CHAPTER 9 UNCERTAINTY：9章不确定性.doc

F Petri Microeconomics for the critical mind chapter9uncertainty 06/03/2019 p. 1 1 Fabio Petri - Microeconomics for the critical mind - provisional CHAPTER 9 UNCERTAINTY - Part I (incomplete) This chapter deals with how economic theory analyzes the implications, for a number of relevant economic issues, of the existence of uncertainty about the results of choices. This is a field in rapid transformation. The most popular approach assumes that the possible outcomes of each choice can be listed exhaustively, and that the decision maker has preferences over choices that imply an opinion on the probability of occurrence of each of the outcomes of each choice. Then choices are described as choices among lotteries, and if a number of axioms are satisfied, preferences can be represented by a utility function additively separable over outcomes, called Von Neumann-Morgenstern (VNM) expected utility, which assigns to each possible choice a utility equal to the sum of the expected values (in the statistical sense) of the utilities of the possible outcomes. Then the empirically relevant phenomenon called risk aversion is explained as due to a decreasing marginal utility of income or wealth, and implications are derived for decisions concerning insurance, risk pooling, risk diversification, portfolio choice, outputs of firms. After a presentation of this approach, Part I of the chapter mentions empirical evidence suggesting that decisions do not always satisfy the axioms behind VNM expected utility, and presents an empirically based alternative approach, prospect theory, which is gaining increasing acceptance; but it is still unclear whether this approach implies significantly different conclusions about the economic decisions listed above. Part II illustrates how uncertainty is dealt with in modern general equilibrium theory. It presents the notion of Arrow-Debreu general equilibrium with contingent commodities, its equivalence with a Radner equilibrium, and the issue of incomplete markets. It is argued that these notions do not avoid the deficiencies of general equilibrium theory discussed in ch. 8. The alternative treatment of uncertainty in the theory of value and distribution, implicit in traditional long-period analyses both classical and neoclassical, is then explained. LOTTERIES AND EXPECTED UTILITY Let us start by enlarging the set of things, among which a consumer chooses, to include lotteries. These are lists of possible outcomes, or payoffs, or prizes, consequent on the choice of that lottery, each one of them with a probability attached to it, the probabilities summing to 1. An example: suppose you consider whether to bet one dollar on number 1 at a roulette spin. Then the lottery will consist of two possible outcomes: 1) you have paid one dollar, number one comes out, you win 36 dollars, with probability (for European roulette) 1/37; 2) you have paid one dollar, number one does not come out, you win nothing, so you obtain a negative payoff of one dollar, with probability 36/37. F Petri Microeconomics for the critical mind chapter9uncertainty 06/03/2019 p. 2 2 Outcomes of lotteries can be vectors of consumption goods, or sums of money, or happenings (e.g. that someone gets married; an accident), anything really; but we will generally consider payoffs that are consumption bundles, or sums of money. A lottery with n outcomes can be represented in different ways: one is as a double vector, that lists first all the outcomes x1,.,xn and then, in the same order, their probabilities of occurrence p1, p2, . , pn. (Actually only the first n-1 probabilities need be listed, since the n probabilities must sum to 1; thus if the payoffs are only two, it is customary to indicate only the probability p of the first payoff, the probability of the second payoff being 1p.) When several lotteries must be compared or combined, one way to proceed is to assume a given set X of all possible outcomes, univocally indexed, and to assume that each lottery has a probability distribution over X, which identifies the lottery: e.g. if there is a finite number N of outcomes, with the outcomes ordered and numbered 1,.,n,.,N, each lottery can be specified simply as an N-list of probabilities p1,.,pN summing to 1. This does not exclude lotteries actually contemplating just a few of the outcomes in X, it only means that all other outcomes in X are assigned zero probability; but if the outcomes in X are numerous, this procedure can make for very cumbersome representations of lotteries. So for simple examples it is preferable to adopt the representation (Varian 1992): p1x1p2x2.pnxn . This means: the lottery is among n different outcomes x1,.,xn; it offers outcome x1 with probability p1, outcome x2 with probability p2, etcetera. The example above of roulette lottery is represented as 1/37$(W+35)36/37$(W1) where W is the initial wealth in dollars1. Among the outcomes or prizes of a lottery there can be other lotteries (some of whose outcomes can in turn be lotteries); e.g. one can have, with A, B, C three outcomes: L = p1Ap2(p3Bp4C) where p1+p2=1, p3+p4=1. A lottery where some prizes are lotteries is said compound, otherwise simple. The completely reduced lottery corresponding to compound lottery L is defined as the simple lottery L stating the total probabilities of occurrence of each final payoff, for example for the above L: L=p1Ap2p3Bp2p4C. If it had been A=C, the reduced lottery would have been L”=(p1+p2p4)Ap2p3B. Reduction can also be only partial. Conversely a lottery with more than two outcomes can always be transformed into a compound lottery with only two outcomes, and reducible to the original lottery, by having the two outcomes consist of compound lotteries; for example a lottery L=p1Ap2Bp3Cp4D (with p1+p2+p3+p4=1) is the reduced form of