Precautionary Principle - CiteSeerX

Learning and irreversibility: An economic interpretation of the "Precautionary Principle" Christian Gollier Universit¶e de Toulouse Bruno Jullien Universit¶e de Toulouse Nicolas Treich 1 Universit¶e de Toulouse June 22, 1998

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We are grateful to P.-A. Chiappori, N. Doherty, J. Drµeze, L. Eeckhoudt, F. Ewald, C. Henry, D. Kessler, J.-J. La®ont, N. Ladoux, E. Malinvaud, P. Pestieau, R. Guesnerie, J.-C. Rochet, D. Ulph and members of the french Commissariat µa l'Energie Atomique (CEA) for valuable discussions. We thank FFSA and CEA for their ¯nancial supports.

Abstract We consider the problem of the optimal use of a good whose consumption can produce damages in the future. Potential damages are proportional to the accumulated lifetime consumption of the good. Scienti¯c progress is made over time that provides information on the distribution of the intensity of potential damages. Applications of the model are the greenhouse e®ect, the "mad cow" disease, asbestosis, electromagnetic ¯elds, radon gas exposures, low radiological exposures and genetic manipulations, to give just a few examples. Under which conditions on preferences do more initial scienti¯c uncertainty lead to a lower socially e±cient level of early consumption of the good? This would be compatible with the so-called "precautionary principle" appearing now in several international treaties. We show that the precautionary principle can be e±cient only if absolute prudence is larger than twice absolute risk aversion. This condition is necessary and su±cient if relative risk aversion is constant. Our model contains an irreversibility constraint. Keywords: Irreversibility, comparison of experiments, precautionary principle, greenhouse e®ect, mad cow disease, prudence. D81, D91, Q25, Q28.

If you can avoid a decision do so without delay. Murphy's 3rd law of decision making

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Introduction

Recent international events made it clear that an important source of uncertainty in our Society is due to our imperfect scienti¯c knowledge. For example, we know that CO2 emissions can have an impact on our climate in the long run. But scientists are still debating about the intensity of potential damages in 100 years from now. Current estimates vary by a factor 20. We also have some scienti¯c doubts about whether the so-called "mad cow" disease can be transmitted to human beings. Thirty years ago, we had doubts about the possibility of asbestos to be responsible for some forms of cancer. More recently, in the mid eighties, physicians had doubts about the possibility of blood products to transmit AIDS. We can also mention uncertainty associated to electromagnetic ¯elds, radon gas exposures and low radiological exposures. There is a clear link between all these examples. Namely, the intensity of potential damages depends upon the accumulation of earlier exposures to the risk, or consumption. For example, climate change relates to the stock of CO2 in the atmosphere, not just to the current °ow of CO2. This hysteresis e®ect is due to the relatively long lifetime of this gas (approximately 200 years). Also, the risk of developing asbestosis seems to depend upon the duration of the exposure to asbestos. The same observation holds for low radiological exposures and other risks associated to repeated exposures. The hysteresis e®ect implies that those problems are intrinsically dynamic. Another obvious link between all these examples is that the uncertainty prevailing at some time is resolved, at least partially, over time. Learning takes place which induces decision makers to adapt their strategy. There are ongoing researches on the above-mentioned issues that will provide information about the intensity of potential damages. Formally, experiments are performed that allow for a Bayesian updating of the prior distribution of potential damages. Also, History tells us that scienti¯c progresses are made about our understanding of Nature. The expectation that uncertainty will resolve over time, together with the phenomenon of hysteresis, implies that the existing literature of optimal dynamic risk management is not very satisfactory to provide e±cient rules to deal with such situations. The objective of the paper is to provide guidelines to deal with risks when one expects a °ow of information on the actual distribution of potential damages to arrive over time. Because of the emergence of many such uncertain situations worldwide, new guidelines for decision makers have been advocated by international organizations. The "Precautionary Principle" emerged in the mid eighties as a clause in

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international treaties, as in the Maastricht Treaty, or at the Conference of Rio.1 The Precautionary Principle in the Maastricht Treaty states that "the absence of certainty, given our current scienti¯c knowledge, should not delay the use of measures preventing a risk of large and irreversible damages to the environment, at an acceptable cost". Our interpretation of the Precautionary Principle is that more (scienti¯c) uncertainty on the distribution of a future risk should induce society to take stronger prevention actions today. More uncertainty means that a better information on the distribution of the risk is expected.2 We compare two economies such that the information structure is better in the sense of Blackwell (1951) in economy B than in economy A. To simplify, consider two economies that are equivalent except for the °ow of information. In economy A, no information is expected in the future which would allow for a Bayesian revision of the distribution of future risks. The risk borne in economy B is the same ex ante, but some scienti¯c information is expected to °ow over time which will improve our knowledge about the distribution of the risk. In which of economies A and B is it optimal to invest more in prevention in the short run? The Precautionary Principle would get some theoretical support if it would be socially e±cient to make no less prevention e®ort in economy B than in economy A. Two contradictory e®ects are taking place in economy B. The classical argument would be to invest less in prevention in economy B than in economy A, because delaying sacri¯ces allows for an e±ciency gain generated by the better information. Society will adapt its e®ort to the severity of the risk. This is the "learn then act" strategy. However, if this strategy is implemented, the risk borne in the future will be larger ex ante, i.e. before knowing the information. This is due to the sensitivity of the future prevention e®ort to the information obtained. It seems intuitively appealing then to compensate this increase in risk by an increase in prevention at earlier dates. Only this second e®ect is compatible with the precautionary principle, which is to bias decisions in favor of the "act then learn" strategy. Thus, we cannot a priori predict the impact of the °ow of future information on the optimal timing of prevention e®orts. We show that prudence, a concept introduced by Kimball (1990), plays an important role in determining the optimal decision in the face of dynamic uncertainty. Prudence is equivalent to the positiveness of the third derivative of the utility function. A prudent individual increases her savings in the face of an increase in risk associated to future revenues. The link of this concept with our problem is that the second e®ect mentioned above deals with an increase in early sacri¯ces to forearm oneself against an increase in future risks. This is thus not so much a surprise that prudence is at play when considering the e®ect of an 1

For more information on the Precautionary Principle, see the book edited by O'Riordan and Cameron (1995). 2 The relation between a "better information structure" and "more uncertainty" is examined by Jones and Ostroy (1984).

