Does Ambiguity Aversion Raise the Optimal Level of Effort?

Transcription

1 Does Ambiguity Aversion Raise the Optimal Level of Effort? Loïc Berger This version: June 2012 Abstract I consider two-period self-insurance and self-protection models in the presence of ambiguity and analyze the effect of ambiguity aversion. I show that in most common situations, ambiguity prudence is a sufficient condition to observe an increase in the level of effort. I proposes an interpretation of the model in the context of climate change, such that self-insurance and selfprotection are respectively seen as adaptation and mitigation efforts a policymaker should provide to deal with an uncertain catastrophic event, and interpret the results obtained as an expression of the Precautionary Principle. Keywords: Non-expected utility, Self-protection, Self-insurance, Ambiguity prudence, Precautionary Principle. JEL Classification: D61, D81, D91, Q58. I thank Philippe Weil, Christian Gollier, David Alary and François Salanié for helpful comments and discussions. The research leading to this result has received funding from the FRS-FNRS. FNRS Research Fellow, ECARES Université libre de Bruxelles, and Toulouse School of Economics, address: loic.berger@ulb.ac.be 1

2 1 Introduction Self-insurance and self-protection, in the terminology of Ehrlich and Becker (1972), are two risk management tools used to deal with the risk of facing a monetary loss when market insurance is not available. In both situations, a decision maker (DM) has the opportunity to furnish an effort to modify the distribution of a given risk. In particular, the effort in a self-insurance model corresponds to the amount of money invested to reduce the size of the loss occurring in the bad state of the world, while in a self-protection model (or prevention model), the effort is the amount invested to reduce the probability of being in the bad state. Though these two models have received a great deal of attention in the literature during the last few years, it is noteworthy to mention that this literature has, until now, generally only focused on simple one-period, two-state models remaining in the expected utility framework. In particular in those setups, the effect of risk aversion has been shown to have different impacts depending on the model considered: it always increases the optimal demand for self-insurance, but has an ambiguous effect on the demand for self-protection 1 (Dionne and Eeckhoudt (1985); Briys and Schlesinger (1990)). It is therefore necessary to have access to wider information such as the level of the initial optimal probability of loss (Jullien, Salanié, and Salanié (1999)), or to make further assumptions on the utility function (Dachraoui et al. (2004); Eeckhoudt and Gollier (2005)) to obtain a general result from the analysis of risk aversion. Although single period risky models are well adapted to describe a certain number of situations, they seem to be too restrictive in at least two dimensions to describe a large number of other important issues. First, there exist many situations requiring self-insurance or self-protection in real life in which the decision to invest and the realization of uncertainty does not take place at the same point in time. A long period of time may pass between these two events, leading to the necessity of taking intertemporal considerations into account. To illustrate this assertion, think for example of an individual living for two periods. This individual faces the risk of heart attack when he becomes old and has the choice, when he is young, to practice sport or not. Sport is costly but has the advantage of either reducing the probability of a heart attack with which a potentially important fixed loss is associated, 1 Since an increase in effort in this model does not correspond to a decrease in risk in the sense of Rothschild and Stiglitz (1970). 2

3 or of reducing the severity of an attack that happens with a fixed probability 2. In this example, it is clear that many years may separate the moment at which the effort decision is taken and the moment the uncertainty realizes. A single period setting may therefore not be very appropriate to model such a problem. In Berger (2010), I extended the analysis of self-protection and self-insurance to a two-period environment. I found that the single-period results could be easily extended in the self-insurance case, while in the most usual situations concerning self-protection (i.e. for events characterized by a low probability of accident occurrence), the notion of prudence plays a central role 3. The second limitation of the self-insurance and self-protection models studied in the literature is that they remain in the expected utility framework, and are therefore unable to deal with other kinds of uncertainty besides risk 4. In many real-life problems however, the nature of the uncertainty considered cannot be limited to risk. Think for example of the high level of uncertainty that characterized environmental economics problems, and more specifically climate change policy issues. Due to our imperfect scientific knowledge, we do not know exactly for instance what the benefits associated with a reduction in environmental damage are, or we what the costs are for human beings of an increase in the average temperature. This means that, when considering the right thing to do to preserve the environment, the decision making process has to be carried out without knowing the probabilities associated with random events perfectly. Instead, what the decision maker usually has at his disposal is a panel of predictions from different scientific models, or confidence intervals. As noted by Lange and Treich (2008), the role of the decision maker is therefore to aggregate the findings from competing models, and this is generally not done by taking the average value to end up with a single probability distribution, as it is the case in the (subjective) expected utility framework. Indeed, as first shown by Ellsberg (1961) and later confirmed by a number of experimental studies (see Einhorn and Hogarth (1986), Viscusi and Chesson (1999), and Ho et al. (2002) among others), the uncertainty on the probabilities of a random event (called ambiguity) often leads the decision maker to violate the axiom of reduction of compound lotteries in 2 Think for example that doing sport enables to lower recovery costs, thanks to a better physical condition. 3 In particular, I show that prudent and risk-averse expected utility maximizers exert more effort than the risk-neutral agent. 4 The probabilities associated with the different outcomes are therefore all assumed to be known with certainty. In particular, those models implicitly assume the absence of any kind of ambiguity, or equivalently, assume that agents are ambiguity-neutral (and therefore behave as subjective expected utility maximizers in the sense of Savage (1954)). 3

