Abstract
This chapter consists of two parts; first, a standard introduction to choice under uncertainty, and second, how uncertainty is introduced into general equilibrium theory. It starts with lotteries and utility over lotteries, it defines Von Neumann–Morgenstern expected utility and the axioms that justify its form; then it introduces the Arrow–Pratt measure of absolute risk aversion, CARA and CRRA utility functions, stochastic dominance, Jensen’s inequality and discusses how risk aversion explains insurance, risk pooling and portfolio selection among risky assets, including Tobin’s preference for liquidity. Then the chapter mentions state-dependent utility and subjective expected utility. A discussion of Allais’ and Ellsberg’s paradoxes is followed by an introduction to prospect theory. The notion of informational cascades allows an enlightening use of Bayes’ Rule. Then the chapter moves to how uncertainty is introduced into general equilibrium theory; after a section on traditional marginalist authors, the chapter describes the intertemporal general equilibrium with contingent commodities and introduces the Radner equilibrium with uncertainty. There follows a short introduction to incomplete markets and sunspot equilibria. The chapter ends with a summing up of conclusions on general equilibrium theory, which is found difficult to defend. The online Appendix explains the Anscombe–Aumann approach to the existence of subjective expected utility; then it reports Wilson’s example of non-existence of equilibrium with incomplete markets, and finally it reports an interesting page on insurance by Alfred Marshall that gives a taste of how traditional authors dealt with uncertainty.
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Notes
- 1.
An additively separable function is an additive function of several variables where in each addendum there appears only one variable. E.g. f(x,y) = x + xy2 is additive but not additively separable.
- 2.
It must be said at the outset that the empirical evidence does not strongly support expected utility; we will see later some counterexamples, and one attempt (out of several) to construct an alternative theory more in accord with the empirical evidence, prospect theory. Just to give a hint of other possibilities, I mention here worst-case-scenario preferences: a lottery is preferred to a second lottery if, given a utility function over outcomes, the lowest utility that can be yielded by the first lottery is greater than the lowest utility that can be yielded by the second lottery. But VNM utility has been so dominant that most of game theory and of models in industrial economics and in labour economics that introduce uncertainty (some will be mentioned in Chaps. 12 and 13) assume VNM utility; certainly an enormous reconstruction would be necessary if VNM utility were discarded. For the moment, at least at textbook level not much can be told a critical mind beyond ‘keep this caveat in mind, VNM utility might need replacement’. Of course the doubts about the empirical validity of VNM utility theory imply analogous doubts about the axioms, but it is difficult to understand which one of several axioms is responsible for the predictions that raise doubts about the theory.
- 3.
The assumption of completeness is here a stronger assumption than for choices under certainty, because 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 a lottery (C) equal to B except that p = 0.502; but the increment of p from 0.5 to 0.502 may be above the threshold, so the same person may declare she strictly prefers (C) to (A), violating transitivity. This case too seems to have limited relevance in economic choices and I will neglect it.
- 4.
This axiom concerns continuity in probabilities which is not the same continuity of preferences as in Chap. 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 on preferences over sure consumption bundles, among them continuity in the sense of Chap. 4.
- 5.
Replacing C with A, one obtains A ~ B if and only if A ~ p◦B (1 − p)◦A, which shows that the Independence axiom excludes the possibility that the presence of uncertainty in a lottery may be found by itself attractive, perhaps because one likes adrenalin. This is an important limitation of expected utility theory which makes it unfit to analyse phenomena like addiction to gambling.
- 6.
And yet this axiom too has been criticized, see below Sect. 9.25.
- 7.
A ‘book’ is a bet (hence ‘bookmakers’). From the time when the Dutch were shrewd merchants competing with the British in international markets, the British seem to have derived a hostile popular image of the Dutch as swindlers, which explains the denomination.
- 8.
‘Axiom’ is in inverted commas because here it is a derived result, but many treatments take it as one of the axioms (in place of the continuity axiom, which is then derived from the independence axiom and the Archimedean axiom). The qualification ‘Archimedean’ appears to have derived from the analogy with ‘Archimedes’ principle’: given a number x > 0 no matter how small, and a number y > 0 no matter how big, there is an integer n such that nx > y.