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increase in uncertainty on the level of early consumption. This is in spite of the lack of a strong relationship between the notions of a Rothschild-Stiglitz (1970) increase in risk and of a Blackwell (1951) increase in uncertainty.3 We exhibit in this paper the necessary and su±cient condition on the utility function to sign the e®ect of a better information structure on the optimal level of prevention. There exist many results in the economic literature on the role of uncertainty and learning in dynamic optimization problems. The literature on certainty equivalence established conditions to guarantee that uncertainty does not a®ect the optimal course of action. Simon (1956) showed that this result holds if the objective function is quadratic. Malinvaud (1969) relaxed this assumption but at the cost of limiting the analysis to small risks. These two results suggest that the direction of the e®ect of uncertainty on optimal decisions will depend on properties of the third derivative of the utility function. Other papers emphasized the role of learning. Spence and Zeckhauser (1972), Drµeze and Modigliani (1972) and Kreps and Porteus (1978) pointed out that individuals regard uncertainties resolved at di®erent times as being di®erent. It is thus a widespread belief that the e±cient dynamic management of uncertainties should be di®erent if we expect new scienti¯c knowledges to emerge in a near future rather than in a distant one. Epstein (1980) examined the direction in which the optimal decision changes with learning. He developed a method to investigate the comparative statics of a better information structure. In particular, he provided an elegant proof that the prospect of more information increases the incentive to preserve °exibility.4 In the area of environmental irreversibilities, earlier studies showed that an agent should take stronger actions to prevent future irreversible risks if he expects obtaining information (see Arrow and Fisher (1974) and Henry (1974)). Thus, learning should in°uence action in favor of the environment. However, the irreversible nature of our decision today is only one of the elements entering the picture of environmental problems. Most existing models dealing with learning and irreversibility are such that the decision today a®ects future welfare only through the constraint of the opportunity set available later. To illustrate, for the problem of global warming, the irreversibility constraint is our inability to emit a negative quantity of greenhouse gas, i.e., to use energy to remove the excess of gas. We believe that the problem is not so much that we will not be able to remove the excess of CO2 in the future, but rather that we will have to reduce the emission of CO2 if we emit too much of it today. The prospect that one will have to reduce emissions so much that one will want to emit a negative volume of gas seems rather unlikely. But we do not deny this possibility in this paper. Our point is that most environmental problems deal 3

An increase of uncertainty µa la Blackwell can be seen as a mean-preserving spread in the set of distribution function. 4 See also Jones and Ostroy (1984).

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with the management of a limited stock of a good. Varying this stock through early consumption generates an environmental externality for future generations. In our paper, we consider the problem of the consumption of a substance which may turn out to be toxic in the future. The ¯nal payo® for society depends upon the product of the accumulated consumption and a random variable which represents the intensity of the risk. As noticed by Freixas and La®ont (1984) the irreversibility e®ect can go in the counter-intuitive direction when the welfare depends directly on the ¯rst period decision. In particular, Ulph and Ulph (1997) and Kolstad (1996) showed that su±cient conditions established by Epstein (1980) are not directly applicable when taking into account the accumulation phenomenon mentioned above.5 Because of the di±culty to derive analytical conditions in this case, recent contributions used numerical simulations, as in Nordhaus (1994), Ulph and Ulph (1997) and Torvanger (1997). We believe that this paper is an important contribution to the existing literature. We are the ¯rst to provide a complete description of the necessary and su±cient condition that is tractable for this problem with both accumulation and irreversibility. This is of course at the cost of considering a speci¯c payo® function. But we believe that it has many potential applications. The model is presented in Section 2. A second contribution of this paper is to provide a useful decomposition of the basic problem examined in the literature in primitive components. The e®ect of more uncertainty on the current action entails ¯rst what we call a "precautionary e®ect" that takes place even without any sort of irreversibility. In short, a change in the learning that is anticipated will a®ect the optimal exposure to risk in the future. According to Kimball (1990), this change in the future risk induces the decision makers to take precautionary actions today. This e®ect is examined in Section 3. The existence of an irreversibility constraint adds another e®ect. The fact that the decision maker knows that he will be constrained in his future actions is expected to a®ect his current actions if he is not myopic. Moreover, since the intensity of this constraint in the future depends in general upon current actions, more uncertainty should induce the agent to let more options opened for the future. This "irreversibility e®ect" is considered in Section 4. An important result is that the comparative statics analysis is basically the same with or without the irreversibility constraint. 5

Kolstad (1996) observed that another kind of irreversibility may be involved in the context of stock externalities: "If one invests in pollution control and then learns that the damage is low one cannot instantly reduce the abatement capital stock". See also Pindyck (1991) and Dixit (1992) for an analysis of the irreversible nature of some investments.

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2 2.1

The model The economic problem

To illustrate our basic model, let us consider the speci¯c case of the greenhouse e®ect. Suppose that carbon and oxygen are at free disposal. Combining them leads to the production of energy and CO2: C + O2 ! CO2 + 1 unit of energy

(1)

The use of energy yields utility for Society, but the accumulation of carbon dioxide is likely to lead to global warming, which is potentially detrimental for Society. The following decision problem involving the sequence of decisions c1 and c2 is the simplest one to examine the greenhouse e®ect: max u(c1 ) + Ey~ fmax Ex~j~y v(c2 ¡ x~(±c1 + c2 ))g: c1 ¸0

c2 ¸0

(2)

Variable ct is the quantity of energy produced in period t by using carbon. This energy combined optimally with other non-polluting sources yields welfare u(c1 ) in period 1. But the stock of CO2 in the future generates in turn climate changes that a®ect the economy. The economic cost of these changes is measured in energy-equivalents. It is assumed to be proportional to the stock ±c1 + c2 of CO2 in the atmosphere. The parameter ± denotes the fraction of the current gas emission that survives to the next period, i.e., 1 ¡ ± is the rate of natural decay of CO2 in the atmosphere. The greenhouse e®ect is an example in which ± < 1. For some other applications, one may expect ± > 1. In many instances, as for low radiological exposures, adverse e®ects on health appears randomly only in the long run. For this kind of applications in which e®ects appear with delays, early consumptions may have a larger impact on health than late exposures. Random variable x~ is the damage to society per unit of accumulated carbon dioxide in period 2. Society has some prior beliefs on the distribution of x~. It is known that some information y~ will °ow over time on the distribution of x~. Current estimates of the greenhouse e®ect provides an a priori distribution for x~. Researchers worldwide work on re¯ning these estimates by performing experiments whose outcomes will allow society to revise the distribution of x~ according to the Bayes' rule. Program (2) is the problem of a social planner who has to determine the socially e±cient level of consumption at every date given the ambiguity on the distribution of the future risk generated by this consumption. We assume time separability of preferences. Utility functions u and v are assumed to be three times di®erentiable, with u0 > 0, u00 < 0, v0 > 0 and v 00 < 0. The concavity of v implies that the second-order condition for c2 is satis¯ed. We assume that there always exists a solution to that problem. It implies that the maximization

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problem on c1 is also well-de¯ned. We also assume that this program is concave, with a unique positive solution.6 Some technical constraints may limit the choice of c2 in period 2. For example, it is well-known that the chemical reaction (1) cannot be reversed at an acceptable economic cost, i.e., the reaction is irreversible. It implies that c2 is constrained to be non-negative. The program includes such a constraint. More generally the constraint could take the form c2 2 L(c1 ), where L(c1) is the opportunity set in period 2. Most of the literature on irreversibility and learning focuses the attention on the impact of the relation between the ¯rst period decision and the second period constraint.7 It is crucial that, in our model, the ¯rst period decision also a®ects second period welfare directly. The gas accumulates in the atmosphere, which increases the exposure to greenhouse risk in the future. To simplify the notation, we will hereafter rewrite problem (2) as max u(c1 ) + Ey~ max Ez~j~y v(¡±c1 + z~C); c1 ¸0

C¸±c1

(3)

where C = ±c1 +c2 is the accumulation of carbon dioxide in period 2 and z~ = 1¡ x~ is the marginal bene¯t of using energy, net of the loss due to the greenhouse e®ect that it generates. The irreversibility constraint take the form C ¸ ±c1 under this notation.