4 the sense that it makes him over-evaluate less desirable outcomes. It is therefore important to take this individual behavior into account when considering problems in the presence of ambiguity. In this the paper, I present two models of self-insurance and self-protection that do not suffer from these two limitations. Each model takes the form of a simple twoperiod model incorporating the theory Klibanoff, Marinacci, and Mukerji (2005, 2009) developed to deal with ambiguity. The timing of the decision process is very simple: in a first period, a DM chooses the level of effort he wants to exert to either reduce the probability of being in the bad state in second period (self-protection), or to reduce the loss associated with this bad state (self-insurance). This model may be used in the context of climate change to analyze the properties of the instruments available to fight the global warming phenomenon. In particular, self-protection and self-insurance may be respectively seen as mitigation and adaptation efforts 5 in the sense that the former refers to action taken to permanently eliminate or reduce the risk of climate change to human life, while the latter refers to the ability to adapt to climate change to moderate potential damage or to cope with the consequences. If one considers there are only two states of the world, one good state in which the temperature does not increase too much and in which economic activities are not really altered, and one catastrophic bad state which corresponds to a very unfavorable situation in which the economic environment is deeply damaged 6, the situation corresponds exactly to the self-protection or self-insurance model. In that view, it becomes clear that the two limitations imposed by the one-period expected utility model that I discussed above need to be overcome. The various IPCC reports strengthen this position by noting that climate change decision-making is not a once-and-for-all event [...] rather it is a process that will take place over decades (Halsnæs et al. (2007)), and by making a clear distinction between risk and uncertainty: in most instances, objective probabilities are difficult to estimate [...] where we cannot measure risks and consequences precisely, we cannot simply maximize net benefits mechanically. In such a context, the traditional decision 5 The definitions given in the IPCC Third Assessment Report are the following. Mitigation: An anthropogenic intervention to reduce the sources or enhance the sinks of greenhouse gases. Adaptation: Adjustment in natural or human systems in response to actual or expected climatic stimuli or their effects, which moderates harm or exploits beneficial opportunities (IPCC (2001)). 6 Think for example of situations resulting from an abrupt climate change such as the disintegration of the West Antarctic Ice Sheet, that could raise the sea-level by 4-6 meters, the increase in the frequency of droughts or of important tropical cyclones that could have extreme ecological and agricultural consequences (see Meehl et al. (2007)). Note that those situations are generally characterized by a very low probability of occurrence. 4

5 making process consisting in choosing the policy set that maximizes the expected (monetary) value of the outcomes might not be appropriate and non-conventional criteria might be required to make robust decisions (Halsnæs et al. (2007)). The question whether the ambiguous nature of an outcome leads an ambiguity averse decision maker to exert more effort than another who does not take this ambiguity into account may therefore be assimilated to the one referring to the Precautionary Principle in the context of climate change. Should the lack or the incompleteness of information lead environmental policy to be precautionary in the sense of favouring a more intense intervention? In this paper, I show 7 that in the most usual situations, the economic theory is consistent with the claim when an activity raises threats of harm to human health or the environment, precautionary measures should be taken even if some cause-andeffect relationships are not fully established scientifically (1998 Wingspread Statement on the Precautionary Principle) 2 The model The concept of ambiguity is now widely accepted to refer to situations in which the probabilities of a random event are not objectively known. Instead, what the decision maker (DM) has at his disposal is a subjective prior distribution of what these probabilities might be. In what follows, I express this idea by considering that the probability an outcome realizes consists in a set of probabilities, depending on an external parameter for which the DM has prior beliefs. In that sense, the uncertainty on the relevant priors makes the probabilities of the outcomes uncertain and in turn creates ambiguity. Ambiguity therefore modifies the final probability distributions and thus has an impact on the ex ante evaluations of the possible alternatives. In their simplest versions, the self-protection and self-insurance models consider only two states of the world: a loss and a no-loss state. 7 Notice that this paper proposes an economic interpretation of the Precautionary Principle that is different from the one proposed by Gollier, Jullien, and Treich (2000). 5

6 2.1 Ambiguity in a binary world In an ambiguous world, the probability p(e, θ) of being in the bad state of the world is not known with certainty but depends on the level of effort e, and on the value of a parameter θ for which the DM has subjective prior beliefs 8. Ambiguity may therefore be interpreted as a multi-stage lottery. A first lottery determines the value of the parameter θ, and a second one determines whether a loss occurs or not. Using theory Klibanoff, Marinacci, and Mukerji (2005, 2009) (KMM) developed to take into account the DM s attitude towards ambiguity, the general form of the decision maker s problem is given by { max u(w 1 e) + βφ {Eφ 1 p(e, θ)u(w 2 L(e)) + (1 p(e, θ))u(w }} 2 ). (1) e Remark that KMM model does not remain in the expected utility framework, however it has the advantage of being fairly tractable and enables to consider the effects of ambiguity 9, ambiguity attitude and risk attitude separately. Problem (1) is a problem of self-insurance when p(e, θ) = p(θ) for all levels of effort e, and a problem of self-protection when L(e) = L for all e. In this formulation, w i is the exogenous wealth in the beginning of period i = 1, 2, u represents the period vnm utility functions, and β [0, 1] is the discount factor. The probability of loss p depends on the value of the stochastic parameter θ which can take n different values, and on the level of effort in the self-protection case, while the loss L may also be a function of e. It is assumed that p e (e, θ) p(e,θ) 0 for all θ, and that L (e) L(e) 0. e e Finally, function φ represents attitude towards ambiguity, and E, the expectation operator taken over the distribution of θ, conditional on all information available during the first period. The function φ is assumed to be three times differentiable, increasing, and concave under ambiguity aversion, so that the φ-certainty equivalent in equation (1) is lower in that case than when the individual is ambiguity neutral characterized by a linear function φ 10. In that sense, an ambiguity averse DM dis- 8 It would be the case for example if the climate scientists of a country are able to compute the law linking effort to the probability of a catastrophe occurrence, but there exists a stochastic external parameter that the government cannot control (it can be the actions taken by governments of other countries, or the natural evolution of the environmental system,). Notice that in an unambiguous world, the value of this parameter would be objectively known. 9 Even if this notion embodies in fact both objective ambiguity and ambiguity perception, an individual characteristic. See the discussion on this subject in Berger (2011) for more details. 10 Notice that for simplicity, I assume that φ is only defined for non-negative values. Any value inside the second bracket must therefore be non-negative, which should not be a problem since any positive affine transformation of u represents the same preferences over risky situations. KMM consider for example the unique continuous, strictly increasing function u with u(0) = 0 and u(1) = 1 that represents any given preferences. 6

