Stochastic OLG Models, Martket Structure, and …apps.eui.eu/Personal/Gottardi/SOLG.pdfStochastic OLG Models, Market Structure, and Optimality1 Subir Chattopadhyay Departamento de

Journal of Economic Theory 89, 21�67 (1999)

Stochastic OLG Models, Market Structure, and Optimality1

Subir Chattopadhyay

Departamento de Fundamentos del Ana� lisis Econo� mico, Universidad de Alicante,Alicante 03071, Spainsubir�merlin.fae.ua.es

and

Piero Gottardi

Dipartimento di Scienze Economiche, Universita� di Venezia, Italy, and Cowles Foundation,Yale University, New Haven, Connecticut

gottardi�unive.it

Received January 26, 1998; revised March 31, 1999

For a general class of pure exchange OLG economies under uncertainty, weprovide a complete characterization of the efficiency properties of competitive equi-libria when markets are only sequentially complete and the criterion of efficiency isconditional Pareto optimality. We also consider a particular case in which marketsfail to be even sequentially complete and provide a characterization when thecriterion of efficiency is weakened to ex post Pareto optimality. Journal of EconomicLiterature Classification Numbers: D52, D61. � 1999 Academic Press

1. INTRODUCTION

We consider a general class of pure exchange, two-period overlappinggenerations (OLG) economies under uncertainty and characterize theoptimality properties of competitive equilibria under different marketstructures and with respect to alternative criteria of optimality.

Following Cass' [6] characterization of efficient growth paths in termsof the properties of their supporting competitive prices (later generalized inBenveniste [3]), Balasko and Shell [2] and Okuno and Zilcha [20]provided similar results for the competitive equilibria of standard,

Article ID jeth.1999.2543, available online at http:��www.idealibrary.com on

210022-0531�99 �30.00

Copyright � 1999 by Academic PressAll rights of reproduction in any form reserved.

1 We thank F. Marhuenda and H. M. Polemarchakis for useful discussions. We are alsograteful to an associate editor and a referee for helpful comments. Chattopadhyay gratefullyacknowledges financial support from the IVIE and DGICyT Project PB94-1504; Gottardi isgrateful to CNR and MURST for financial support.

deterministic OLG economies.2 These authors considered pure exchangeeconomies under certainty with a fixed number of goods available in everyperiod, a fixed number of two-period lived agents born in every period, anda complete set of markets at the beginning of history where all agents arefree to trade.3 They showed that (i) if aggregate endowments are uniformlybounded above and preferences are locally nonsatiated, strictly convex andthe curvature of the indifference sets is bounded below uniformly acrossagents, then divergence of the series obtained in the ``Cass criterion'' (wherethe terms appearing in the series are the reciprocals of the norms of theprice vectors) implies that the associated competitive allocation is Paretooptimal; (ii) if preferences are strictly monotonic, the curvature of the indif-ference sets is bounded above uniformly across agents and the equilibriumallocation is interior, then convergence of the same series implies that thecompetitive allocation is not Pareto optimal. The argument exploits a pairof curvature conditions for each agent; these are inequalities which relatechanges in the composition of expenditure across the two periods of theagent's life, relative to the composition at the equilibrium level of consump-tion, to changes in the agent's utility level. The interaction between thecurvature conditions and the feasibility conditions lead to the characteriza-tion result. This result has been subsequently extended, by Burke [5] andGeanakoplos and Polemarchakis [12] among others, in particular toeconomies where the number of agents born, and the number of com-modities available, vary over time.

Our focus is on OLG economies with uncertainty. No special assump-tions are made on the structure of the uncertainty; it is simply assumed tobe represented by a date�event tree where each node has a finite numberof successors. As a consequence, the number of (contingent) commoditiesat each date increases over time, and will typically tend to infinity. If wewere to maintain the assumption that agents are free to trade on a com-plete set of markets at the initial period of the economy, so that agentsevaluate their consumption plans in terms of their ex ante utility (i.e.,before any realization of the uncertainty), then the only complication thatwe would face is the fact that the dimension of the commodity spaceincreases over time (and, typically, tends to infinity). In this setup, agents'preferences are strictly monotone in all commodities available at a givendate and the characterization of those complete market competitive equi-librium allocations which are ex ante Pareto optimal is easily obtainedfrom the results proved in [5] and [12] for general deterministic OLG

22 CHATTOPADHYAY AND GOTTARDI

2 See also Bose [4].3 As is well known, the equilibrium allocations with this market structure are the same as the

equilibria with a sequential structure of markets��a system of spot markets and one asset, possiblyin non-zero net supply (e.g., fiat money), allowing transfers of income across periods.

economies; it is given by the divergence of a series whose terms are thereciprocals of the norms of the vector of all prices at a date.

Under uncertainty we cannot find a sequential structure of marketswhere agents trade only after they are born and which supports the sameequilibrium allocations as when agents have unrestricted access to a com-plete set of markets at the initial date; this is in contrast with the case ofcertainty. When trading is sequential, agents will in fact be unable to insureagainst the realization of the uncertainty at the time of their birth so thatmarkets are necessarily incomplete. The ``most complete'' market structurewhich is compatible with the demographic structure of the economy andthe sequential nature of trading, where, in each period and for each realiza-tion of the uncertainty, spot markets exist for every commodity and thereis also a complete set of one-period contingent claims whose payoff iscontingent on all possible events next period, conditional on the currentevent, is called sequentially complete markets. By trading in these markets in thetwo periods of his or her life, each agent can fully insure against the realizationof the uncertainty, but only conditionally on the event at his or her birth.

When markets are only sequentially complete, typically, competitive equi-libria will not be ex ante Pareto optimal. A more appropriate notion for eva-luating the efficiency properties of allocations when trades take place sequen-tially is the notion of conditional Pareto optimality (CPO), first proposed byMuench [19].4 According to this criterion, agents' welfare is evaluated byconditioning their utility on the event at the date of their birth. Agents arethus distinguished not only according to their type and their date of birth butalso according to the event at that date, and an agent's preferences aredefined over a subset of all the commodities available in the two periods ofthat agent's lifetime since the commodities which enter the utility function ofan agent born at date t are the date t and t+1 commodities whose availa-bility is contingent on the occurrence of the particular event at the agent's birth.

The problem of evaluating a possible reallocation of resources in termsof the CPO criterion, where the reallocation is relative to a competitiveequilibrium obtained with sequentially complete markets, cannot bereduced to a one-dimensional problem in which only the change in thetotal expenditure at a date matters; this is in contrast to the case of thedeterministic OLG model (or, more generally, to the complete marketscase). With uncertainty, a reallocation of resources induces sequences oftransfers (of income or ``value'' ) along distinct paths in the date�event treewith the feasibility condition satisfied at each date�event. When marketsare only sequentially complete, agents of the same type born at the same datebut in different events are considered to be distinct so that nonzero levels of

23STOCHASTIC OLG AND OPTIMALITY

4 See Dutta and Polemarchakis [11] for a discussion of the various criteria for optimality inOLG economies under uncertainty.

transfers conditional on more than one event at a given date lead to an extreme-ly rich set of redistributional possibilities since they affect different agents.

Moreover, as each agent cares about only some of the commoditiesavailable at a date, the curvature conditions that one obtains areinequalities which consider changes in expenditure in the event at whichthe agent is born and the changes in expenditure in the set of immediatesuccessor events. Whether or not a reallocation is improving depends onthe interaction, across a collection of paths, among the transfers that thereallocation generates; in particular, the changes in expenditure at differentevents at the same date, and not just the value of the change at a givendate, need to be considered. Hence the issue of characterizing those equi-librium allocations, obtained with sequentially complete markets, whichare efficient relative to the CPO criterion is far more complex than theanalogous problem in the deterministic case (and its generalizations aspresented in ([5, [12]) and those earlier results cannot be easily extendedto the economies considered in this paper.

One of the main results of this paper is the derivation of a necessary andsufficient condition for the conditional Pareto optimality of competitive equi-libria with sequentially complete markets under general assumptions on theeconomy; this is established in Theorems 1 and 2. The condition we obtainrequires us to consider, for each given collection of paths (or moreprecisely, for each ``sub-tree''), the behaviour of all the price seriesassociated with these paths. The terms appearing in each price series arethe reciprocals of the norms of the vectors of prices at a node, multipliedby the values of a weight function defined for each node in the sub-tree. Aswe argued above, the consideration of possible redistribution of resourcesrequires us to examine the interaction among transfers defined across acollection of paths; hence, we need to evaluate the relative `ìmportance'' ofvarious nodes at each date. This is indeed the role of the weight functionwhose values can be interpreted as a conditional probability of reaching anode from the root of the sub-tree.

The form of the condition that we obtain is rather different from the onederived for the deterministic OLG model (or, equivalently, for the case ofcomplete markets) though it reduces to that condition in the special casein which there is no uncertainty. At a more abstract level, our resultsindicate how a characterization of efficiency can be obtained in the moregeneral case in which agents care for only a subset of the set of com-modities available at each point in time.5


5 Deterministic OLG economies in which agents live for more than two periods provide anotherexample where the composition of transfers at a date across agents born at different dates needs tobe considered. The reformulation of such an economy as an economy with two-period lifetimesresults in agents caring about only a subset of commodities available in each period in the trans-formed economy.

The necessary and sufficient condition presented in Theorems 1 and 2 isnot easy to verify and to work with. Hence, in Theorem 3 and Corollary 1we provide alternative sufficient conditions for CPO which are stronger buteasier to verify. These conditions require the consideration, again for eachgiven collection of paths, of a single series whose terms are the reciprocalof the sum of the norms of the vectors of prices across nodes at a givendate.

When we consider a restricted class of economies, and, in particular, amore specific uncertainty structure, our characterization result takes asimpler form. For the special case of one-good OLG economies withstationary Markov uncertainty, we show (in Theorem 4) that the sufficientcondition derived in our Corollary 1 is also necessary for stationary equi-libria with sequentially complete markets to be CPO, and reduces to a verysimple condition on the value of the dominant root of the matrix of con-tingent claim prices.

When markets are not sequentially complete, i.e., when agents are unableto insure against all sources of uncertainty affecting them after their birth,competitive equilibria are typically not even CPO. A general investigationof the efficiency properties of competitive equilibria in this case goesbeyond the scope of this paper. Here we will only examine the case ofa simple asset structure with a single one-period asset, a bond with aconstant real return, and examine its efficiency properties with respectto a notion of optimality which is weaker than CPO, ex post Paretooptimality (EPPO). According to this criterion, agents' welfare is evaluatedby conditioning not only on the state when young but also on thestate realized in the second period of their lives. Agents are then dis-tinguished by the realization of the uncertainty over their entire lifetime.In this situation efficiency concerns only the allocation of commodities ateach given date�event, and over time, but not across states (i.e., not theallocation of risk).

In Theorem 5 we provide necessary and sufficient conditions for the expost Pareto optimality of competitive equilibria when a bond is the onlyasset. The conditions we obtain are again on collections of series ofweighted reciprocals of the norms of price vectors, as in Theorems 1 and 2.A crucial difference, however, is the fact that the weighting factor is nowunivocally determined by the average of the agents' Lagrange multipliers(thus, to evaluate the efficiency of competitive equilibrium allocations weneed to know not only the supporting prices but also the Lagrange multi-pliers of all agents).

There has been some earlier work on the efficiency properties,with respect to the CPO criterion, of one commodity OLG economieswith stationary uncertainty (Peled [21], Manuelli [17], Zilcha [24],etc.). We discuss in detail the relation of our work to this literature and


show that all these earlier results can be obtained as special cases of ourresults.

The rest of the paper is structured as follows. Section 2 presents themodel and notation. In Section 3 we present a characterization of equi-librium allocations that are CPO when markets are sequentially complete,while in Section 4 we focus our attention on stationary equilibria. InSection 5 we introduce a (particular) incomplete asset structure and wecharacterize the efficiency properties of equilibria obtained with that assetstructure with respect to the weaker EPPO criterion of optimality. Allproofs are collected in Section 6.

2. THE MODEL

We consider a general, two-period, pure exchange overlapping genera-tions (OLG) economy under uncertainty. The economy evolves in discretetime with uncertainty described by an abstract date�event tree as inChapter 7 of Debreu [9] or in Radner [22].

