Precautionary decision rules under risk - A general representation result.

Transcription

1 Precautionary decision rules under risk - A general representation result. Preliminary Version, October 1, 2005 Christian P. TRAEGER Department of Economics Research Center for Environmental Economics University of Heidelberg Abstract: This paper formalizes a general class of decision rules conforming with the precautionary principle. To this end a new notion of risk aversion for the multicommodity case (risk aversion on welfare) is introduced. A general representation for preferences satisfying the von Neumann-Morgenstern axioms, time consistency and additive separability of welfare over time on certain outcomes is derived. By explicitly allowing for distinct choices (gauges) of the evaluation function on individual outcomes (Bernoulli utility) my representation theorem allows to relate different setups in the literature. I show that the introduced notion of risk aversion coincides with precautionarity of decision rules. In the one-commodity setting it corresponds to a specified difference between standard risk aversion and intertemporal substitutability. In the general case risk aversion on welfare is shown to be the main determinant of preference for the timing of risk resolution. Keywords: uncertainty, welfare, precautionary principle, expected utility, recursive utility, risk aversion, intertemporal substitutability, certainty additivity, temporal lotteries, gauge-freedom Correspondence: Permanent: AWI - Umweltoekonomie Bergheimer Str Heidelberg Germany traeger@uni-hd.de Present: Department of Agricultural & Resource Economics 207 Giannini Hall #3310 University of California Berkeley, CA

2 Precautionary decision rules under risk 1 Introduction Recently Hahn & Sunstein (2005) predicted in the Economists Voice that Over the coming decades, the increasingly popular precautionary principle is likely to have a significant impact on policies all over the world. However there is an ongoing debate between and among economists, environmental scientists and policy makers about the merit and meaning of the precautionary principle. The usual formulations of the principle are rather vague, as discussed for example in Turner & Hartzell (2004) and Sandin (2004). This paper suggests an axiomatic formalization of decision-making under risk that takes up some of the concerns of the precautionary principle. For doing so I introduce a new notion of risk aversion in the multi-commodity case and connect it to the idea of precaution. However, there is a second, quite different point of view motivating the line of thought of this paper. That is, I work out the general time consistent model that falls back to expected utility in the atemporal (or one period) setting and to additivity over time when restricted to certain outcomes. I consider this question particularly interesting as these are the two predominantly used specifications in the respective framing scenarios. Merging the assumptions underlying these respective representations into an overall time consistent framework will not bring about intertemporal expected utility, but a much more general class of representations that also accommodate precautionary decision rules. The third major contribution of this paper is to generalize Epstein & Zin s (1989) approach to distinguish between risk aversion and intertemporal substitutability to a multi-commodity setting. I show that in a world with many commodities absolute values of risk aversion and intertemporal substitutability are good-dependent while a specified difference between the two is an invariant. This invariant closely relates to the introduced notion of risk aversion on welfare (and hence precautionarity). Moreover I establish a general relation between the above mentioned quantities and Kreps & Porteus s (1978) preference for the timing of risk resolution. This answers a question raised by Epstein & Zin (1989, 952 et seq.) on the interlacement of (standard) risk aversion, intertemporal substitutability and the preference for timing of risk resolution. Moreover my representation suggests that risk aversion on welfare is the primitive for determining the timing-preference for risk resolution. My paper relates to the seminal work of Kreps & Porteus (1978) who extend the atemporal von Neumann-Morgenstern setting for choice under uncertainty to a temporal structure. Under the assumption of intertemporal consistency they obtain a recursive 1

3 Introduction representation that uses expected utility evaluation within each period and a generally nonlinear time aggregation from one period to the next. Kreps and Porteus show that an agent behaving in accordance with their axioms generally exhibits a preference for the timing of risk resolution. In terms of the representation result their paper can be seen as an extension of Koopmans s (1960) recursive utility model under certainty to a recursive model for risky settings. My paper shows that even when starting out with a timeadditive model for certain outcomes the general time consistent model for evaluation of risky outcomes will exhibit recursivity. While Kreps & Porteus s (1978) model is attractively general, it seems to invite at the same time for more structure in order to enhance an economic interpretation. Contributing to the intricacy of interpreting their representation is also the fact that it crucially depends on the nonlinear aggregation of utility over time. Working with a utility (or welfare) function that is additive over time on certain outcomes in my view greatly simplifies its interpretation and thus the move from mathematical representation to economic intuition. In such a time additive representation a welfare gain of one unit today and a welfare gain of another unit in the next period is just as good as a welfare gain of two units in a third period (talking about certain welfare gains and taking the pure rate of time preference to be zero). Such a reasoning usually does not hold true anymore in the representation of Kreps & Porteus (1978). Epstein & Zin (1989) reduce 1 Kreps & Porteus s (1978) representation to one commodity in order to extract from it information on the attitudes with respect to risk and with respect to intertemporal substitution. Their (non-axiomatic) setup enables them to decompose (recursive) utility in a way to distinguish between risk aversion and intertemporal substitutability which is not possible within a standard intertemporal expected utility model. My paper will analyze to what extend this specification of risk aversion and intertemporal substitutability can be maintained when taking the model back into the multi-commodity world. It will be found that there is no canonical risk aversion or intertemporal substitutability coefficient anymore as these quantities vary between goods. However a specified difference between the quantities describing risk aversion and intertemporal substitutability will be shown to be an invariant. In fact this difference characterizes risk aversion on welfare. Complementary attempts on formalizing the concept of precaution have been carried out by Gollier, Jullien & Treich (2000), Gollier (2001) and Gollier & Treich (2003), 1 In their main setup Epstein & Zin (1989) only look at one commodity, no history dependence and time aggregation that exhibits constant elasticity of substitution. However they allow for a more general evaluation of uncertainty then implied by the von Neumann-Morgenstern axioms. 2