7 likes any mean-preserving spread in the space of conditional second period expected utilities. Notice that under KMM specification, the concavity of u and φ does not guarantee that the maximization problem (1) is convex, so additional assumptions are needed for the solution of this program to be unique. These conditions are summarized in the following proposition. Proposition 1 The maximization program of a two-period self-insurance or selfprotection problem under ambiguity as described by (1) is convex if: function φ has a concave absolute ambiguity tolerance: φ (U)/φ (U) is concave in U, and L(e) is convex in e in the self-insurance case: 2 L(e)/ e 2 0, or p(e, θ) is convex in e in the self-protection case: 2 p(e, θ)/ e 2 0 for all θ. Proof Relegated to the Appendix. Under the special case of ambiguity neutrality, problem (1) becomes a simple intertemporal expected utility problem consisting in finding the level of effort e that maximizes [ ] IEU(e) = u(w 1 e) + β p(e)u(w 2 L(e)) + (1 p(e))u(w 2 ), (2) where p(e) Ep(e, θ) 11. In that case, the optimal level of effort e is the solution of the first-order condition 12 (FOC) [ ] u (w 1 e ) β p (e )[u(w 2 ) u(w 2 L(e ))] L (e )p(e )u (w 2 L(e )) = 0. The first term of this expression is negative and represents the marginal cost of effort, while the first and second terms in the brackets are both positive and represent 11 In this case, utility function u incorporates both attitude towards risk and towards intertemporal substitution, but remark that for IEU to remain in the expected utility framework, the coefficient of relative risk aversion must be the inverse of the elasticity of intertemporal substitution. 12 Notice that problem (2) is convex if p (e) 0 and L (e) 0, meaning that the marginal benefits of self-protection and of self-insurance are decreasing in the level of effort. 7

8 the marginal benefits of self-protection and of self-insurance respectively. Ambiguity aversion raises the optimal level of effort if the FOC of problem (1) evaluated at e is positive. By letting U(e, θ) denote the conditional second period expected utility for a level of effort e and parameter θ: U(e, θ) p(e, θ)u(w 2 L(e))+ (1 p(e, θ))u(w 2 ), and by denoting by U e the first derivative of this expression with respect to variable e, it is easy to see that this will be the case if and only if [ E φ {U(e, θ)}u e (e, θ) ] { φ φ {Eφ{U(e 1, θ) }} EU e (e, θ). (3) The interpretation of this condition is simple: since ambiguity only affects variables during the second period, the marginal cost of effort, which takes place in first period, is unaffected and the condition indicates that the marginal benefit of protection or insurance must be higher under ambiguity aversion. The following lemma is useful to pursue. Lemma 1 Let φ be a three times differentiable function reflecting ambiguity aversion. If φ exhibits DAAA (Decreasing Absolute Ambiguity Aversion) then Eφ { x} φ {φ 1 {Eφ{ x}}}. Proof φ is DAAA is equivalent to saying that φ equivalently that φ /φ is more concave than φ, or φ /φ. Since (φ ) 1 is a decreasing function, the proof follows from the fact that the certainty equivalent of function φ is larger than that of function φ. Corollary 1 If φ exhibits CAAA (Constant Absolute Ambiguity Aversion), then Eφ { x} = φ {φ 1 {Eφ{ x}}}. Using this lemma, it is easy to see that condition (3) is equivalent to condition ( cov φ {U(e, θ)}, U e (e, θ) ) 0 (4) in the case of constant absolute ambiguity aversion, and that it is always respected in the case of decreasing absolute ambiguity aversion if condition (4) holds 13. The expressions U(e, θ) and U e (e, θ) must therefore covary negatively in θ since φ is a 13 Notice that the assumption of decreasing absolute ambiguity aversion does not seem too restrictive when modeling individuals preferences. Gierlinger and Gollier (2008) for example mentioned that DAAA is a reasonable property of uncertainty preferences. 8

