Flexibility and Uncertainty (Ostroy)

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The Review of Economic Studies, Ltd.

Flexibility and UncertaintyAuthor(s): Robert A. Jones and Joseph M. OstroyReviewed work(s):Source: The Review of Economic Studies, Vol. 51, No. 1 (Jan., 1984), pp. 13-32

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Review of EconomicStudies(1984) LI, 13-32 0034-6527/84/00020013$00.50? 1984 The Societyfor EconomicAnalysisLimited

Flexibility a n d UncertaintyROBERT A. JONESUniversity of British Columbia

andJOSEPH M. OSTROYUniversity of California, Los Angeles

The preserving f flexibilitywhenfacedwith uncertaintys a neglectedaspectof behaviourunder risk. Yet it is an importantactor n decisions o hold liquidassets or delayirreversibleinvestment.Thispaperformalizes he notion of flexibilityn a sequentialdecisioncontext,andrelates ts value to the amountof information nagent expects o receive. A rudimentarymoneydemandmodelis developedembodyingheseideas,and the historyof flexibility s an economicconcept s traced.1. INTRODUCTION

Choices are frequentlymade between alternatives hat imply differentdegreesof futurecommitment-between a short-term investmentthat leaves future options open, forexample,and a long-termone that, by its very nature, forecloses those options. Therelative attractivenessof the two dependson their probabilitydistributionsof payoffsover time. A basic consideration n this choice is the recognition hat beliefs about therisks governing hese payoffsmay change. Currentdoubts may be partiallyresolved inthe near future. This prospect decreases the attractiveness f the longer term commit-ment, in that one is able to respond less fully to new information,and, even if it doesnot directlyaffect the risks associatedwith shorterterm choices, enhancestheir appeal.Thispaper formalizes he above remarksby establishingconnectionsbetween thefollowingtwo (partial)orderings:The first s an orderingbased on variabilityof beliefs.One set of beliefs is more variable than anotherif more final risk is resolved at anintermediate tage. The more one expectsto learnby an intermediateperiod, relativeto what one knowstoday, the more variationone is anticipatingn beliefs about the finaloutcome. The second is an orderingof currentactions,or positions,based on flexibility.One position is more flexible than anotherif it leaves availablea largerset of futurepositionsat any given level of cost. These two orderingsare incorporatedn a simplesequentialdecision model to suggest the following behaviouralprinciple:The morevariableare a decision-maker's eliefs, the more flexibleis the positionhe will choose.This principlepotentially applies whenever (i) there will be opportunities o act afterfurther nformation s received,and(ii)currentactions nfluenceeitherthe attractivenessor availability f different uture actions.The applicationof this principleto macroeconomicphenomenais of particularinterest. Consider he newspaperheadline:"Decrease n confidence eadsto cutback ncapital goods orders". The decreasein confidencecanbe interpretedas an increaseinthe variability fbeliefs-the less confidentarecurrentbeliefs,thegreaters the likelihoodof substantial evision n the near future.As a consequence, here is a fall in the demandfor inflexiblepositions (commitmentso new capitalgoods)anda risein the demandforflexiblepositions (postponement,waiting,or holding iquid assets).At the other end ofthe liquidity spectrum, Section 5 below shows that variabilityof beliefs generatesademandfor moneyeven whenmoneyis dominatedby all other assetsin termsof yield

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14 REVIEW OF ECONOMIC STUDIESand the cost of reversingpositions n other assets s modest.Decreasedconfidence nducesa temporarypremiumof liquidityand discounton illiquidity.The analysis of choices under uncertainty conventionally revolves around thedecision-maker's ttitude towardrisk. The demand for flexibility,however, is basicallyunconnectedwith riskaversion.This is because having many ratherthan few positionsavailablefor future choice implies nothing about the variabilityof final payoffs. Oneindividual might value flexibility because, by appropriatelyadapting choices to theinformationreceived, it permits a more nearly certain pecuniaryreward;but anothermight value it because it allows the makingof informed higher risk bets at the lastmoment. The way flexibility s usedto exploit forthcomingnformationmay be dictatedby attitudestowardrisk; but flexible positionsare attractivenot because they are safestoresof value, but because they aregood storesof options.The next two sections define the orderingof beliefs based on variabilityand theorderingof choices based on flexibility. Section 4 states qualified versions of thepropositionrelating these two orderings.Section 5 illustrates he principle n a simpleasset choice context. Discussionof the work of others and the extensive historyof theseconcepts s reserved or Section6.

2. COMPARISON OF BELIEFS BASED ON VARIABILITYLet S be a set of states, withelementss, and Y an indexset of messages/observations,with elements y. The sets of possible probabilityvectors on S and Y are, respectively,

As -{7r = (r): sr, Z, = 1}and

Ay {q = (qy): q, ' O, Eq = 1}.The set of possibleconditionaldistributions f s given y is AY=(As)Y. An elementIrE Aswill be called a belief (about the possibilities in S), with Ir(y)e As denoting a beliefconditionalon observing he message y and11a matrixwhose columnsare Ir(y), y E Y.Let

(n, q) E AYX Ay.(II, q) is a set of beliefs, one for each possible observation n Y, together with a vectorof probabilities.that ach of those beliefs will be held. In the terminologyof Marschakand Miyasawa, (II, q) is an information structure. The mean of (II, q) is

r- q,7 (y)andwill be called the priorbelieffor (II, q)-the beliefbeforeany message s observed.Throughoutour discussion he observations,Y, and the structureof beliefs, (LI,q),are exogenousto the individual.They are not objectsof choiceas they wouldbe if theindividualwere able to select the experiment o perform.A partial ordering of belief structures, denoted by (1, q) - (LI', q'), is defined by thecondition hat

ZqyD(7T(y)) _ E q(D(7r'(y)) for all convex functions D:As -- R. (1)Regarding7r(y)asa vector-valued andomvariableof y, (1) maybe interpretedassayingthat the beliefs {7(y)} are more dispersed than are {1r'(y)}. Note that if (nI, q) can becomparedto (I',q') via -, then their means, or prior beliefs, must be identical:(LI,q) !(LI',q') implies

qy.7(y q ',7T'(y) (2)