2.2

Information structure

The random variable z~ is assumed to be discrete with an arbitrary number m of atoms (z1 ; z2 ; :::; zm ); and will be referred to as the "underlying risk". In period t = 1, society has some prior beliefs about the distribution of z~, which is characterized by a prior probability distribution on support (z1 ; :::; zm ). After deciding on c1, but before taking decision on C, the decision maker observes the realization y of a discrete random variable y~. In the terminology of Blackwell (1951), y~ describes an "experiment". Because y~ and z~ can be statistically related, this observation may change the beliefs on z~, which in turn has an impact on the e±cient level of consumption c1 . We denote ¼y (z) for the probability that z~ = z conditional to receiving signal y, and ¼y = (¼y (z1 ); :::; ¼y (zm )). Let us Pm de¯ne S = f¼y 2 Rm + j i=1 ¼y (zi ) = 1g as the set of conditional distributions on z~. Together with Epstein (1980), we are interested in determining the comparative statics of a better information structure. In our model, it means determining how an experiment y~ that is better than experiment y~0 in the sense of Blackwell (1951) a®ects the e±cient level of consumption prior to the observation of the outcome of the experiment. 6

This is assumed for expositional simplicity. The results are valid for any function u(c1 ) provided that c2 is ¯nite and uniquely de¯ned. 7 An exception is Ulph and Ulph (1997).

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We compare two economies with the same characteristics except for the experiment they use to update their belief on z~. In the ¯rst (resp. second) economy, the experiment leads to the distribution of signals y~ (resp. y~0 ), with conditional distribution for z~ characterized by vectors ¼y (resp. ¼y0 0 ) for every potential realization y of y~ (resp. y~0 ). Following Blackwell (1951) and Marschak and Miyasawa (1968), we say that experiment y~ is more informative than experiment y~0 if and only if: for any ½ convex on S : Ey~½(¼y~) ¸ Ey~0 ½(¼y0~0 ):

(4)

¼ = Ey~¼y~ = Ey~0 ¼y0~0 ;

(5)

P

Function ½ in (4) generalizes the entropy function ¡ i ¼y (zi ) log ¼y (zi ) which is commonly used in information theory. The entropy function is concave on S and is an inverse measure of informativeness. Notice that condition (4) implies that

i.e. the two economies have the same prior beliefs on z~. Only the experiment di®ers. As shown for example by Marschak and Miyasawa (1968), condition (4) is equivalent to the fact that all expected-utility maximizers prefer to live in economy y~ than in economy y~0 .8 It implies in particular that : Ey~ max Ez~j~y v(¡±c1 + z~C) ¸ Ey~0 max Ez~j~y 0 v(¡±c1 + z~C): C¸±c1

C¸±c1

(6)

for any c1 and any function v. Let us denote j(c1; ¼y ) = maxC¸±c1 Ez~jy v(¡±c1 + z~C) = Ez~jy v(¡±c1 + z~C(c1 ; ¼y ))

(7)

as the value function of the dynamic problem obtained from solving the second period problem, where C(c1 ; ¼y ) is the optimal risk-taking decision if the ex post distribution of z~ is ¼y . Inequality (6) can then be rewritten as: Ey~j(c1; ¼y~) ¸ Ey~0 j(c1 ; ¼y0~0 )

(8)

Two extreme cases are often examined in the literature. First, the information is perfect when z~ conditional on y~ = y is degenerate for any y. Second, if y~ and z~ are statistically independent, then experiment y~ is non informative. Another situation often encountered in the Bayesian literature is when y~ and z~ are normally distributed. In this case, a better information such as the one de¯ned in (4) is equivalent to rz2~y~ ¸ rz2~y~0 , where rz~y~ and rz~y~0 are the correlation coe±cients for the corresponding pairs of random variables. 8 The equivalence with the notion of a ¯ner partition of the states of nature is also wellknown. Notice that Marschak (1951) showed that a less informative structure is obtained by "garbling" the messages of the initial information structure. See Hirshleifer and Riley (1992) for more details.

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Just a few words on the terminology. There are three equivalent terms available to refer to two economies satisfying property (4). We will use in this paper the concept of "more information". Epstein (1980) used the concept of an "early resolution of uncertainty". Finally, Jones and Ostroy (1984) emphasized the equivalence between a better information structure and more uncertainty. A better experiment implies that the beliefs will be more sensitive to the information. This increase in variability of beliefs leads to more uncertainty/ambiguity about the distribution of potential damages ex ante.

2.3

Some preliminary results

The e±cient level of early consumption in the economy y~ is the solution of program (3), which is rewritten as max u(c1 ) + Ey~j(c1; ¼y~): c 1

(9)

Since the objective function is assumed to be concave in c1 , it is immediate that the e±cient level of emissions in period 1 decreases (resp. increases) for all u(:) with any better information structure if and only if Ey~jc1 (c1 ; ¼y~) is smaller (resp. larger) than Ey~0 jc1 (c1; ¼y0~0 ), where ¼y~ and ¼y0~0 satisfy condition (4). This result should be used for c1 being the optimal solution of program (9). But because concave function u is arbitrary, so is c1 . The next Lemma provides another way to determine whether better information decreases early consumption. It directly follows from the combination of the above remark and the de¯nition (4) of a better information structure. This is a particular case of Theorem 1 in Epstein (1980). Lemma 1 The e±cient level of emissions in period 1 decreases (resp. increases) with any better information structure if and only if jc1 (c1; ¼) is concave (resp. convex) in ¼. The remaining of the paper is organized as follows. We ¯rst determine the solution of the problem with no irreversibility constraint, i.e., when the constraint C ¸ ±c1 is not binding. This is done in Section 3. In Section 4, we examine the same problem, but with the irreversibility constraint.

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The comparative statics without the irreversibility constraint

In this section, we examine problem (3) when C(c1 ; ¼y ) > ±c1 and C(c1; ¼y0 0 ) > ±c1 : That is to say when the information received cannot be disastrous enough

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to induce a complete collapse of the production. Our objective here is to isolate the "precautionary e®ect" due to the accumulation of carbon dioxide in the atmosphere. By the envelope theorem, we have jc1 (c1; ¼) = ¡±Ev 0 (¡±c1 + z~C(c1; ¼));

(10)

E z~v0 (¡±c1 + z~C(c1 ; ¼)) = 0;