In this section (and in most of the paper) we will consider the case inwhich markets are sequentially complete so that agents are able to insureagainst all risks that arise after they are born but are unable to insureagainst the risk of being born in a particular event. This situation is usuallydescribed by assuming that at every date�event, there exists a complete setof spot markets and one-period forward markets, contingent on all thepossible realizations of uncertainty at the next date given the currentdate�event. Here we consider the following equivalent structure of markets:at the initial date, there exists a complete set of Arrow�Debreu contingentcommodity markets, but each agent is allowed to trade only on themarkets for delivery contingent on the event at his birth and on the eventsthat can occur in the next period given the event at birth. It is easy to showthat the agent's opportunities for trade are the same in the two marketstructures.

We turn to a formal description of the model and the notation used.Time is discrete and dates are denoted by t=1, 2, 3... .Uncertainty is described by a date�event tree which is defined by (i) _0 ,

the root of the tree, (ii) finite sets 7t , for t�1, and (iii) a set of functionsf 0: 71 � _0 and f t: 7t+1 � 7t , for t�1, where each function is surjective.Define 7 :=�t�1 7t and 1 :=7 _ [_0]; we will abuse notation and use 1to denote a generic date-event tree. Elements of 1 are called nodes (to bethought of as the ``date�events'' or simply `èvents''), and a generic node isdenoted by _. 7t is the set of all nodes at date t; thus if _ # 7t we say that_ is a node at date t (sometimes the notation _t will be used to stress the


fact that we are referring to a node at date t), and the function f t&1(_)identifies the node at t&1 which is the unique predecessor of _.6

Given a node _ # 7, t(_) denotes the value of t at which _ # 7t ; the setof immediate successor nodes of _ will be denoted by _+ (thus, if _ # 7t ,_+ is the set of nodes, at date t+1, such that _ is their immediate prede-cessor). Similarly, the unique predecessor of a node _, given by f t(_)&1(_),will also be referred to as _&1 . The total number of nodes at t, >7t , willbe denoted by St ; S(_) :=>_+ is then the number of immediate successorsof _ (or the branching number of _).7

A path is defined by an infinite sequence of nodes [_t]t�1 such that, forall t�1, _t= f t(_t+1), and will be denoted by _�.

L commodities are available for consumption at each node _ # 7.At each node _ # 7 a generation of H agents is born. Each agent lives at

two dates, t(_) and t(_)+1. The fact that agents are unable to trade inmarkets offering insurance against the event at their birth is captured byrequiring that the consumption plan of an agent specify the level of con-sumption in the event at birth and in its immediate successor nodes, andthat his preferences be defined over such plans. So, an agent can be dis-tinguished according to his type h and the node identifying the event at hisbirth; consequently, a member of generation _ of type h is denoted by(_, h).

In addition, there is a set of H one-period lived agents who enter theeconomy at each node _ # 71 at date 1; they constitute the generation ofthe `ìnitial old,'' and are indexed by (_, h, o), where _ # 71 .

Each agent (_, h) is described by a consumption set, X_, h , an endow-ment vector, |(_, h), and a utility function, u_, h( } ) (for the initial old,X_, h, o , |(_; h, o), and u_, h, o( } )). A consumption plan for agent (_, h) willbe denoted by x(_, h) (x(_; h, o) for the initial old).8

For all (_, h) # 7_H, the elements of the endowment vector |(_, h) ofagent (_, h), describing the endowment at birth and in all successor nodes,will be written as follows: (|(_; _, h), (|(_$; _, h))_$ # _+). Similarly, theelements of the consumption vector x(_, h) are (x(_; _, h), (x(_$; _, h))_$ # _+).


6 This definition of a date�event tree leads directly to another standard definition in which theordered pairs ( f t(_t+1), _t+1) are called arcs and induce a partial order on 1 with the propertiesthat (i) each node traces its origin, by the partial order of precedence, to _0 , and (ii) each node,except _0 , has exactly one predecessor which is also an element of 1.

7 Since all the functions f t( } ) are surjective, S(_)�1 for all nodes. We treat this restriction (i.e.,the economy ``never ends'') as part of the definition of a tree. Given that our interest is inoptimality, this is without loss of generality.

8 The fact that the number of commodities available at each node and the number of agents bornat each node are date�event independent is without loss of generality; our results can easily beextended to more general specifications so long as the total numbers of commodities and of agentsborn before a given date are finite, and this for all dates. Our specification prevents the notationsfrom getting more cluttered.

Denoting by |(_) the total endowment at node _, we have then:

|(_) := :h # H

|(_; _, h)+ :h # H

|(_; h, o) for _ # 71 ,

|(_) := :h # H

|(_; _, h)+ :h # H

|(_; _&1 , h) for _ # .t�2

7t .

Agents' preferences and endowments are assumed to satisfy the followingstandard conditions:

Assumption 1.

(i) 1�L<�, 1�H<�, and 1�S(_)�S� <� for all _ # 1.

(iia) For all (_, h) # 71_H, X_, h, o=RL+ , |(_; h, o) # RL

+�[0],u_, h, o : X_, h, o � R is C2, strictly monotone, and differentiably strictly quasi-concave.

(iib) For all (_, h) # 7_H, X_, h=RL(1+S(_)+ , |(_; _, h) # RL

+�[0]and ((|(_$; _, h))_$ # _+) # RLS(_)

+ �[0], u_, h : X_, h � R is C2, strictly mono-tone, and differentiably strictly quasi-concave.

(iii) For all _ # 7, |(_) # RL++ .

Definition 1. A feasible allocation x is given by an array((x(_; h, o))(_, h) # 71_H , (x(_, h)) (_, h) # 7_H) such that x(_; h, o) # X_, h, o forall (_, h) # 71_H, x(_, h) # X_, h for all (_, h) # 7_H, and

:h # H

x(_; _, h)+ :h # H

x(_; h, o)�|(_) for all _ # 71 ,

:h # H

x(_; _, h)+ :h # H

x(_; _&1 , h)�|(_) for all _ # .t�2

7t .

Applying the notion of Pareto efficiency to the economy describedabove, where agents are distinguished by the event at their birth, yields thecriterion of conditional Pareto optimality, first proposed by Muench [19]:

Definition 2 (CPO). Let x be a feasible allocation. x is conditionallyPareto optimal (CPO) if there does not exist another feasible allocation x̂such that

(i) for all (_, h) # 71_H, u_, h, o(x̂(_; h, o))�u_, h, o(x(_; h, o)), for all(_, h) # 7_H, u_, h(x̂(_, h))�u_, h(x(_, h)),

(ii) either for some (_$, h$) # 71_H, u_$, h$, o(x̂(_$; h$, o))>u_$, h$, o

(x(_$; h$, o)), or for some (_$, h$) # 7_H, u_$, h$(x̂(_$, h$))>u_$, h$(x(_$, h$)).


In addition to allocating commodities optimally at each node, a CPOallocation requires that the risk carried in the second period of the agents'lives be allocated optimally.

We will now define competitive equilibria for the economy we described.A complete set of contingent commodity markets is available at the initialdate. Let p(_t) be the vector of prices, quoted at the initial date, for con-tingent delivery of the L commodities at the node at _t at date t. A pricesystem is then defined by a non-negative sequence [ pt]t�1 where pt=(( p(_))_ # 7t

) # RStL+ . Prices are normalized as follows: p l (_)=1 where l=1

and _ # 71 .

Definition 3 (S-CE).9 (x*, p*) is a competitive equilibrium withsequentially complete markets (S-CE) if x* is a feasible allocation, and

(i) for all (_, h) # 71 _H, p*(_) } x*(_; h, o)�p*(_) } |(_; h, o),

u_, h, o(x(_; h, o))>u_, h, o(x*(_; h, o)) O p*(_) } x(_; h, o)>p*(_) } |(_; h, o)

(ii) for all (_, h) # 7_H, ( p*(_), ( p*(_$))_$ # _+) } x*(_, h)�( p*(_),( p*(_$))_$ # _+) } |(_, h),

u_, h(x(_, h))>u_, h(x*(_, h))

O ( p*(_), ( p*(_$))_$ # _+) } x(_, h)>( p*(_), ( p*(_$))_ # _+) } |(_, h).

Remark 1. The date�event tree structure, together with Assumption 1,implies that the set of nodes is countable; hence, so is the set of agents. Theeconomy defined satisfies also the other assumptions of Corollary 1 inGeanakoplos and Polemarchakis [12] so that existence of S-CE is guaran-teed. Furthermore, our assumption of strict monotonicity of preferencesimplies that at an S-CE allocation or at a CPO allocation, the feasibilitycondition holds as an equality at all nodes.

3. SEQUENTIALLY COMPLETE MARKETS AND CPO

In this section we examine the efficiency properties of competitiveequilibria with sequentially complete markets with respect to the CPOcriterion. We derive a necessary and sufficient condition on equilibriumprices under which the equilibrium allocation is CPO.


9 It is easy to show that if, at each node _ # 7, there are L spot commodity markets and LS(_)markets for delivery contingent on every possible realization _$ # _+ (i.e., at every successor node),then the specification of the agents' budget constraints is the same as in Definition 3; as a conse-quence, the set of equilibrium allocations is also the same.

As a preliminary step to the results we derive an implication of thecurvature conditions imposed on preferences by Assumption 1.

Definition 4.10 Let x(_, h) solve the utility maximization problem ofagent (_, h) # 7_H at prices ( p(_), ( p(_$))_$ # _+) such that &p(_)&>0,&p(_$)&>0 for all _$ # _+. For k>0, let

P�

_, h(k) :={\ # R } for all x̂(_, h) # X_, h satisfying (i) &x̂(_, h)&x(_, h)&�k

and (ii) u_, h(x̂(_, h))�u_, h(x(_, h)), we have

:_$ # _+

$2(_$, _, h)�&$1(_, h)+\($1(_, h))2

&p(_)& = ,

where $1(_, h) :=p(_) } [x̂(_; _, h)&x(_; _, h)], $2(_$, _, h) :=p(_$) }[x̂(_$; _, h)&x(_$; _, h)], for _$ # _+.

(a) Given k>0, \�

_, h(k) is the lower curvature coefficient of agent(_, h) at ( p(_), ( p(_$))_$ # _+) where \

�_, h(k) :=sup [\ # P

�_, h(k)] if

P�

_, h(k){< and \�

_, h(k) :=&� if P�

_, h(k)=<.11

(b) The preferences of agent (_, h) satisfy the non-vanishing Gaussiancurvature condition at ( p(_), ( p(_$))_$ # _+) if there exists k

� _, h>0 for which\�

_, h(k� _, h)>0.

Definition 5. Let x(_, h) solve the utility maximization problem ofagent (_, h) # 7_H at prices ( p(_), ( p(_$))_$ # _+) such that &p(_)&>0,&p(_$)&>0 for all _$ # _+. For k>0, let

P� _, h(k) :={\ # R } for all x̂(_, h) # X_, h satisfying (i) &x̂(_, h)&x(_, h)&�k,

(ii) p(_) } [x̂(_; _, h)&x(_; _, h)]<0, and

(iii) :_$ # _+

$2(_$, _, h)� &$1(_, h)+\($1(_, h))2

&p(_)&,

we have u_, h(x̂(_, h))�u_, h(x(_, h))= ,


10 Here, as well as in the rest of the paper, & }& will denote the Euclidean norm unless otherwisenoted.

11 The coefficient \�

_, h depends on x(_, h) but this dependence has been suppressed in order tosimplify the notation (hence the qualifier at ``( p(_), ( p(_$))_$ # _+)'').

where $1(_, h) :=p(_) } [x̂(_; _, h)&x(_; _, h)], $2(_$, _, h):=p(_$) }[x̂(_$; _, h)&x(_$; _, h)], for _$ # _+.

(a) Given k>0, \� _, h(k) is the upper curvature coefficient of agent(_, h) at ( p(_), ( p(_$))_$ # _+) where \� _, h(k) :=inf[\ # P� _, h(k)] if P� _, h(k){< and \� _, h(k) :=� if P� _, h(k)=<.

