This chapter deals with three issues related to consumption decisions under uncertainty, namely, (i) the determinants of risk aversion for future consumption; (ii) the impact of uncertainty about future resources on current consumption and (iii) the separability of consumption decisions and portfolio choices. These issues are discussed in the context of a simple model introduced, together with our assumptions, in Section 1. The first issue is motivated and treated in Section 2, the conclusions of which are summarised in proposition 2.5. The other two issues are treated in Section 3 under the assumption that there exist perfect markets for risks, and in Section 4 under the converse assumption.

Some technical results needed in the text are collected in Appendices A, B and C; a simple graphical illustration of our major result, theorem 3.3, is given in Appendix D.

The model and the assumptions

Model

Following Fisher (1930), we study the problem faced by a consumer who must allocate his total wealth y between a flow of current (or ‘initial’) consumption c1 and a residual stock (y – c1) out of which future consumption c2 (including bequests) will be financed. We restrict our attention to the aggregate values of present and future consumption, or equivalently to a single-commodity, two-period world.

We conceive of the consumer's wealth y as being the sum of two terms:

1. The (net) market value of his assets, plus his labour income during the initial period, to be denoted altogether by y1.

2. The present value of his future labour income, plus additional receipts from sources other than his current assets.

Forms of risk-sharing

The economic theory of decision-making under uncertainty and riskbearing rests on the assumption that individual agents behave consistently. If preferences do not depend upon the state of nature and if there exist opportunities for gambling at fair odds (say, on the stock market, or through private bets), then consistent behaviour implies aversion, or at least neutrality, towards economic risks (cf. Drèze (1971)). Casual empiricism confirms that risk aversion is indeed the rule: most individuals seem eager to shed their risks, even at unfair odds.

These risks take many forms, and affect an individual's health, his wealth, his liability, etc. Risk-shedding is achieved mainly through diversification, exchange or insurance. Each one of these three devices is limited in scope.

Diversification consists in splitting a risk into components with limited stochastic dependence, thereby reducing the total variance. Asset portfolios provide an example. But diversification entails costs of information and transactions. And an element of stochastic dependence is always present on the level of general economic activity: collective risks (as opposed to individual risks) cannot be eliminated through diversification.

Exchange sometimes enables strongly risk-averse individuals to sell their risks to other agents who are more tolerant or endowed with complementary risks. Futures markets enable buyers and sellers to shed price uncertainties (but not quantity uncertainties). Equity financing is a way of selling business risks. But exchange must take place before the relevant information becomes available (once a lottery is drawn, there is no market any more for its tickets).

Most of the papers collected in this volume rely, explicitly or implicitly, upon (i) a formal description of uncertainty situations in terms of the concepts of events, acts and consequences; and (ii) an axiomatic theory of individual choices, which justifies the representation of preferences among acts by their expected utility.

The purpose of this introductory essay is to review these concepts and axioms against the background of their economic applications. The essay is not a systematic exposition of the theory (results will be stated without proofs); neither is it a survey of contributions to the theory (only a few key references will be used). Rather, it is a review of the main properties of the concepts and of the axiomatic theory, with a discussion of their usefulness and limitations in economic applications.

Section 1 provides an element of historical perspective. Section 2 reviews the concepts of events, acts and consequences. Section 3 reviews the axioms of a normative theory of individual choice under uncertainty and their main implication (the moral expectation theorem). Section 4 reviews some objections that have been raised against the normative appeal of the axioms. Section 5 is devoted to a general discussion of the usefulness and limitations of the theory. Section 6 relates the concepts of the general theory to those underlying economic applications presented elsewhere in this volume.

The firm fits into general equilibrium theory as a balloon fits into an envelope: flattened out! Try with a blown-up balloon: the envelope may tear, or fly away: at best, it will be hard to seal and impossible to mail…. Instead, burst the balloon flat, and everything becomes easy. Similarly with the firm and general equilibrium – though the analogy requires a word of explanation.

General equilibrium theory – GET for short – has two attributes. First, it defines clearly the boundary between economic analysis and the exogenous primitive data or assumptions from which it proceeds; that is, it defines a precise, self-contained ‘model’. Second, it verifies the overall consistency of the economic analysis. A natural step in verifying overall consistency is to exhibit sufficient conditions for the existence of the proposed solutions, or ‘equilibria’. This step is usually amenable to mathematical reasoning.

Still, I do not mean to identify general equilibrium theory with that potent cocktail of economics and mathematics known as mathematical economics. (To some, mathematical economics is merely a pleonasm; to others, it is a branch of mathematical pornography; the word cocktail, with its element of pornographic pleonasm, is purposely neutral.) Work in mathematical economics lacking the GET–attributes is abundant.

