Allocation under Uncertainty: Equilibrium and Optimality

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.

In the theory of optimum growth it has been found that models with discrete time are easier to treat rigorously than models with continuous time. But continuous-time models often have the advantage of providing simpler results. I shall illustrate this tension in the present paper by discussing the model for optimum growth under uncertainty that has received most attention in the literature (Phelps [6], Levhari and Srinivasan [4], Hahn [2], Hakansson [3], Brock and Mirman [1]). An existence theorem will be proved for the discrete-time case. By a heuristic argument, I obtain an equation for the optimum under continuous-time which makes possible results about the effects of uncertainty on the optimum policy more general than are available in discrete time. These latter results are somewhat surprising. By way of prelude I outline the reasons for research into optimum growth under uncertainty, and offer a classification of models. The model discussed in this paper is less appealing than some others; but it seems to be the easiest one.

An extension of the theory of allocation of resources to the case of uncertainty has been realised recently. After the pioneering article of Arrow [2], who with Allais [1] initiated this analysis, the field was explored by Baudier [4], and Debreu [7, 8] whose results have been somewhat generalised by Radner [14]. One can find stimulating discussions about the implications of the theory in Arrow [3], applications in Borch [5] and Hirshleifer [11], or critical comments on various aspects in Drèze [6].

In the classical model of resource allocation, often called the Arrow-Debreu model, all transactions are determined at the outset; the contracts are final and the history of the economy is in accordance with the initial plans; all possibility of reopening the markets can be dispensed with; it would serve no real purpose.

In a market with ‘many’ traders who bear risks, there is the possibility of pooling their independent risks and in this way to eliminate traders’ risks. There is a benefit from trade, and the way this benefit is divided between the traders depends on the system of exchange.

This paper is an attempt to introduce stochastic events in consumer behaviour and the related equilibrium theory. The idea of stochastic behaviour is not very new as may be seen from the literature. Nevertheless, it is not introduced in the current theory of general equilibrium. Indeed, one always supposes that people behave rationally (i.e. their preferences can be represented by a continuous utility function). In psychology such a model is not accepted and one tries to replace the ‘homo deterministicus’ by the ‘homo stochas-ticus’ (this is done keeping in mind that they are both ‘homo sapiens’). The introduction of stochastic individuals in general equilibrium theory is an idea of W. Hildenbrand [9]. This paper generalises his results, since we use only asymptotic independence instead of independence. See also [3] for another application to general equilibrium analysis.

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 [1] and pursued further by Debreu [7].²

The simplest model of a productive economy is based on the following assumptions: agents live only for one period, they take input and output prices as given, and production involves no risks.

The problem of efficient allocation of capital in a world of uncertainty has played a major role in the debate on the social rate of discount. One view, which has been advanced by Hirshleifer [7, 8] and supported by Diamond [6], is that differences in rates of return on capital in the private sector of the economy reflect differences in riskiness among alternative lines of investment, and that these differences are of normative significance for the allocation of capital in the public sector. Thus, when discounting costs and benefits of a particular type of public investment, the government should take as its discount rate the rate of return on capital in a private industry of similar riskiness. Another view, which counts Samuelson [17] and Vickrey [22] among its supporters, is that because of the extremely large and diversified investment portfolio held by the public sector, the marginal return from public investment as a whole is practically risk free and should be equated to the market rate on riskless bonds. In an important recent contribution Arrow and Lind [2] come to the same conclusion for a somewhat different reason; the total risk carried by the public sector is shared among so many that each individual’s risk burden becomes negligible.

A model of an exchange economy is presented where money is the only asset. It is shown that, under some assumptions, a short-run equilibrium exists if the traders’ price expectations do not depend ‘too much’ on current prices.

This paper is devoted to proving a more general version of proposition 2.1 of [6]. For any unexplained notion we refer to Dieter Sondermann’s paper [6].