(b) The preferences of agent (_, h) satisfy the bounded Gaussiancurvature condition at ( p(_), ( p(_$))_$ # _+) if there exists k� _, h>0 for which�>\� _, h(k� _, h)�0.

The fact that in Definitions 4 and 5, x(_, h) is assumed to solve theutility maximization problem of agent (_, h) # 7_H at prices ( p(_),( p(_$))_$ # _+), together with local nonsatiation of preferences, implies thatthe agent's preferred set is contained in the half-space defined by the budgetconstraint (i.e., x(_, h) minimizes expenditure) and hence \

�_, h(k)�0 and

\� _, h(k)�0 for all k>0; (b) in Definition 4 requires that the inequality bestrict, while (b) in Definition 5 imposes an upper bound. Lemma 1 showsthat Assumption 1 guarantees both of these properties for interior maxima.

Lemma 1. Let (x*, p*) be an S-CE. If x*(_, h) # RL(1+S(_))++ then, under

Assumption 1, the preferences of the agent (_, h) # 7_H satisfy both (i) thenon-vanishing Gaussian curvature condition and (ii) the bounded Gaussiancurvature condition at prices ( p*(_), ( p*(_$))_$ # _+). In fact, for all k>0 wehave \

�_, h(k)>0 and for all k>0 we have �>\� _, h(k)�0.12

Part (i) of Lemma 1 says that, if the alternative consumption planx̂(_, h) improves with respect to the optimal choice x*(_, h) of agent (_, h)at prices p*, then the difference in value of the alternative plan must obeya quadratic relation; it requires that preferences be locally nonsatiated andthat they satisfy a differentiable form of strict quasiconcavity (both ofwhich are imposed in Assumption 1). Part (ii) of Lemma 1 provides uswith a condition on the income transfers along the lifetime of an agentwhich ensures that we have an improvement in the agent's welfare; thisresult requires a minimum degree of substitutability among goods in theagent's preferences (ensured by the strict monotonicity and the smoothnessof the utility functions imposed in Assumption 1).

Notice that the terms appearing in the two inequalities which defineP�

_, h(k) and P� _, h(k) in Definitions 4 and 5 are the norm of the price vectorand the change in the expenditure in the two periods of an agent's life,where both are indexed by the events in which they occur. This, as pointed


12 Clearly, in Lemma 1 and the other results to follow, the condition that the equilibrium alloca-tion is interior for all the agents can be replaced by a standard assumption on the boundarybehaviour of preferences.

out in the introduction, constitutes a crucial difference relative to the caseof certainty (or of complete markets) as it implies that transfers to andfrom agents must be identified not only by the date but also by the eventin which they take place since, for the agent, transfers received at anarbitrary pair of events at the same date are not substitutable unless bothevents succeed the event in which he is born.

In order to state the main results, we need some additional notation.Given _̂ # 1, we define a sub-tree (of the tree 1 ) with root _̂ # 1, denoted

by 1_̂ , as a collection of nodes such that (i) 1_̂ /1, (ii) 1_̂ is itself a treewith _̂ as its root. Hence the tree itself, 1, is a sub-tree; so is any path, _�.

Given a sub-tree 1_̂ , a weight function is a function *1_̂: 1_̂ � [0, 1]

such that �_$ # _+ & 1_̂*1_̂

(_$)=1 for all _ # 1 _̂ .This function associates to the immediate successors (in the sub-tree 1_̂)

of each given node nonnegative weights such that the weights sum to one.Hence, the weights *1_̂

( } ) can be interpreted as a subjective conditionalprobability of reaching, from each given node, the different immediate suc-cessor nodes in the specified sub-tree.

Given a pair (1_̂ , *1_̂), the induced weight function, denoted *� 1_̂

: 1_̂_*1_̂

� [0, 1], is defined by *� 1_̂(_̂)=1, *� 1_̂

(_)=*1_̂(_) } *� 1_̂

(_&1) for_&1 # 1 _̂ . It associates to each node _ the product of the values *1_̂

( } )along the chain of nodes from _̂ (the root of the sub-tree) to the given node _.

Since, for t�t(_̂), �_ # 7t & 1_̂*� 1_̂

(_)=1,13 the function *� 1_̂can also be

interpreted as the subjective probability of reaching the different nodes_ # 7t & 1_̂ conditional on having reached _̂, the root of the sub-tree.

Given a path _�, let _�t denote the tth coordinate of the path. Given a

sub-tree 1_̂ , define a path in the sub-tree 1_̂ , denoted by _�(1_̂), as a pathwith the property that for t�t(_̂) all the nodes are elements of the sub-tree,i.e., _�(1_̂)/[_�

1 , ..., _�t(_̂)&1] _ 1_̂ .

Notice that there is an obvious way of associating a node to the tth coor-dinate of a specified path; in what follows we shall use _t and _�

t inter-changeably when referring to a node.14

The following results identify necessary and sufficient conditions for anS-CE to be conditionally Pareto optimal.


13 This follows easily since �_ # 7t & 1_̂*� 1_̂

(_)=�_ # 7t&1 & 1_̂[*� 1_̂

(_) } �_$ # _+ & 1_̂*1_̂

(_$)]=�_ # 7t&1 & 1_̂

*� 1_̂(_)= } } } =1, where we repeatedly use the fact that �_$ # _+ & 1_̂

*1_̂(_$)=1 for all

_ # 1_̂ .14 This notational convention will be used throughout without further mention. It implies that

if we are given a pair (1_̂ , *1_̂), then we are also given the values of the weight function on the coor-

dinates t�t(_̂) of each path in the sub-tree 1_̂ .

Theorem 1 (Sufficiency). Let (x*, p*) be an S-CE and suppose Assump-tion 1 holds.15 Assume that x*(_, h) # RL(1+S(_))

++ for every (_, h) # 7_H, andthat there are numbers K>0 and \

�>0 such that for all nodes _ # 7

(i) |l (_)�2KH for all l=1, ..., L

(ii) for all h # H, \��\

�_, h(2KHL(1+S� )).16

If the equilibrium allocation is not conditionally Pareto optimal then thereexists a pair, given by a sub-tree and a weight function (1_~ , *1_~

), and anA<�, such that, for every path _�(1_~ ) in the sub-tree,

A(_�(1_~ ); (1_~ , *1_~)) := :

�

t=t(_~ )

*� 1_~(_�

t )

&p*(_�t )&

�A.

Theorem 2 (Necessity). Let (x*, p*) be an S-CE and suppose Assump-tion 1 holds. If there exist numbers =>0, \� >0, k� >0, A<�, and a pair,given by a sub-tree and a weight function (1_̂ , *1_̂

), such that at every node_ # 1_̂ , there exists an agent h_ # H for whom

(i) x*(_, h_) # RL(1+S(_))++ and x*(_; _, h_)�= } 1L_1

(ii) \� _, h_(k� )�\� ,

and for every path _�(1_̂) in the sub-tree,

A(_�(1_̂); (1_̂ , *1_̂)) := :

�

t=t(_̂)

*� 1_̂(_�

t )

&p*(_�t )&

�A,

then the equilibrium allocation is not conditionally Pareto optimal.

Clearly, the characterization provided by the two theorems is tight.The sufficient condition for CPO obtained in Theorem 1 says that if for

every pair (1_̂ , *1_̂), given by a sub-tree and a weight function defined on


15 The result also holds if the restrictions on preferences, imposed as Assumption 1, and the inte-riority condition on the equilibrium allocation are replaced by the assumption that, at the S-CEbeing considered, the preferences of all the agents being improved satisfy the nonvanishingGaussian curvature condtion. The assumptions we made are only sufficient conditions for the cur-vature conditions to hold; that is the content of Lemma 1. Similar considerations apply to theother results presented in the paper. In particular, Theorem 1 holds even if indifference sets haveunbounded curvature (``kinks'') while Theorem 2 below holds even if they have zero curvature(``flat'' segments).

16 Definition 4 specifies the way in which the term \�

_, h(k), for k=2KHL(1+S� ), is obtainedfrom the solution to the optimization problem faced by agent (_, h) at prices ( p*(_), ( p*(_$))_$ # _+);similarly for \� _, h(k� ) in Theorem 2. Lemma 1 shows that, under the assumptions of the theorems,the two curvature conditions hold for each agent.

it, and every real number B, there exists a path in the sub-tree, _�(1_̂), forwhich A(_�(1_̂); (1_̂ , *1_̂

))>B, then the S-CE allocation is CPO. HereA(_�(1_̂); (1_̂ , *1_̂

)) is a series whose tth element is given by the terms*� 1_̂

(_�t )�&p*(_�

t )&, i.e., by the reciprocal of the norm of the price vector ata node, where the node is given by the tth coordinate of the path _�(1_̂),weighted by the associated value of *� 1_̂

( } ).The structure of the proof of Theorem 1 is the following. The first step

consists in showing that if there exists a CPO improvement then necessarilythere exists a sub-tree (which could be just a path) with the property thatthe per capita transfer (of income or ``value'') to the old agents at eachnode in the sub-tree is strictly positive. We then construct a weight functionby associating to each node _ the proportion of the total per capita transferwhich is received at that node by the old agents, i.e., the per capita transferto the old agents at that node divided by the total per capita transferreceived by those agents at different nodes when old. But then, by thedefinition of the weight function, the total per capita transfer received whenold can also be written as the per capita transfer received at one node whenold divided by the weight assigned to that node. This allows us to isolatethe relation between the transfer at a node and the (weighted) transfer atonly one successor node. Now, by invoking the feasibility condition at eachnode, and iteratively applying the nonvanishing Gaussian curvature condi-tion, we are able to show that along every path, the sequence of these percapita transfers multiplied by the induced weight increases according to aquadratic function. Given the uniform bound on endowments, theadmissibility of the improvement requires that the norm of the price vectordivided by the value of the induced weight function grow at an appropriaterate. Hence we find that the existence of a CPO improvement requires thatthe series ��

t=t(_~ ) (*� 1_~(_�

t )�&p*(_�t )&) converges, and this must happen on

every path in the sub-tree.Turning to necessity, the condition in Theorem 2 says that a CPO

improvement can be constructed if we can find a sub-tree (which couldsimply be a path) with the property that along every path in this sub-tree,the price norm at a date-event does not go to zero faster than the inducedweight function at the date-event. This condition ensures the existence oftransfer sequences with the property that the transfer that an agent makeswhen young (in the event at his birth) is more than compensated (in utilityterms) by the transfer that he receives when old (in possibly more than oneevent).

The argument of the proof of Theorem 2 is as follows. First a collectionof transfers (of income) to agents along the sub-tree is constructed. Nextwe show that there exists a corresponding collection of vectors of com-modity transfers, supporting these income transfers, which give rise to


vectors which are in the consumption set of each agent and are feasible, i.e.,compatible with the endowment of the economy. Finally, we show that theproposed collection of commodity transfers generates a Pareto improve-ment by verifying that the associated collection of transfers satisfy thebounded Gaussian curvature condition. The proportions according towhich the total transfer received by an agent when old should be dis-tributed over each of the possible events when old are determined accord-ing to the values of the function *� 1_̂

.The logic of the proof of both the theorems is similar to the argument

usually given in the case of deterministic OLG economies (in particular,Benveniste [3] and Balasko and Shell [2]); not surprisingly, the necessaryand sufficient conditions established in the two theorems are in the form ofcriteria like the one obtained by Cass (in fact, in the special case in whichthere is no uncertainty, they reduce to the standard Cass criterion).

