Flexibility is a concq)t that has croppd up a number of timcs in the dcvelopmcnt of micro-economic theory. The notion that good current actions may ~C J those which permit good later responses to later observations has appeared in discussions about choice of plant (Stigler [I:{], Hart (4, 51) and of portfolio (1. Marschak (91). Recently the iiotiori lias appearecl again in an analysis of clecisionmaking in thc conduct of licscrirch ant1 Ilcvclopmcnt (171, 181, [IO]).In cach of these discussions it is conjectured or a r g u d that flexibility (of plant, portfolio, or X and D policy) is good in an uncertain world.The diversity of the cascs discusscd tempts one to broaden the conjecture to cover all of them (and others) at once.As additional problems in sequential decision-making under uncertainty are investigated (and wliat economic prohlcms facing firm or household arc not csscntially of this kind?) the notion of flexibility seems likely to play a larger role in economic theory.It is perhaps time, thcrefore, to propose some precise definitions (or measures) of flexibility, to formulatc the above conjecture using the definitions, and to see if the coiijccture is truc in some simple cases.We attempt to do so licre. Section I formulates some dcfinitioiis of flexibility, Sectioii I1 makes a conjecture about the value of flexibility as uncertainty grows, and Section 111-V apply the definitions and test the conjccture in three simple cascs. Two of the caws are the classic oncs niciitioned above and the third has to do with the conduct of Research ant1 I)cvclopnicnt. 'rO\YAHD A 1)IJFINITION OF FLEXIBILITT.Many decision problcms are charac terizecl by tlic following structurc:a) The payoff to the dccision maker is a function of a sequciice of actions takcn by hiin, at times t, , t, , ... , t, , ... , a d a sequence of states of thc world bcyond his control. b) At any point in the decision sequence, the decision maker has less than pcrfect infornration about what the future sequence of states of thc world will be.
This paper obtains finite counterparts of previous results that showed the informational efficiency of the Walrasian mechanism among all mechanisms yielding Pareto-optimal individually rational trades in an exchange economy while using a continuum of possible messages. Such mechanisms lack realism, since it is not possible to transmit or announce all points of a continuum, and it generally takes infinite time to find an equilibrium message, among all the messages in a continuum. Accordingly, the paper studies approximations of the continuum Walrasian mechanism, in which the number of messages is finite. It applies general results from a companion paper, which considered finite approximations of continuum mechanisms in general organizations, with exchange economies as a particular example. For classic exchange economies, we compare the continuum Walrasian mechanism with alternative continuum mechanisms that also find a Pareto-optimal and individually rational allocation. There are many of them, and some of them, like the continuum Direct Revelation mechanism, do not use prices at all. A finite approximation to a continuum mechanism will have an error. Its ovemll error for a given class of economies is the worst distance (over all members of the class) between the continuum mechanism's final allocation and the approximation's final allocation. We measure a finite mechanism's cost by the number of its (equilibrium) messages. We consider exchange economies in which traders' utility functions are quasi-linear and strictly concave. We find that the overall error of a sufficiently fine finite approximation of the Walrasian mechanism is arbitrarily close to the overall error of a not more costly approximation of an alternative continuum mechanism that has the same number of message variables. The former overall error is smaller than the latter if the alternative continuum mechanism has a larger number of message variables. A continuum Direct Revelation mechanism is an example of an alternative mechanism with a larger number of message variables than the Walrasian mechanism. Thus the informational superiority of the Walrasian mechanism emerges again when we approximate it and take the finite number of messages as our cost measure. [11]. 2While the paper by Hayek ([2]) is frequently cited as an articulate statement of the superiority of price mechanisms, an extensive discussion of the issue is found earlier, in the debate about the possibility of socialism. See, for example, Lange [7].
W e study a newsvendor who can acquire the services of a forecaster, or, more generally, an information gatherer (IG) to improve his information about demand. When the IG's effort increases, does the average ex ante order quantity rise or fall? Do average ex post sales rise or fall? Improvements in information technology and in the services offered by forecasters provide motivation for the study of these questions. Much depends on our model of the IG and his efforts. We study an IG who sends a signal to a classic single-period newsvendor. The signal defines the newsvendor's posterior probability distribution on the possible demands and the newsvendor uses that posterior to calculate the optimal order. Each of the possible posteriors is a scale/location transform of the same base distribution. When the IG works harder, the average scale parameter drops. Higher IG effort is always useful to the newsvendor. We show that there is a critical value of order cost. For costs on one side of this value more IG effort leads to a higher average ex ante order and for costs on the other side to a lower average order. But for all costs, more IG effort leads to higher average ex post sales. We obtain analogous results for a "regret-averse" newsvendor who suffers a penalty that is a nonlinear function of the discrepancy between quantity ordered and true demand.
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