(11)

where ¼ is the distribution of z~. Even without an irreversibility constraint, determining whether jc1 (c1 ; ¼) is concave in ¼ is technically complex. Indeed ¼ ¯rst appears through the expectation operator in a linear way. But it also appears as an argument in C, the solution of the second period problem when z~ is distributed as ¼. This link between C and ¼ is implicitly de¯ned by the ¯rst-order condition for problem (7):

where z~ has a probability distribution ¼. In most cases, unambiguous comparative statics results can be obtained only when restricting the model.9 As it will appear later on, two functions play an important role in the analysis. The ¯rst one is the well-known Arrow-Pratt coe±cient of absolute risk aversion, denoted A(:) = ¡v00 (:)=v0 (:). The second one is the coe±cient of absolute prudence, denoted P (:) = ¡v000 (:)=v00 (:). It has been introduced by Kimball (1990) to measure the strength of the precautionary saving motive. The larger P is, the larger is the willingness to save to forearm oneself against future exogenous risks. The necessary and su±cient condition for this to be true is positive prudence, i.e. P (s) ¸ 0 for any s. A stronger condition than prudence is decreasing absolute risk aversion A0 · 0, which is equivalent to P ¸ A. In order to explore the e®ect of a better information structure on c1 , let us assume for the time being that z~ is a "small risk" in the sense of Samuelson (1970). Using a ¯rst order Taylor approximations for v0 (zC ¡±c1 ) around v0 (¡±c1 ) in condition (11), we obtain: ¡v 0 (¡±c1 ) E z~ C(c1 ; ¼) » : = 00 v (¡±c1) E z~2

(12)

Similarly, using a second order Taylor approximation for v0 in (10) yields h

i

jc1 (c1 ; ¼) » = ¡±v0 (¡±c1 ) ¡ ±E C z~v 00 (¡±c1 ) + 0:5C 2z~2 v000 (¡±c1 ) : Replacing C in (13) by its expression in (12), we ¯nd : h

(E z~)2 jc1 (c1; ¼) » = ¡±v0 (¡±c1 ) + ± E z~2 v 0 (¡±c1 ) ¡ 0

= ¡±v (¡±c1 ) +

z~)2 (v0 (¡±c1 ))2 ± (E E z~2 ¡2v 00 (¡±c1 )

(v0 (¡±c1 ))2 v000 (¡±c1 ) 2(v00 (¡±c1 ))2

i

(13)

(14)

[2A(¡±c1 ) ¡ P (¡±c1 )] :

9 For more details on the di±culty to derive unambiguous comparative statics of a better information structure, see Epstein (1980).

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z~) One can easily verify that (E is a convex function of ¼. Then jc1 is concave E z~2 in ¼ for small risks if and only if P (¡±c1) is larger than 2A(¡±c1 ). Because c1 and u are arbitrary, this condition should hold for any c1 . It yields the following result:

Proposition 1 Suppose that the irreversibility constraint is never binding. A necessary condition for a better information structure to reduce (resp. increase) the e±cient level of emissions in period 1, independent of the ¯rst-period utility function u and the distribution of potential damages, is that absolute prudence be larger (resp. smaller) than twice the absolute aversion to risk (P ¸ 2A). This condition is su±cient if all posterior distributions of potential damages are small risks in the sense of Samuelson. The condition on v is necessary in the sense that if it is violated, then one can ¯nd a concave function u, and two information structures, one being ¯ner than the other such that one emits more greenhouse gas in the economy with the ¯ner information structure. A potential intuition for this result is as follows: because EC z~ is convex in ¼,10 a better information structure increases the expected consumption of society which follows its optimal signaldependent strategy. By consumption, we mean the monetary value of using energy net of the damages due to global warming. This increase in expected consumption is obtained from the more e±cient level of emissions due to better knowledge of the risk. It generates a wealth e®ect: the increase in consumption generated by the better information structure tends to induce society to worry less about the future. It tends to increase the willingness to use energy in period 1. The intensity of this e®ect depends upon the degree of concavity of v, as measured by A. But another e®ect takes place. More information in the future implies the optimal exposure to risk to be more sensitive to signals. It yields an increase in risk ex ante. It a®ects the expected marginal utility of future consumption, which in turn tends to reduce the optimal level of early consumption if marginal utility is convex. The strength of this e®ect thus relies on the convexity of v 0 , which is measured by P . This is a precautionary e®ect. Proposition 1 states that the Precautionary Principle can be justi¯ed only if the precautionary e®ect has a bigger impact than the wealth e®ect. Notice that the condition P ¸ A, i.e. decreasing absolute risk aversion is weaker than condition P ¸ 2A. Thus, decreasing absolute risk aversion is necessary, but not su±cient, to economically justify the Precautionary Principle. To determine whether this necessary condition is also su±cient, let us ¯rst explore this problem when the utility function belongs to the well-known class of HARA functions such that 10

This is true for small risks.

10

"

1 x v(x) = ´+ 1¡° °

#1¡°

:

(15)

Some restrictions on the parameters are required to make v increasing and concave. In particular, we require ´ + x° to be positive. This is the set of functions whose absolute risk tolerance is linear in x. Indeed, we can easily verify that ¡v0 (x)=v00 (x) = ´ + °x . If ´ = 0, we get the standard Constant Relative Risk Aversion (CRRA) functions, with constant relative risk aversion °. We get Constant Absolute Risk Aversion (CARA) functions if ° tends to in¯nity. We get the quadratic utility function for ° = ¡1. Since "

x A(x) = ´ + °

#¡1

"

1+° x and P (x) = ´+ ° °

#¡1

;

(16)

P > 2A if and only if ° 2]0; 1[: This necessary condition for the Precautionary Principle to be economically justi¯ed in our model is indeed su±cient for our problem. Proposition 2 Suppose that the irreversibility constraint is never binding. Suppose also that v is HARA. Then, a better information structure reduces (resp. increases) the e±cient level of emissions in period 1 if P > 2A (resp. P < 2A). In the case with ° = 1 (P = 2A), i.e. v(x) = ln(´ + x), the structure of the information has no e®ect on the e±cient level of emissions. Proof. It is easy to check that if v is de¯ned as in (15), then the optimal C is proportional to ´ ¡ ± c°1 . More precisely, we obtain C(c1 ; ¼) = c(¼)(´ ¡ ± c°1 ), with z ¡° ) = 0. It yields c(¼) de¯ned by E z~(1 + c~ ° j(c1; ¼) =

1 c1 (´ ¡ ± )1¡° g(¼) 1¡° °

(17)

z 1¡° with g(¼) = E(1 + c(¼)~ ) . By the de¯nition of the concept of a better infor° mation structure, we know that j is convex in ¼. Thus, g is convex in ¼ if ° is less than 1, whereas it is concave in ¼ if ° is larger than 1. But we have

± c1 jc1 (c1 ; ¼) = ¡ (´ ¡ ± )¡° g(¼): (18) ° ° We conclude that jc1 is concave in ¼ only if ° is strictly positive and less than unity. Lemma 2 concludes the proof. The case of the logarithmic utility function is let to the reader. In the special case of ´ = 0, we get the standard power utility functions with constant relative risk aversion. Thus, to determine whether a better experiment reduces early consumption in this case, one should check whether relative risk