4 Precautionary decision rules under risk Immordino (2000), Immordino (2003) and Barrieu & Sinclair-Desgagné (2005). Thereby Gollier et al. (2000), Gollier (2001) and Gollier & Treich (2003) look at a simple onecommodity, two-period model in which consumption causes potential damage in the second period. In this model they analyze the effect of newly arriving information between first and second period consumption taking place. The authors label a decision rule precautionary if it implies that whenever welfare improving information about the future is expected, the decision rule implies a reduction in the (potentially harmful) first period consumption as compared to a situation where no information is expected at all. They derive a criterion for such an understanding of precaution to hold in terms of absolute prudence 2 dominating (twice absolute) risk aversion. However Gollier (2001, 312) points out that for decision makers exhibiting constant relative risk aversion this condition is usually regarded unlikely to hold. Influences of irreversibility, market distortions and time inconsistency are discussed in the various papers. While the above model looks at the reaction of a decision-maker in terms of reducing the potential level of harm by reducing first period consumption, Immordino (2000) and Immordino (2003) explore the decision makers willingness to invest in a reduction of the probability that the harmful event takes place. Using the terminology of Ehrlich & Becker (2005) Immordino calls actions that reduce the harm level in case it occurs (like analyzed by the above authors) self-insurance and actions that reduce the probability of the potential harm to take place self-protection. In a similar setup as Gollier et al. (2000) Immordino analyzes when decision rules exhibit precautionarity in the sense of self-protection. Barrieu & Sinclair-Desgagné (2005) define a precautionary strategy as an action that either is selfinsuring or self-protecting and derive a set of rules that a decision maker confronting a potential threat with a described evaluation rule has to satisfy. My paper is structured as follows. The upcoming section 2.1 will motivate my formalization of the precautionary principle. Then section 2.2 precisely defines the concept of general and precautionary uncertainty aggregation rules. 2.3 revisits the atemporal von Neumann-Morgenstern setting. Here I pay special attention to different possibilities of fixing (gauging) the originally ordinal value function over the certain outcomes (Bernoulli utility). Finally section 2.4 briefly looks at the other framing scenario of additively separable preferences over certain outcomes. Section 3 derives the general representation theorems for the intertemporal uncertain setting. First this happens in section 3.1 for the simple certain uncertain case used in 2 A concept introduced by Kimball (1990). It relates to the positiveness of the third derivative of the utility function which can cause an individual to increase (precautionary) savings when facing an increase of risk on future revenues. 3

5 Preliminaries sections 2.1 and 2.2 to motivate my understanding of precaution. Section 3.2 works out the effect of different gauges for Bernoulli utility on the representational form. Finally section 3.3 gives the extension to the general multiperiod framework. Section 4 works out the economic content of the representations. In section 4.1 I discuss Epstein & Zin s (1989) distinction between intertemporal substitutability and risk aversion in the one-commodity case. I show that in the multi-commodity setting only a specified difference between the two is gauge-invariant. Section 4.2 identifies this difference as a notion of risk aversion called risk aversion on welfare and shows that it coincides with the interpretation of precautionarity motivated above. Finally in section 4.3 I extend Kreps & Porteus s (1978) criterion for the preference for the timing of risk resolution to the general gaugeable setting. This allows to relate it to the notions of (standard) risk aversion, intertemporal substitutability and risk aversion on welfare as introduced in this paper. The appendix gives sketches of the proofs. 2 Preliminaries 2.1 The Precautionary Principle The most frequently cited definition of the precautionary principle was given at the Wingspread Conference, agreed upon in 1998 by 32 participants with different academic and professional backgrounds. They state that it is necessary to implement the Precautionary Principle: Where an activity raises threats of harm to the environment or human health, precautionary measures should be taken even if some cause and effect relationships are not fully established scientifically (Raffensperger & Tickner 1999, 8) 3. Now any reasonable economic model depicting uncertainty will take into consideration a threat of harm to human welfare. In a standard model such a threat would be displayed by a positive probability of yielding low welfare. Note that these do not have to be objective probabilities 4 and thus do not have to be based on a complete scientific 3 Iteration added, the Wingspread declaration can also be found online at 4 The empirical definitions of probability by frequency or symmetry are usually referred to as objective probabilities. In the situation described by the Wingspread declaration an epistemologic approach to probabilities seems to better fit the situation. Here probabilities are seen as elements of a (nonbinary) logic or as beliefs. Compare also Kyburg & Smokler (1964). However a public decision maker should should somewhat connect his probabilistic beliefs on scientific evidence. As there will never be an infinite sequence of observation and the symmetry arguments does not apply in these situations, 4