9 decreasing function under ambiguity aversion. From now on, I also assume that the θ may be ranked in such a way that p θ (e, θ) > 0 for all levels of effort, meaning that the probability of being in the bad state is higher for higher values of parameter θ. This assumption ensures that the level of ambiguity is a monotonic function of effort, or equivalently that the order between the different values of θ must be the same when e evolves. Since second period conditional expected utility U(e, θ) is decreasing in θ, a sufficient condition to observe a higher level of effort under CAAA or DAAA than under ambiguity neutrality therefore simply becomes that the marginal benefit of effort U e (e, θ) is increasing in θ. The analytical form of this expression is the following U e (e, θ) = p e (e, θ) [ U(w 2 ) U(w 2 L(e )) ] L (e )p(e, θ)u (w 2 L(e )). (5) Since by assumption the probability of loss does not depend on the effort furnished (p e = 0) in the self-insurance model, it is easy to see that U eθ (e, θ) 0, and therefore condition (4) is respected, in the case of self-insurance. To analyze the self-protection model, the crucial element is the sign of p eθ (e, θ), so a positive sign implies a higher level of prevention due to ambiguity aversion. Simple examples To illustrate what precedes, consider two particular forms of loss probability function. To keep things simple, both are linear in the ambiguity parameter θ, the first is additive and takes the form p(e, θ) = p(e) + θ, while the second is multiplicative and is written p(e, θ) = θp(e). In the additive case, U e (e, θ) = p (e) [ u(w 2 ) u(w 2 L(e)) ] and it is easy to see that an increase in θ has no effect on U e (e, θ). This means that the level of self-protection is exactly the same for any individual manifesting constant absolute ambiguity aversion. In particular, an ambiguity neutral individual and a maxmin expected utility maximizer à la Gilboa and Schmeidler (1989) both choose to selfprotect precisely the same way. Moreover, if the individual manifests decreasing absolute ambiguity aversion, he will always choose a higher level of protection. On the contrary, if for some reason ambiguity is made smaller when the effort is higher, p e (e, θ) will be higher for higher value of θ and hence condition (4) will hold. This is the case with the multiplicative form described above. In that case U e (e, θ) = θp (e) [ u(w 2 ) u(w 2 L(e)) ], and it is clear that an increase in θ will have a positive impact on U e, therefore that any individual manifesting nonincreasing absolute ambiguity aversion chooses a higher level of protection. Figure 1 illustrates the situation when there are two possible values of θ: θ 1 and θ 2, 9

10 and when the ambiguous loss probability is linear in θ. p(e, θ) 1 0 p(e) + θ 2 θ 2 p(e) p(e) θ 1 p(e) p(e) + θ 1 e Figure 1: Linear ambiguous loss probability As can be seen in Figure 1, when θ increases, from θ 1 to θ 2 14, different scenarii are possible. In the additive case, the slopes of the two dashed lines are exactly the same for any given level of effort. Ambiguity in this case is therefore constant for any level of effort. On the contrary, with the multiplicative form it is easy to see that the slope of the dotted curve for any given level of effort is higher with θ 2 than with θ 1. Intuitively, this corresponds to a situation in which ambiguity decreases with the effort furnished and condition (4) is therefore respected. The intuition behind these two examples is simple. In the absence of ambiguity, we know that a key determinant of the optimal level of self-protection is the slope of p(e). Now when ambiguity is introduced, the DM does not know exactly in which situation he is: if his prior beliefs are equal, he has one chance out of two to be confronted with probability p(e, θ 1 ), and one chance out of two to have p(e, θ 2 ). If the individual is ambiguity neutral, this situation does not affect him and the decision is taken by considering the expected law p(e). However, if the agent is ambiguity averse, he will over-evaluate the less desirable outcome (i.e. the law p(e, θ 2 )) and hence take a decision by considering a law somewhere above the line p(e). In the special case of infinite ambiguity aversion, corresponding to the maxmin model of Gilboa and Schmeidler (1989), the DM takes his decision by considering the worst scenario p(e, θ 2 ). 14 Remark that in this example, the DM associates the same prior belief to each value of θ, in such a way that θ 2 = θ 1 in the additive case, and θ 2 = 2 θ 1 in the multiplicative case. 10

11 The study of these two particular cases in which the probability is linear in parameter θ emphasizes the differences there are between the single and the two-period models. In the single period model, when both the marginal cost and the marginal benefit of self-protection are affected by the introduction of ambiguity, it is indeed impossible to sign the effect ambiguity aversion has on the optimal prevention, even when the probabilities are linear in the ambiguity parameter. In particular, in that situation, the DM will always choose to reduce his demand of self-protection if the probability law is additive, while he will choose a higher level of protection if the probability law is multiplicative (Snow (2011)). As explained by Alary, Gollier, and Treich (2012), this inability to obtain a general result is due to the fact that both the marginal cost and the marginal benefit of self-protection increase under ambiguity aversion. The net effect therefore depends on which one is more affected. In the two-period model analyzed in this paper however, ambiguity aversion only affects the marginal benefit, making it possible to draw general conclusions. Finally, remark that if exerting an effort has an effect on both the loss and the probability(this is the case of a self-insurance-cum-protection model, in the words of Lee (1998)), it is possible to combine the effects described above to obtain a result in the case where ambiguity is not increasing in the level of effort. In reality however, it is clear that there is no reason for p(e, θ) to be linear in θ. We a priori have no information on the way the ambiguity parameter enters the loss probability. The only information the DM has at his disposal is the way in which his beliefs evolve when the level of effort is altered. If for example a DM finds a situation less ambiguous when the level of effort is high than when it is low, there is a higher chance to be in a situation like the multiplicative than the additive one. Putting aside the most implausible case in which ambiguity increases with the level of effort, the results of this section can be summarized by the following proposition. Proposition 2 If ambiguity is not increasing in the level of effort, an individual manifesting decreasing absolute ambiguity aversion (DAAA) will always choose (a) a higher level of effort than an expected utility maximizer if the state of the world is ambiguous in a two-period model of self-protection, (b) a strictly higher level of effort than an expected utility maximizer if the state of the world is ambiguous in a two-period model of self-insurance. (c) a a strictly higher level of effort than an expected utility maximizer if the state of the world is ambiguous in a two-period model of self-insurance-cumprotection. 11