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aversion is less than unity. There is much controversy on this, either on the theoretical perspective, and on the empirical one. Econometric estimates of ° vary from 0:1 to 100. Drµeze (1981) suggested that relative risk aversion be something between 2 and 3. In order to solve the equity premium puzzle, one would require ° to be around 30. If this is true, then a better experiment should rather increase early emissions of carbon dioxide! Because constant absolute risk aversion implies P = A < 2A, Proposition 1 suggests that a better experiment should rather increase early emissions under CARA. Proposition 2 con¯rms this conjecture, since CARA implies ° = +1 > 1. Observe that this result goes the opposite direction than the one recommanded by the Precautionary Principle. Our conjecture has long been that condition P ¸ 2A is necessary and su±cient in the general case. This is not true, as proven by the following result. Proposition 3 Suppose that the irreversibility constraint is never binding. If the utility function v is not HARA, then the e®ect of a better information structure is ambiguous in the following sense: there exist an underlying risk z~ and three experiments y~; y~0 and y~00 such that y~0 and y~00 are each less informative than y~, and such that the optimal c1 for y~0 exceeds the optimal c1 for y~, but the optimal c1 for y~00 is less than the optimal c1 for y~. Proof. Consider two random variables x~1 and x~2 such that E x~i v0 (¡±c1 + x~i ) = 0:

(19)

De¯ne z~i = ¹i x~i , i = 1; 2. Let also ¼i denote the distribution of z~i . From (19), we know that C(c1; ¼i ) = ¹1i . Let z~ be a risk equal to z~1 with probability 12 ; and z~2 with probability 21 : Consider two experiments. In experiment y~, two signals may be received with equal probability, leading to posterior distributions z~1 and z~2 respectively. The other experiment correspond to no information. Consider the following program: (

max H(¹1 ; ¹2 ) = ¡± ¹ ;¹ 1

2

2 X 1 i=1

2

)

0

Ev (¡±c1 + ¹i x~i )

¡

(

¡±

2 X 1 i=1

2

0

)

Ev (¡±c1 + x~i )

(20)

subject to constraint 2 X 1 i=1

2

E¹i x~i v 0 (¡±c1 + ¹i x~i ) = 0:

(21)

Condition (21) states that the agent selects C = 1 in the non informative system, whereas the sign of H determines the e®ect of the better information structure on early consumption. Following Lemma 1, a necessary condition for a 12

better information structure to reduce early consumption is H(¹1 ; ¹2 ) ¸ 0 for any x~i satisfying condition (19) and ¹i sastifying condition (21). The symmetric condition holds for the opposite comparative statics property. Since H(1; 1) = 0; ¹1 = ¹2 = 1, must be a global extremum for program (20). The ¯rst order condition for this problem is written as n

o

E x~i v00 (¡±c1 + ¹i x~i ) = ¡º E x~i v0 (¡±c1 + ¹i x~i ) + ¹i E x~2i v00 (¡±c1 + ¹i x~i ) ; (22) for i = 1; 2. Thus a necessary condition is that (22) be satis¯ed for ¹1 = ¹2 = 1. If this condition is not sati¯ed, then H alternates in sign in the neighborhood of (1; 1), and the comparative statics will be ambiguous in the sense presented in the Proposition. Using condition (19), this condition is equivalent to requiring that E(~ xi + º x~2i )v 00 (¡±c1 + x~i ) = 0:

(23)

Given that this condition must be satis¯ed for all pair of random variables verifying condition (19), the value of the coe±cient º must be the same for all such random variables. In short, one must check whether there exists º such that: E x~v0 (¡±c1 + x~) = 0 =) E(~ x + º x~2)v 00 (¡±c1 + x~) = 0:

(24)

This is possible only if there exists a couple (º; ¯) such that for all x: x(1 + ºx)v00 (¡±c1 + x) = ¯xv0 (¡±c1 + x), or, equivalently, ¡

v0 (¡±c1 + x) 1 º = ¡ ¡ x: 00 v (¡±c1 + x) ¯ ¯

(25)

This is possible only if absolute risk tolerance is linear in wealth, i.e. if v is HARA. Proposition 3 shows that, apart from the case of HARA preferences (which is quite restrictive), one cannot defend the Precautionary Principle on basis of the attitude toward risk only, irrespective of the nature of the risk faced. One direction of research is thus to focus on the risk itself, viewing the Principle as applying to particular types of risk. Suppose for example that the posterior distributions of z~ have a ¯xed two-point support. Either the severity of the damage to society is "low" | no damage at all is a possibility | or the severity of the damage is "large". This captures many situations, and in particular the risk of major catastrophies.

Proposition 4 Suppose that the irreversibility constraint is never binding. Suppose also that the distribution of potential damages has a two-atom support. A 13

better information structure reduces (resp. increases) the e±cient level of emissions in period 1 if and only if absolute prudence is larger (resp. smaller) than twice absolute risk aversion: P (s) ¸ 2A(s) (resp. P (s) · 2A(s)) for all s. Proof. See the Appendix. The consequences of Proposition 4 are twofold. First, it provides an alternative proof for the necessity of P ¸ 2A to justify the Precautionary Principle, i.e. for Proposition 1. In addition, it shows that this condition is also su±cient for the Precautionary Principle when the risk of damage to society is of the 0-1 type. It con¯rms that the natural extension of the condition obtained for power utility functions, i.e. relative risk aversion be less than unity, is whether absolute prudence be larger than twice absolute risk aversion. The limit case of 1 for the coe±cient of relative risk aversion is not new in the theory of ¯nance.11 Determining whether P is larger or smaller than 2A appeared more recently in di®erent contexts. Notice that this condition is equivalent to 1=v0 being concave. Gollier and Kimball (1996) showed that this condition is necessary and su±cient for opening a new asset market to increase savings. Drµeze and Modigliani (1972) examined the reduction in the level of savings due to the introduction of complete insurance markets. They proved that the level of savings 2 decreases in such a circumstance if and only if d (Udc1 2=U2 ) ¸ 0 where Uj denotes 2 partial derivatives of a non time-separable utility function U (c1; c2 ). Assuming U(c1 ; c2 ) = u(c1 ) + v(c2 ) this condition becomes 1=v0 convex. Condition P ¸ 2A has also been used in the economics of asymmetric information. Gabel and Sinclair-Desgagn¶e (1996) for example considered the problem of optimal audit in the principal-agent problem. If the principal is risk-neutral and the agent has utility function v on his incomes, the expected wage after an audit occurs is larger than the wage when no audit takes place if and only if P ¸ 2A. Dionne and Fombaron (1996) considered the problem of the e±cient allocation of risks in an economy with adverse selection. They showed that P ¸ 2A is su±cient for the e±cient frontier in the income-states space to be convex. To sum up, we examined in this section problem (3) when there is no constraint on the exposure to risk. We have shown that a better information structure reduces the ¯rst period emission of carbon dioxide only if absolute prudence is larger than twice the absolute aversion to risk. This is su±cient if the utility function is HARA, or if the future risk has a two-point support. 11 See for example Grossman and Vila (1992) who showed that a liquidity constraint a®ects the optimal portfolio of a non-myopic investor one way or another depending upon whether constant relative risk aversion is larger or smaller than 1.