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JONES & OSTROY FLEXIBILITYAND UNCERTAINTY 15We shalldescribecondition(1) as sayingthatthe beliefsin (II, q) are more variable hanin (LI', ').To understandhisordering,consider ts extremes. At one extreme, et (I*, q*) besuch that q*(y)>0 impliesthat 7r*(y)) ,-. There is no variabilityof beliefs; they arethe same whichevermessage s received. At the otherextreme, et (Il*, q *) be suchthatq *(y)>0 impliesthat eachelementof 7 *(y) is either0 or 1. Thisrepresentsmaximumvariability f beliefs;receiptof a messageconveys certaintyaboutwhichstatewilloccur.The definitionof more variable beliefs is taken from earlier work in statisticaldecision theory. Let B be a finite set of actions and u(b, s) be a payofffunctiondefinedon B x S. Bohnenblust,Shapleyand Sherman(1949) showedthat (1) is equivalenttothe followingdefinitionof (II, q) as morevaluablethan (LI', '):

Y.q, maxb.B Z.srT(y)u(b, s)-Y,q'y maxbsB ,sr'(y)u(b, s)for all bounded u (b, s). (3)An individual'spreferencefor risk is embodiedin the curvatureof his utilityfunction.But, since the expressionsin (3) can be viewed as attainableexpected utilities, allindividualsprefergreatervariabilityn beliefs, regardlessof theirattitudetoward risk.Another perspective s provided by regarding II,q) as an "experiment"where ywill be observedwith probabilityqy, after which one's beliefs about S will be 7r(y).Blackwell(1951, 1953)showedthat(1) is equivalent o the followingdefinitionof (II,q)asmore informative han,or sufficient or, (HI', '):There exists a non-negativen x n matrixM with columnssumming o 1, wherenis the numberof elementsin Y,suchthat

I '= IM and q=Mq'. (4)If (4)is satisfied,one couldconstructa "blackbox"that acceptsy as inputsandgeneratesoutputs abelledy' that have exactly he samejointdistributionwith s as they' associatedwith the experiment(II',q'). Thus greatervariabilityof beliefs is desirable becauseitmeansthatmessagesconveymoreinformationabout s.Proposition5 below utilizes a restrictedversion of-. We shall say that beliefs(n, q) area star-shaped preadingof (L', q'), indicatedby (H, q) 's (L', q'), if flq = 1'q',q = q', and there exists0 'Ac _ 1 for eachy E Y such that

7r'(y)= Ay7 (y) + (1 - Ay)fr. (5)A star-shaped preadingof beliefs impliesmore variablebeliefs,but the conversedoesnot generallyhold. Figure 1(a) displaysthe beliefs of two information tructures hatdo satisfy (II,q) 's (LI',q'), withthreepossiblestates. Figure1(b) depictsa casewhere(I, q) - (LI',q') without star-shaped spreading, and Figure 1(c) a pair of informationstructuresnot rankableby eitherordering.The two orderingsareequivalent,however,when either (i) (LI', ') conveysno information,or (ii) q = q' and has only two positive

state 3A. B. C.

07r(4) +/ ~ ~~~ ~0r3 71(3731(2)__0 0 ~ ~ ~4(7 )371(1 O7r2) rb3 Q + + +\+72 MD D(12)M 71D1 7r(i)

state 1 state 2FIGURE 1

Beliefs plotted in the unit simplex



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16 REVIEW OF ECONOMIC STUDIESelements (i.e. there are just two possible observations, as in the example of Section 5below). A further instance of star-shaped spreading occurs when the messages' usefulnessfor prediction is contingent on the validity of some theory, which if false renders themvalueless. Then, letting A= Aybe the probability that the theory is valid, any rise in Aincreases variabilityin the required fashion. The ordering is useful when certain functionsof 7rare quasiconvex but not convex.

Remark 1. Formally, the definition of more variable beliefs is closely related tothe Rothschild and Stiglitz (1970) definition of beliefs that are more risky, an orderingon As. Assume that states can be associated with realizations of a random variable, x (s).Then ir is more risky than ir' if for all convex functions qf:R - R,ZrVso(x(s)) ?Z i'fr(x(s)). (6)

The similarities help to point out the differences. In the older view contrasting riskand uncertainty, see Section 5.1, beliefs that were thought to be objectively fixed were"risks" while beliefs that were subject to fluctuations were "uncertainties". Just ascondition (6) defines an ordering of risks, condition (1) establishes an ordering ofuncertainties.

Remark 2. In economic contexts there are two distinct sources of increased variabil-ity of beliefs. First, there may be an improvement in the information content of availableobservations. Surveys can be based on larger samples; econometric forecasts can becomemore accurate. Such changes correspond to performing "better experiments" in statisticaldecision theory. tSecond, there can be a change in the confidence with which prior beliefsare held. The revised belief, ir(y), combines the information contained in y with theinformation on which prior beliefs were based. If the amount of information embodiedin 7ris large and regarded as relevant for s, then the individual is likely to make onlysmall revisions upon observing y. Conversely, if the prior information is limited or itsrelevance becomes questionable, then subsequent observations carry more weight andthe same observations result in greater variation in beliefs (see Jones (1981)). Forexample, suppose some event occurs (e.g. a government policy change) that causes anindividual to "lose confidence" in his estimates of the parameters of his model of theeconomic environment. The change reduces the relevance of past experience andincreases the anticipated impact of future observations on his beliefs. Changes arisingfrom this source are relevant for macroeconomics since such a "loss of confidence" inbeliefs is likely to be widely shared. It is this second possibility that led us to adopt themore neutral terminology, variability of beliefs, over the standard terminology, moreinformative information structure.

3. COMPARISON OF ACTIONS BASED ON FLEXIBILITYConsider an information structure joined to a sequential decision problem. In periodone the individual chooses an initial position a e A. In period two, after observing y,there is an opportunity to choose a second period position b EB. A and B are assumedfinite. In period three s is revealed. The consequence for the individual is described bya payoff function f: A x B x S - R.

Flexibility is a property of initial positions. It refers to the cost, or possibility, ofmoving to various second period positions. To rank positions by their flexibility somepart of the total payoff f(a, b, s) must be imputed to the move from a to b, as distinctfrom having been in positions a and b. The payoff must be decomposed into a formf(a,b,s)=r(a,s)+u(b,s)-c(a,b,s), (7)



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JONES & OSTROY FLEXIBILITY AND UNCERTAINTY 17where r(a, s) is the direct return on the first period action, u (b, s) is the return on thesecond period action, and c (a, b, s) is the cost of "switching" from a to b.The nature of the switching cost function is summarized by the following construction.Let G:A xSxR -28 be defined by

G(a,s,a)l{b: c(a,b,s)?a}a (8)G (a, s, a) is the set of second period positions attainable from a at a cost that does notexceed a in state s.Structure is added to the concept of switching costs by the following restrictions:Non-negative switching costs:

G(a,s,a)=0 foralla


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18 REVIEW OF ECONOMIC STUDIESprohibitivelycostly or requirean expendituregreater than a(a). Note that both theperfectly lexibleand irreversible ositionshave the propertyof extremeswitching osts.The ordering?!F is complete if for any a, a' E A either a ZF a' or a' >F a. In the specialcase where !F is both completeand extreme,weshallsay that switching osts are nested.