The truly novel features of our results are the fact that transfers aredefined on nontrivial sub-trees, and the role played by the weight function.We see that the difficulty in the proofs, relative to the arguments given inthe case of the deterministic OLG model, comes from the fact that in ourset-up the preferences of each agent are defined on a subset of the set ofcommodities available at each point in time; moreover, the number of com-modities and agents increases over time. In both the theorems we have toallow for the possibility that transfers are made at more than one event ata given date; this leads naturally to the consideration of sub-trees whosepresence is a manifestation of the rich redistributional possibilities in thepresent model. In addition, the Gaussian curvature conditions relate onenode at date t to one or more nodes at date t+1 (but typically a strict sub-set of the set of all nodes at date t+1). So, in order to evaluate whethera sequence of transfers is feasible and improving we have to distinguishthem according to the event in which they take place in addition to thedate of the transfer. Using the weight function, we are able to disaggregatetransfers across the different immediate successor nodes of a given node;hence, we can isolate the pattern of transfers across pairs of successivenodes so that the admissibility of the transfers, which is defined in terms ofpaths, can be verified directly along all the paths in a sub-tree. We caninterpret the weight as the importance given to the transition from a givennode to one of its immediate successors, i.e., as a subjective conditionalprobability.17


17 The system of subjective conditional probabilities defined by the weight function need notbear any relationship with the system of local ``subjective'' probabilities on the date�event treeimplicit in the specification of agents' preferences.

The sufficient condition for CPO obtained in Theorem 1 is rather cum-bersome to verify because one has to check that for every pair given by asub-tree and a weight function (1_̂ , *1_̂

), and every real number B, thereexists a path in the sub-tree, _�(1_̂), for which A(_�(1_̂); (1_̂ , *1_̂

))>B.We now provide another sufficient condition for CPO, Theorem 3, that isstronger but that is somewhat easier to verify.

For any _� # 1_ , we denote by 1(_� , 1_) the sub-tree that has _� as its rootand includes all the nodes that are successors of _� and are elements of 1_ .

Theorem 3 (Sufficiency). Let (x*, p*) be an S-CE and suppose Assump-tion 1 holds. Assume that x*(_, h) # RL(1+S(_))

++ for every (_, h) # 7_H, andthat there are numbers K>0 and \

�>0 such that for all nodes _ # 7

(i) |l (_)�2KH for all l=1, ..., L(ii) for all h # H, \

��\

�_, h(2KHL(1+S� )).

The equilibrium allocation is conditionally Pareto optimal if for every sub-tree 1_ there exists a node _� # 1_ , for which

A(_� , 1_) := :�

t=t(_� )

1� _̂ # 7t & 1(_� , 1_) &p*(_̂)&

diverges.

The sufficient condition for CPO stated in Theorem 3 requires, for everysub-tree 1_ , the existence of a node _� for which the series defined byA(_� , 1_) diverges. The terms of the series A(_� , 1_) are constructed bytaking the reciprocal of the sum of the norm of prices at all nodes at agiven date which are successors of _� , and are elements of the sub-tree 1_ .

As in the proof of Theorem 1, the first step in the proof of Theorem 3consists in showing that if there exists a CPO improvement then necessarilythere exists a sub-tree 1_~ (which could be just a path) with the propertythat the per capita transfer to the old agents at each node _ is strictlypositive. For each date we consider the sum of these transfers across nodesin the sub-tree at that date; due to the non-vanishing Gaussian curvaturecondition the sequence of these terms increases according to a quadraticfunction. Consider the sequence we obtain starting from the date of theroot of the sub-tree, t(_~ ). Given the uniform bound on endowments,feasibility requires that the sum of price norms across all nodes in the sub-tree at a given point in time increases sufficiently fast, so that the existenceof a CPO improvement requires that the series

:�

t=t(_~ )

1�_ # 7t & 1_~

&p*(_)&(1)


converges, for the sub-tree 1_~ . Since the same argument can be made start-ing from any other node in the specified sub-tree, we find that the existenceof an improvement implies that a whole family of series must converge.This directly leads to the sufficient condition for CPO stated in Theorem 3that, for every sub-tree, at least one of the series should diverge.18

To understand the difference between the sufficient conditions inTheorems 1 and 3, and the fact that the latter are stronger, notice that inTheorem 1 we define weights which allow us to disaggregate the totaltransfer at a date, obtained from the curvature conditions, in terms of thetransfer at a node, and hence to identify the pattern of the transfers alongpaths in a sub-tree. In Theorem 3, on the other hand, we aggregate thetransfers of agents born at different date�events into a single term by takingthe sum. Still, the condition in Theorem 3 also deals with the limitingbehaviour of a whole family of series; note, however, that unlike inTheorem 1, there is no claim that the family of series is uniformly bounded.

As a corollary to Theorem 3 we obtain an even simpler, but stronger,sufficient condition for CPO; Example 1 below illustrates the differencebetween the two conditions. The condition in the corollary rules out theexistence of a special type of CPO improvement, one which requirespositive average transfers at every node in the date�event tree 1. But it iseasy to show that if the series in Corollary 1 diverges then the series in (1)also diverges for every sub-tree; since the latter is a sufficient condition forCPO, as noted in Footnote 17, the condition in Corollary 1 also impliesthe nonexistence of any improvement.

Corollary 1. Under the assumptions of Theorem 1, the equilibriumallocation x* is conditionally Pareto optimal if, for the corresponding pricesequence p*, the series

:�

t=1

1�_ # 7t

&p*(_)&

diverges.

Since sub-trees include paths as special cases, our previous results alsocharacterize the conditions under which improvements can be achieved bytransfers along only one path. By Theorem 1, if an improvement exists on


18 Of course, the divergence of the series in (1) for every sub-tree 1_ is also a sufficient conditionfor CPO. Note, however, that the convergence of the series in (1), obtained when we start summingfrom the root of the sub-tree, does not imply the convergence of the series we get when we startsumming from some other node in the sub-tree. Hence, the sufficient condition given in Theorem 3is tighter than requiring the divergence of the series in (1) for every sub-tree.

only a path then, the series ��t=1 1�&p*(_�

t )& must converge along thatpath. Conversely, Theorem 2 shows that if the series ��

t=1 1�&p*(_�t )& con-

verges on some path then the allocation is not CPO; in this case, animprovement can be constructed via transfers which are different from zeroonly along the specified path. Clearly, the divergence of the series��

t=1 1�&p*(_�t )& along every path is not sufficient to ensure that the

allocation is CPO (see Example 2 below).

Remark 2. Here we examine the relationship between existing resultsfor the general deterministic OLG model (Burke [5] and Geanakoplosand Polemarchakis [12]) and our characterization of CPO. We will arguethat these results allow us to derive the necessary and sufficient conditionsfor ex ante Pareto optimality when markets are complete, since in that casethe economy is isomorphic to a deterministic OLG economy in which thenumber of commodities varies with time, but do not have any directimplication as far as CPO is concerned.

With complete markets, an agent is able to trade in the full menu of con-tingent commodity contracts so that he can insure against the uncertaintyboth at the date of his birth and in the second period of his life; conse-quently, his consumption plan specifies his consumption at the date of hisbirth, for all possible realizations of the event at birth, and at the subse-quent date, again for all possible realizations of the uncertainty. Hence anagent can be distinguished simply by the date of his birth, t, and his type,h, and is identified by the pair (t, h). His consumption set is given byXt, h :=RL(St+St+1)

+ , with generic element x(t, h), his endowment vector is|(t, h) # Xt, h , and his utility function ut, h : Xt, h � R which is strictlyincreasing in every coordinate. The agent's budget restriction takes theform ( pt , pt+1) } x(t, h)�( pt , pt+1) } |(t, h). In all other respects, equi-libria with complete markets are as in Definition 3. Let ( p**, x**) denotea competitive equilibrium with complete markets.

The ``natural'' criterion for evaluating the efficiency properties of com-petitive equilibria when markets are complete is ex ante Pareto optimalitywhich evaluates the welfare of every agent before any realization of the uncer-tainty; at an ex ante Pareto optimum, risk is allocated optimally in bothperiods of the agents' lives. Evidently, ex ante Pareto optimality implies CPO.

As in Lemma 1(i), it can be shown that the agent's preferences satisfy thefollowing non-vanishing Gaussian curvature condition if an assumptionanalogous to Assumption 1 is made: for x**(t, h) # R (St+St+1)

++ , and k**=4KHL, there exists \

�t, h(k**)>0 such that for &x̂(t, h)&x**(t, h)&�k**,

ut, h(x̂(t, h))�ut, h(x**(t, h))

O :_ # 7t

:_$ # _+

$2(_$, _, h)�& :_ # 7t

$1(_, h)+\(�_ # 7t

$1(_, h))2

&pt**&


for all 0�\�\�

t, h(k**). Similarly, one obtains a bounded Gaussiancurvature condition.

From [5, 12] it follows that, under conditions analogous to those ofTheorems 1 and 2 a competitive equilibrium allocation with completemarkets is ex ante Pareto optimal if and only if the series ��

t=1 1�&pt**&diverges, i.e., the competitive prices satisfy the classical Cass criterion witha variable number of commodities.19

The condition for ex ante Pareto optimality cannot be compared to thecondition for CPO derived in Theorems 1 and 2 since the results refer todifferent market structures and prices ( p* and p**) obtained with differentequilibrium concepts.

We now argue that the results in [5, 12] do not have any direct implica-tion as far as a characterization of CPO is concerned; this is because thoseresults utilize the fact that preferences are strictly monotonic in all the com-modities available at a given date, a condition which cannot be satisfiedwhen agents are distinguished by the event at their birth (as required bythe CPO criterion). In fact, if the curvature coefficients, described in Defini-tions 4 and 5, satisfy the uniform bounds specified in Theorems 1 and 2then divergence of the series ��

t=1 1�&pt*& is neither a necessary conditionnor a sufficient condition for CPO with sequentially complete markets: asExample 1 below shows, an allocation could be CPO even though theseries ��

t=1 1�&pt*& converges, while Example 2 shows that a CPOimprovement may exist even though the series ��

t=1 1�&pt*& diverges.Furthermore, if the curvature coefficients are as in Theorems 1 and 2 thendivergence of the series ��

t=1 1�St } &pt*& is a sufficient condition for CPObut not a necessary condition; this criterion obtains from [12] when thetotal number of agents born at date t is HSt while the total number ofcommodities is LSt , as in our set-up (where agents are distinguished accor-ding to the event at their birth). Divergence of the series ��

t=1 1�St } &pt*&necessarily implies that the series ��

t=1 (1��_ # 7t&p*(_)&) diverges,20

which, by Corollary 1 of this paper, is a sufficient condition for CPO; butExample 1 below shows how an allocation could be CPO even though��

t=1 1�St } &pt*& converges.

The following example shows that equilibrium prices could satisfy thecriterion in Theorem 3 even though both the series ��

t=1 1�&pt*& and theseries ��

t=1 1�St } &pt*& converge. Moreover, it shows that the condition inTheorem 3 is different from the condition in Corollary 1 since the series��

t=1 (1��_ # 7t&p*(_)&) converges; this follows from the inequality

��t=1 (1��_ # 7t

&p*(_)&)��t=1 1�&pt*&.


19 It is worth emphasizing that in this case for sufficiency we need |l (_) St(_) , rather than |l (_),to be uniformly bounded for all l # L and for all _ # 7.

20 Since &pt*&>max_ # 7t&p*(_)& we have St } &pt*&>St } max_ # 7t

&p*(_)&>�_ # 7t&p*(_)&.

Example 1. Consider an economy where H=1, L=1, S(_)=2 for all_ # 1; also, 71=[_a , _b]. Let prices be

p*(_)=1 if t(_)=2, 4, 6, ... and _ succeeds _a

or if t(_)=1, 3, 5, ... and _ succeeds _b

p*(_)=(1�3)t(_) otherwise

so that at even dates all the nodes in the ``top half '' of the tree are assignedprice one, at odd dates all the nodes in the ``bottom half '' of the tree areassigned price one, and at all other nodes prices decline at a geometric rate;so we have a perverse form of nonstationarity in the prices. It is easy tospecify preferences and endowments so that the prices at an S-CE take thisform.

Given any node _� , consider the sub-tree 1_� that has _� as its root andincludes every successor of _� in 1. The sub-tree has 2t&t(_� ) nodes at datet and

:T

t=t(_� )

1�_ # 7t & 1_�

&p*(_)&= :

T

t�t(_� ) and t even

12t&t(_� )+ :

T

t�n and t odd

12t&t(_� ) } (1�3)t .

As T � �, this sum diverges since the second term on the right diverges.So the criterion in Theorem 3 is satisfied, indicating that the equilibriumallocation is CPO (if the additional conditions specified in the theoremhold).