6 Precautionary decision rules under risk understanding. Such a threat of harm would obviously reduce the expected welfare. As implied by the cited phrase a precautionary measure in the Wingspread definition is an action taken before the observed impact on welfare. Hence to formally depict it at least two periods have to be considered. Let me lay out my formal intuition of precautionarity in such a simple model. Let there be a set of multidimensional outcomes giving the level of consumption, effort 5 and harm for a particular state of the world that a decision maker envisions within a period. In the first period such outcome is assumed to be certain and denoted by x 1. For the second period I consider two different outcomes perceived possible. One is a standard or unharmed outcome x 2 and the other an outcome where society suffers serious harm x 2. Furthermore let each of the two second period states be associated with probabilities p( x 2 ) and p(x 2 ) = 1 p( x 2 ). A function u characterizing societies welfare (or an individuals utility) within a period and state of the world is called a Bernoulli utility function. The standard evaluation of such a scenario would be depicted by the following equation: u(x 1 ) + p( x 2 )u( x 2 ) + p(x 2 )u(x 2 ) = u(x 1 ) + E p u(x 2 ) u(x 1 ) + E(p,u). (1) The pure rate of time preference is set to zero for simplicity. 6 Such a model translated into real terms is also the standard cost-benefit-analysis answer for a two period setting with uncertainty, compare for example Brent (1996, 167 et seq.) or Johansson (1993, 142 et seq.). Now in equation (1) with u( x 2 ) > u(x 2 ) the threat of harm p(x 2 )u(x 2 ) obviously diminishes overall welfare. Hence there will always be some willingness on sides of the decision maker to undergo efforts that decrease or prevent the threat of harm p( x 2 )u( x 2 ). In accordance to the Wingspread definition precautionary measures are undergone in the first period to reduce or eliminate the threat of harm in the second period. If these measures come at no cost they would obviously be carried out. So the interesting case is when such precautionary effort lowers the welfare in the first period. A decision maker using equation (1) for his evaluation is willing to accept a reduction of first period welfare in order to eliminate the threat of harm up to u( x 2 ) E u(x 2 ). such probabilities could never be fully established scientifically. 5 Effort stands for various ways and amounts of effort that are undergone in the first period in order to avoid a potential threat of harm in the second period. Usually increasing effort will be assumed to reduce welfare. 6 A positive rate of time preference would not change the general insights. Obviously the below mentioned cost-benefit approaches do work with a positive rate of discount seeking numerical results. 5

7 Preliminaries Now this effort does not seem to suffice the advocates of the precautionary principle. The authors of the Wingspread declaration state explicitly that We believe existing environmental regulations and other decisions, particularly those based on risk assessment, have failed to adequately protect human health and the environment, as well as the larger system of which humans are but a part. However some sort of assessment for the uncertainty is needed. The minimal information would be that some harming scenario is deemed possible while others are not. Such a situation is formalized by Arrow & Hurwicz (1972) who show that decision rules coping with that little information have to be based only on evaluation of the extreme outcomes in order to satisfy certain rationality properties. With respect to a precautionary evaluation it seems to be more reasonable to base the decision on the evaluation of the worst possible outcome than basing it on the best possible outcome. This reasoning is supported by Bossert, Pattanaik & Xu (2000) who show that in such a situation the minimum rule is the only one conforming with uncertainty aversion. 7 However an evaluation that only takes into account the worst possible outcome is usually considered as too extreme. If one is willing to add a little more structure concerning the evaluation of uncertainty, models of ambiguity such as Gilboa & Schmeidler (1989) and its recent generalization by Ghirardato, Maccheroni & Marinacci (2004) can be used. These models work with sets of probability distributions instead of unique probabilities. In Gilboa & Schmeidler s (1989) axiomatization decision makers use the worst probability distribution deemed possible to assess an uncertain situation. With respect to the relation between ambiguity and precaution Gollier (2001, 310 et seq.) criticizes such an attitude as too extreme. However Ghirardato et al. (2004) recently gave a more satisfactory axiomatization which allows for a much broader and more reasonable class of ambiguity attitudes. In my opinion this resolves the criticism and supports that when decision makers are not willing or able to assign a single probability distribution to outcomes, but rather a set of such distributions, there is evidence that some sort of more precautionary decision rule can be needed in order to represent general preferences. What I will show in this paper is however that it is by no means necessary to abandon unique probability distributions in order to have decision rules representing reasonable preferences that yield a more precautionary evaluation than equation (1). As motivated above I will call a decision rule more precautionary than the standard evaluation rule if the decision maker s willingness to reduce welfare in the first period in order to prevent a 7 For a definition of what it means to be uncertainty averse in such a setting compare Bossert et al. (2000, Definition 3). The authors also refine the minimum rule for situations where the extreme outcomes of different scenarios coincide. 6

8 Precautionary decision rules under risk threat of harm p(x 2 )u(x 2 ) is bigger than u( x 2 ) E u(x 2 ). Any decision rule satisfying this criterion within the above evaluation setup will be called a precautionary decision rule. The extreme case of taking the minimum over all second period outcomes perceived possible would imply a willingness to reduce welfare in the first period in order to avoid the threat of harm by u( x 2 ) min x2 u(x 2 ) = u( x 2 ) u(x 2 ). In a scenario where u(x 1 ) = u( x 2 ) this would imply that the decision maker is willing to reduce welfare in the first period to the harm-level u(x 2 ) just to prevent that such welfare level could come up in the second period. To me this extreme seems to be unreasonable. Therefore I will be looking for uncertainty aggregation rules that render an evaluation of the uncertain second period lying somewhere between expected value and the minimum over the possible outcomes. 2.2 Uncertainty Aggregation Rules This section defines the concept of uncertainty aggregation rules. Let X be the set of outcomes x. It is assumed to be a connected compact metric space. The space of all continuous functions from outcomes into the reals is denoted C 0 (X). An element u C 0,u : X IR is called a value function. Its range will be denoted U. Let P s be the set of all simple probability measures on X, i.e. those probability measures having finite support. I refer to the elements p of P s as lotteries. Where no confusion arises the degenerate lottery δ x giving weight 1 to outcome x is written as x P s. Moreover the lottery giving probability p(x) = λ to outcome x and probability p(x ) = 1 λ to outcome x is written as λx + (1 λ)x P s. 8 I call a functional M : P s C 0 IR an uncertainty aggregation rule. It takes as input the decision makers perception of uncertainty expressed by the probability distribution p and his evaluation of certain outcomes expressed by his value function u. For certain outcomes uncertainty aggregation rule s are imposed to return the evaluation function u: M(x,u) = u(x). An obvious example of an uncertainty aggregation rule is given by the expected value operator through: E(p,u) E p u = p(x)u(x). x However, any weighted mean over U IR induces 9 an uncertainty aggregation rule. 8 Note that X is just a compact metric space. In case it is additionally equipped with the structure of a vector space, the vector addition will not coincide with the + used here. 9 I will only look at uncertainty aggregation rules that are induced by weighted means in the following sense. Let p u denote the probability distribution induced from p on X by means of the value function u C 0 on its range U. Then an uncertainty aggregation rule M is said to be induced by a mean M : P s (U) IR whenever M(p,u) = M(p u ) p P s. Mean inducedness implies that only the probability of x is used to weigh u(x). 7