14

4

The irreversibility e®ect

We now examine problem (3) when condition C ¸ ±c1 can be binding. This irreversibility constraint will a®ect the level of emissions in period 2. It has in turn an e®ect on the value function j that determines the optimal c1 . As before, we want to determine whether more uncertainty leads to less emissions in period 1. Since the initial information structure is arbitrary, it is clear that the necessary conditions obtained in the reversible case are still necessary in the irreversible case. To prove that, take two information structures where the posterior distributions of z~ are favorable enough so that constraint C ¸ ±c1 is nowhere binding. Thus, our work here simpli¯es to determining whether the su±cient conditions obtained in the reversible case are also su±cient in the irreversible case. De¯ne set ¦(c1 ) = f¼ 2 S j E z~v 0 (¡±c1 + ±c1z~) ¸ 0g:

(26)

¦(c1) is the set of probability distributions for which constraint C ¸ ±c1 is not binding, for a given level of early emission c1 . Thus, if ¼ is in ¦(c1 ), then the investment C(c1 ; ¼) that is optimal for the constrained problem satis¯es condition (11). Otherwise, C(c1; ¼) = ±c1. We want to know whether jc1 (c1; ¼) =

(

¡±Ev0 (¡±c1 + C(c1 ; ¼)~ z ); if ¼ 2 ¦(c1); ¡±E(1 ¡ z~)v0 (¡±c1 + ±c1 z~); if ¼ 2 = ¦(c1)

(27)

is concave in the distribution ¼ of z~, i.e., whether jc1 (c1; ¸1¼1 + ¸2 ¼2) ¸ ¸1 jc1 (c1; ¼1 ) + ¸2 jc1 (c1; ¼2 );

(28)

for all ¼1 ; ¼2 in S, and all positive ¸1; ¸2 such that ¸1 + ¸2 = 1. Observe that ¸1 ¼1 + ¸2 ¼2 is in ¦(c1 ) (resp. in ¦(c1)) if ¼1 and ¼2 are in ¦(c1 ) (resp. in ¦(c1 )). Thus, we may decompose the problem into four parts. Suppose ¯rst that ¼1 and ¼2 are in ¦(c1 ). The concavity of jc1 in the region where the constraint is not binding is examined in the previous section. Suppose alternatively that ¼1 and ¼2 are in ¦(c1). In this case, jc1 is linear in ¼. Third, suppose that ¼1 is in ¦(c1), that ¼2 is in ¦(c1 ), and that ¸1 ¼1 + ¸2 ¼2 is in ¦(c1 ). Inequality (28) is rewritten here as ¸1 E(1 ¡ z~1)v 0 (±c1(¡1 + z~1 )) + ¸2 E(1 ¡ z~2 )v0 (±c1 (¡1 + z~2 ))

(29)

· ¸1 Ev0 (¡±c1 + ®~ z1) + ¸2E(1 ¡ z~2 )v0 (±c1 (¡1 + z~2 ));

where z~i is distributed as ¼i , and where ® = C(c1 ; ¼1 ) ¸ ±c1 . This is equivalent to Ev0 (¡±c1 + ±c1 z~1) + [¡E z~1 v0 (¡±c1 + ±c1z~1 )] · Ev0 (¡±c1 + ®~ z1 ) 15

(30)

Notice that the second term in the left-hand side of the above inequality quanti¯es the \quasi-option" e®ect that is central in the literature on irreversibility. It measures the bene¯t (in Ev units) of relaxing constraint C ¸ ±c1 by one unit of energy. By de¯nition (26) of ¦(c1 ), this term is always negative. Thus, the quasi-option e®ect is always compatible with the Precautionary Principle. Thus, we would be done if E z~1v 0 (¡±c1 + ®~ z1 ) = 0 =) Ev0 (¡±c1 + ±c1 z~1) · Ev0 (¡±c1 + ®~ z1 )

(31)

whenever ® ¸ ±c1 . By eliminating the \quasi-option" value from the picture, we decomposed the irreversibility e®ect into its two components. The second one is called the rigidity e®ect. By the rigidity e®ect, we mean the e®ect of exogenously limiting the choice available for the future on current decisions. Signing this e®ect in the direction compatible with the precautionary principle requires condition (31) to hold. In words, condition (31) determines whether allowing a °exible exposure to risk increases the willingness to reduce early emissions. This condition does not hold in general. However we show in Appendix B that this condition holds when P ¸ 2A. The fourth possible case is when ¼1 is in ¦(c1 ), ¼2 is in ¦(c1 ), and ¸1 ¼1 +¸2 ¼2 is in ¦(c1 ). Inequality (28) is rewritten here as ¸1 Ev 0 (¡±c1 + C z~1 ) + ¸2 Ev 0 (¡±c1 + C z~2 )

(32)

· ¸1 Ev0 (¡±c1 + ®~ z1) + ¸2 E(1 ¡ z~2 )v0 (¡±c1 + ±c1z~2 );

where C = C(c1 ; ¸1 ¼1 + ¸2 ¼2 ) ¸ ±c1, ® = C(c1; z~1 ) ¸ ±c1 . We show in the Appendix that, again, the e®ect of irreversibility can be decomposed into a \quasioption" e®ect and a rigidity e®ect. The quasi-option e®ect is always compatible with the precautionary principle. A su±cient condition for the rigidity e®ect to go the same direction is decreasing absolute risk aversion, a condition that is weaker than P ¸ 2A. The important message of this analysis is that condition P ¸ 2A together with the su±cient conditions for the reversible case are su±cient in the irreversible case. Since P ¸ 2A was already necessary in the reversible case, nothing must be added to the su±cient conditions obtained in the previous section, as stated in the next Proposition. Proposition 5 Consider the global warming problem (2) with the irreversibility constraint c2 ¸ 0. Expecting a more precise information on the risk of global warming reduces the current e±cient emission of carbon dioxide if absolute prudence is larger than twice the absolute aversion to risk, and if either the utility function is HARA or the risk is binary. These conditions are necessary in the sense that if they are not ful¯lled, then one can ¯nd a prior distribution of the risk and a better information structure such that the latter increases the e±cient emission of greenhouse gas. 16

Notice that the same result would hold under the more general irreversibility constraint c2 ¸ k(c1 ), with k 0 (c1) · ±.

5

Conclusion

Our interpretation of the Precautionary Principle is that more uncertainty on the distribution of a future risk should induce Society to take stronger prevention actions today. Three arguments may justify the Precautionary Principle. The ¯rst argument is based on the irreversibility of early actions, which entails a "quasi-option" e®ect. More uncertainty should induce the decision maker to favor taking more conservative actions today to value more these options in the future. Arrow and Fisher (1974) and Henry (1974) showed that irreversibility should induce a risk-neutral society to favor current decisions that allow for more °exibility in the future. In a more general model, we isolated this "quasi-option" e®ect in the last section of the paper, and we showed that it always goes the intuitively appealing direction. But decisions in the past do not only in°uence the current opportunity set. They also a®ect welfare directly: past emissions of CO2 increase the current exposure to greenhouse risks due to the accumulation of gas in the atmosphere. This accumulation yields two other e®ects that are not easy to sign, as noticed by Malinvaud (1969), Freixas and La®ont (1984) and Ulph and Ulph (1997). The two additional e®ects that take place under accumulation are respectively a \precautionary e®ect" and a \rigidity e®ect". The precautionary e®ect is the one that occurs when the irreversibility constraint is never binding. We showed in this case that more actions should be taken to prevent future risks when the situation is more uncertain | in accordance with the Precautionary Principle | only if absolute prudence is larger than twice the absolute risk aversion. Or, equivalently, if the inverse of marginal utility is concave. Under constant relative risk aversion, this requires relative risk aversion to be less than unity. Finally, the rigidity e®ect takes place when a constraint on future actions is introduced into the model, but this constraint is independent of the current action in order to separate this e®ect from the quasi-option e®ect. We showed that the conditions to add to the model to control for the rigidity e®ect do not di®er from the ones to control the precautionary e®ect. We obtained some su±cient conditions guaranteeing that more uncertainty in the future generates a more conservative action today. For example, this is the case if the utility function exibits linear risk tolerance together with an absolute prudence exceeding twice the absolute aversion to risk. Notice that if absolute risk tolerance is not linear, one can ¯nd a prior distribution for the risk and a re¯nement of the information structure such that the latter induces Society to take a less conservative action. If one view the linearity of risk tolerance as very restrictive, our result means the Precautionary Principle does not follow only from 17