Remark 3. In some decisionproblems,particularlywhen the choices available nA are identicalto B, there is a naturaldecompositionof f into (7) where c satisfies henon-negativityand existence of zero-cost alternativeassumptions.For example, whena and b are portfoliosof assets, r(a, s) and u (b, s) can be the portfolioyields over thetwo time periods and c (a, b, s) can be the cost of liquidating hose assets in a that arenotinb. WhenthesetsA andB are quitedifferent, he selectionof g(a) mayneverthelessbe clear from the context.For example, if the initialchoice is the technologyto installin a plant and the second periodchoice is the output evel at whichthe plant is operated,one might associate with each technology, a, that output level yieldinglowest averagecost for a and call thatg(a). In othercontexts,a decompositionof f into (7) may resultin a seemingly arbitrary mputationof switchingcosts. The imputationdoes not affectthe optimal strategy,of course, but it does affect any rationalization f the strategy ntermsof flexibility.

4. RELATION BETWEEN THE VALUE OF FLEXIBILITY AND THEVARIABILITY OF BELIEFSThe three period two-choice sequentialdecision problem thus gives rise to two partialorderings:One ranks nformationtructures ccording o variability f beliefs, indicatingthe amount to be learned from future observation; the other ranks initial positionsaccording o flexibility, ndicating he rangeof alternatives eft open at any given levelof switching ost. We explorethe relationbetween these two orderings.An optimal strategyfor the sequential decision problem consistsof a first periodposition, a, and a set of second period positions, {by}, to be taken depending on theobservationy E Y received,which maximize the expectedtotal payoff. The expectedpayoffso obtainedcanbe expressedrecursivelyusing the maximumprincipleof dynamicprogramming:

J(H7, ) = maxaEAj{by}EB y Is qyirsy)f(a, b, s)= maxaEAZyqy maxbB Zsirs y)f(a,b, s). (10)

Decomposingf(a, b, s) into r(a, s)+u(b, s)-c(a, b, s), utilizing #rs yZqym,(y) (see (2)),and separating terms givesJ(1I, q) = maxa [ s 4rsr(a, ) +y qymaxb Es rs y)(u(b,s) -c(a, b, s))]

= maxa [F(a) +y qyv a; ir(y))]= maxa [F(a)+ V(a; [I,q)] (11)The expected firstperiodreturnto position a, 1(a), dependsonly on the priorbeliefii. Since #r must be the same for all informationstructurescomparable n terms ofvariability, he (opportunity)cost of flexibilityis independentof what the individualexpectsto learn. The function v(a; ir) is the maximumexpectedsecondperiod return,including witching osts,fora givenbeliefand nitialposition. V(a; HI,q) istheexpectedsecondperiodreturnto takinginitialpositiona with information tructure I1,q)-i.e.the expectedvalueof v(a; ir).We now pose the followingquestion:Supposethat a (respectivelya') is optimal nthe firstperiodwhen the structureof beliefs is (H,lq) (respectively LI', ')). If (LI,q) -(H',q'), and a and a' are rankedby zF, must we have a -Fa'? In other words,does



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JONES & OSTROY FLEXIBILITYAND UNCERTAINTY 19the prospectof an improvementn the informationo be receivedbetween periodsresultin a more flexiblefirst-perioddecision?Stated more formally, et A (H, q) be the set of optimal initialpositions, the onesfor which f(a) + V(a; H,q) is a maximum. We shall say that - induces an order-preserving elationon tF whenever

(ii)a q)AHt, q) A)(ii) a(HE)1H,q)anda'EA) (', q')(iii) either a A(Hi',q') or a' A(H, q) implya _Fa'.(iv) a anda' are F-comparableIf a and a' satisfy (i)-(iii), the condition stipulates that either a and a' are not-F-comparableor else thata' Fa'. Thatis, an increase n the variability f beliefs neverleadsto the choice of a less flexibleposition. It will be shown what this requires.Forour purposes, he three period decisionproblem s summarizedby {r,v, (HI, )},where r:AxS -R and v:AxAs -R are as defined above. Note that v(a;lr), as afunctionof r, is a maximumof a finite numberof linear unctionsand is thereforeconvexin v. It followsfrom the definitionof (H1, ) - (H',q') that V(a; H, q) ' V(a; H',q')-thevalue of any initial position is a non-decreasing unctionof the variabilityof beliefs(compare hiswith the notionof being morevaluable n equation(3)).However,for anincrease nvariabilityo causea shifttowarda more flexible(strictlyspeaking,not less flexible)position, thisgain in value must be greaterthe more flexiblethe position. The requiredrelationships

(HI, )-(tI', q') and a ZFa'V(a; H,q) - V(a; Hi',q') = V(a'; l, q) - V(a'; Hl',q'). (12)

Remark4. Notice that such a relationship,when valid,alsosays that any incrementin flexibilityraises the value of any increment n information.Thus, if the individual'schoice were how much information o purchase,with his flexibilitybeing exogenouslygiven-the reverseof the conceptualexperimentwe have interest n-we could say thatan increasein flexibilityinduces the rational agent to purchasea larger quantityofinformation (assuming, hat is, that the total payoff is additivein the price paid forinformation ndthe profitobtained romexploiting t). The relationship etweenflexibil-ity andinformation s thus much ike thatbetween complementaryactorsof production.Rewriting 12) asZyq,[v(a; vr(y))-v(a'; v (y))] Eqy[v (a; r'(y))-v(a'; r'(y))] (13)

leads to the following result that it is the convexity of the difference [v (a; v) - v (a'; i)]that is critical or an order-preservingelationto be induced. This result is independentof ourparticular efinitionof flexibilityn the sense that ZF can be any orderingover A.Proposition 0. (a) If a -FaF' implies that [v (a; v) - v (a'; i)] is convex on As forall a, a' E A, then : induces an order-preserving elationon ZF.(b) If there exists a zF a' such that a' F a and [v (a; v) - v(a'; i)] is not convex onAs, then there exists (H, q)->


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20 REVIEW OF ECONOMIC STUDIESinformation tructuresand payoff functions in variousways, describe circumstancesnwhichthe desired relationshipmay be obtained.The firsttwo propositions, ncludedfor historical nterest (see 6.2), emphasize hatfor some payoff structures he amount to be learned in the future has no effect on theinitial choice.