However, the series

:�

t=1

1&pt*&

= :�

t=1

1(2t&1(1+(1�3)2t))1�2< :

�

t=1

12 (t&1)�2

converges. Clearly, the series ��t=1 1�St } &pt*& also converges since St=2t.

The next example shows how equilibrium prices could satisfy the condi-tion that ��

t=1 1�&pt*& diverges even though the allocation is not CPO.Furthermore, it shows that the condition that there is no improvementalong only one path is weaker than CPO.

Example 2. Consider an economy in which H=1, L=1, S(_)=2 forall _ # 1; p*(_)=(1�2)t�2 for all _ # 7. Clearly, the series ��

t=1 1�&p*(_�t )&

diverges along every path; however, the allocation is not CPO. To see this,consider the series ��

t=1 (1�(2t } 1�2t�2))=��t=1 1�2t�2, which is the series

defined in Theorem 2 with the weight function *(_)=1�2 for all _ # 7; theseseries are uniformly bounded across paths so that a CPO improvementexists (if the additional conditions specified in the theorem hold).


Also

:�

t=1

1�_ # 7t

&p*(_)&= :

�

t=1

12t } (1�2t�2)

converges; in fact, for _� an arbitrary node, ��t=t(_� ) (1��_ # 7t & 1_�

&p*(_) &)converges where 1_� is the sub-tree that has _� as its root and includes everysuccessor of _� in 1.

Furthermore,

:�

t=1

1&pt*&

= :�

t=1

1(2t((1�2)t))1�2

diverges.

Remark 3. Zilcha [24] provides a characterization of the efficiencyproperties of allocations in the framework of a one good OLG model withuncertainty and production according to a different efficiency criterion. Heprovides necessary and sufficient conditions both for allocations to beSOSD-efficient and FOSD-efficient, i.e., to be not dominated, for anyagent, in terms of second order stochastic dominance (respectively, firstorder stochastic dominance), conditional upon the event at birth. Note thatCPO is a stronger welfare criterion than SOSD-efficiency and, a fortiori ofFOSD-efficiency.21

Peled [21] on the other hand provided a sufficient condition for a com-petitive equilibrium to be CPO in an economy with one commodity andone agent per generation, stationary Markov uncertainty, and money asthe only asset. It is well known that with H=1 a market structure withmoney as the only asset is equivalent to having sequentially completemarkets; so we can convert Peled's formulation into our set-up. We canthen show that (see Section 6 for the details) if Peled's sufficient conditionfor CPO holds then, for some =>0, the inequality 1��_ # 7t

p*(_)�= holdsfor all t. Note that this condition implies the validity of the condition given


21 A precise study of the relationship between the results in [24] and this paper is difficult heresince the structure of the economy considered in [24] is rather different from the one examinedhere, and so is the efficiency notion. The precise relationship between our characterization of theconditions for CPO and the conditions for SOSD-efficiency derived in [24] is given in Chat-topadhyay [8], where the current analysis is applied to the case in which uncertainty is describedby a filtered probability space instead of a date�event structure. Results analogous to Theorems 1and 2 of this paper are shown to hold with the weights replaced by a sequence of densities; thecriterion in [24] is obtained when the densities are restricted to take the value one everywhere.This suggests that SOSD efficiency is much weaker than CPO since the path sums could divergeunder the criterion in [24] even though a subset of the sums are uniformly bounded for densitieswhich are not identically equal to one, indicating that CPO improvements exist.

in Corollary 1 above; the converse is not true though (consider for examplethe case where �_ # 7t

p*(_)=t). Hence Peled's condition is stronger thanthe condition in Corollary 1 (which we saw was in turn stronger than theconditions provided in Theorems 1 and 3).

4. STATIONARY EQUILIBRIA

In this section we restrict attention to stationary equilibria, characterizedby the fact that for each agent the equilibrium consumption level onlydepends on the realization of the uncertainty during his lifetime, not onpast realizations nor on the date of his birth. The market structure isunchanged, characterized by sequentially complete markets.

We will assume here that the uncertainty in the environment is generatedby the realizations of a time homogeneous Markov process with fixed finitestate space S (where >S=S). The structure of the date�event tree inducedby all possible realizations of the Markov process from an initial date t=0is as follows. The initial node _0 is an element of S; set 71 :=[_0]_S,and iteratively 7t :=7t&1_S for t=2, 3, ... . In this framework a node _t

can be identified with the string (_0 , s1 , s2 , ..., st), where s{ # S denotes therealization of the Markov process at date {, {=1, ..., t (_0 is then therealization at the initial date).22 The date�event tree 1_0

thus induced hasat every node the same number of successors, >S(_)=S for all _, and>7t=S t.

For stationary equilibria to exist additional conditions are needed.We will assume that the economy is also stationary, i.e., that the endow-ments and utility functions of each agent only depend on the realiza-tions of the Markov process during his lifetime, not on time nor on pastrealizations: |(_t , h)=|(st , h)=(|(st ; st , h), (|(st , s; st , h))s # S), andu_t , h=ust , h , for all h, _t , st , t such that _t=(_0 , s1 , s2 , ..., st). In additionwe will assume that a single commodity is available at each date, i.e.,L=1.

Under the above stationarity conditions of the environment, each agentcan simply be identified by the triple (st , t, h) # S_[1, 2, ...]_H, describ-ing the state and the date at his birth as well as the agent's type (or(_0 , 0, h) for the initial old).

Formally, we say a competitive equilibrium with sequentially completemarkets (x*, p*) is stationary if, in addition to the properties stated in


22 In the present set-up, st identifies the arc leading from node (_0 , s1 , s2 , ..., st&1) to node(_0 , s1 , s2 , ..., st). It is a general property of date�event trees that nodes can also be identified bythe sequence of arcs leading to them from the root of the tree��see Footnote 5 in Section 2.

Definition 3, the following condition also holds: x*(_t , h)=x*(st , h) for all(_t , h) # 7_H (i.e., the consumption allocation of each agent only dependson the state at the date of his birth and the states at the next date). Underthe above conditions, and Assumption 1, stationary equilibria with S-CEalways exist.23

To characterize efficiency we will use the fact that the system [ pt] t�1

of prices of all contingent commodities at the initial date implicitly de-fines a system of spot prices for the commodities and one-period contingentprices at all nodes (given the equivalence already noticed in Section 2).Since L=1, spot prices can be ignored here; the vector q(_t) of theprices at a generic node _t of one period claims, contingent on all possiblerealizations of the uncertainty at the next date, conditional on _t , isthen given by (( p(_$)�p(_t))_$ # _t

+) . It is easy to see that at a stationaryequilibrium q(_t) only depends on st ; hence the system of one-periodcontingent prices is simply described by a matrix Q, of dimensionS_S, with generic element qss$ , describing the price quoted in state s fora claim promising the delivery of one unit of the good at the next date instate s$.

Let *(Q) denote the dominant root of the matrix Q. The next re-sult shows that a complete characterization of the efficiency proper-ties of stationary equilibria can be given in terms of the value of*(Q):

Theorem 4. Let (x*, p*) be a stationary competitive equilibrium (withsequentially complete markets) and Q* be the matrix of the associatedone-period contingent claim prices. Under Assumption 1, if the environment isstationary, L=1, and x*(s; h) # R1+S

++ (i.e., the equilibrium allocation isinterior for each agent), we have:

(i) the equilibrium allocation x* is conditionally Pareto optimal if*(Q*)�1;

(ii) if *(Q*)>1, x* is not conditionally Pareto optimal; moreover, itis Pareto dominated (in the CPO sense) by another stationary allocation.24

The result is to a large extent an implication of the following:


23 See e.g., Gottardi [13]. When L>1, as shown by Spear [23], stationary equilibria do notexist, generically (in endowments and utility functions).

24 In Theorem 4 we not longer need to impose uniform upper and lower bounds on the cur-vature coefficients as in Theorems 1 and 2. The stationarity of the environment and of the equi-librium considered ensure in fact that the set of equilibrium consumption levels is described by afinite collection of vectors, so that under Assumption 1 uniform bounds always hold.

Lemma 2. Let p* be a system of prices at a stationary competitive equi-librium (with sequentially complete markets) and Q* be the matrix of theassociated one period contingent claim prices. Then we have:

(i) ��t=1 (1��_ # 7t

p*(_))<� O there exists a weight function *1_0and a scalar A<� such that, for every path _�(1_0

),

A(_�(1_0); (1_0

, *1_0)) := :

�

t=1

*� 1_0(_�

t )

p*(_�t )

�A.

(ii) *(Q*)>1 � ��t=1 (1��_ # 7t

p*(_))<�.

The result in part (i) of Lemma 2 says that, for the special case of thestationary equilibria of a stationary economy, the sufficient condition forCPO obtained in Corollary 1 is also a necessary condition for CPO (itsviolation implies in fact the validity of the necessary condition for CPO ofTheorem 2). Part (ii) of Lemma 2 then establishes the equivalence betweenthis condition��the convergence of the series appearing in Corollary 1,whose terms are given by the reciprocals of the sum of the prices p*(_) atall nodes at a given date��and the condition that the dominant root of thematrix of the associated one-period contingent prices q*ss$ is greater than 1.Combining these two results with Corollary 1 and Theorem 2 we obtainthe characterization provided in Theorem 4: the necessary and sufficientcondition for stationary equilibria to be CPO is *(Q*)�1. The theoremactually establishes a slightly stronger result as it shows that if *(Q*)>1a CPO improvement exists even if we limit our attention to stationaryallocations.

Remark 4. Aiyagari and Peled [1] examine a stationary stochasticOLG economy, similar to the one of this section. By studying the solutionsof a finite dimensional planner's problem for such an economy, they showthat an interior stationary equilibrium allocation is optimal, according tothe CPO criterion, in the class of all stationary allocations if and only if*(Q)�1. The result in Theorem 4 is significantly stronger as it shows thatif the same condition (*(Q)�1) holds, stationary equilibria are optimal, inthe CPO sense, in the class of all allocations, i.e., even if we allow for non-stationary reallocations.

Manuelli [17] considers the case of a stationary stochastic OLGeconomy where uncertainty is described by a stationary Markov processwith infinite support, described by a compact set. Under the additionalcondition that there is a single agent per generation, i.e., H=1, he showsthat the necessary and sufficient condition for a stationary competitiveallocation to be CPO in the class of all allocations is *(Q)�1. The argu-ment of the proof of his result is rather different from ours and is based onthe Martingale Convergence Theorem.


5. SEQUENTIALLY INCOMPLETE MARKETS: A SPECIAL CASE

We consider in this section the case in which agents are no longer ableto fully insure against every possible realization of the uncertainty whenold; markets are then said to ``sequentially incomplete.'' A general analysisof the efficiency properties of competitive equilibria with sequentiallyincomplete markets is beyond the scope of this paper;25 here we onlyexamine the special case which arises when there is only one asset, ariskless bond, available at each date�event.

CPO appears to be too demanding as an efficiency criterion for com-petitive equilibrium allocations when markets are sequentially incomplete.Consequently, we consider a weaker criterion, ex post Pareto optimality(denoted EPPO).26 According to this notion the agents' welfare isevaluated by conditioning on the realization of the uncertainty not only inthe first period of their lives but also the second period. Efficiency then onlyconcerns the allocation of commodities at each date�event pair, and betweenthe two periods of the agents' lifetime, not the allocation of risk. Such anotion appears to be particularly well suited for the analysis of efficiency insituations where there is no asset whose payoff is contingent on the state.

In order to be able to apply this criterion, we need to be able to evaluateagents' preferences over consumption conditional upon both the eventwhen young and the event when old. To this end, we will impose here theadditional condition that preferences are additively separable over time andacross the various possible events when old:

Assumption 2. For all (_, h) # 7_H, u_, h(x(_, h))=u_, _, h(x(_; _, h))+�_$ # _+ u_$, _, h(x(_$; _, h)).

Under this assumption the preferences of agent (_, h) naturally induce aspecification of preferences over consumption conditional upon both theevents, that when young and that when old.