the attitude toward risk. We showed that if we restrict the set of distributions of risk, this conclusion may be relaxed. One line of extension is thus to focus on the nature of the risk faced in situations where the principle is supposed to apply. Another line of research, which may capture the motives for the principle, is to view the Precautionary Principle as a safe guard against the opportunism of decision makers in situations of asymetric information or imperfect monitoring by the Society.

18

References [1] Arrow, K.J. and A.C. Fischer, (1974), Environmental preservation, uncertainty and irreversibility, Quarterly Journal of Economics, 88, 312-319. [2] Blackwell, D., (1951), Comparison of Experiments, in J. Neyman (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 93-102. [3] Chichilnisky, G. and G. Heal, (1993), Global environmental risks, Journal of Economic Perspectives, 7, 65-86. [4] Dionne, G. and N. Fombaron, (1996), Non-convexities and e±ciency of equilibria in insurance markets with asymmetric information, Economics Letters, 52, 31-40. [5] Dixit, A.K., (1992), Investment and Hysteresis, Journal of Economic Perspectives, 6, 107-132. [6] Drµeze, J.H., (1981), Inferring risk tolerance from deductibles in insurance contracts, Geneva Papers on Risk and Insurance, 20, 48-52. [7] Drµeze, J.H. and F. Modigliani, (1972), Consumption decisions under uncertainty, Journal of Economic Theory, 5, 308-335. [8] Epstein, L.S., (1980), Decision-making and the temporal resolution of uncertainty, International Economic Review, 21, 269-284. [9] Freixas, X. and J.-J. La®ont, (1984), On the irreversibility e®ect, in M. Boyer and R. Kihlstrom (eds.) Bayesian Models in Economic Theory, Ch. 7, Elsevier, Dordrecht. [10] Gabel, H.L. and B. Sinclair-Desgagn¶e, (1997), Environmental auditing in management systems and public policy, Journal of Environmental Economics and Management, 33, 331-46. [11] Gollier, C., and M.S. Kimball, (1996), Toward a systematic approach to the economic e®ects of uncertainty II : Characterizing utility functions, mimeo, University of Michigan. [12] Grossman, S. and J.-L. Vila, (1992), Optimal investment strategies with leverage constraints, Journal of Financial and Quantitative Analysis, 27, 151-168. [13] Henry, C., (1974), Investment decisions under uncertainty: the irreversibility e®ect, American Economic Review, 64, 1006-1012.

19

[14] Hirshleifer, J. and J.G. Riley, (1992), The analytics of uncertainty and information, Cambridge University Press. [15] Jones, J.M. and R.A. Ostroy, (1984), Flexibility and uncertainty, Review of Economic Studies, 13-32. [16] Kimball, M.S., (1990), Precautionary saving in the small and in the large, Econometrica, 58, pp 53-73. [17] Kolstad, C.D., (1996), Fundamental irreversibility in stock externalities, Journal of Public Economics, 60, 221-233. [18] Kreps, D.M., and E.L. Porteus, (1978), Temporal resolution of uncertainty and dynamic choice theory, Econometrica, 46, 185-200. [19] Malinvaud, E., (1969), First order certainty equivalent, Econometrica, 37, 706-718. [20] Marschak, J., and K. Miyasawa, (1968), Economic comparability of information systems, International Economic Review, 137-174. [21] Nordhaus, W.D., (1994), Managing the global commons, The MIT Press, Cambridge. [22] O'Riordan, T. and J. Cameron, (1995), Interpreting the Precautionary Principle, Earthscan Publications, London. [23] Pindyck, R., (1991), Irreversibility, uncertainty and investment, Journal of Economic Literature, 29, 1110-1148. [24] Rothschild, M.J., and J. Stiglitz, (1970), Increasing risk I: A de¯nition, Journal of Economic Theory, 2, 225-243. [25] Samuelson, P.A., (1970), The fundamental approximation theorem of portfolio analysis in terms of means, variances, and higher moments, Review of Economic Studies, 37. [26] Simon, H., (1956), Dynamic programming under uncertainty with a quadratic criterion function, Econometrica, 24, 74-81. [27] Spence, M., and R. Zeckhauser, (1972), The e®ect of timing of consumption and the resolution of lotteries on the choice of lotteries, Econometrica , 40, 401-03. [28] Torvanger, A., (1997), Uncertain climate change in a intergenerational planning model, Environmental and Resource Economics, 9, 103-124. [29] Ulph, A. and D. Ulph, (1997), Global warming, irreversibility and learning, The Economic Journal, 107, 636-50. 20

Appendix A: Proof of Proposition 4 Suppose that z~ has its support in fz1 ; z2g, with z1 > 0 > z2 . Let wi denote ¡±c1 + C(c1 ; ¼)zi , together with p1 = ¡z2=(z1 ¡ z2 ) and p2 = z1 =(z1 ¡ z2). Parameter pi can be interpreted as the pseudo state-price of state i, with p1 +p2 = 1 and p1 w1 + p2 w2 = ¡±c1 . The ¯rst order condition (11) is rewritten as: ¼(z1 )v 0 (w1 ) ¼(z2 )v0 (w2 ) = = ¸(¼): p1 p2

(33)

The ¯rst equality determines C(c1 ; ¼), whereas the second equality is a de¯nition for function ¸(¼). Observe that jc1 (c1; ¼) = ¼(z1 )v0 (w1) + ¼(z2)v 0 (w2 ) = p1 ¸(¼) + p2 ¸(¼) = ¸(¼): ¡±

(34)

Our problem thus simpli¯es in ¯nding the conditions under which ¸(¼) is convex in ¼. Because w1 and w2 are related through condition p1 w1 +p2 w2 = ¡±c1, using condition (33) implies that ¸(¼) is implicitly de¯ned by X i

pi v 0¡1[¸(¼)pi =¼(zi )] = ¡±c1 :

(35)

Let us de¯ne function à such that Ã(w) = v0¡1(1=w). It follows that the above condition can be rewritten as X i

pi Ã(

¼(zi ) ) = ¡±c1 : ¸(¼)pi

(36)

Notice that Ã0 (w) = ¡v0 (Ã(w))2 =v 00 ((Ã(w)) > 0 and à 00 (w) = ¡v0 (Ã(w))4 =v 00 ((Ã(w))2 f¡2 000

v 00 (Ã(w)) v 000 (Ã(w)) + 00 g: v0 (Ã(w)) v (Ã(w))