Proposition 1. When all positions are perfectly flexible, the optimal initial positionis that which offers the highest expected firstperiod return (determinedby priorbeliefs alone).Proposition 2. When all positions are economically irreversible,the optimal initialposition depends only on prior beliefs r.

Propositions1 and 2 imply that the same initial position would be chosen for any twoinformation structures (H, q) > (H', q'). In the circumstances of Proposition 1, but notin those of Proposition 2, the optimal second period position depends on aspectsof(H1, ) other thanf.

Proposition 3. An increase in the variability of beliefs raises the value of any positionrelative to any economically irreversibleposition.This proposition mpliesthat the prospectof more information an induce an agentto changefroman irreversible nitialpositionto one thatis at leastpartially lexible,butnever the other way around. Moreover, the disadvantageof irreversiblepositionsincreases monotonicallywith the degree of uncertaintyabout future beliefs. Capitalformationdecisionsare frequently conomically rreversible.Proposition3 thus suggestsan inverserelationshipbetween investmentand "lackof confidence" n beliefs.The next two propositionsprovide counterparts o Proposition 3 at the oppositeendof the flexibility pectrum: ncreases n the amount o be learnedby waitingenhancethe relativevalueof perfectly lexible positions. Since holding iquid assets, particularlymoney,providesgreatflexibility n manysituations, hese results uggestadirectrelation-shipbetween the demand or liquidityand the expectation hat beliefswillchange.Leta denote,forthe remainder f thepaper,anyelementattainingmaxaEA (a; #).Position a offers the highest expected second period returnfrom the perspectiveofprior beliefs. It is the choice that would be made if, for example, first period returnswereindependentof the action chosenand no further nformationwasexpected.Proposition 4. Anticipating some change in beliefs, as opposed to none, raises thevalue of any perfectly flexible position relative toposition a.Since Proposition4 says nothing about positionsother than a and a* (perfectlyflexible)thatmightbe chosen,it appliesmost usefullyto situationswhere the choice isbetween going ahead with what seems best at the moment versus waitingfor moreinformation. Its significance ies in the fact that irreversibilitys not essential for arelationshipbetween learningand the valueof flexibility.Proposition5 strengthens he resultsof 4 in certain directions. By requiring hecost of "undoing"a to be independentof both s and the positionswitchedto, andbyrequiringhatbeliefs be comparableaccordingo the star-shapedordering,a monotonic

relationships obtainedbetween the amountto be learned and the advantageof perfectflexibilityover a.Proposition 5. Let switching costs satisfy c (d, b, s) = c (d) for all b $ g (ad).Then astar-shaped spreading of beliefs raises the value of any perfectly flexible position relativetoposition a-.



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JONES & OSTROY FLEXIBILITYAND UNCERTAINTY 21The finalPropositioncomesclosestin spirit o the generalconjecture.It establishesa monotonicrelationshipbetweenthe variability f beliefsand the flexibilityof the initialpositionforparticular ayoffstructures.Recall that switchingcosts are nested if (i) for each a EA, there is a Ba such that

for all s anda > 0, B_ = G(a, s, a) and (ii) for all a, a' eA, eitherBa is contained n Ba'or vice versa. We shallsay that the payoffstructure atisfiesConditions1 and 2 ifCondition1. Switching ostsare nested.Condition2. Foreach pair (a, b)e A x B, thereexistsb1 Ba such thatfor all ir,

either, maxb'EBa E iriu(b', s) irsu(b,s)or, maXb ,eB. E,rsu (b' s) Y.rsu b, s)

TointerpretCondition2, it statesthatwhenever omepositionb wouldbe preferableto those available n Ba, it is alwaysthe same positionbEBa, which may dependon b,that is the best availableposition. Simplyknowingthat b is preferred o any elementin Ba is enough to determine the optimal second period decision, without specificknowledgeof ir.

Proposition6. Let thepayoffstructureatisfyConditions1 and 2. Then - inducesan order-preservingelationon !F-Remark5. Conditions1 and 2 aretrivially atisfiedwhen therearejusttwo initialpositions,one of which is irreversible so the right-handside of (9) is the emptyset),which reveals Proposition3 to be a special case of 6. But there is another class ofdecision problemsthat meet its requirements. Assume that the total payoff has anadditive form, r(a, s) + u(b,s), and that the second period choice is the level of areal-valuedcontrolvariablesubject to an inequalityconstraintdeterminedby the firstperiodchoice:b _ z (a). Initialpositionsarecompletelyordered n the mannerrequiredfor Condition 1 by their levels of z (a), with z (a) z (a') implyinga >F a'. Furtherassumethat u (b, s) is concave in b for each s. Then, wheneverthe expected secondperiodreturncouldbe increasedby removing he constrainton b, b = z(a) mustbe thebestconstrained hoiceregardless f thevalueof r. Condition2 is thusmet. Inparticularapplications, he constraintz (a) mightsignifythe maximumcapacityof an otherwiseconstantmarginalcost plant, a numberof delivery options acquired,or quantityof a

naturalresourceleft unextracted.In such circumstances, n increasein the variabilityof beliefs leads a rationaldecision-maker o choose a less bindingconstrainton thissecondperiodchoice.The strengthof Condition2 is apparentwhenconsideringa plausibleextensionofthisproblem. Supposeb is a vector chosen subjectto x(b) z (a), withotheraspectsoftheproblemunchanged.Knowing hat aparticular violating his constraintspreferredto all those satisfying t tells us that the best availablechoice is on the boundaryof Ba,but the particularpointon the boundarygenerallydependson r, violatingCondition2.5. LIQUIDITY AS FLEXIBILITY:AN EXAMPLE

Marketsprovide flexibility by allowing assets to be transformed,throughsale andpurchase, nto other assets. In a monetaryeconomythese transformationsre effectedin two stages:the initialasset is exchanged or money, the moneyis exchanged or thedesired good or asset. The liquidity(saleability)of an asset describes the ease, orcostlessness,withwhichthe firststage is accomplished.Moneyis the most liquidassetsincecostsassociatedwiththe firststageare avoidedcompletely.This sectionillustrates