L= (p1+p2)p1/(p1+p2)Ap2/(p1+p2)B(p3+p4)p3/(p3+p4)Cp4/(p3+p4)D. We assume the so-called Equivalence (or Reducibility) Axiom, that states that the decision maker treats a compound lottery and all the partially or completely reduced lotteries derivable from it as the same lottery, or at least is indifferent among them. On this basis, if one assumes that all 1 Sometimes the same lottery would be represented as 1/37$3536/37$1, indicating that one is only interested in the net payoffs. F Petri Microeconomics for the critical mind chapter9uncertainty 06/03/2019 p. 3 3 lotteries in the problem under examination assign probabilities to all the N elements of a common set X of indexed outcomes and therefore can be represented simply by N-vectors of probabilities (see above), then a compound lottery L(1-)L (also called a mixture of the two lotteries) can be represented as L+(1-)L where L and L now stand for the respective vectors of probabilities, because the vector L+(1-)L correctly describes the reduced lottery corresponding to the compound lottery2. A lottery can also be over a continuum of outcomes, with their probabilities defined by a distribution function. Let us assume that all the alternatives among which a consumer chooses are lotteries, and that the decision maker has a preference order over them; actions with a single sure outcome can be represented as degenerate lotteries assigning probability 1 to that outcome and zero to any other outcome, so traditional consumer theory can be considered a special case of choice among lotteries. I will now describe a particular utility function over lotteries and then inquire what assumptions imply that form of utility function. Definition: The decision maker has preferences (over lotteries) representable by a utility function of expected utility form if her preference order can be described by a utility function such that the utility of a lottery that assigns probability p1 to payoff x1, probability p2 to payoff x2, ., probability pn to payoff xn, can be represented as 9.1 U(x1,.,xn; p1,p2,.,pn) = p1u(x1)+p2u(x2)+.+pnu(xn). The function u() appearing on the right-hand side means different things according as it has as argument (i) an object like in traditional consumer theory (e.g. a bundle of consumption goods, an amount of money), or (ii) a sub-lottery; in case (i) u() is a standard utility function as defined in ch. 4, for example if the payoffs are amounts m of money it might be u(m)= log m; in case (ii) u() has itself expected utility form, it is the sum of the expected values of the payoffs of that sub- lottery. (For this reason the same symbol u() can also be used for the overall utility function on the left-hand side.) When the lottery is simple (or completely reduced), the xis are not lotteries and then the utility function u(xi) on the right-hand side is distinguished from the overall expected utility function U by calling u() the basic utility function, or felicity function. The overall utility function is called expected utility function or also Von Neumann-Morgenstern (VNM) utility from the name of the originators of the notion. Of course if a VNM utility function correctly represents preferences, any increasing monotonic transformation of it correctly represents the same preference order, but if one wants to maintain the expected utility form (thus maintaining its convenient additive form) then only affine positive transformations are acceptable, from U() to V()=aU()+b with a, b scalars and a0. 2 An implication of this framework is that if xiX indicates an outcome and if one decides that the same symbol also indicates the degenerate lottery that assigns probability 1 to this outcome and zero to all other outcomes, then a lottery L=(p1,.,pN) can also be indicated as L=p1x1+.+pNxN. F Petri Microeconomics for the critical mind chapter9uncertainty 06/03/2019 p. 4 4 If preferences satisfy certain axioms then they admit representation via a VNM utility function. I proceed to list a number of axioms that suffice to guarantee the expected utility form. Some of these axioms are so universally accepted that some authors do not call them axioms, but rather assumptions; these are: - that the decision maker doesnt care about the order in which a lottery is described, the lottery p1Ap2B and the lottery p2Bp1A are treated by the decision maker as the same lottery; - that the decision maker considers the degenerate lottery that assigns probability 1 to one outcome as the same as getting that outcome for certain; - the Equivalence Axiom3. Empirical evidence is not always in accord with these assumptions; sometimes people choose differently depending on how a choice among lotteries is presented. We will neglect this fact, that seems of limited relevance for normal economic choices. To these assumptions, which are about the decision makers perception of the lotteries open to her, axioms are added that specify the structure of preferences. It is normally assumed - that preferences over lotteries are complete, reflexive, and transitive4. From here on, different authors take different routes, starting from different axioms, and sometimes to make things easier for students assuming as axioms some conditions that can be deduced from more primitive axioms. I will indicate how the axioms I will list differ from those listed by Varian 1992. The minimal number of additional axioms appears to be two: - 1. Continuity axiom - 2. Independence axiom. Continuity axiom: The preference relation on the space of simple lotteries is continuous, that is, for any three lotteries A, B, C the sets p0,1: pA(1p)B C and p0,1: C pA(1p)B are closed. This axiom5 states that if a lottery pA(1p)B is strictly preferred to another lottery or outcome, a sufficiently small change in p will not invert the preference order. An example that 3 The Equivalence Axiom appears as assumption L3, p. 173, in Varian 1992. 