- 9.
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 curvilinear segment consisting entirely of points of the set. Each point of a continuous curve is a point of accumulation along the curve from either direction; hence if a continuous curvilinear segment 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 F H = S, then any point of the curvilinear segment not in F is in H, and there must be a point of it 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.
- 10.
To give the reader a feeling of the possibility of different derivations of expected utility, online Appendix to the chapter, in the book’s website, in its Sect. 1 reports a different proof of Lemma 3.
- 11.
Exercise: Prove that this implies that if A ~ B then u(A) = u(B).
- 12.
These symbols E1, E0 should not be confused with the expectation operator.
- 13.
Consider a lottery L = pA◦A pB◦BpC◦C where pC = 1 – pA – pB; it was shown that L can be seen as the reduced form of a compound lottery M = pA◦A pD◦D where pD = pB + pC and D is a lottery with B and C as outcomes with respective probabilities pB = pB/pD and pC = pC/pD. If preferences satisfy the continuity and independence axioms then u(M) = pAu(A) + pDu(D) and, D being a lottery, u(D) = pBu(B) + pCu(C), which reduces to u(M) = u(L) = pAu(A) + pBu(B) + pCu(C).
- 14.
Cf. footnote 3.
- 15.
Then does risk aversion not apply to roulette gamblers? It may well apply, and yet be compensated by the pleasure of gambling. We are implicitly assuming that only the money result of the lottery affects preferences, but this is often untrue.
- 16.
Note that being risk loving is not the same as deriving pleasure from gambling.
- 17.
There is reason to suspect that people tend not to calculate expected values when the gamble involves a very small loss versus a very large (although highly improbable) gain; most people have only the vaguest idea of the probability of winning the big prizes of important lotteries. The anticipation of a regret if one completely gives up the possibility, however improbable, of becoming very rich is probably an important reason why one buys lottery tickets.
- 18.
So second-order stochastic dominance does not imply first-order stochastic dominance. Exercise: Does first-order stochastic dominance imply second-order stochastic dominance?
- 19.
This result requires finding the value to which the series ∑nxn converges, and then setting x = 1/2 in it. Consider the geometric series 1 + x + x2 + … + xn + … = 1/(1 – x) for 0 < x < 1; differentiate both sides to obtain 0 + 1 + 2x + 3x2 + 4x3 + … + nxn–1 + … = 1/(1 – x)2; rewrite the left-hand side as 1 + x + x + 2x2 + x2 + 3x3 + x3 + … = (1 + x + x2 + … + xn + …) + ∑nxn, hence ∑nxn = 1/(1 – x)2 – 1/(1–x) = x/(1 – x)2 = 2 if x = 1/2.
- 20.
One can also see x* as initial wealth x1 minus the expected value of the damage x1-x2, i.e. x* = x1 – (1 – p)(x1 – x2). In an example below the expected value of the damage is indicated as d.
- 21.
The term ‘premium’ can create confusion: it is used (without adjective) in commercial language to indicate what the customer pays for the insurance coverage, the downpayment, x1–x^ in our example, it must not be confused with the ‘risk premium’. To add to the risk of confusion, ‘risk premium’ is also used to mean a very different notion in portfolio theory, see below.
- 22.
In many cases the probability of the damage depends on the individual’s actions: e.g. a house fire, or falling ill and in need of medical assistance, can depend on how careless one is. This problem of moral hazard (the decrease of the insuree’s incentive to avoid actions favouring a bad event if the damage deriving from it is reduced by insurance, which makes the probability of the bad event depend on the extent of insurance), that requires a different approach and will be discussed in Chap. 11, is the reason why most insurance contracts do not offer ‘full coverage’ (full reimbursement of the damage); the difference between damage and reimbursement, called franchise clause, necessarily borne by the insuree, is intended to discourage her from acting carelessly.
- 23.