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22 REVIEW OF ECONOMIC STUDIESthe relation between flexibilityand the prospect of information n a sequentialassetchoice problem, nterpreting he demand ormoneyas a desirefor flexibility.Suppose an individual must choose between three non-diversifiedportfolios:M, A1, A2 (money, asset 1, asset 2). In periodone he chooses whichasset to holduntilperiod two; in periodtwo, afterfurther nformation s received,he chooseswhichassetto hold until periodthree. Thereare two ultimatestatesthatcanoccur, S = {s1,s2}, andtwoobservations hat canbereceived,Y = {y1, Y }. Letthe payoffstructurebe asfollows.The return on portfolioM is 0 in both periods with certainty. In periodone A1 andA2 both returnf > 0 with certainty; n period two they yield 1 and0 respectivelywhens1 occurs (whichfavors asset 1), 0 and 1 respectivelywhen s2 occurs. No costs areincurred f the individual witchesfrom M in the firstperiodto either A1 or A2 in thesecond,or if he continues o hold the sameportfolioas before. But a "liquidation ost"of e >0 is incurred f he switches from either A1 or A2 to a differentportfolioin thesecondperiod. ThusM is more flexiblethanbothA1 andA2 in the sense of Section3.The totalpayofffor eachsequenceof actionsand state is givenin Table I.

TABLE IPayoff f(a, b, s)Initial position Second position , I I(a) (b) State 1 State 2

A1 r+1 fAl m ~~~~~F-e -A2 F-cr rc+1A1 1 0


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JONES & OSTROY FLEXIBILITY AND UNCERTAINTY 23informativeness

p

A A

X ~~~~~~~~~~~pobabilityO 1/2 1-r/ 1

I I~~~~~~~~A/ / pA1 + (to

FIGURE 2Parameter alues orwhichAl, M, andA2 arethe optimalassetsto holdinitially

assets. Region M vanishes if eitlier r->cj/2 or > 1/2-money is never held if itsopportunity cost overshadows either the alternatives' switching costs or the maximumsecond period yield at stake. Holding M can be rational because if Al or A2 is choseninitially, and subsequent observation indicates that the opposite position promises higherexpected returns, then either cost c is incurred or the agent passes up the opportunityto profit from the information.Varying the parameters has plausible effects on the demand for money. Reducingr, the yield on alternative assets, moves outward the vertical boundaries and downwardthe lower boundaries of region M, enlarging the set of beliefs for which money is theoptimal first period asset. Raising c, the illiquidity of alternative assets, has a similareffect. Moving ax toward 1/2, increasing prior uncertainty about which asset has thehighest yield, can move one into region M but not out of it. Increasing p, the informationcontent of y, never causes a switch out of M.An alternative way to see how anticipated information affects the demand for moneyis to ask: at what r is the decision-maker indifferent between all three assets? Lettingax= 1/2,, so he is indifferent between A, and A2, the three regions intersect at p = 2F.The short term yield the individual is willing to forgo by holding money is thus r = p/2(up to F= c/2, beyond which it stays constant to keep region M from vanishing). Thegreater is the information expected in the near term, the higher is the yield required forless liquid assets to be held.

Although we assumed risk neutrality for this illustration, one can verify that the effectof risk aversion on the value of flexibility is ambiguous. Suppose the agent is extremelyrisk averse, concerned only with maximizing his minimum possible payoff. If y conveysless than perfect information, p < 1, then he must hold either Al for both periods orA 2; only in that way is he guaranteed at least r (see Table I). Alternatively, if y promisesperfect information, p = 1, then he must hold M initially; only in that way is he guaranteed



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24 REVIEW OF ECONOMICSTUDIESa returnof 1. Since there are points in Figure2 where M is held althoughp < 1, andpoints whereA1 is held althoughp = 1, it is apparent hat risk aversionhas in one caseenhancedand in the othercase diminished he value of flexibility.This examplewas constructed o distinguishts motivefor holdingmoney as muchas possiblefrom the motivesembodied n existingtheoriesof moneydemand. Risk wasessential,butnot risk aversebehaviour;differential sset liquidation osts wererequired,butnotcompulsoryiquidationstomeet,forexample,unforeseen"cashrequirements");yieldson alternativeassetswere uncertain,but moneywas dominated, n termsof bothimmediate periodone) andfuture (period wo)yields,by all otherassets-none yieldedless than0 in eachperiod. Liquidityhas valuebecause it permitsprofitableexploitationof informationnot yet received.Finally,notice that witha renamingof the positionsM, A1, A2, the structureof theexampleappliesto the heterogeneouscapital nvestmentproblem. Let A1 and A2 referto twodifferent ypesof capitala firmmightacquire,andM referto acquiringno capitalat all (postponingchoice to period two). Both investmentscould be unambiguouslyprofitable,but the firmmay be unsure whichwill be the most profitable. If it expectsthisuncertaintyo be partiallyresolved by periodtwo, it mayrationallyrejectinvestingcurrently n either typeof capital. Investment alls becauseof the expectation hat morewill be learned.

6. CONNECTIONSWITH OTHER WORKSRisk and uncertaintyOur distinctionbetweenthe risk embodied n beliefs andthe variabilityof those beliefsovertime invitescomparisonwiththe distinctionbetweenriskanduncertaintymaintainedby some writers. The most well-known uxstapositionof risk anduncertaintys thatofKnight(1921). He reserves the term "uncertainty"or those events which cannot beassignednumericalprobabilities,and "risk"for those homogeneous,repetitiveeventswhose relativefrequenciescan be ascertained. The distinctionappears o be basedonthe differencebetweenobjectivelyandsubjectivelyormedestimates,withKnightunwill-ing to considernumericalprobabilitiesattachedto events if there is no statisticalbasisfor theirestimation.Keynes too believed that economic risks involved more than just well definedchances. Propositionsand eventsvaryin their "appropriate egree of rationalbelief".The highest degree is knowledge,or certainty;althoughthat certaintymay involvenumericalprobabilities, uch as those assignedto the outcomesof a spin of a roulettewheel knownto be fair-what Knightmighthave calledrisk. Keynesview was similarto Knight's n thathe did not believe thatdegrees-of-beliefneed be numericallycaled;but, unlikeKnight,he was concernedwithbuildinga theory that involvedcomparisonof degrees-of-belief. (Keynes(1936) laterreplacesthe term"degreeof rationalbelief"with "confidenceof beliefs".) Ourapproachs similar o Keynes'if for no other reasonthan his conceptof degree-of-belief nvitesinterpretationn termsof ourrankingbasedon variability.We wouldsay that the degree-of-belief n a priordistribution ver statesincreasesas the variabilityof (11,q) decreases,and that the stateof perfectcertaintyorknowledgecorresponds o the belief that there is nothingmore to learn (i.e. 7r(y)=4forally). Furthermore, ariability nly partially rders nformationtructures ndcannotbe numericallycaled.Distinguishingrisk from uncertaintyn this sense is not new. It is explicitin theterminologyof Marschak 1938, 1949), Tinter(1942) and Hart (1942), among others,whouse the termuncertaintyo describe he prospectof learning.Whatwe have addedis the characterizationf more informative xperimentsby Blackwell (1951, 1953) andby Bohnenblust,ShapleyandSherman 1949) (seealsoMarschak ndMiyasawa 1968),DeGroot (1962) and Kihlstrom 1973))to describechanges n uncertainty.