Introduction

General equilibrium under uncertainty

The theory of equilibrium and efficiency of resource allocation, initially developed for a world of certainty, has been reinterpreted for a world of uncertainty, thanks to a suggestion made by Arrow (1953) and pursued further by Debreu (1959).

An economy is defined by (i) a set of commodities, with the total resources (quantities of these commodities) initially available; (ii) a set of consumers, with their consumption sets and preferences; (iii) a set of producers, with their production sets. The resources, consumption sets and production sets define the physical environment. In a world of certainty, the environment is given. In a world of uncertainty, the environment depends upon uncertain events. Let these be determined by ‘the choice that nature makes among a finite number of alternatives’ (Debreu, 1959, p. 98).

The reinterpretation consists in defining a commodity not only by its physical properties (including the time and place at which it is available) but also by an event conditional upon which it is available. An allocation then specifies the consumption of every consumer and the production of every producer, conditional on every event. Uncertainty means that these consumptions and productions may vary with the event that obtains.

Consumer preferences are defined over commodity vectors, that is, over plans specifying fully the consumption associated with every event. These preferences are introduced as a primitive concept.

Introduction and preview

Introduction

The economic theory of resource allocation was initially developed for a given environment, defined by (i) a set of commodities; (ii) a set of consumers, with their initial resources, consumption technology and preferences; (iii) a set of producers, with their initial resources and production technology. Actually, the environment is not given, but depends upon uncertain events. Research introduces new commodities and new technologies, resources are discovered or accidentally destroyed, consumer preferences are subject to unpredictable changes, the yield of production processes is affected by meteorological and random circumstances, and so on. An important conceptual clarification, introduced in the early fifties by Arrow (1953) and Savage (1954), consists in considering a set of alternative, mutually exclusive ‘states of the environment’, among which ‘nature’ will choose. This approach provides a more natural starting point for the economic theory of uncertainty than earlier formulations in terms of probability distributions for environmental characteristics or economic variables.1 In particular, individual decisions and overall resource allocation remain amenable, under the new approach, to a unified treatment, into which the deterministic theory fits as a special case.

Over the past 25 years, theoretical developments within the new framework have been conclusive on some issues, while other issues remain debated. Broadly speaking, the theories of consumer decisions, and of competitive equilibria with complete markets, have received lucid expositions, with successive contributions fitting neatly together.

The impact of demand fluctuations on expected or average profits for a competitive firm was underlined by Walter Oi (1961), who showed the following: If a firm producing a single output under increasing marginal cost, equates (ex post) marginal cost to price, then expected (or average) profits increase with the variance of price. An obvious implication of this result is that increased uncertainty of demand, because it increases expected profits, will attract new firms in the industry if there exist entrepreneurs motivated by profit expectations. This implication was pursued in 1967 by Drèze and Gabszewicz (henceforth D–G), who considered an industry consisting of identical firms supplying a single commodity under demand uncertainty, and defined a competitive equilibrium by the condition that expected profits for each firm be zero.

Under specific assumptions (a cubic cost function, a random demand with known symmetric density, and zero-price elasticity), they derived the following propositions:

1. In competitive equilibrium there is an optimum number of firms, each of which is operating with excess capacity on the average.

2. In competitive equilibrium the expected price is less than minimum average cost, and could not be reduced further without causing expected profits to become negative.

3. In competitive equilibrium expected marginal cost is equal to expected average cost when and only when the expectations are weighted by quantities.

4. Under price rigidity a price equal to expected marginal cost (unweighted) exceeds minimum average cost when expected profits are zero.

Introduction

Scope and example

In the theory of utility, as presented in modern terms by von Neumann and Morgenstern (1947) and their followers, one starts from a set of prizes, say the finite set B with elements bq, q = 1,…, t. One then defines probability mixtures of prizes, by means of numerical probability vectors β on B, and observes the preferences of a decision-maker among such probability mixtures. When these preferences satisfy three simple axioms (complete order, independence and continuity – see Section 3 below), there exists a real-valued function u on B, called utility, such that preferences among probability mixtures of prizes are isomorphic with expected utilities. That is, the mixture β is preferred to the mixture if and only if

In the theory of games against nature, as presented in modern terms by Savage (1954) and his followers, one starts from a set of alternative, mutually exclusive states of the world, say the finite set S with elements s1,…, sn. Using, as before, a set B of prizes (‘consequences’), acts f, f′,…, are defined as mappings of S into B – that is, as state distributions of prizes (instead of the probability distributions of prizes considered in utility theory). One then observes the preferences of a decision-maker among acts.