12 2.2 Generalization to a world with S states If the analysis above is interesting to describe a given type of situations, there are other cases in which the states of the world cannot be modeled using a Bernouilli distribution. Instead, many different states may hypothetically materialize, and the probability that each one happens may potentially depend on the level of effort exerted. The generalization of the two-period self-protection problem under ambiguity to S states of the world is the following max u(w 1 e) + βφ {Eφ 1 e { S }} p(e, s, θ)u(w 2,s ) where p(e, s, θ) is the probability that state s realizes, given level of effort e and parameter θ. Notations are kept the same as before and it is assumed that all the standard assumptions hold. Note that it is clear that, for any value of θ, S s=1 p(e, s, θ) = 1, and hence S p(e,s,θ) s=1 S e s=1 p e(e, s, θ) = 0. Proceeding in exactly the same way as in the binary case, yields the following. Proposition 3 In the general two-period model of self-protection under ambiguity, ambiguity aversion raises the optimal level of effort exerted by an individual manifesting non-increasing ambiguity aversion if S s=1 p(e, s, θ)u(w 2,s ) and S s=1 p e(e, s, θ)u(w 2,s ) are anti-comonotone in θ. Again, e is the level of effort optimally chosen by an ambiguity neutral decision maker. s=1 (6) Since the conditions under which Proposition 3 is satisfied are not easy to handle, I here make further assumptions. Without loss of generality, assume that: w 2,1 < w 2,2 <... < w 2,S. Moreover, consider the three following restrictions, that hold for all e, s and θ, and that may be justified in many contexts: i. p(e, s, θ) < p(e, s + 1, θ) when p e < 0 p(e, s, θ) > p(e, s + 1, θ) when p e > 0. (7) which means that, for events considered as (un)favorable, i.e. those for which exerting an effort enables to increase (reduce) the probability of occurrence, 12

13 the higher (lower) the second period outcome, the lower the probability. ii. p e (e, s + 1, θ) p e (e, s, θ), (8) which means that the marginal impact of effort on the probability is increasing in the level of outcome. iii. p e (e, s, θ + 1) p e (e, s, θ) when p(e, s, θ + 1) p(e, s, θ) p e (e, s, θ + 1) p e (e, s, θ) when p(e, s, θ + 1) p(e, s, θ) (9) which is basically the same assumption as in the binary case, stipulating that the level of ambiguity decreases with the level of effort (curves p(e, s, θ) and p(e, s, θ + 1) become closer as e increases). Imagine for example that there exist n probability distributions of w 2, that can be ranked according the first-order stochastic dominance (FSD) criterion. Say for example that w 2 1 FSD w 2 2 FSD... FSD w 2 n, where wθ 2 corresponds to the distribution (w 2,1, p(e, 1, θ);...; w 2,S, p(e, S, θ)). Then, it is clear that S s=1 p(e, s, θ)u(w 2,s ) is decreasing in θ. The only thing needed for ambiguity aversion to induce a higher level of self-protection is therefore that S s=1 p e(e, s, θ)u(w 2,s ) is increasing in θ. It can easily be proven that this will be the case if conditions i. to iii. above are satisfied and if p(e, s, θ + 1) p(e, s, θ) for all θ and s such that p(e, s, θ) 0 (states that I called unfavorable ), and p(e, s, θ + 1) p(e, s, θ) for all θ and s such that p(e, s, θ) 0 ( favorable states). Figure 2 below illustrates the situation for S = 6. Ambiguous losses Another case in which Proposition 3 is verified is the case that I call ambiguous losses. This corresponds to a situation in which the probability of suffering from a loss is known to be p(e), but then the size of the loss is ambiguous. Imagine there exist S 1 different loss situations L s (e) for s = 1,..., S 1, and that each of them is associated with a probability q s (θ). Again, think for example that this ambiguity results from the inability to obtain a single estimate for the economic impact of a catastrophic climate change, due to insufficient scientific knowledge. 13

14 Unfavorable states Favorable states p(e,s,θ) e p(e, 3, θ 2 ) p(e, 3, θ 1 ) p(e, 2, θ 2 ) p(e, 2, θ 1 ) p(e, 1, θ 2 ) p(e, 1, θ 1 ) e p(e,s,θ) e p(e, 4, θ 1 ) p(e, 4, θ 2 ) p(e, 5, θ 1 ) p(e, 5, θ 2 ) p(e, 6, θ 1 ) p(e, 6, θ 2 ) e Figure 2: Example of a situation leading to an increase in the level of effort The DM therefore can rely on a range of values that he associates with some probabilities 15. As before, I assume that the agent has a prior distribution over θ, and that I can rank the S 1 possible laws, say from the least to the most favorable, implying that L s (e) L s+1 (e) for all e and for all s = 1,..., S 2. The schedule of the multi-stage lottery is now the following: a first stage determines whether the DM suffers from a loss or not, a second determines the value of the prior and a third one determines the size of the loss 16. To illustrate the situation, imagine there are only two possible loss functions L 1 (e) and L 2 (e), and two θ, θ 1 and θ 2, associated respectively to the beliefs µ and 1 µ. In this case, the DM is confronted to the lottery illustrated in Figure 3. Such a situation corresponds for example to the case of a big country, which knows that, by reducing its greenhouse gas emissions by investing an amount e of its GDP in mitigation, is able to reduce the probability of an extreme increase in average temperatures to p(e). However, if temperatures still drastically increase, leading to the partial or complete disintegration of the West Antarctic Ice Sheet, it is not clear whether the sea-level will increase by 3 or by 6 meters, making the destructive power of the catastrophic event ambiguous. 15 For example, Stern (2007) estimates the total loss for a high-climate scenario with non-markets impacts and risk of catastrophe to be between 2.9% and 35.2% in GDP per capita in 2200 (see Figure 6.5 in Stern (2007) for more details). 16 Notice that we could alternatively have a lottery in which the value of the prior is determined in the first stage, the size of the loss in the second, and the possibility of suffering from the loss or not in the third one. However, it can be shown that both setups lead to the same comparative statics results when investigating the effect of ambiguity aversion. 14