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JONES & OSTROY FLEXIBILITY AND UNCERTAINTY 25Flexibilitywithand withoutuncertaintyThe notion of flexibilityhas arisen in numerous economic contexts. Without risk,flexibilityconsiderationscan still be important. Makinginvestment rreversiblealtersthe optimalpathof capitalaccumulation Arrow,Beckmannand Karlin(1958), Arrow(1968), Nerlove and Arrow (1962)); asset liquidationcosts influenceportfolio choiceeven whencash needs areperfectly oreseen(Baumol 1952), Grossman 1969), Niehans(1978)). That individualsmighthavea distinctpreference or "postponement f choice"in the absenceof risk anduncertaintys exploredby Koopmans 1964). Thatpreferencesfor flexibilitycan be treatedaxiomaticallywithoutreferenceto probabilities,althoughthey may be equivalentto ones derivedfrom expected utility theory, is demonstratedby Kreps (1979). Marschakand Nelson (1962) recognizethe usefulnessof flexibilityasan economic concept andconsiderhowit mightbe formalized.A connectionbetween random changes and the value of flexibility s drawnbyLavington(1921), who providesa superbearly discussionof what he terms "the riskarisingfrom the immobilityof invested resources". It re-emergesin the context ofbehaviourof the firm in Kalecki(1937) and in Stigler(1939), who describesone plantas beingmore flexible than another f it has a flatteraveragecost curve(thisis pursuedfurtherbyTisdell (1968)).The effect of changes in risk on investment has been studied by Smith (1969),Rothschildand Stiglitz 1971), Hartman 1972) and Nickell(1977),amongothers. Thesestudies supporta basicallyambiguousrelation between risk and investmentdemand.We focussedon the possible inverserelationshipbetween uncertaintyand investment(inflexibility).Although this difference in emphasisreflects, in part, the distinctionbetween changesin riskand changesin uncertainty riskheld constant),it also hingeson thecharacterizationf flexibilityn investmentdecisions. Acquiringadditional apitalcanrepresenta choiceof moreflexibility,orexample,whenit increasesplant capacity-investment oday permits he firmto producemoreas well as less tomorrow.Regardinginvestment as one-dimensionalvariationof a homogeneouscapitalstock is certainlyapossible specification.But if one regards he investmentdecision underuncertaintyasessentiallya choice between no investment and various postponableadditionsto aheterogeneouscapitalstock, then the illiquidityof specificcapitalbecomes a centralconsideration,with more investmentassociatedwith less flexibility see remarksat theend of Section 5).The connectionbetween flexibilityand the prospectof learning s explicitin Hart(1942). He distinguishes isk from uncertaintyas we have, and takes the position that,compared o uncertainty,"rriskascomparativelyittleimportanceneconomicanalysis".Hart points out that uncertaintycan be ignoredwhen all choices are either perfectlyflexibleor economicallyrreversible-ourPropositions1 and2. (Thisobservations alsomadeby Hirshleifer 1972) andHicks (1974).) Concerning he importanceof attitudestoward risk when learning is involved, he states: " . . . the central problems of uncertaintycan be posed and largelysolved under the assumptionof 'riskneutrality"'.Hart alsoanticipated ecentqualificationsf the Simon(1956)-Theil (1957) certainty-equivalencetheorem.In the context of environmentalpreservation,Henry (1974a, 1974b) and Arrowand Fisher (1974) (see also Fisher, Krutillaand Cicchetti (1972)) show that it issub-optimalto replace probabilitydistributionsby their mean values when choosingbetween irreversibleand perfectlyflexiblealternatives,even thoughall other require-ments for the certainty-equivalenceheorem mightbe fulfilled. Propositions3 and 4extend these results.In a relatedpaper,Hirshleifer 1972) measures he "illiquidity" f investmentsbythe timerequired o completea technologicallyrreversibleprocess (i.e. the timefortheinvestment o mature),and showsthatthe prospectof emerging nformation anexplainthe lowerequilibrium ieldon shorter ermassets. BaldwinandMeyer (1979),who cast



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26 REVIEW OF ECONOMICSTUDIEStheiranalysis n termsof the sequentialarrivalof potentiallyattractivenew investmentopportunitiesratherthan new informationaboutfixedexistingopportunities, imilarlyderivea positiverelationbetween the durationof investmentsand the liquiditypremiumneeded to inducetheir acceptance. l3ernanke's 1979) study of the optimaltimingofirreversiblenvestment,by rationalizinghe postponingof projectsduringperiodswhenmuchcanbe learnedby waiting, trongly uggests hatthe observed nstability f aggregateinvestmentover the businesscyclecanbest be explained ntermsof the fluctuating alueof flexibility.Recentwork by Epstein(1980)relatesthe choiceof a to uncertaintywhena is realvaluedand the derivative