9 Preliminaries The one above corresponds to the arithmetic mean. Others examples for uncertainty aggregation rules are G(p,u) = x for U IR + 10 corresponding to the geometric mean and min x u(x) u(x) p(x) induced by the (maxi-) min principle. All of the above uncertainty aggregation rules are contained for U IR + in the one induced by the power mean M α (p,u) = [ x p(x)u(x) α ] 1 α defined for α IR. 11 It is M 1 (p,u) = E p u and defined by limit M 0 (p,u) lim α 0 M α (p,u) = G(p,u), M (p,u) lim α M α (p,u) = max x u(x) and M (p,u) lim α M α (p,u) = min x u(x). Let me take M α as an example to illustrate the intuition of uncertainty aggregation rules. Assume that an exogenously given u specifies some cardinally measurable value information for the outcomes x X. It will be a major task of the following sections to render a sound basis to this value function assumed here. Now consider a lottery yielding ū = u( x) = 100 with probability p = 0.9 and u = u(x) = 10 with probability p = 0.1. Then an expected value maximizer will evaluate the lottery by the certainty equivalent u c E = u c α=1 = 91. On the contrary a person who is extremely precautious might value the lottery only as high as the worst of its outcomes, that is u c min = uc α= = 10. However, as discussed in the preceding section this is a (too) extreme assessment. As I motivated in that section a general precautionary decision rule in the respective setup should go along with an uncertainty aggregation rule for the second period that renders an evaluation lying somewhere between expected value and the minimum over the possible outcomes. Rewriting the scenario evaluation of equation (1) with a general uncertainty aggregation rule brings about the decision rule u(x 1 ) + M(p,u). (2) The decision rule is considered precautionary if it yields a higher willingness to reduce first period welfare in order to avoid a threat of harm than does equation (1). Within the setup of section 2.1 and with the set P th = {p P s : p( x 2 ),p(x 2 ) > 0, 10 IR + = {x IR : x 0},IR ++ = {x IR : x > 0}. 11 IR denotes the extended real line. For α = 0,,+ define M α by the respective limits. 8

10 Precautionary decision rules under risk p( x 2 ) + p(x 2 ) = 1 with x 2,x 2 X,u( x 2 ) > u(x 2 )} describing the set of potential threat of harm lotteries this corresponds to an uncertainty aggregation rule satisfying: u(x 2 ) < M(p,u) < E(p,u) p P th (3) Such an uncertainty aggregation rule will be called a precautionary uncertainty aggregation rule. 12 Let me assume that the lottery given above describes the threat of harm in the second period. For the uncertainty aggregation rule M α it can be shown that the smaller is α, the lower is the certainty equivalent the respective power mean brings about (e.g. Hardy, Littlewood & Polya 1964). Hence, within this setup, a precautionary decision-maker would be expected to choose a parameter α between and 1 rendering certainty equivalents between 10 and 91. Note that already α = 10 yields an uncertainty evaluation that most would consider very precautious. For the lottery discussed above it brings about a certainty equivalent of u c α= 10 = With an uncertainty evaluation by M α= 10 the decision maker is willing to reduce first period welfare in order to avoid the threat of harm by 87.4 units as opposed to 9 on sides of an expected welfare maximizer. 13 Finally define a more general uncertainty aggregation rule for any strictly monotonic, continuous function f : U IR as [ ] M f (p,u) = f 1 p(x)f u(x), x where f u denotes the usual composition of two functions. The composition sign will often be omitted. This shall not create confusion as usual multiplication of two functions does not appear within this paper. The uncertainty aggregation rule M f contains M α for f(z) = z α and will appear in my representation theorem. The corresponding mean (compare footnote 9) is sometimes known as the generalized (f-)mean. Within this class the precautionary uncertainty aggregation rules are characterized by f being increasing and strictly concave The extreme case, where the individual only looks at the worst outcome to evaluate a lottery, will be excluded in what follows for technical reasons (continuity) and because it is not considered an interesting evaluation rule as pointed out above. That implies excluding α =, in what follows. However note that the (maxi-) min rule can be approximated arbitrarily close with α IR. 13 Let me repeat that a major task of this paper is to render a sound basis to this value function and its cardinality. 14 For all p P s with 0 < p( x 2 ),p(x 2 ) < 1 it holds E(p,u) > M f p( x 2 )u( x 2 ) + p(x 2 )u(x 2 > f 1 [p( x 2 )f[u( x 2 )] + p(x 2 )f[u(x 2 )]] f[p( x 2 )u( x 2 ) + p(x 2 )u(x 2 ] > p( x 2 )f[u( x 2 )] + p(x 2 )f[u(x 2 )]. But 9