The previous definitions of feasible allocations remain unchanged. Weare now ready to define the new welfare criterion:

Definition 6 (EPPO). Let x be a feasible allocation. x is ex post Paretooptimal (EPPO) if there does not exist another feasible allocation x̂ such that

(i) for all (_, h) # 71_H, u_, h, o(x̂(_; h, o))�u_, h, o(x(_; h, o)), for all(_, h) # 7_H, _$ # _+,

u_, _, h(x̂(_; _, h))+u_$, _, h(x̂(_$; _, h))

�u_, _, h(x(_; _, h))+u_$, _, h(x(_$; _, h)),


25 See Cass et al. [7] and Gottardi [14] for some preliminary steps on this issue.26 Lucas [16] was the first to use such a criterion to evaluate the welfare properties of com-

petitive equilibria in OLG models under uncertainty.

(ii) either for some (_~ , h� ) # 71_H, u_~ , h� , o(x̂(_~ ; h� , o))>u_~ , h� , o(x(_~ ; h� , o))or for some (_~ , h� ) # 7_H, _~ $ # _~ +,

u_~ , _~ , h� (x̂(_~ ; _~ , h� ))+u_~ $, _~ , h� (x̂(_~ $; _~ , h� ))

>u_~ , _~ , h� (x(_~ ; _~ , h� ))+u_~ $, _~ , h� (x(_~ $; _~ , h� )).

Thus EPPO applies the usual Pareto criterion to the larger set of agents(relative to the CPO criterion) where every agent is distinguished by theevent faced at each of the two dates at which he is alive. When carrying outthe EPPO optimality analysis, we will use the notation (_$, _, h) to identifyan agent.

To obtain an improvement according to the EPPO criterion, an agentwho receives a negative transfer when young must be compensated with apositive transfer in every possible realization of the uncertainty when old.By iterating the argument we see that we must have a nonzero transfer atall nodes which are successors to a node at which a transfer is made. Con-sequently, we define a full sub-tree with root _~ , denoted by 1(_~ ), as a sub-tree with the property that every node _ # 1 which is a successor of _~ is anelement of 1(_~ ).

We can relate the transfers that generate EPPO and CPO improvementsby comparing Definitions 2 and 6. Consider a transfer defined on some fullsub-tree; call it the `ìnitial'' transfer. For each path in the sub-tree, inducea ``new'' transfer which takes the value zero at all nodes off the path thathas been fixed and takes the same value as the `ìnitial'' transfer at all nodeson the path. This procedure generates a collection of ``new'' transfers, onefor each path in the sub-tree. Suppose that each of these ``new'' transfers isCPO improving; then the `ìnitial'' transfer must be EPPO improving.27

Remark 5. To properly evaluate the consequences of the sequentialincompleteness of markets, we provide first a necessary and sufficient con-dition for competitive equilibria with sequentially complete markets to beEPPO. So, let (x*, p*) be an S-CE. By an immediate extension of theargument of Lemma 1 we obtain that, under Assumptions 1 and 2, forx*(_, h) # RL(1+S(_))

++ , the following version of the non-vanishing Gaussiancurvature condition holds for the agent (_$, _, h): for k*=4KHL, thereexists \

�_$, _, h(k*)>0 such that for &(x̂(_; _, h)&x*(_; _, h)), (x̂(_$; _, h)&

x*(_$; _, h))&�k*,

u_, _, h(x̂(_; _, h))+u_$, _, h(x̂(_$; _, h))

�u_, _, h(x*(_; _, h))+u_$, _, h(x*(_$; _, h))

O $2(_$, _, h)�&$1(_, h)+\($1(_, h))2

&p*(_)&


27 Clearly, this is not the same as requiring a CPO improvement on only one path.


_$, _, h(k*) (where we recall that $1(_, h) :=p*(_) }[x̂(_; _, h)&x*(_; _, h)], $2(_$, _, h) :=p*(_$) } [x̂(_$; _, h)&x*(_$; _, h)],for _$ # _+). A similar expression can be obtained for the boundedGaussian curvature condition.

One can show that, under conditions analogous to Theorems 1 and 2, anS-CE allocation is ex post Pareto optimal if and only if there exists a fullsub-tree 1(_~ ) and an A<� such that, for every path a�(1(_~ )) in the fullsub-tree, ��

t=t(_~ ) 1�&p*(_�t )&�A.

The proof of this result follows closely the proofs of Theorems 1 and 2since, as was noted earlier, the conditions obtained for EPPOimprovements at an S-CE are simply the conditions for having a CPOimprovement on every path contained in a full sub-tree.

It is easy to see that the necessary and sufficient conditions for an S-CEallocation to be EPPO are less demanding than the corresponding condi-tions for CPO. As we noticed, EPPO is in fact a weaker notion ofoptimality than CPO.

We turn to the particular sequentially incomplete market structure inwhich there is only one asset, a riskless bond, promising the delivery of afixed amount (set equal to one without loss of generality) of the numerairecommodity in the subsequent period, independent of the realization of theuncertainty.

In order to be able to directly compare the conditions obtained here withthe results obtained in the earlier sections of this paper, we use the noarbitrage condition to write the price of the asset in terms of a vector ofstate prices and then substitute away the portfolio holdings in the budgetconstraints. It follows that the variables appearing in an agent's budgetconstraints are, as before, the contingent commodity prices ( p(_),( p(_$))_$ # _+) and the consumption bundle x(_, h), but, given marketincompleteness, the agent now faces a multiplicity of budget constraints.

Definition 7 (B-CE). (x� , p� ) is a competitive equilibrium with completespot markets and a riskless bond (B-CE) if x� is a feasible allocation, and,

(i) for all (_, h) # 71 _H, p� (_) } x� (_; h, o)�p� (_) } |(_; h, o)

u_, h, o(x(_; h, o))>u_, h, o(x� (_; h, o)) O p� (_) } x(_; h, o)>p� (_) } |(_; h, o),

(ii) for all (_, h) # 7_H,

( p� (_), p� (_$)) } (x� (_; _, h), x� (_$; _, h))

�( p� (_), p� (_$)) } (|(_; _, h), |(_$; _, h)) for all _$ # _+,


u_, h(x(_, h))>u_, h(x� (_, h))

O ( p� (_), p� (_$)) } (x(_; _, h), x(_$; _, h))

>( p� (_), p� (_$)) } (|(_; _, h), |(_$; _, h)) for some _$ # _+.

In what follows we provide a characterization of the efficiency properties,according to the EPPO criterion, of equilibria with complete spot marketsand a riskless bond, in terms of the prices p� .

A crucial step in the argument is the derivation of appropriate curvatureconditions, a more subtle problem since agents now face a multiplicity ofbudget constraints. We begin by noting that under Assumptions 1 and 2,for all (_, h) # 7_H, the optimal choice x� (_, h), if it is interior, i.e.,x� (_, h) # RL(1+S(_))

++ , satisfies the first order conditions

Du_, _, h(x� (_; _, h))=\ :_$ # _+

+� (_$, _, h)+ } p� (_),

Du_$, _, h(x� (_$; _, h))=+� (_$, _, h) } p� (_$) for all _$ # _+, and

( p� (_), p� (_$)) } [(x� (_; _, h), x� (_$; _, h))&(|(_; _, h), |(_$; _, h))]=0,

where Du_, _, h( } ) (respectively, Du_$, _, h( } )) is the vector of derivatives ofu_, _, h( } ) (respectively, u_$, _, h( } )) with respect to its L arguments, andwhere ((+� (_$, _, h))_$ # _+) # RS(_)

++ is the vector of Lagrange multipliersassociated with the S(_) constraints faced by (_, h) (positivity follows fromthe strict monotonicity of preferences).

From the above expressions we see that the ``personalized'' price vector((�_$ # _+ +� (_$, _, h)) } p� (_), (+� (_$, _, h) } p� (_$))_$ # _+) supports the same choicex� (_, h) of agent (_, h) were he to maximize utility subject to a single inter-temporal budged constraint (as he does when markets are sequentiallycomplete). This allows us to obtain the non-vanishing Gaussian curvaturecondition in terms of these ``personalized'' prices by essentially the sameargument as in the proof of Lemma 1: under Assumptions 1 and 2, forx� (_, h) # RL(1+S(_))

++ , and k*=4KHL, there exists \�

_$, _, h(k*)>0 such thatfor &(x̂(_; _, h)&x*(_; _, h)), (x̂(_$; _, , h)&x*(_$; _, h))&�k*,

u_, _, h(x̂(_; _, h))+u_$, _, h(x̂(_$; _, h))

�u_, _, h(x� (_; _, h))+u_$, _, h(x� (_$; _, h))

O +� (_$, _, h) $2(_$, _, h)

�&\ :_$ # _+

+� (_$, _, h)+ $1(_, h)+\((�_$ # _+ +� (_$, _, h)) $1(_, h))2

(�_$ # _+ +� (_$, _, h)) &p� (_)&



_$, _, h(k*) (where $1(_, h) :=p� (_) } [x̂(_; _, h)&x� (_; _, h)],$2(_$, _, h) :=p� (_$) } [x̂(_$; _, h)&x� (_$; _, h)], for _$ # _+). Since�_$ # _+ +� (_$, _, h)>0, we can rewrite the second inequality in the aboveexpression as

+� (_$, _, h)(�_$ # _+ +� (_$, _, h))

$2(_$, _, h)�&$1(_, h)+\($1(_, h))2

&p� (_)&


_$, _, h(k*).In a similar manner, we can obtain the bounded Gaussian curvature

condition.Let +� (_$) denote the average of the agents' Lagrange multipliers at a

given node _$:

+� (_$) :=1H

:h # H

+� (_$, _, h)(�_$ # _+ +� (_$, _, h))

for _$ # .t�1

7t .

In the following result, we obtain separate necessary and sufficient condi-tions for competitive equilibria with complete spot markets and a risklessbond to be EPPO.

Theorem 5. Let (x� , p� ) be a B-CE and let [+� (_$, _, h)]_ # 1, _$ # _+, h # H bethe system of associated Lagrange multipliers. Suppose Assumptions 1 and 2hold and assume that there are numbers K>0, \

�>0, \� >0, =>0, and k� >0,

such that

(i) |l (_)�2KH for all l=1, ..., L for all nodes _ # 7,

(ii) for every (_, h) # 7_H, and every _$ # _+, x� (_, h)�= } 1L(1+S(_))_1 , \

��\

�_$, _, h(4KHL) and \� _$, _, h(k� )�\� .

(A) If the equilibrium allocation is not ex post Pareto optimal then thereexist a full sub-tree 1(_~ ) and an A<� such that, for every path _�(1(_~ ))in the full sub-tree,

A(_�(1(_~ ))) := :�

t=t(_~ )

1H t&t(_~ )(> t

{=t(_~ )+1 +� (_�{ )) &p� (_�

t )&�A.

(B) If there exists a full sub-tree 1(_~ ), with an agent h_ # H at everynode _ # 1(_~ ), and there exists an A<� such that, for every path _�(1(_~ ))in the full sub-tree

A(_�(1(_~ ))) := :�

t=t(_~ ) \ `t

{=t(_~ )+1

�_$ # (_�{&1)+ +� (_$, _�

{&1 , h_�{&1

)

+� (_�{ , _�

{&1 , h_�{&1

) + 1&p� (_�

t )&�A,

then the equilibrium allocation is not ex post Pareto optimal.


In the present set-up, the prices appearing in the Gaussian curvatureconditions are agent-specific and, since markets are incomplete, will typi-cally vary in a non-trivial way with h. Hence, the sufficient condition withmany agents per generation is stated in terms of the average of these agent-specific prices.28 Evidently, this issue does not arise when there is a singleagent per generation as in this case we can simply use the prices supportingthe choice of that agent. So with H=1, the two conditions in Theorem 5reduce to a single necessary and sufficient condition for a B-CE not to beEPPO,

A(_�(1(_~ ))) := :�

t=t(_~ ) \ `t

{=t(_~ )+1

�_$ # (_�{&1)+ +� (_$, _�

{&1)

+� (_�{ , _�

{&1) + 1&p� (_�

t )&�A,

where [+� (_$, _)]_ # 1, _$ # _+ are the agent's Lagrange multipliers at the B-CE.Notice that in the present set-up, with one agent per generation, markets

are effectively sequentially complete; hence, the equilibrium allocation at aB-CE is also an S-CE equilibrium, supported by the ``personalized'' prices((�_$ # _+ +� (_$, _, h)) } p� (_), (+� (_$, _, h) } p� (_$))_$ # _+). Clearly, once prices arereplaced by these ``personalized'' prices, the condition above (for B-CE tobe EPPO when H=1) reduces to the necessary and sufficient conditiongiven in Remark 5 for S-CE to be EPPO.