va(a, T) limh +o (v(a +h)-v(a))/hexists. In particular, e shows thatwhen Va is convex(concave) n 7r, he optimala rises(falls)with the uncertainty f (II, q). This is similar o ourProposition0 with the naturalorderingof the realnumbersreplacing F. Epsteinillustrates his resultwithexampleshaving a flexibility nterpretation.Freixas and Laffont (1979) examine the situationwhere the firstperiodchoice imposesan inequalityconstrainton secondperiodchoices,a case of nested switchingcosts, demonstrating in their Theorem 1) the corollarydescribed n Remark5.Flexibility nd liquidityThe term"liquidity" asbeenusedto referbothto an asset'scertaintyof yield,includingcapitalgains, and to the differencebetween its purchaseand sale price, includingalltransaction osts. Keynes(1930, p. 67) leaves some ambiguitywhenhe introduces hetermby callingone assetmore liquidthananother f it is "morecertaintyrealizableatshortnotice withoutloss". Makowerand Marschak 1938) take care to distinguishanasset's"safety" romits "plasticity", r futuresaleability,usingliquidity o describe helatterproperty.Certaintyof yield is singledout in the Tobin (1958)-Markowitz(1957) approachto moneydemand. The title of Tobin'spaperaptlyexpressesthe viewpoint:"LiquidityPreferenceasBehaviourTowardRisk".Flexibilitys not an issuesincechoiceis confinedto assetsfree of switching osts.In his recent contributionto monetary theory, Hicks (1974) outlines anotherapproach,encompassinga broaderclassof assetsthat differin termsof saleability. Itrepresents he application o monetary heoryof Hart'sframework,and,in comparisonwith Tobin,couldbe entitled:"LiquidityPreferenceas BehaviourTowardUncertainty".The exampleof Section5 fillsin the formal detailsof an illustration ketchedby Hicks.A similarconnectionbetweenemergingnformation ndthe demand orliquid(saleable)assets is suggestedby Marschak1949), Goldman(1974, 1978) andCropper 1976).Hick's essay, even more than Hart's, indicates the range of macroeconomicphenomena hatmaybe treatedwiththis approach o liquiditypreference.He remarksthat the separationof determinantsof financialand real asset equilibrium, he twincuttingedgesof his earlierIS-LM analysis,mayneedreworking; nd that with thismorerecent approach, n which"the balancesheet must be consideredmore generally,... itis desirablefor the marginalefficiencyof capitaland the theoryof money to be takentogether".

We thushave a sixty year traditionof isolatedrecognition hat flexibilitychoice isa componentof a widerangeof economicdecisions. The difficulty f defining lexibilityin such a way as to have universalapplication,and the difficultyof obtainingformalresultswithout model-specificqualifications,mayaccount or its limitedrole in conven-tionalmicroeconomicheory. Froma macroeconomicperspective,however,the tantal-izingprospectof portrayinghe connectionbetweenbusinesscyclesandpublicconfidence



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JONES & OSTROY FLEXIBILITY AND UNCERTAINTY 27as a relation between flexibilityinduced shifts in asset demands (awayfrom capitalinvestmentand towards more liquid assets, especiallymoney) and uncertainty s toocompelling o be ignored.

APPENDIXProofs of Propositions0-6

Proposition0. (a) v(a, 7r) v(a', 7r)is convex on As for all a, a' E A such thata Fa' implies a'Fa' for all a, a'eA such that (i) (H,q)-(H',q') (ii) aEA(H,q),a' EA(H', q') (iii) a 0 A(H', q') or a'i A (17,q) (iv) a, a' are-F-comparable.Proof. Suppose - does not inducean orderpreserving elationon zF. Thenthereexists (H, q), (H',q'), a, a' satisfying (i)-(iv) and a''Fa. The hypothesisassertsthatv(a', Ir)-v(a, ir) is convexin ir. Hence, from definition 1) of-,

V(a'; II, q) - V(a; H, q) = Zqy[v(a', ir(y)) - v(a, 7r(y))]_ qy v(a', 7'(y ))- v(a, r'(y))]

= V(a'; H', q')- V(a; HI',q'). (A.1)But a EA(H, q) implies 7(a) + V(a; H, q) - F(a') + V(a'; H, q), by definition,and a' EA (HI', ') implies F(a')+ V(a'; H', q') ?_r(a) + V(a; H1', '), with condition (iii) requiringstrict nequality n at least one case.Therefore

V(a'; HI, ) - V(a; HI, ) < V(a'; HI', ')- V(a; HI', '), (A.2)contradictingA. 1). 11

(b) Existence of a, a'eA such that a Fa', a';Fa and v(a, -7r)-v(a',7r) is notconvex on As implies existence of (H, q), (EL', ') and r: A x S - R such that {a'} = A(HI, q)and {a} = A(fl', q'). (Le. - does not induce an order-preserving elation on zF.)Proof. Define d(7r)= v(a, 7r) v(a', 7r). d(7r) is not convex implies there exists

IrT,ITr2EAs andAE[0, l]suchthatAd(I7r)+ (1-A) d(7r ) < d(Air'+ (1 -A )7r 2). (A.3)

Define(H1,q)= ([Tt .]'1-A)) ir-Hq, and (H',q')= ([ -r,i] (11-A))

Clearly (H, q) - (H',q') by (4) since [TM = H' andMq' = q withM=-A 1-Al

Finally,definer: A x S -> R as follows:F(a) = r(a, s) = 0, Vs E Sf(a') = r(a', s) = [d(#r) +Ad(i7r1) (1 -A)d(,7r2)]/2, Vs ESr(a") = r(a", s) < -k Vs ES anda"EA\{a, a'}

where k is sufficiently large to make all other positions a" non-optimal for (H, q) or



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28 REVIEW OF ECONOMICSTUDIES(11',q'). I.e. A(HI,q) c {a, a'} andA(JI', q')c {a, a'}. Now,

(F(a)+ V(a; [I,q))-(F(a')+ V(a'; [I, q))= F(a)-F(a')+Ad(irT)+(1-A)d(i,-2)- -(d(#) -Ad(irT)-(1 -A)d(r2))/2< 0 by(A.3).Hence a'eA( H, q) andaiA( H, q).Similarly,

(r(a) + V(a; LI', '))- (F(a') + V(a'; LI',q')) = f(a)-f(a') +Ad(#) +(1-A)d(#)= (d(#) -Ad(IT1)-(1 -A)d(1T2))/2>0 by(A.3).

Hence a EA(H', q') and a'o A(H', q'). Thus A(H, q) = {a'} and A(H', q') = {a}. 1Inequalitiesusedin proving he nextfourpropositionsarecollected n the followingLemma.Lemma. For all (11,q)-(11', q'),

(L.1) V(a*;L, q) V(a;Lt,q) for all a, a*eA(L.2) V(a;HI,q) V(a;HI',q') for allaEA(L.3) V(a*; I, q)= V(a*; HI',q') for all a* EA(L.4) V(a*; HI,q)=V(a*'; HI, q) for all a*, a*'eA.

Proof. (L.1) follows immediately rom the definitionof V, the non-negativityofswitchingcosts,andthe definitionof perfectflexibilityn Section3.(L.2) follows from the definition of V, the convexityof v(a;7r) in 7r (it is themaximumof a finite collectionof bounded inearfunctions),andthe definitionof>.(L.3) follows from the fact that economic irreversibilitymeans that c(a*, b, s)u(b, s) - u(g(a*), s) for all b, s. Hence v(a*; 7r) = Z,s7rsu(g(a*),s), implyingV(a*; [I,q)=EZs isu(g(a*),s). Since (1,q)_ (11',q') implies 4r=4r', it follows thatV(a*; [I,q) = V(a*; HI', ').(L.4)is obtainedby applying L.1)to both a* anda*' in turn. IIProposition 1. F(a*)-F(a*') implies f(a*)+ V(a*; H q)>f-(a*')+ V(a*'; H,q)for all (H, q).Proof. Immediate rom(L.4). ||Proposition2. f(a*) + V(a*; H,q) > F(a*)+ V(a*; HI, ) implies

F(a*)+ V(a*; II',q') _ F(a*) + V(a' ; 'q')whenever Hq = H'q'.