4 The assumption of completeness is here a stronger assumption than for choices under certainty, since one may have to compare rather complex compound lotteries, and it is more likely that one may find it impossible to compare their desirability. The assumption of transitivity can also be questioned, because people seem to have fuzzy preferences, i.e. to be unable to distinguish between lotteries that differ by less than a certain threshold; so a person may declare indifference between (A): obtaining 1000 dollars for sure or 3000 dollars with probability p=0.5, and (B): the same but with p=0.501; and she may declare indifference between (B), and (C): the same but with p=0.502; but these small increments of p may have now gone above the threshold, so the same person may declare she strictly prefers (C) to (A), violating transitivity. F Petri Microeconomics for the critical mind chapter9uncertainty 06/03/2019 p. 5 5 illustrates its concrete meaning is the following: if a trip with zero probability of a serious accident (e.g. death) is strictly preferred to no trip, the same trip with a positive but sufficiently small probability of a serious accident is still preferred to no trip. Independence axiom: The preference relation on the space of simple lotteries satisfies independence, that is, for any three lotteries A, B, C and for p(0,1) it is AB if and only if pA(1p)C pB(1p)C. This is sometimes called the substitution axiom. It can be shown (cf. Appendix 1) that from this axiom one can derive the strong-preference and the indifference versions of the same axiom: 9.2 AB if and only if pA(1p)C pB(1p)C. 9.3 A B if and only if pA(1p)C pB(1p)C6. The Independence axiom and its implications 9.2 and 9.3 assume that if we have two lotteries L and M with L M then for any positive probability p and any third lottery C it is pL(1p)C pM(1p)C, and the same preservation of ranking holds if L M. Replacing the third lottery with any other one, even with L or M, does not alter this preservation of ranking. This axiom is justified as follows: consider a lottery px(1p)y and a third outcome z such that the consumer is indifferent between x and z; if x and y formed a usual consumption bundle the consumer might well be indifferent between x and z but not between the consumption bundles (x,y) and (z,y) owing to the possibility of a different complementarity between x and y and between z and y; but whether the lottery px(1p)y is preferred to another lottery should not be affected by replacing x with z, because the consumer is going to consume either x or y, and either z or y, so if the probability of y is the same in the two cases, it seems plausible that the consumer be indifferent between px(1p)y and pz(1p)y if x z7. From these axioms one derives: Monotonicity-in-Probabilities (M-in-P)Lemma: if AB and p p with p, p (0,1), then (i) A pA(1p)B, (ii) pA(1p)B B (iii) pA(1p)B pA(1p)B, 8 5 It appears as axiom U1, p. 174, in Varian 1992. This axiom concerns continuity in probabilities which is not the same continuity of preferences as in ch. 4; but remember that the basic utility u() when defined over consumption bundles is a standard utility function, so it is assumed to satisfy the usual assumptions of preferences over sure consumption bundles, among which continuity in the sense of ch. 4. 6 9.3 appears in a slightly different form (because referred to a best lotter b and a worst lottery w, whose existence is assumed by Varian but is not necessary here) as U2, p. 174, in Varian 1992. 7 And yet this axiom too has been criticized, see below on prospect theory. 8 Result (iii) (referred to lotteries with the best and the worst outcome as prizes) is assumed for simplicity by Varian 1992 as an axiom, U4 p. 174; but Varian admits that it could be derived from the earlier axioms he has listed. F Petri Microeconomics for the critical mind chapter9uncertainty 06/03/2019 p. 6 6 and conversely if pp with p, p(0,1) and ApA(1p)B or pA(1p)BpA(1p)B, then AB. Proof. The trick is to use B or A in place of C in 9.2 and to remember that A = pA(1p)A. Inequality (i) derives from the fact that AB and 9.2 imply A = (1p)ApA (1p)BpA for all p(0,1). Inequality (ii) analogously derives from pA(1p)BpB(1p)B=B. Now define a probability = (pp)/(1p) (0,1), admissible since p0 and how big is another number y0 there is an integer n such that nxy. 10 A set is connected if it is possible to connect any point of the set to any other point of the set with a continuous curve consisting entirely of points of the set. Each internal point of a continuous curve is a point of accumulation along the curve from either direction; hence if a continuous curve in a connected set S goes from a point internal to a subset F of S to a point not in F but in another subset H of S, with F and H both closed and connected and such that FH=S, then any point of the curve not in F is in H, and there must be a point of the curve which is a frontier point of F and also an accumulation point for H, so by the definition of closed set it belongs to H, and therefore to both subsets. 11 Many economists would write indifferent to C, using the adjective indifferent in sentences like “x is indifferent to y” to mean “the chooser is indifferent between x and y”. Varian is one example. This ugly deformation of English is avoided by other authors who write in the same sense “x is equivalent to y”; I prefer to use equipreferred, i.e. equally ranked in the preference order, that clarifies that one is talking about preferences. F Petri Microeconomics for the critical mind chapter9uncertainty 06/03/2019 p. 7 7 Proof.13 For any number s in 0,1 different from p*, if p*s then, by result (iii) in the M- in-P Lemma, C p*A(1p*)B sA(1s)B ; and if sp* then sA(1s)B C. Now I build a utility function for preferences satisfying