11 Preliminaries In what follows I will work with the space of Borel probability measures on X, denoted by P = (X), rather than with its subset P s. The more general elements p P will also be called lotteries. For this purpose extend the above definitions by defining M f : P C 0 IR through 15 M f (p,u) = f 1 [ X ] f u dp. (4) 2.3 Atemporal Uncertainty This section revisits the atemporal or one-period setting of von Neumann & Morgenstern (1944). Their well known representation theorem for choice under uncertainty is slightly extended to allow for a broader class of representations. A useful perspective on the subject matter is that von Neumann and Morgenstern use the originally ordinal character of utility on certain outcomes to single out the additive representation by gauge fixing. A gauge is a degree of freedom within a theory that has no observable effect. If however a cardinal evaluation of certain outcomes is given and gauge freedom no longer prevails, additive representations no longer suffice to represent all decision rules conforming with the vnm-axioms. Such a situation can be given when there is additional information on welfare, for example stemming form intertemporal considerations as carried out in the next section. This idea is made precise below. Let preferences over lotteries be represented by the binary relation. An uncertainty aggregation rule is said to represent the preference relation over lotteries if p q M(p,u) > M(q,u) for all p,q P (5) and some u C 0. It is said to represent for ū C 0 iff equation 5 holds with u = ū. For the following equip the space P of Borel probability measures with the Prohorov metric (generating the topology of weak convergence). The theorem by von Neumann & Morgenstern (1944) in the version of Grandmont (1972, 49) states the following. the latter is just the definition of strict concavity of f. Note that f decreasing and strictly convex describes the same set of uncertainty aggregation rules (compare appendix A). 15 Note that by continuity of f u and compactness of X Lesbeque s dominated convergence theorem (e.g. Billingsley 1995, 209) ensures integrability. 10

12 Precautionary decision rules under risk Theorem (von Neumann-Morgenstern): The axioms A1 (weak order) is transitive and complete, i.e.: transitive: p,q,r P : p q and q r p r complete: p,q P : p q or q p A2 (independence) p,q,r P : p q ap + (1 a)r aq + (1 a)r a [0, 1] A3 (continuity) p P : {q P : q p} and {q P : p q} are closed in P hold if and only if there exists a continuous function u: X IR such that p,q P : p q E p u E q u. (6) Moreover if u represents in the sense of (6) then u : X IR also represents in this sense if and only if there exist a,b IR,a > 0 such that u = a u + b. The theorem states that accepting axioms A1-A3 there exists a value function u on the outcomes such that the uncertainty aggregation rule is of the expected utility form. Now I will ask what happens in the situation when a decision maker has a given evaluation for the certain outcomes. 16 The minimal requirement for a value function to expresses evaluation of certain outcomes is that it coincides with the ordinal requirement that a certain outcome x is preferred over a certain outcome y if and only if the value for x is higher than that for y. I want to call this subset of value functions representing preferred choice on the certain outcomes the set of Bernoulli utility functions B = {u C 0 (X) : δ x δ y u(x) u(y) x,y X}. Obviously the value function in von Neumann-Morgenstern s theorem is a Bernoulli utility function, as well as any strictly increasing transformation of it. Now let me specify the uncertainty aggregation rules that represent the decision makers preference over lotteries in the sense of (5) with a given value function u B satisfying the von Neumann-Morgenstern setup. Proposition 1: Let there be given a binary relation on P and a Bernoulli utility function u B with range U. The relation satisfies axioms A1-A3 if and only if there exists a strictly monotonic continuous function f : U IR such that for all p,q P p q M f (p,u) M f (q,u). (7) 16 Where such a cardinal evaluation can come from will be subject of the following sections. 11

13 Preliminaries Moreover if f represents in the above sense then f : U IR represents in this sense if and only if there exist a,b IR,a 0 such that f = a f + b. 17 Note that the indeterminacy of f up to affine transformations does not translate into the unique functional M. A function f = a f + b with a,b IR,a 0 renders the same uncertainty aggregation rule as f, that is M f (, ) = M f (, ), because the inverse f 1 cancels out the affine displacement of f relative to f. In what follows the group of nondegenerate affine transformations will be denoted A = {a : IR IR : a(z) = a z + b, a,b IR,a 0} with elements a A and the group of positive affine transformations will be denoted A + = {a + : IR IR : a + (z) = a z+b, a,b IR,a > 0}. Let me come back to the perspective given in the beginning of this section. Choice under certainty only renders ordinal information on the value function u which is represented by all members of B. Proposition 1 states that this gauge freedom for Bernoulli utility u translates into the representing uncertainty aggregation rule M through the form of the parametrizing function f. Taking this correspondence the other way round one gets Corollary 1: For any strictly monotonic, continuous function f : U IR the following equivalence holds: A binary relation on P satisfies axioms A1-A3 if and only if M f represents. The latter is: there exists a continuous function u : X U such that p,q P : p q M f (p,u) M f (q,u). (8) Moreover if u represents in the sense of equation (8) then u : X IR represents in this sense if and only if there exist a + A + such that u = f 1 a + f u. Note that obviously u will be a member of B, as for any f it holds that M f (δ x,u) = u(x). The uniqueness of the value function u is no longer up to affine transformations. Indeterminacy of the value function u corresponds to those transformations of u which result in affine transformations of f that leave the uncertainty aggregation rule unchanged. For example in the geometric mean representation the remaining gauge freedom for u after fixing f will be expressed by the group of transformations u u = c u d, c IR ++,d IR\{0} The theorem can also be stated using only increasing versions of f. Then M α would be included in a less obvious way in M f than through f(z) = z α. Strictly decreasing functions are also allowed in the proposition due to the fact that the inverse in (4) cancels out any non-degenerate affine transformation. 18 The easiest way to see this is to note that lim α 0 z α 1 α = ln(z). For any α > 0 the function 12