A conditional good is a good to be delivered to the purchaser only if a certain event or state of nature obtains. Cf. below, Sect. 9.28.
- 24.
If the insurance company is risk neutral and can decide both q and π, what is the choice that maximizes its profit?
- 25.
This is so if the consumer finds it convenient to buy at least some insurance coverage; it is also possible that π be so high that the consumer does not insure at all: this is the case corresponding in Fig. 9.4 to an ‘insurance line’ through x not steeper there than the indifference curve.
- 26.
One implicit assumption is that the capacity of money to yield utility is given, which implies that money prices are given, at least the money prices in markets other than the one under study (cf. in Chap. 4 the analysis of quasilinear utility and of consumer surplus). The analysis is necessarily a partial-equilibrium one. If the agents involved are firms, wealth effects are always absent because it is as if firms had quasilinear utility functions.
- 27.
Remember that if Y = a + bX then Var(Y) = b2Var(X), that Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y), and that YA and YB are assumed independent so Cov(YA,YB) = 0.
- 28.
‘Allocation of risk’ is a common expression, which however can be given a precise meaning only under restrictive assumptions, for example it has no quantitative meaning when probabilities are not defined (see, e.g., Sect. 9.33 below).
- 29.
For example, in all likelihood a decision to buy a house with the help of a mortgage stimulates savings; then if a higher rate of interest discourages from buying a house because the mortgage has become too expensive, this reduces savings.
- 30.
See, for example, McKenna (1986).
- 31.
For example, customers are sometimes ready to wait even for a couple of months to get their preferred car model. Clearly, the good must have no very close substitutes more readily available.
- 32.
A more complete analysis would have to determine W on the basis of the overall utility maximization problem, which will determine which portion of the individual’s income is allocated to immediate consumption and which portion is left for financial investment. We concentrate on a given W because for any W the consumer has the problem of getting out of it as much wealth as possible in the second period.
- 33.
To help intuition, imagine the safe asset to be gold, with the money price of one ounce of gold at t and at t + 1 equal to 1 dollar because it is legal tender; the attempt at t to short sell great amounts of titles promising the delivery of an ounce of gold at t + 1 would cause the price of these titles to fall below 1 dollar.
- 34.
Exercise: Show graphically how to build a riskless portfolio from two risky assets, using short sales if necessary.
- 35.
This point is found by finding the point (a*(0)r1(1 + h), a*(0)r2(1 + h)) on the Ox line, and tracing from it toward the s-x segment a line parallel to the certainty line i.e. with slope 1.
- 36.
There is no universally accepted definition of riskiness; the definition is implicitly supplied by how one measures it. One approach is to define the riskiness of a lottery by the risk avoidance price, or better (in order to be able to compare lotteries with different sizes of prizes) by the ratio of the risk avoidance price to the expected value of the lottery. Then which one, of two lotteries, is riskier can depend on the utility function one uses.
- 37.
I leave to texts on finance the task of a more detailed introduction to portfolio choice and of a discussion of the limits of mean-variance analysis. A healthy dose of realism is certainly very useful in this field, inside stories written by financial market operators can be of help, for example, MacDonald (2009).
- 38.
Convex indifference curves now mean that the consumer will prefer to a lottery the lottery with the same expected value but where the slope of the indifference curve equals the slope of the constant-expected-value line (like point z relative to initial point x in Fig. 9.10), which means that the marginal utility of money is the same in the two states: this condition takes the place of certainty in the determination of the initial lotteries from which the consumer will refuse to move by accepting a fair bet.
- 39.
This definition of prospect has no necessary connection with prospect theory which is a specific theory of the utility function in the presence of uncertainty and is briefly discussed later.
- 40.
The continuity axiom specified in the derivation of VNM expected utility is unnecessary for the moment because prospects are not lotteries (i.e. do not include a specification of probabilities).
- 41.