00

(w) (w) Thus à is convex if and only if ¡v ¸ 2 ¡v , i.e. P (w) ¸ 2A(w). We now v 00 (w) v0 (w) prove that ¸(¼) is convex in ¼ if and only if à is convex in w. Take any pair of conditional distributions ¼1 and ¼2 in the support fz1 ; z2 g, and any t 2 [0; 1]. Let ¤(t) such that

X

pi Ã(

i

t¼1 (zi ) + (1 ¡ t)¼2 (zi ) ) = ¡±c1 ; ¤(t):pi

viz ¤(t) = ¸(t¼1 + (1 ¡ t)¼2 ). Suppose that à is convex. Then, we obtain X i

pi Ã(

t¼1(zi ) + (1 ¡ t)¼2(zi ) 1 ) t¸(¼1 ) + (1 ¡ t)¸(¼2 ) pi 21

(37)

= · =

P

i pi Ã

P

i pi

h

n

t¸(¼1 ) ¼1 (zi ) t¸(¼1 )+(1¡t)¸(¼2 ) ¸(¼1 )pi

+

t¸(¼1 ) ¼1 (zi ) Ã( ¸(¼ ) t¸(¼1 )+(1¡t)¸(¼2 ) 1 ):pi

t¸(¼1 ) t¸(¼1 )+(1¡t)¸(¼2 )

= ¡±c1 :

P

i

¼2 (zi ) (1¡t)¸(¼2 ) ¸(¼1 )+(1¡t)¸(¼2 ) ¸(¼2 )pi

+

¼1 (zi ) pi Ã( ¸(¼ ) + 1 ):pi

o

i

¼2 (zi ) (1¡t)¸(¼2 ) Ã( ¸(¼ ) t¸(¼1 )+(1¡t)¸(¼2 ) 2 ):pi (1¡t)¸(¼2 ) t¸(¼1 )+(1¡t)¸(¼2 )

P

i

¼2 (zi ) pi Ã( ¸(¼ ) 2 ):pi

The inequality is the consequence of the convexity of Ã, whereas the last equality is obtained by using condition (36) for respectively ¼1 and ¼2. Now compare this result with condition (37). Since à is increasing in its argument, we have ¤(t) = ¸(t¼1 +(1¡ t)¼2 ) · t¸(¼1 )+(1¡ t)¸(¼2). This concludes the proof of su±ciency. Necessity is obtained by contradiction: if à is locally concave, there will exist a support fz1 ; z2 g and a distribution ¼ on this support such that ¸(¼) is locally concave. Appendix B: Proof of su±ciency of P ¸ 2A for condition (31) Let us use notation s = ¡±c1 , k = ¡s. We have to prove that E z~v 0 (s + ®~ z ) = 0 and ® ¸ k =) Ev0 (s + k~ z ) · Ev 0 (s + ®~ z)

(38)

if P ¸ 2A. Using the di±dence theorem presented in Gollier and Kimball (1996), condition (B1) holds for all s, ®, and z~ if and only if v0 (s + kz) ¡ v0 (s + ®z) ¡

(k ¡ ®)v00 (s) 0 zv (s + ®z) · 0 v0 (s)

(39)

for all s, z, ® ¸ k. De¯ne Á(x) = 1=v0 (x). Observe that Á0 (x) = ¡v00 (x)=(v0 (x))2 and ¡v000 (x)(v 0 (x))2 + 2(v 00 (x))2 v 0 (x) Á (x) = : (v0 (x))4 00

(40)

Thus Á is concave if and only if P (x) ¸ 2A(x) for all x. Let us rewrite condition (39) as Á(s + ®z) ¡ Á(s + kz) +

Á0 (s) (k ¡ ®)zÁ(s + kz) · 0: Á(s)

(41)

Let us de¯ne z1 = kz and z2 = ®z. Given ® ¸ k ¸ 0, we have either 0 · z1 · z2 or z2 · z1 · 0. Condition (41) is rewritten as 22

Á(s + z2 ) ¡ Á(s + z1 ) ·

Á0 (s) (z2 ¡ z1)Á(s + z1 ): Á(s)

(42)

We now show that P ¸ 2A is su±cient. The concavity of Á implies that Á(s + z2 ) ¡ Á(s + z1 ) · (z2 ¡ z1 )Á0 (s + z1 ):

(43)

By concavity of Á again, and because z1 (z2 ¡ z1 ) is positive by construction, we have that (z2 ¡ z1 )Á0 (s + z1 ) · (z2 ¡ z1 )Á0 (s). It yields Á(s + z2 ) ¡ Á(s + z1) · (z2 ¡ z1 )Á0 (s):

(44)

Finally, by monotonicity of Á, and because z1 (z2 ¡ z1 ) is positive, (z2 ¡ z1 )Á0 (s) 1) 0 is smaller than (z2 ¡ z1 ) Á(s+z Á (s). Combining this result with condition (44) Á(s) yields the su±ciency of P ¸ 2A. Appendix C: Analysis of condition (32) Let denote ¯ = C(c1 ; z~2 ) · ±c1 . We would be done if E z~v0 (¡±c1 + ¯ z~2 ) = 0 =) E(1 ¡ z~2 )v0 (¡±c1 + ±c1 z~2 ) ¸ Ev0 (¡±c1 + ¯ z~2 ) (45) whenever ¯ · ±c1 . Indeed, suppose that the unconstrained jc1 be concave. It would yield ¸1 Ev 0 (¡±c1 +C z~1 )+¸2Ev 0 (¡±c1 +C z~2 ) · ¸1Ev 0 (¡±c1 +®~ z1 )+¸2Ev 0 (¡±c1 +¯ z~2); (46) which would in turn imply inequality (32) by using condition (45). The \quasi-option" value ¡E z~2 v0 (¡±c1 +±c1z~2 ) is always positive (since ±c1 ¸ ¯). Thus, a relaxed version of (45) is: E z~v0 (¡±c1 + ¯ z~2) = 0 =) Ev0 (¡±c1 + ±c1 z~2) ¸ Ev0 (¡±c1 + ¯ z~2)

(47)

In the following we show that this condition holds under decreasing absolute risk aversion. We would be done if @ Ev0 (¡±c1 + k~ z ) = E z~v 00 (¡±c1 + k~ z) @k is nonnegative for all k ¸ C(c1 ; ¼) where ¼ is the distribution of z~. Suppose that P ¸ A which means that v is DARA. Obviously, it also means that ¡v 0 is more concave than v. Thus, ¡v0 (x) = Ã(v(x)) with à increasing and concave. It yields 23

E z~v00 (¡±c1 + k~ z ) = ¡E z~Ã 0 (v(¡±c1 + k~ z ))v0 (¡±c1 + k~ z ):

(48)

E z~v00 (¡±c1 + k~ z) ¸ ¡Ã0 (v(¡±c1 ))E z~v0 (¡±c1 + k~ z ) ¸ 0:

(49)

Since à is concave, it implies that zà 0 (v(¡±c1 + kz)) · zà 0 (v(¡±c1 )) for all z. It yields

The second inequality comes from the assumption that k ¸ C(c1 ; ¼) which is equivalent to E z~v 0 (¡±c1 + k~ z ) · 0.

24