14 Precautionary decision rules under risk Corollary 1 points out how value functions and representations always come in pairs. For f(z) = z α, α < 1 corollary 1 reproduces von Neumann-Morgenstern s theorem with expected value replaced by a precautionary uncertainty aggregation rule (in the sense of the introduction). For the example M α it states in particular: In the atemporal framework a dispute on whether to apply a precautionary uncertainty aggregation rule or expected value cannot be distinguished from (or can be stated as) a disagreement on the valuation functions over the certain outcomes. This statement will stay true for the general class of precautionary uncertainty aggregation rule s derived in section 4.2. The meaning of choosing a aggregator-value pair will be further discussed in later sections. In this context, the time structure introduced in the next section will play an important role. 2.4 Certainty This short section treats the other framing scenario of additively separable preference over certain outcome paths. Time is discrete with planning horizon T IN such that t {1,...,T }. x t X specifies consumption in period t. A (planned) consumption path is denoted by x = (x 1,x 2,...,x T ). Let x 0 X denote a benchmark consumption. It is arbitrarily fixed and serves to define the shorthand notation [x] (x,x 0,...,x 0 ) for the consumption path that yields the specified consumption x in the first period and the benchmark consumption ever after. In this section the binary relation depicts the preference relation over certain consumption paths. To focus the model I will not introduce time preference in the sense of a positive discount rate. 19 As mentioned before the the model sought for shall be additive over time with respect to certain outcomes. Axiomatizations for this can be found for example in Koopmans (1960) and Radner (1982). I will take it as an assumption itself. A4 (certainty additivity) There exists u : X IR such that for all x,x X T T T x x u(x i ) u(x i). (9) i=1 i=1 f α (z) = zα 1 α is an affine transformation of f(z) = z α so that both parametrize the same uncertainty aggregation rule. 19 However footnotes 26 and 28 will indicate changes in the representation from introducing positive (exponential) discounting. However time preference concerning the timing of the the resolution of uncertainty will naturally arise within my setting. 13

15 The Representation Theorems Note that this axiom also includes the assumptions of stationarity 20 and history independence 21. Again the different value functions representing preferred choice on certain one-period outcomes are called Bernoulli functions and the respective set of functions is defined by the straight-forward extension B = {u C 0 (X) : [x] [x ] u(x) > u(x ) [x], [x ] X T }, coinciding for T = 1 with the definition given previously in section 2.3. Note that axiom A4 ensures that the definition of B does not depend on the choice of x In analogy to proposition 1 I seek for a representation of with a given Bernoulli utility function u. Proposition 2: Given a Bernoulli utility function u B with range U, a binary relation on X T satisfies axioms A3 restricted to X P and A4 if and only if there exists a strictly monotonic, continuous function g : U IR such that for all x,x X T [ 1 x x g 1 T ] [ T 1 g u(x t ) g 1 t=1 T ] T g u(x t). (10) t=1 Moreover if T 2 and g represents in the sense of equation (10) then g : U IR represents in this sense if and only if there exist a A such that g = ag. 3 The Representation Theorems 3.1 Certain Uncertain Combining the thoughts of sections 2.3 and 2.4 I now combine time structure and uncertainty. Section 3.1 tackles the simplest such case, a two period setting where consump- 20 Stationarity implies that the mere passage of time does not have an effect on preferences. In a particular example it states that I will not (anticipate in my plans to) prefer Beck s beer over Budweiser in 2010 and Budweiser over Beck s beer in If u would be allowed to vary arbitrarily over time, it would be difficult to give it an interpretation of value which is sought for in this paper to render it helpful in a decision process. This is not the same as excluding a specified history dependence which might well change preferences over time in an anticipated way ( if I drink Beck s all through 2010 I might not like it anymore and prefer to drink Budweiser in 2011 ). 21 While I consider it an interesting task to integrate rules for history dependence in an extended model, I do not consider it that helpful to allow for arbitrary dependence on history like in Kreps & Porteus (1978). This implies too little structure and allows in a finite horizon setting for the same preference changes as unstationarity, rendering it difficult to give a value interpretation to u. 22 Nor does it depend on the fact that the defining paths [ ] have constant future consumption streams. 14