An important consequence is that, as Wakker (2008, p. 431) notes, differently from objective probabilities that ‘are stable, and readily available for analyses, empirical tests and communication in group decisions’, radical-uncertainty subjective probabilities ‘can be volatile and can change at any time by mere further thinking by the agent’. Thus, like the expectations about future prices discussed in Chap. 8, subjective probabilities suffer from the indefiniteness and the impermanence problems.
- 42.
Starting from this impossibility, Herbert Simon has stressed the need to abandon the assumption of rational optimization and to replace it with the study of procedural rationality, the set of procedures considered fruitful for the search for a satisfactory decision, a search that will stop at a certain point because further analysis would always be possible (as in scientific research!) but it would be costly in time and resources. Simon’s insistence on limited rationality and satisficing rather than optimization has deeply affected the study of organizations (March and Simon 1958).
- 43.
Here this term has no negative connotation, it does not imply excessive weight, it simply means a greater weight than standard expected utility would assign to it.
- 44.
This is almost not the case in the formal example given, where the probability of being in a correct cascade is very little above 1/2, it is only 0.5133 according to the calculation in Bikhchandani et al. (1992); but in real situations people have private information from multiple sources and the start of a cascade that overpowers contrary private signals generally requires the observation of many concordant choices. Also, if the sequence of decisions is long, people will most probably realize that previous choices are not very informative, and will then look for additional information, or will decide to trust their private signal more than the choices of others; a ‘wrong’ cascade will then probably come to an end. Indeed the theory of information cascades concludes that ‘wrong’ information cascades are generally fragile.
- 45.
Marshall's short period is actually rather long, it refers to period lengths in which only the factors that require a rather long time for considerable alterations of their endowments (e.g. experienced fishermen, ships, fixed plants) are in given supply: to think of it as a couple of years will be often reasonable.
- 46.
Similar considerations apply to uncertainty about demand for one's product: the average inventories or average extra costs incurred to face random changes of demand, or possibly the more expensive wage contracts including an agreement by the worker to accept variations in labour time when needed, are implicitly included in normal costs. In Appendix 1 to this chapter, this time at the end of the chpater and not in the website, you find a long quotation from Alfred Marshall’s Principles, which illustrates the traditional approach to these issues.
- 47.
Marshall notes that the payment of managerial activity is often out of ‘profits’, so activities that require more intense or more skilled managerial activities to take care of risk and of unexpected events will on average show a higher rate of profit because of accounting conventions, and this is another reason for the observation of persistent differences in the rates of profit. Adam Smith's excess of profit over interest to cover the ‘risk and trouble’ of enterprise analogously includes both risk coverage, and remuneration of entrepreneurial activity directly out of profits.
- 48.
The general equilibria discussed in Ch. 8 are accordingly described by modern neoclassical theorists as based on certainty. The term ‘commodity’ is used in this literature not in the old classical sense (good produced to be sold) but as broader than ‘goods’ in that also including contingent commodities, bonds, Arrow securities, etc. A physically specified type of good, e.g. potatoes, or gold, is sometimes called a basic good when it must be distinguished from a good or commodity also described by its date and state.
- 49.
This last example illustrates the danger of ambiguity in the specification of a contingent commodity. What if—a frequent situation—no official declaration of war is issued but fighting is vigorous? In recent years, the danger of ambiguity has been particularly evident in determining when Credit Default Swaps become payable. Note also that not all differences among states of nature can be the basis for contingent contracts: for example, one’s future mood, ‘if I will be in a happy mood tomorrow’, certainly is an important difference between possible future states, but it indicates a cause of uncertainty that cannot be a basis for contingent contracts, because not objectively ascertainable.
- 50.
It is not necessary that the state, on which delivery of a good at date t is conditional, be of date t; it can be of an earlier date, in which case the date-t good is delivered whatever the state of nature at date t.
- 51.
The price pNs is not 1 even if good N is the numéraire, because it is the price for delivery of one unit of good N if state s is realized, while if good N is chosen as numéraire then 1 is the price of the firm delivery of one unit of good N in the last period, i.e. whatever the state of nature in the last period.
- 52.