15 p(e) Bad state µ 1 µ θ 1 θ 2 q(θ 1 ) 1 q(θ 1 ) q(θ 2 ) 1 q(θ 2 ) L 1 (e) L 2 (e) L 1 (e) L 2 (e) 1 p(e) Good state Figure 3: Example of lottery for ambiguous losses In a situation with ambiguous losses, the problem faced by a DM under KMM specification is the following { [ S 1 ] max u(w 1 e) + β (1 p(e))u(w 2 ) + p(e)φ {Eφ 1 q s ( θ)u(w 2 L s (e))}}. e I again assume that this maximization problem is convex 17. In this formulation, the notations are the same as before, and I denote by W(e, θ) S 1 s=1 q s(θ)u(w 2 L s (e)), the conditional second-period expected utility obtained in the loss situation. If the individual is ambiguity neutral, he will choose a level of effort e that satisfies the condition [ u (w 1 e ) β p (e ) [ u(w 2 ) s s=1 q s u(w 2 L s (e )) ] +p(e ) s (10) ] q s u (w 2 L s (e ))L s (e ) = 0. where q s Eq s ( θ) for all s. As before, the first term is the marginal cost of effort while the first and second terms in the brackets represent the marginal benefits of self-protection and self-insurance respectively. The condition to observe an increase in the optimal level of effort due to ambiguity aversion is the following { +p(e )(φ 1 ) {Eφ W(e, θ) { }}Eφ W(e, θ) } p (e ) s q s u(w 2 L s (e )) + p(e ) s s { p (e )φ {Eφ 1 W(e, θ) }} q s ( θ)u (w 2 L s (e ))( L s (e )) q s u (w 2 L s (e ))( L s(e )). (11) 17 It can be shown that the conditions obtained in Proposition (1) are also sufficient to guarantee that problem (10) is convex. 15

16 This condition will always be satisfied in a self-protection model. To see this, remember that L s(e) = 0 for all s and all e when self-protection is considered, and use Jensen s inequality. In the self-insurance model, p (e) = 0, and it is possible to use Lemma 1 under DAAA so the only thing to check is whether the following inequality is respected Eφ {W(e, θ)} s q s( θ)u (w 2 L s (e ))( L s(e ))) Eφ {W(e, θ)} E q s ( θ)u (w 2 L s (e ))( L s (e )). s (12) If this is the case, or equivalently if ( cov φ {W(e, θ)}, ) q s ( θ)u (w 2 L s (e ))( L s (e )) 0, s ambiguity aversion also increases the optimal level of self-insurance for an individual manifesting non-increasing absolute ambiguity aversion. Though this condition is not easy to handle, it is very close to the one needed to observe ambiguity prudence (see Appendix C), and it is therefore still possible to draw some general conclusions. Note for example the role played by the slope L s (e) in the expression above. As was the case when confronted to ambiguous probabilities, the way the curves L s (e) are relative to each other is important. If they come closer when the level of effort increases (i.e. if L s+1(e) < L s(e) for all e), the discrepancy between the possible losses becomes smaller when the individual exerts more effort, so the importance ambiguity has on the final wealth distribution is relatively lower. This situation will give an ambiguity averter extra incentive to increase his level of effort. On the contrary, if the difference between the possible loss is independent of e (i.e. if the curves are parallel), this extra benefit of increasing effort is zero and it is not necessary to take the terms L s (e) into account in the covariance expression above. To illustrate this, imagine there are only two ambiguous outcomes, say L 1 (e) and L 2 (e) associated respectively to the big and the small catastrophe. We need to check that U(w 2 L 2 (e)) U(w 2 L 1 (e)) and U (w 2 L 2 (e))( L 2 (e)) U (w 2 L 1 (e))( L 1 (e)) have opposite signs for inequality (12) to be respected under DAAA. This will be the case if L 1(e) L 2(e). If the loss functions are parallel ( L 1(e) = L 2 (e)), this result is due to the fact that that the marginal utility increases with the size of the loss. To make things concrete, imagine that if the West Antarctic Ice Sheet begins to melt, we can either face an increase of the sea-level of 3 or of 6 meters. If a small coastal country does not make any investment to construct dikes, 16

17 it is confronted with the following situation: under the first scenario, half of the area would be under water, while if the second scenario materializes, the entire country would be flooded. On the contrary, if the entire coastline of this country is equipped with dikes, the first scenario will have no impact and the second will only put half of the country under water. On the other hand, if effort reduces the difference between the two losses ( L 1 (e) > L 2(e)), an additional effect appears which makes the marginal benefit under L 1 higher than the one under L 2. As an illustration of this situation, imagine that the houses in a city would be completely destroyed by a big storm if the city is not protected, while only the roofs would be damaged if the city underwent a small storm without protection. Now if the houses were fully barricaded, only the roofs may be potentially damaged by a big storm while the houses will not be affected at all by a small storm. Note that in a more general context, when there are more than two ambiguous outcomes, it is not necessarily obvious that ambiguity aversion raises the optimal level of self-insurance. It is easy to see that, if the loss functions are parallel, condition (12) is respected if the individual is ambiguity prudent. This will also be the case if the loss functions become closer, since a reinforcing effect enters the formula. The following proposition summarizes the results obtained in this subsection. Proposition 4 Ambiguity aversion always leads the individual to choose a higher level of effort in (a) the two-period model of self-protection with ambiguous losses (b) the two-period model of self-insurance with ambiguous losses if the individual manifests ambiguity prudence and the different loss functions are such that L s (e) L s+1 (e). Ambiguity on a specific state Finally, following an idea developed in Alary, Gollier, and Treich (2012), another way to proceed is to assume that ambiguity is concentrated on a specific state i. In this case, the ambiguous probability to be in state i is p i (e, θ), while the probability to be in any other state s i is given by p s (e, θ) = (1 p i (e, θ))π s, 17