Proof. From the definitionr(a) E i7r-r(a, ) in (2), and the demonstration hatV(a*; H,q)= Z r,su(g(a*),s) in the proofof (L.3), it followsthat f(a*)+ V(a*; H, q)=, ir,(r(a*,s)+u(g(a*),s))=F(a*)+V(a*; ',q'),where iir=Hq =H'q'. HTo shorten he statementsof the remainingpropositions,we use the fact that the changein value of a positionis the same as the changein V(a; H, q) when the priorbelief #ris fixed (i.e. when (H, q) > (H', q')).



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JONES & OSTROY FLEXIBILITY AND UNCERTAINTY 29Proposition3. (II,q) : (HI', ') implies

V(a; II,q) - V(a l', q')>: V(a*; r, q) - V(a*; 11',q').Proof. The left side of the inequality is non-negative by (L.2). The right side is 0by (L.3). 1IProposition4. (1L, ) > (Il*,q*) implies

V(a * R, q)- V(a* fl* q*)-i"V(a; 1, tq) V( a' II*, q*).Proof. (H*,q*) indicatesno variability n beliefs, and (Il, q) ! (HI,q*) implies171q IIq* = ir. Thereforelr*(y) =r for all y. By (LA1)V(a*; H,*,q*) > V(d, Il*, q*).But

V(a; 11*,q*) = Y *yv(d, #) = v (a. r) maxaeA v(a, r)>-v(a, * iY q*yv(a, v) (a*,f* q*).The first and last equalities follow from the definition of v and the fact that ir*(y) = Xfor all y; the second and fifth from Y,qy = 1 for any information structure, the third isthe definition of a. Hence

V(a*; H*, q*) = V(d, H*, q*). (A.4)Applying (L.1) again,

V(a* H, q) > V(a; 171,). (A.5)Subtracting (A.4) from (A.5) yields

V(a*; Hl,q)- V(a*; H*, q*~) V(a; H,q)- V(d; 17I,q*). IIProposition S. c(a,bs)=c(a) for all s and b #g(a), and (H,Iq)>s (171',') imply

V(a*; I,q)- V(d; H,Iq)> V(a* ',q')- V(d; H',q')-Proof. Define b- g(d). The definition of v(a; ir) and fact that c(a*, b, s) = 0 forall b,s imply the first equality in

v(a* ir) - v(d; ir) = maxb ,ir,u (b, s) -maXb EXr.(u(b, s)- c (J, b, s))- mi {c (a), maxb X, 'ire u (b, s) - u(1 s))}. (A.6)

The second equality follows from c (d' b, s) ={0 for b = b, c (d) for b1 -b}. Since(H, q) os (H', q') requires that ir'(y) = Ayir(y)+ (1 -Ay)r, where 0 c Ayc1 for each y E Y,

maxb Y., irT(y)(u (b, s) - u(b, s))cAY maxb Esr, (y)(u (b, s) - u(b, s))+ (1-Ay) maxE, ii (u(b, s) - i(b, s))

maXb Es r. (y) (u (b, s) - u (b, s)). (A.7)The first nequality ollows fromthe convexityof maXbY,7r (u (b,s) - u(b, s)) in ir, as itis the maximum of a finite collection of linear functions of ir. The definition of a impliesmaxb 2;#r(u ( s) - u b, s)) =0 since a* was available; this, together with Ay 1 andmaxb E3 rTs y)(u (b, s) - u (b, s))- 0, gives the second inequality.Substituting ir = ir(y) and ir'(y) in turn into (A.6), and utilizing (A.7), gives

v(a*- 1r(y))-v('- ir()):-:v(- a * 7r'(y))-v(a'-, srfv))



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30 REVIEW OF ECONOMIC STUDIESfor each y E Y. Since 's requires that qy= q' for each y, this means that

V(a*;fl,q)-V(a;fl,q) ' V(a*;W',q')- V(a;W,q') ||Proposition 6. Condition 1. For each a e A, G(a, s, a) = BaC B for all s e S, a-' ;

the ordering ZF on A is complete (nested switching costs). Condition 2. For each(a, b) e A x B, thereexists b E Ba such thatfor all 7r eithermaXb' EBa EsVsU(b Xs ) _E s Tsub,s)

ormaXb'eBa Es VsU(b s) = sIT5U(b,).

Condition 1 and Condition 2 implies ' induces an orderpreserving relation an oZF-Proof. Condition 1 implies that v (a, ir) = maxbeBa Esisu (b,s) for all a and Vr.Thedefinition of ZF and Condition 1 require that Ba DBa' for all a 'F a'. Hence, for all

a 'Fa' and Ir,v(a; ix) - v(a'; rr)maxbEBa EsTsu (b, s) - maxb'eBa'Es 7rsub', s)

=maxdEBa {maX bE{Ba, d} Es Tsu (b, s)-maxb'EBa Es su (b', s)}= maxdEBa {max {0, Es 7rsu d, s) - EsrSu (b, s)}}. (A.8)

The position b in the last expression is the fixed b EBa asserted to exist for each(a, d) EA x B in Condition 2, i.e. b(a, d) does not vary with r.The second and third expressions are equal since Ba DBa'; the last equality followsfrom Condition 2. Since the innermost maximum in the last expression in (A.8) isbetween two linear functions of r, it is a convex function of r. The outer maximum isthus a maximum of convex functions, and hence v (a, r) v (a', vr) is convex in r.Proposition 0 then allow us to assert that - induces an order preserving relation on !F.Requiring that the flexibility ordering on A be complete (i.e. nested switching costs)allows one to strengthen the implication of increased variability of (fl, q) from "theoptimal position is not less flexible" to "the optimal position is at least as flexible". |

First version receivedJuly 1981; final versionaccepted September1982 (Eds.).An earlier version of this paper entitled "Liquidity as Flexibility" was presented at the Third WorldCongress of the Econometric Society, Toronto, August 1975.

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Flexibility and Uncertainty (Ostroy)