16 Precautionary decision rules under risk tion in the first period is certain and consumption in the second period is uncertain. Section 3.2 will look at the effect of choices of Bernoulli utility on the representation and section 3.3 extends the model to multiple periods. In the terminology applied by Kreps & Porteus (1978) which I will adapt in my multiperiod setting in 3.3 the model treated right now is only a one-and-a-half periods. However I consider it useful to familiarize with the structure of the representation and the idea of gauging in this simplified framework first. It extends in a straight forward way to any finite time horizon. Anticipating later representations and avoiding notational confusion I will denote the first period in this section by t = F and the second and last period by t = T. 23 Elements x F X denote certain consumption in the first period. Degenerate lotteries δ xt yielding certain consumption x T in the second period are also denoted by x T X. General objects of choice in the second period are the lotteries p P just as in section 2.3. The preference relation over these objects is denoted by T. Objects of choice in the first period are combinations of certain consumption in the first period and lotteries faced in the next: (x,p) X P. Preferences over these objects are given by the relation F. The set of preferences in both periods will be denoted by = ( F, T ). I demand that preferences restricted to certain consumption paths satisfy certainty additivity (A4) and that lotteries are evaluated on basis of the von Neumann-Morgenstern axioms (A1-A3). In addition the preferences in period one and two should be connected by the consistency axiom A5 (time consistency) x X, p,p P : (x,p) F (x,p ) p T p This is time consistency in the sense of Kreps & Porteus (1978). 24 It simply states that the decision maker in period one should prefer a consumption plan over another that does not differ for the first period if and only if he will prefer its uncertain entry as a lottery in the second period. Again I am interested in finding a representation for for a given valuation on the certain one-period outcomes u B B F. The latter definition is justified by the fact that certainty additivity A4 and time consistency A5 imply that B F = B T Due to backward recursion in the derivation of the general representation what is now the structure of the second period representation will later in the multiperiod setting coincide with that of the last period representation. A full time-step back from T will be when uncertainty is introduced also for the preceding period. 24 Adapted to the one-and-a-half period setting of this section. 25 This is shown in the proof of theorem 1. For the first period it is [x] = (x,x 0 ) = (x,δ x 0). 15

17 The Representation Theorems Denote with F X X the restriction of F to the set of certain consumption paths. The following representation theorem holds. Theorem 1: Let there be given a set of binary relations = ( F, T ) on (X P,P) and a Bernoulli utility function u B with range U. The set of relations satisfies i) A1-A3 for T (vnm setting) ii) A4 for F X X (certainty additivity) iii) A5 (time consistency) if and only if there exist strictly monotonic, continuous functions f : U IR and g : U IR such that v) (x,p) F (x,p ) g 1 [ 1 2 g u(x) g Mf (p,u) ] g 1 [ 1 2 g u(x ) g Mf (p,u) ], vi) p T p M f (p,u) M f (p,u). Moreover g and f are unique up to nondegenerate affine transformations. In period T lotteries are evaluated just the same way as in proposition 1. In the first period these second period lottery-values are aggregated by means of the time aggregator g 1 [ 1 2 g( ) g( )] with the values of the certain outcomes x F the same way as in proposition 2. Now the interesting part will be to look again at the gauge-freedom of the representation. 3.2 Gauging Like in section 2.3 there is some gauge freedom rendered to the model by the freedom to choose the Bernoulli utility function in theorem 1. Given some u B any other Bernoulli utility function is some strictly increasing continuous transformation of u and any strictly increasing continuous transformation of u yields an element of B. Moreover the following lemma holds. Lemma 1: If the triple (u,f,g) represents the set of preferences in the sense of theorem 1, then so does the triple (s u,f s 1,g s 1 ) for any s : IR IR strictly increasing and continuous. Now like in section 2.3 I can gauge the uncertainty aggregation rule in the representation of theorem 1 to any desired form parametrized by a strictly monotonic continuous f by choosing s = f 1 f in lemma 1. This yields the following corollary. 16

18 Precautionary decision rules under risk Corollary 2 (f-gauge) : For any strictly monotonic, continuous function f : IR IR the following equivalence holds: A set of binary relations satisfies i iii) of theorem 1 if and only if there exists a continuous function u : X IR with range U and a strictly monotonic, continuous function g : U IR such that v vi) of theorem 1 hold Moreover the pair (u, g) is unique up to simultaneous transformations of the form (u,g ) = (a g f 1 a + f,f 1 a + 1 f u) with a A,a + A +. The gauge used implicitly by Kreps & Porteus (1978) is obtained for f = id where v) and vi) become: Kreps-Porteus-gauge (f = id gauge) : v) (x,p) F (x,p ) g 1 [ 1 2 g u(x) g E p u ] g 1 [ 1 2 g u(x ) E p u] vi) p T p E p u E p u. Note that Kreps & Porteus (1978) get a slightly more general intertemporal aggregation rule for they do not demand certainty additivity in the sense of axiom A4. In the notion of Johnsen & Donaldson (1985) my axiom corresponds to unconditional strong independence over time for certain outcomes while the analog in their setting would be conditional strong independence, which is slightly weaker. However axiom A4 allows for a special gauge that will prove most helpful for discussing the meaning of welfare and precautionarity in section 4.2. This gauge is a special case of the following Corollary 3 (g-gauge) : For any strictly monotonic, continuous function g : IR IR the following equivalence holds: A set of binary relations satisfies i iii) of theorem 1 if and only if there exists a continuous function u : X IR with range U and a strictly monotonic, continuous function f : IR IR such that v vi) of theorem 1 hold Moreover the pair (u,f) is unique up to simultaneous transformations of the form 17

