can be seen in Fig. 1, Fig. 2 and Fig. 3 respectively.
In all the countries examined, the industrial production index was adversely affected by the pandemic, and a temporary decline was observed. The findings show that the proposed smooth transition regression model could estimate the pit shape impact of the Coronavirus and the trend lines with high precision for the IPI series. On the contrary, the time series path of consumer prices in the G8 The nominal size is 5%. The results are based on 10000 replications.

M. Ö zcan
countries are different from the industrial production index series, and the pit shape could not be observed in all countries. Canada, Germany, Italy and the United States experienced a temporary decrease in 2020 caused by the pandemic, and this situation could be captured by the proposed smooth transition model. However, the estimated non-linear trends show that there have been more serious structural changes in consumer prices in France, Japan, the United Kingdom and Russia at different dates than that which was caused by the pandemic. Finally, the sudden and temporary increase in unemployment rates observed during quarantine periods could be successfully modelled with the method proposed in this study. Among the G8 countries, the fluctuation in France's unemployment rates due to the pandemic has not been as severe as expected and it is also a significant surprise that the unemployment rate in Italy has experienced a sharp and temporary decline contrary to expectations; both of these structural changes have been successfully modelled. Another important point to be considered here is that the smooth transition trend model revealed in this study successfully captures the inverse "U" and/or "V" shaped structural breaks observed in the macroeconomic indicators. Considering all these structural breaks and non-linear trends revealed in the empirical study, the unit root test results in Table 1 can be interpreted. The unit root null could be The nominal size is 5% and T is 100. The results are based on 10000 replications. to serious econometric problems in applied studies.

Conclusion
The main question of this paper was: does the shape of economic recovery after the Coronavirus pandemic matter? To answer this Although it is not a direct alternative of linear DF test statistics, the new test statistics that are introduced in this paper have more power than linear DF if the DGP is a stationary asymmetric smooth transition model. The monthly industrial production index, the consumer price index and the unemployment rate series, which belong to G8 countries, are chosen for empirical study and (most of) their time series path clearly show a pit or inverse pit shape recovery pattern. The empirical findings thus indicate that the shape of economic recovery which could be observed in industrial production, consumer prices and unemployment, econometrically matters for G8 countries. The motivation of this study was born from pandemic experience. However, the proposed method of this study could be used for any specific recession-recovery cases like the Coronavirus pandemic causes of 2020.

Funding
The author declare that no funds, grants, or other support were received during the preparation of this manuscript.

Declaration of competing interest
The author has no relevant financial or non-financial interests to disclose.

Data availability
The dataset sample files and the R Programming Language codes are available at the following link: https://github.com/mehmetozcan/Shape_of_economic_recovery/releases/tag/published.