6. PROOFS

Proof of Lemma 1

We follow Benveniste [3] and Balasko and Shell [2].We will prove a more general result from which Lemma 1 follows.For A and B positive integers, let RA

+_RB+ be an agent's consumption

set, U: RA+_RB

+ � R be his utility function, and ( pA , pB) } (xA , xB)�m behis budget restriction, where p=( pA , pB) is a price vector, x=(xA , xB) isa consumption vector, and m>0.

Also, let Vx :=[ y # RA+_RB

+ | U( y)�U(x)], the weak upper contourset at x, and, for k # R+ , \ # R, let K(x, k) :=[ y # RA

+ _RB+ | &y&x&�k],

K(x, k, p) :=[ y # K(x, k) | pA } ( yA&xA)<0], and R(\, x, p) :=[ y # RA_RB | pB } ( yB&xB)�& pA } ( yA&xA)+\( pA } ( yA&xA))2�&pA &],a paraboloid of curvature \ which has x on its boundary and p as the sup-port vector at x.


28 Notice that the number of agents did not play any role in our earlier results.

Lemma 1A. Let x=(xA , xB) # RA++_RB

++ be the solution to theagent's utility maximization problem at prices p=( pA , pB) # RA

++_RB++ .

Under the assumption that U( } ) is C2, strictly monotone, and differentiablystrictly quasi-concave, so that \z # RA_RB, z{0, z } {xU=0 O z$D2

xUz<0,we have

(i) for any k>0, \�>0 where \

�:=sup[\ | Vx & K(x, k)/R(\, x, p)

& K(x, k)];

(ii) for any k>0, �>\� �0 where \� :=inf[\ | R(\, x, p) & K(x, k, p)/Vx & K(x, k, p)].

Proof. (i) Under the assumptions of the lemma, for any k>0, there isa closed ball of radius r

�, for 1�r

�>0, with center at c=x+(r

��&p&) p, denote

it by B(c, r�) , which is (a) tangent to the budget plane at x, (b) tangent to

the upper contour set, Vx , at the point x, and (c) contains the setVx & K(x, k) (see, e.g., Mas-Colell [18, page 40]).

Now take any point y=( yA , yB) in the set Vx & K(x, k); so y # B (c, r�)

and &y&c&�r�. Let *{0 be such that &*( y&x)&(r

��&p&) p&=r

�; since c=

x+(r��&p&) p, we have &*( y&x)+x&c&=r

�, and hence *�1 as

&y&c&�r�. Now

*( y&x) } \*( y&x)&2r

�&p&

p+=\*( y&x)&

r�

&p&p+

r�

&p&p+ } \*( y&x)&

r�

&p&p&

r�

&p&p+

="*( y&x)&r�

&p&p"

2

&" r�

&p&p"

2

=r�2&r

�2=0.

Thus * &y&x&2=( y&x) } (2r��&p&) p, which implies p } ( y&x)�

(1�2r�) &y&x&2 &p& since *�1. Since &( pA , pB)&�&pA&, and &y&x&�

&yA&xA&, we have p } ( y&x)�(1�2r�) &yA&xA&2 &pA &. But &yA&xA&

&pA &�| pA } ( yA&xA)|, so we have &yA&xA&2 &pA&�| pA } ( yA&xA)| 2�&pA &, and p } ( y&x)�(1�2r

�) | pA } ( yA&xA)|2�&pA&. Letting \=1�2r

�we

see that for y # K(x, k),

U( y)�U(x) O pB } ( yB&xB)�& pA } ( yA&xA)+\( pA } ( yA&xA))2

&pA &.

So, for \=1�2r�>0, Vx & K(x, k)/R(\, x, p) & K(x, k), implying that

sup[\ | Vx & K(x, k)/R(\, x, p) & K(x, k)]>0. Thus, \�>0.


(ii) For any k>0, by smoothness and strict monotonicity ofpreferences there exists a paraboloid with curvature \ which is locally con-tained in the set Vx and is tangent to Vx at x (see, e.g., Benveniste [3]).So, for any k>0, there is a \ for which R(\, x, p) & K(x, k, p)/Vx &K(x, k, p); furthermore, by local non-satiation, \�0 (in fact, by (i) above,\>0). Hence, inf[\ | R(\, x, p) & K(x, k, p)/Vx & K(x, k, p)] is bounded.Thus, �>\� . K

Proof of Theorem 1

As indicated in the text, the proof of Theorem 1 starts by showing thatif there exists a CPO improvement then necessarily there exists a sub-treewith the property that the per capita transfer to the old agents at each node_ is strictly positive. We then construct a weight function, by disaggregat-ing the per capita transfer across nodes when old, and then we show thatthe non-vanishing Gaussian curvature condition implies that the per capitatransfer, transformed by the weights, increases according to a quadraticfunction along every path. Finally, we use the uniform bound on endow-ments to show that this quadratic increase can occur only if the weightedreciprocal of the norm of the price increases sufficiently fast along every path.

Since the competitive equilibrium allocation x* is, by assumption, notCPO, there must exist an improving allocation; let x̂ denote such an improv-ing allocation. For (_, h) # 7_H, construct the sequences $1(_, h)= p*(_) }[x̂(_; _, h)&x*(_; _, h)], $2(_$, _, h)= p*(_$) } [x̂(_$; _, h)&x*(_$; _, h)],for _$ # _+; similarly $2(_, h, o) :=p*(_) } [x̂(_; h, o)&x*(_; h, o)]. Define$2(_, o) :=(1�H) �h # H $2(_, h, o), $1(_) :=(1�H) �h # H $1(_, h), $2(_$, _) :=(1�H) �h # H $2(_$, _, h).

By strict monotonicity (Assumption 1(ii)), and the fact that x̂ improvesover x*, we have that for every agent (_, h, o) of the initial generation$2(_1 , h, o)�0, while for every agent (_, h) of the successive generations$1(_, h)+�_$ # _+ $2(_$, _, h)�0; a strict inequality holds if x̂(_, h){x*(_, h) (by strict quasi-concavity). So, averaging across h # H,

$2(_, o)�0 for all _ # 71 ,

$1(_)+ :_$ # _+

$2(_$, _)�0 for all _ # 7,

and the inequality is strict if at least one of the agents born at node _ isbeing strictly improved.

Feasibility of x̂ and x* implies that

$1(_)+$2(_, o)�0, for _ # 71 ,

$1(_)+$2(_, _&1)�0, for _ # .t�2

7t .


Furthermore, since x̂ and x* differ, there is a finite t�, and a set

7t _H@ �7t_H, for t�t�, such that for every agent (_, h) # �t<t

�(7t_H),

x̂(_, h)=x*(_, h), while for (_t , h) # 7t_H@ , x̂(_t , h){x*(_t , h). Hence forall (_, h) # �t<t

�(7t _H), and _$ # _+, we have $1(_, h)=$2(_$, _, h)=0.

On the other hand, for _t�

such that (_t�, h) # 7t

�_H@ , by feasibility,

$1(_t�)+$2(_t

�, _t

�&1)�0; since $2(_t

�, _t

�&1)=0, we have then $1(_t

�)�0.

Since x̂ was assumed to be an improving allocation, from the strict quasi-concavity and monotonicity of preferences it follows �_$ # _t

�

+ $2(_$, _t�)>0;

hence for some _~ # _+t�

, $2(_~ , _t�)>0. A Pareto improving sequence of

reallocations of resources, starting from generation _t�

must therefore becharacterized by a positive per capita transfer to the members of thisgeneration in at least one state when old, _~ # _+

t�

.Note that _~ # 7t

�+1 . Using again feasibility, we find $1(_~ )<0, so that by

monotonicity, �_$ # _~ + $2(_$, _~ )>0 and hence $2(_", _~ )>0 for some_" # _~ +. Identify all the nodes that are immediate successors of the node _~and at which the old receive a positive per capita transfer. Iterating theargument, we can find a collection of nodes which are successors of_~ # 7t

�+1 and are characterized by (i) a positive level of per capita transfers

to the old agents and (ii) the property that the set of successors to anygiven node in the set includes all nodes at which the per capita transfer ispositive. It is easy to verify that this sequence defines a sub-tree; let it bedenoted by 1_~ . By construction, for all _ # 1_~ , $2(_, _&1)>0.

By strict monotonicity of preferences, p*(_)>>0 at all nodes; so&p*(_)&>0 for all _ # 7. Furthermore, by condition (i) of Theorem 1 andAssumption 1(i), &x̂(_, h)&x*(_, h)&�2KHL(1+S� ) for all (_, h) # 7_H.Hence, by interiority of the allocation and by Lemma 1, for all agents(_, h) # 1_~ _H we have

:_$ # _+ & 1_~

$2(_$, _, h)�&$1(_, h)+\�

_, h(2KHL(1+S� ))($1(_, h))2

&p*(_)&. (2)

Using condition (ii) of Theorem 1 the inequality in (2) continues to holdif we replace \

�_, h(2KHL(1+S� )) by \

�. In addition, notice that the set of

pairs ($1(_, h), �_$ # _+ $2(_$, _, h)) # R2 which satisfy (2) is a convex set;hence, if we consider the per capita transfer to the H members of genera-tion _ # 7, the following inequality is still valid:

:_$ # _+ & 1_~

$2(_$, _)�&$2(_)+\�

($1(_))2

&p*(_)&. (3)


Now, define the function *1_~: 1_~ � [0, 1] by the rule

*1_~(_$) :=

$2(_$, _)�_$ # _+ & 1_~

$2(_$, _).

Given the way in which the sub-tree 1_~ was constructed, the function *1_~

is well defined on its domain, is positive, and satisfies the restriction�_$ # _+ & 1_~

*1_~(_$)=1 for all _ # 1_~ .

So we have a pair (1_~ , *1_~) consisting of a sub-tree and a weight func-

tion, the latter determined by the relative distribution of the per capitatransfers across successor nodes.

For the rest of the proof, we concentrate on _�(1_~ ), an arbitrary pathin the sub-tree.

Using the definition of the function *1_~, the inequality in the curvature

condition, (3), can be rewritten as

1*1_~

(_$)$2(_$, _)�&$1(_)+\

�

($1(_))2

&p*(_)&; (4)

i.e., as a condition at a given pair of nodes, and (4) holds for all _$, _ # 1_~ .Using the fact that, by feasibility, $2(_, _&1)+$1(_)=0, and that$2(_, _&1)>0 for all _ # 1_~ , we can invert (4) to obtain

*1_~(_$)

$2(_$, _)�

1$2(_, _&1)

&1

&p*(_)&\�1+\

�$2(_, _&1)�&p*(_)&

(5)

for all _ # 1_~ .By condition (i) of Theorem 1, the per capita endowment of every com-

modity is bounded at each node. Hence, 0<$2(_, _&1)�2KL &p*(_)& sothat

\�1+2KL\

�

�\�1+\

�$2(_, _&1)�&p*(_)&

. (6)

Substituting (6) into (5) we have

*1_~(_$)

$2(_$, _)+

\�1+2KL\

�

1&p*(_)&

�1

$2(_, _&1). (7)

But by iterating on the inequality (7) along _�(1_~ ), the path in the sub-tree that we fixed, we have

*1_~(_�

T ) } } } *1_~(_�

t(_~ ))

$2(_�T , _�

T&1)+

\�1+2KL\

�

:T&1

t=t(_~ )

*1_~(_�

t ) } } } *1_~(_�

t(_~ ))

&p*(_�t )&

�1

$2(_~ , _~ &1)

(8)


so that the series on the left-hand side of (8), being positive and monotoni-cally increasing, must converge (the other two terms being positive since\�>0). Recalling the definition of the function *� 1_~

, and definingA=: 1�$2(_~ , _~ &1), we have

A(_�(1_~ ); (1_~ , *1_~)) := :

�

t=t(_~ )

*� 1_~(_�

t )

&p*(_�t )&

�A.