18 where π s is the unambiguous probability of being in state s 18 conditional on the information that the state is not i. As mentioned by these authors, it is easy to see that this structure is without loss of generality when there are only two states of the world. The DM s program in this situation is max u(w 1 e) + βφ {Eφ 1 e { p i (e, θ)u(w 2,i ) + (1 p i (e, θ)) }} S π s u(w 2,s ). (13) Under ambiguity neutrality, the optimal level of effort e is implicitly given by [ ( )] u (w 1 e ) + β p ie (e, θ) π s u(w 2,s ) u(w 2,i ) = 0, where p ie (e, θ) E p i(e, θ). Remark that if the second period expected utility, e conditional on the information that the state is not i, is higher than the utility obtained if the state is i (i.e. s i π su(w 2,s ) u(w 2,i ) 0), exerting an effort decreases the probability of being in state i (i.e. p ie (e, θ) 0). In that sense, state i corresponds to the above-mentioned notion of unfavorable state 19. Using the same comparative techniques as above, it is easy to prove the following Proposition. Proposition 5 In the general two-period self-protection problem in favor of state i in which ambiguity is concentrated, ambiguity aversion always leads an individual manifesting decreasing absolute ambiguity aversion to exert more effort if p i (e, θ) and p ie (e, θ) are anti-comonotone in θ. In the words of Proposition 2, this means that DAAA raises the optimal level of effort if ambiguity is not increasing in the level of effort (or graphically, if the curves p i (e, θ), for different values of θ, are getting closer when e increases). s i Again, if DAAA and non-increasing ambiguity in the level of effort are thought to be realistic assumptions, Proposition 5 may be interpreted as an expression of the Precautionary Principle in the context of a self-protection problem. 18 An implicit assumption is of course that s i π s = Note that a favorable state i corresponds to a situation in which s i π su(w 2,s ) u(w 2,i ) and p ie (e, θ) 0. s i 18

19 3 In practise As mentioned above, this specification enables to interpret these models as climate change models, in which self-protection corresponds to mitigation effort, and self-insurance to adaptation effort. In particular, one can imagine the decision maker as being a government that has to choose the percentage of GDP to invest today to fight or deal with global warming. In the case of self-protection, this government represents a big country (i.e. a country responsible for a large portion of global greenhouse gas emissions) so the action it undertakes now has an impact on the average global temperatures in the future 20. Think for example of the US or China. On the contrary in the self-insurance model, the government is that of a small country, whose preventive action alone cannot affect global temperatures (for example the Netherlands). If there are no possibilities of coordination with other countries, the only action this government can take is to invest today in technologies that enable its population to survive and continue its economic activity even if a catastrophic event occurs. A good example of possible action is the construction of dikes to protect against coastal surges resulting from a potential sea-level raise. Self-insurance-cum-protection: Imagine for example the case of mitigation efforts that enable to decrease the probability of catastrophe due to global warming, and that has an impact on the severity of the loss if the catastrophe still occurs. In the context of climate change, this means that taking into account the presence of ambiguity and the attitude agents generally manifest towards it, leads to an increase in the mitigation and adaptation efforts if, as proposed by Halsnæs et al. (2007), the desirability of preventive efforts is measured not only by the reduction in the expected (average) damages, but also by the value of the reduced uncertainties that such efforts yield. This result is in line with the IPCC report which claims that Some uncertainties will decrease with time for example in relation to the effectiveness of mitigation actions and the availability of low-emission technologies, as well as with respect to the science itself. The likelihood that better information might improve the quality of decisions (the value 20 As noted by Schneider et al. (2007), this assumption does not seem unrealistic since recent research indicates that human influence has already increased the risk of certain extreme events such as heatwaves and intense tropical cyclones. There is high confidence that a warming of up to 2 C above levels would increase the risk of many extreme events, including floods, droughts, heat waves and fires, with increasing levels of adverse impacts and confidence in this conclusion at higher levels of temperature increase. 19