An example will help. Suppose there are two dates, 0 and 1, two states at date 1, α and β, and good N is gold of date 1 delivered for sure. Consumer A wants to buy 1 unit of good (2, 1, α), that is 1 unit of good 2 of date 1 conditional on state α occurring, with payment only if the good is delivered because state α occurs. B accepts, for a conditional price consisting of 4 units of good N. The two contingent contracts might be the following ones. A pays a contingent price of 3 units of N to B for the contingent delivery of 1 unit of good (2, 1, α); since this is a firm payment, it means a quantity of N going to B whatever the state at date 1. B pays a contingent price of 3 units of N to A for the contingent delivery of 4 units of N if state α occurs. The two contingent payments of 3 firm units of N cancel out and what is left is that if state α occurs, A delivers 4 units of N to B and B delivers 1 unit of good (2, 1,α) to A. Note that if the general numéraire is, for example, gold of date 0 and the discounted price of a unit of N delivered for sure is, say, 1/2 (an own gold rate of return of 100% for firm delivery), this allows us to say that the discounted value of the contingent payment of 3 firm units of N is 1.5, but can we calculate the discounted value of a unit of good (N, α), that is, N conditional on state α? Yes. We know that A and B are indifferent between 3 units of N for sure and 4 units of (N, α); this must mean that the two have the same discounted value, 1.5, which implies a discounted value of (N, α) equal to 0.333, and by implication a discounted value of (N, β) equal to 0.167.
- 53.
In these formulas, the suffix 1 inx1, π1, p1 refers to state 1 of date 1. I assume non-satiation. Also remember that the spot price of good 1 is normalized to 1 inall states.
- 54.
The unpredictability of novelties takes here the form of inability to list all possible future states of the world; states of the world may be realized that no one knew to be possible – universal contingent markets are then impossible.
- 55.
There isn't complete agreement on the meaning of these terms. The term ‘rational expectations equilibrium’ occurs in the intertemporal macroeconomics literature in connection with economies without universal markets but where agents have somehow arrived at understanding the probabilities of the several states and therefore agents agree on these probabilities and hence on the probability distribution of the several equilibrium price vectors unanimously correctly associated with the different states. This agreement on the probabilities is not assumed in Arrow–Debreu equilibria, nor in Radner equilibria, and implies a rational expectations equilibrium can only be a long-period equilibrium (Sect. 8.18).
- 56.
Laffont (1989) has attempted a list of reasons why markets are incomplete.
- 57.
Formally, securities that promise a unit of money at a future date for sure, that is, regardless of the environmental state at that date.
- 58.
For example, if for each possible aggregate output of apples next year one is to stipulate a different contingent contract for the purchase of apples, how many contingent contracts would have to be signed? As noted by Ciccone (1999, p. 91) ‘Since altering the specification of even one single element of a state of the world defines a different state, and this procedure may in principle be repeated at will for any element, the number of possible states of the world is clearly infinite’. Then there is the impossibility of a complete list of all possible future states, owing to the unpredictability of novelties insisted upon in Chap. 8.
- 59.
So again the theory excludes by assumption unexpectable events, that is, novelties, things that could not have been imagined before they come about (e.g. scientific discoveries, and more generally new ideas). This fact has been used by Davidson (1991) to criticize the assumption that the future is quantitatively predictable at least stochastically: this assumption, he argues, requires that the paths followed by economic variables be ergodic, that is, roughly, that the statistical regularities observed up to the present continue to exist in the future. Davidson argues that this would require the absence of novelties; history does not repeat itself. I agree: some predictability of tendencies does exist (e.g. about the influence of cost on price, about production adjusting to demand, about multiplier processes), but the tendencies of growth and distribution depend on too many unpredictable influences to allow robust predictions reaching out into the future even only ten years.
- 60.
That is, a price vector ensuring intertemporal contingent equilibrium when at the initial date it is possible to trade all contingent goods against each other – universal markets, in my terminology.
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Petri, F. (2021). Uncertainty and General Equilibrium. In: Microeconomics for the Critical Mind. Classroom Companion: Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-62070-7_9
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