19 The Representation Theorems (u,f ) = (a f g 1 a + g,g 1 a + 1 g u) with a A,a + A +. It renders the above mentioned certainty additive gauge for g = id. Certainty-additive-gauge (g = id gauge) : v) (x,p) F (x,p ) u(x) + M f (p,u) u(x ) + M f (p,u), vi) p T p M f (p,u) M f (p,u). Another special gauge is possible if the outcome space is one-dimensional, X IR, and nonsatiation in the interior of X is assumed. Then the value function of theorem 1 can be chosen as the identity, rendering immediately the Epstein-Zin-gauge (u = id gauge, one commodity only) : v) (x,p) F (x,p ) g 1 [ 1 2 g(x) g Mf p ] g 1 [ 1 2 g(x ) g Mf p ] vi) p T p M f p M f p with M f p M f (p, id) = f 1 [ X f dp ]. This gauge is used by Epstein & Zin (1989). The representation they assume to hold slightly differs from the one supported by my axiomatization. With respect to the intertemporal aggregation rule Epstein & Zin (1989) assume the special case where g(z) = z ρ, what renders a CES function. On the other hand they assume a more general uncertainty aggregation rule which does not comply with von Neumann & Morgenstern s (1944) independence axiom. Before I will discuss the economic insights that can be gained from the derived results in section 4 I want to extend the representations to a multiperiod setting. 3.3 Multiperiod Extension The objects of choice in period F of the last section were elements of X T 1 X (X). Now the second-last period t = T 1 will start before period F in the sense that uncertainty over the choice objects of the respective period will not have resolved at t = T 1. Hence preferences in period T 1 are expressed by a relation T 1 on the space of lotteries over X denoted P T 1 ( X T 1 ). In general define X T = X and recursively X t 1 = X ( X t ) for t {1,...,T }. Equip the set of Borel probability measures on X t, P t ( X t ), with the Prohorov metric as to render X t 1 compact in the product topology. Its elements are called (period t-) lotteries. The set of degenerate lotteries in P t is identified with the set X t in the usual way. An uncertainty aggregation rule in period t is a functional M : P t C 0 ( X t ) IR. 18

20 Precautionary decision rules under risk As the arguments clearly indicate the time period I will not explicitly attach a time indice to M. The time consistency requirement for the set of preferences = ( 1,..., T ) now writes as A5 (time consistency) For all t {1,...,T } : (x t,p t+1 ) t (x t,p t+1) p t+1 t+1 p t+1 x t X, p t+1,p t+1 P t+1. Note that (x t,p t+1 ) stands for the degenerate lottery δ (xt,p t+1 ) P t. The interpretation of the axiom is the same as for axiom A5 on page 15. Again time consistency together with certainty additivity will make the set of Bernoulli functions coincide for all t, so that it makes sense to define B B 1. The following representation holds. Theorem 2: Let there be given a set of binary relations = ( t ) t {1,...,T } on (P t ) t {1,...,T } and a Bernoulli utility function u B with range U. The set of relations satisfies i) A1-A3 for all t,t {1,...,T } (vnm setting) ii) A4 for 1 X T (certainty additivity) iii) A5 (time consistency) if and only if there exist strictly monotonic, continuous functions f t : U IR for t {1,...,T } and g : U IR such that with defining the functions ũ t : Xt IR for t {1,...,T } by ũ T (x T ) = u(x T ) and recursively with α t = 1 26 T+1 t ũ t 1 (x t 1,p t ) = g [ 1 α t g u(x t 1 ) + (1 α t )g M ft (p t,ũ t ) ] it holds that for all t {1,...,T } p t t p t M ft (p t,ũ t ) M ft (p t,ũ t ) p t,p t P t. This representation allows the uncertainty aggregation rules to depend on absolute time. 27 This is an unattractive feature of recursive models approaching a finite planning horizon like the one of Kreps & Porteus (1978). In models with infinite planning horizon this is usually avoided by the axiom of stationarity. To capture the point that also in a finite time horizon risk attitude should not depend on absolute time let me introduce 26 In the case of a positive discount rate δ it is α t = (1 + δ + δ δ T t ) 1 = 1 δ 1 δ T+1 t. 27 That is, my uncertainty evaluation might depend on the date when the uncertainty occurs, stays fixed while I approach this date, but might completely differ from the way I evaluate uncertainty for another date (compare also footnote 20). 19

21 The Representation Theorems the following notation. Let x t = ( x, x,..., x) denote the certain constant consumption path that gives consumption x from t until T. Then 1 2 xt x t P t is the lottery in period t that randomizes with p = 1 between the constant consumption stream x and 2 x. The following axiom demands that these randomized consumption streams relate to certain consumption streams the same way in different periods. A6 (quasi-risk-stationarity) For all t {1,...,T }: 2 x 1 t + 2 x 1 t t x t 1 2 x t x t+1 t+1 x t+1 x,x,x X. Adding A6 to the assumptions of theorem 2 yields Theorem 3: Let there be given a set of binary relations = ( t ) t {1,...,T } on (P t ) t {1,...,T } and a Bernoulli utility function u B with range U. The set of relations satisfies i) A1-A3 for all t,t {1,...,T } (vnm setting) ii) A4 for 1 X T (certainty additivity) iii) A5 (time consistency) iv) A6 (quasi-stationarity) if and only if there exist strictly monotonic, continuous functions f : U IR and g : U IR such that with defining the functions ũ t : Xt IR for t {1,...,T } by ũ T (x T ) = u(x T ) and recursively with α t = 1 28 T+1 t ũ t 1 (x t 1,p t ) = g [ 1 α t g u(x t 1 ) + (1 α t )g M f (p t,ũ t ) ] it holds that for all t {1,...,T } p t t p t M f (p t,ũ t ) M f (p t,ũ t ) p t,p t P t. Moreover g and f are unique up to nondegenerate positive affine transformations. As an example for the representation let me write out the first period evaluation functional for a full two period setting explicitly 1 representation for T = 2 : M f ( p 1,g 1 [ 1 2 g u(x 1) g Mf (p 2,u) ]). (11) Furthermore gauging works out the same way as in section 3.1. Lemma 2: If quasi-stationarity A6 is assumed, then lemma 1, corollary 2 and corollary 3 also hold in the multiperiod setup. That is the respective statements hold true with theorem 1 replaced by theorem 3 and i-iii replaced by i-iv. 28 In the case of a positive discount rate δ it is α t = (1 + δ + δ δ T t ) 1 = 1 δ 1 δ T+1 t. 20