20 of information) can support increased investment in knowledge accumulation and its application, as well as a more refined ordering of decisions through time. (Halsnæs et al. (2007)) 4 Conclusion This paper proposes an economic interpretation of the Precautionary Principle based on the concept of ambiguity aversion. I show that scientific uncertainty, characterized by ambiguous probabilities of occurrence of a catastrophic event or by ambiguous losses associated with this event, should induce a decision maker to take stronger measures today to adapt to or to prevent the formation of such an event in the future. This conclusion is similar to the one obtained by Gollier, Jullien, and Treich (2000) but results from a different model in which ambiguity and attitudes agents manifest towards it are specifically taken into account. Through a simple economic model, I gave a justification to the idea that the decision-making tools of risk assessment in the expected utility framework are not appropriate in the presence of scientific uncertainties. As a consequence, the statement stipulating that when an activity raises threats of harm to human health or the environment, precautionary measures should be taken even if some cause-and-effect relationships are not fully established scientifically (January 1998 Wingspread Statement on the Precautionary Principle) should be respected. To arrive to that conclusion, I use a two-period model of self-protection and selfinsurance under KMM preferences to show that in most usual situations, the level of effort chosen by an agent manifesting decreasing ambiguity aversion is always higher than that chosen by an ambiguity neutral individual when the state of the world or the loss is ambiguous. This conclusion may equivalently be interpreted as the impact on the optimal level of effort resulting from the existence of scientific uncertainty. To make things concrete, I connect this model to the climate change policy problem consisting in finding the optimal level of adaptation (actions helping human and natural systems to adjust to climate change) or mitigation (actions reducing net carbon emissions and limiting long-term climate change) efforts a government should provide. Indeed these measures, that can be seen respectively as self-insurance and self-protection, concern both long term horizon and deeply uncertain situations, and can therefore not be analyzed using traditional models in the expected utility framework. I chose to move away from the single period model traditionally used in the self-protection and self-insurance literature to make a clear distinction between 20

21 the cost of effort (which is decided today and is therefore not ambiguous) and the benefit this effort yields in the future (which is ambiguous because of the presence of different scientific uncertainties). In this view, my model is quite different from those in the recent papers of Snow (2011) and Alary, Gollier, and Treich (2012) in which the marginal cost of effort is also affected by ambiguity, since they remain in the single period framework. The drawback of this latter approach is that these models are therefore not able to draw general conclusions because of the conflicting effect ambiguity aversion has on marginal benefit and marginal cost. Alternatively, remark that one could obtain the same results I present in this paper by considering a single-period model in which the cost of effort is be non-monetary and not affected by ambiguity. In the portfolio of measures that exist to respond to climate change, there is also a third instrument that the IPCC describes as the research on new technologies, on institutional designs and on climate and impacts science, which should reduce uncertainties and facilitate future decisions (Fisher et al. (2007)). This instrument on its own is not studied explicitly in this paper, but it should be clear that an effort whose only effect is to modify the ambiguity level would always be desirable for an ambiguity averter if, as assumed throughout this paper, the decision problem is convex. However, when the desirability of the preventive effort measured by the reduction in the loss probability is also measured by the value of the reduced uncertainties that this effort yields, I show that ambiguity aversion always increases the desired effort. Overall, this paper contributes to a better incorporation of the concept of uncertainty into policy design, undoubtedly one of the most important fields in economics in general, but especially in environmental economics. 21

22 Appendices Appendix A: Proofs Proof of Proposition 1. This proof is based on Proposition 1 in Gierlinger and Gollier (2008), and is based on the following Lemma, that can be found in Gollier (2001) (Lemma 8). Lemma 2 Let φ be a twice differentiable, increasing and concave function: R R. Consider a probability vector (q 1,..., q n ) R n + with n θ=1 q θ = 1, and a function f : R n R, defined as f(u 1,...; U n ) = φ 1 { n θ=1 q θ φ {U θ } }. Let T be the function such that T(U) = φ {U} φ {U}. Function f is concave in Rn if and only if T is weakly concave in R. Remark that program (1) is convex if { V (e) = φ {Eφ 1 p(e, θ)u(w 2 L(e)) + (1 p(e, θ))u(w }} 2 ) is concave in e. Self-insurance (p(e, θ) = p(θ) for all levels of e): Consider two scalars e 1 and e 2, and let U iθ denote the second period expected utility conditional on θ, for a level of effort e i : U iθ = p(θ)u(w 2 L(e i )) + (1 p(θ))u(w 2 ). Under the notations above, V (e i ) = f(u i1,..., U in ). Then, under concavity of u and convexity of L, we have, for any (λ 1, λ 2 ) with λ i 0 and λ 1 + λ 2 = 1: λ 1 u(w 2 L(e 1 ))+λ 2 u(w 2 L(e 2 )) u(w 2 λ 1 L(e 1 ) λ 2 L(e 2 )) u(w 2 L(λ 1 e 1 +λ 2 e 2 ). Multiplying the first and the third parts of this chain of inequalities by p(θ) and adding (1 p(θ))u(w 2 ) yields λ 1 U 1θ + λ 2 U 2θ U λθ p(θ)u(w 2 L(e λ )) + (1 p(θ))u(w 2 ) 22

23 for all θ, where e λ = λ 1 e 1 + λ 2 e 2. Because f is increasing in R n if φ is increasing, this implies V (e λ ) = f(u λ1,..., U λn ) f(λ 1 U 11 + λ 2 U 21,..., λ 1 U 1n + λ 2 U 2n ). If φ /φ is concave, by Lemma 2 we have f(λ 1 U 11 + λ 2 U 21,..., λ 1 U 1n + λ 2 U 2n ) λ 1 f(u 11,..., λ n f(u 1n ) + λ 2 f(u 21,..., λ n f(u 2n ) = λ 1 V (e 1 ) + λ 2 V (e 2 ). Combining these two results yields V (λ 1 e 1 + λ 2 e 2 ) λ 1 V (e 1 ) + λ 2 V (e 2 ) implying that V is concave in e. Self-protection (L(e) = L for all levels of e): In this case, the proof is similar but U iθ is now given by U iθ = p(e i, θ)u(w 2 L) + (1 p(e i, θ))u(w 2 ) and we exploit the convexity of p(e, θ) in e to obtain λ 1 U 1θ + λ 2 U 2θ U λθ. 23