This completes the proof of the theorem since the argument above wasmade for an arbitrary path in the sub-tree. K

Proof of Theorem 2

The proof of Theorem 2 consists in (i) proposing a candidate collectionof transfers (of income) which will support an improvement, (ii) verifyingthat there exists a corresponding collection of commodity transfers whichis feasible, i.e., generates commodity bundles in the consumption set ofevery agent and is compatible with aggregate endowments, and (iii) verify-ing that the transfers satisfy the bounded Gaussian curvature condition sothat the associated commodity transfers generate a Pareto improvement.

Step 1. We begin by specifying the transfers at every _ # 1_̂ . Since eachnode can be identified with a coordinate of some path in the sub-tree, itsuffices to define the transfers for each coordinate along each path in thesub-tree. Consequently, for t�t(_̂) we define

$(_�t+1) :=

}(1+}\� ) A2 *� 1_̂

(_�t+1) :

t

{=t(_̂)

*� 1_̂(_�

{ )

&p*(_�{ )&

where } :=(1�(1+S� )1�2) min[=, k� ] and (a) =>0 is as specified inhypothesis (i) of Theorem 2, (b) k� >0 and \� >0 are as specified inhypothesis (ii) of Theorem 2, (c) S� �1 is the maximal number of successornodes (see Assumption 1(i)), and the function *� 1_̂

is induced by the pair(1_̂ , *1_̂

), the sub-tree and weight function, specified in the theorem.Set $(_) :=0 for _ � 1_̂ .

Step 2. We now specify the commodity transfers. For _ # 1, define thevectors 2x_ # RL as follows: 2x_ :=( p*(_)�&p*(_)&) ($(_)�&p*(_)&). Nowlet 2x(_, h) :=(&2x_ , (2x_$)_$ # _+) for agents h_ as defined in the state-ment of Theorem 2, and set 2x(_, h) :=0 # RL for h{h_ .

The consumption vectors induced by the proposed transfers arex̂(_, h) :=x*(_, h)+2x(_, h).

Evidently, these transfer vectors are compatible with the aggregateendowments of the economy. In order to be able to use the inequalitydefined by the bounded Gaussian curvature condition (Definition 5) we


need to check that these vectors lie in the consumption sets of the agents,and satisfy the inequality &2x(_, h)&�k� . We now check these two proper-ties.

By the hypothesis of Theorem 2, *� 1_̂(_�

t )�&p*(_�t )&�A for all nodes

_�t # 1_̂ and � t

{=t(_̂) (*� 1_̂(_�

{ )�&p*(_�{ )&)�A, so that A is a uniform

bound for the set of partial sums over paths in the sub-tree. Hence

A2�*� 1_̂

(_�t+1)

&p*(_�t+1)&

:t

{=t(_̂)

*� 1_̂(_�

{ )

&p*(_�{ )&

. (9)

Furthermore, since }>0 and \� >0,

1>1

1+}\�O }>

}1+}\�

. (10)

Combining (9) and (10), and using the definition of $(_) we have

}>}

1+}\�}

1A2

*� 1_̂(_�

t+1)

&p*(_�t+1)&

:t

{=t(_̂)

*� 1_̂(_�

{ )

&p*(_�{ )&

=$(_�

t+1)&p*(_�

t+1)&.

So, for every node _ and for l=1, 2, ..., L,

2x l_ :=

p l*(_)&p*(_)&

$(_)&p*(_)&

<}.

But by the definition of }, =>} so that the sequence of commoditytransfer vectors, 2x_ , is bounded above in each coordinate by = (it isobviously bounded below by zero). Invoking hypothesis (i) of Theorem 2,we see that x̂(_, h) # X_, h for each agent. Furthermore, we easily see that,in the Euclidean norm,

&2x_ &=" p*(_)&p*(_)&

$(_)&p*(_)&"=

$(_)&p*(_)&

<}.

Hence, in the Euclidean norm,

&2x(_, h)&=&(&2x_ , (2x_$)_$ # _+)&<}(1+S� )1�2�k�

where the first inequality follows from the fact that each of the (1+S� )L-dimensional vectors 2x_ and (2x_$), for _$ # _+, have norms bounded by}, so that the Euclidean norm of 2x(_, h) is bounded by }(1+S� )1�2, andthe second inequality follows from the definition of }. Thus, we have&2x(_, h)&�k� as required.


Step 3. We complete the proof by showing that the allocationgenerated by the above specification of the transfers is improving. We showthat the inequality in the bounded Gaussian curvature condition is satisfiedfor agents who are born at nodes which belong to the sub-tree 1_̂ andreceive a transfer, i.e., the agents h_ ; since all the other conditions imposedin Definition 5 on the vectors of transfers are readily seen to be satisfied bythe vectors 2x(_, h), verification of the inequality guarantees that theagents are being weakly improved. This is sufficient since the old agent atthe node _̂, receives a positive transfer of every commodity when old andis strictly improved by strict monotonicity of preferences.

By construction, the income transfers, corresponding to the commoditytransfers 2x(_, h_t

), received by the agent h_tin the first period of his life

and in the second period at the node _�t+1 , are given by, respectively,29

$1(_�t , h_t

)=&$(_�t )=&

}(1+}\� ) A2 *� 1_̂

(_�t ) :

t&1

{=t(_̂)

*� 1_̂(_�

{ )

&p*(_�{ )&

$1(_�t+1 , _�

t , h_t)=$(_�

t+1)=}

(1+}\� ) A2 *� 1_̂(_�

t+1) :t

{=t(_̂)

*� 1_̂(_�

{ )

&p*(_�{ )&

.

Since, by the defining property of the function *1_̂, we have

�_$ # _+ & 1_̂*1_̂

(_$)=1 for all _ # 1_̂ , and recalling that *� 1_̂(_�

t+1)=*1_̂

(_�t+1) } *� 1_̂

(_�t ), we have

:_$ # _t

+$2(_$, _t , h_t

)=}

(1+}\� ) A2 *� 1_̂(_�

t ) :t

{=t(_̂)

*� 1_̂(_�

{ )

&p*(_�{ )&

,

where we have used the fact that transfers are made only on nodes in thesub-tree 1_̂ , so that

:_$ # _t

+$2(_$, _t , h_t

)= :_$ # _t

+& 1_̂

$2(_$, _t , h_t).

Now it is immediate from the definitions of the transfers that

$1(_t , h_t)+ :

_$ # _t+

$2(_$, _t , h_t)=

}(1+}\� ) A2 *� 1_̂

(_�t )

*� 1_̂(_�

{ )

&p*(_�{ )&

O_$1(_t , h_t)+ :

_$ # _t+

$2(_$, _t , h_t)& &p*(_t)&

=}

(1+}\� ) A2 (*� 1_̂(_�

t ))2.


29 In what follows, we will identify _�t and _t , etc.

Since \� _, h_(k� )�\� and }>0, we have

1�}\� _t , h_t

(k� )

1+}\�

and from the uniform bound over all paths in the sub-tree we have

A� :t

{=t(_̂)

*� 1_̂(_�

{ )

&p*(_�{ )&

.

It follows that

_$1(_t , h_t)+ :

_$ # _t+

$2(_$, _t , h_t)& &p*(_t)&

�}\� _t , h_t

(k� )

1+}\� \� t&1

{=t(_̂) (*� 1_̂(_�

{ )�&p*(_�{ )&)

A +2

}}

(1+}\� ) A2 (*� 1_̂(_�

t ))2

=\� _t , h_t(k� ) \ }

(1+}\� ) A2 *� 1_̂(_�

t ) \ :t&1

{=t(_̂)

*� 1_̂(_�

{ )

&p*(_�{ )&++

2

=\� _t , h_t(k� )($1(_t , h_t

))2.

So the transfers proposed verify the inequality condition in Definition 5. K

Proof of Theorem 3

The proof of Theorem 3 proceeds by contradiction. As in Theorem 1, westart by showing that if there exists a CPO improvement then necessarilythere exists a sub-tree with the property that the per capita transfer to theold agents at each node _ is strictly positive. We then show that given thecriterion that we choose to aggregate the per capita transfers across nodesat the same point in time, the non-vanishing Gaussian curvature conditionimplies that the aggregated per capita transfer increases according to aquadratic function. Next we use the uniform bound on endowments toshow that this quadratic increase can occur only if the sum of the normsof prices at the same point in time increases sufficiently fast. This gives aCass-like criterion. The proof is completed by considering an arbitrarynode in the sub-tree that has been identified and by applying the sameargument to the sequence of aggregated per capita transfers obtained bystarting from that node.

We omit some of the details of the proof of Theorem 3 referring insteadto the proof of Theorem 1.


Suppose that the allocation x* is not CPO so that there exists animprovement; let x̂ denote the improving allocation. As in the proof ofTheorem 1, construct the sub-tree 1_~ . By construction, for all _ # 1_~ ,$2(_, _&1)>0. By the same argument as in the proof of Theorem 1, weobtain, for all _ # 7,

�_$ # _+ $2(_$, _)&p*(_)&

�&$1(_)

&p*(_)&+\

� \$1(_)

&p*(_)&+2

. (11)

If _ # 1_~ , (11) holds a fortiori if on the left hand side of the expression wereplace the numerator with �_$ # _+ & 1_~

$2(_$, _), i.e., if we only consider thesum of average transfers to the agents when old over the subset (defined by1_~ ) of the collection of nodes at which these transfers are strictly positive.

Let _{ be an arbitrary node of 1_~ , at period {=t(_{), and consider1(_{ , 1_~ ). Recall that 1(_{ , 1_~ ) is the sub-tree that has _{ as its root andincludes all the nodes that are successors of _{ and are elements of 1_~ . Ifwe consider the average across all nodes30 at t, _ # 7t & 1(_{ , 1_~ ), for t�{of the transfers ($1(_), �_$ # _+ & 1_~

$2(_$, _)), the inequality in (11) is stillvalid so we get

:_ # 7t & 1(_{ , 1_~ )

�_$ # _+ & 1_~$2(_$, _)

&p*(_)&&p*(_)&

�_ # 7t & 1(_{ , 1_~ )&p*(_)&

�& :_ # 7t & 1(_{ , 1_~ )

$1(_)&p*(_)&

&p*(_)&�_ # 7t & 1(_{ , 1_~ )

&p*(_)&

+\� \ :

_ # 7t & 1(_{ , 1_~ )

$1(_)&p*(_)&

&p*(_)&�_ # 7t & 1(_{ , 1_~ )

&p*(_)&+2

O�_ # 7t & 1(_{ , 1_~ )

�_$ # _+ & 1_~$2(_$, _)

�_ # 7t & 1(_{ , 1_~ )&p*(_)&

�&�_ # 7t & 1(_{ , 1_~ )

$1(_)

�_ # 7t & 1(_{ , 1_~ )&p*(_)&

+\� \

�_ # 7t & 1(_{ , 1_~ )$1(_)

�_ # 7t & 1(_{ , 1_~ )&p*(_)&+

2

.

To simplify notation, let us write $2(t+1, 1(_{ , 1_~ )) for�_ # 7t & 1(_{ , 1_~ )

�_$ # _+ & 1_$2(_$, _), i.e., to denote the total transfer to the

old at time t+1 on the subtree 1(_{ , 1_~ ). Notice that $2({, 1(_{ , 1_~ ))=


30 By Assumption 1(i) the numbers of such nodes is always finite.

$2(_{ , _{&1)>0 since _{ # 1_~ . In addition, notice that, by feasibility&$1(_)=$2(_, _&1), so that the above inequality can be rewritten as

O $2(t+1, 1(_{ , 1_~ ))�$2(t, 1(_{ , 1_~ ))+\�

($2(t, 1(_{ , 1_~ )))2

�_ # 7t & 1(_{ , 1_~ )&p*(_)&

O1

$2(t+1, 1(_{ , 1_~ ))�

1$2(t, 1(_{ , 1_~ ))

&1

�_ # 7t & 1(_{ , 1_~ )&p*(_)&

\�1+\

�($2(t, 1(_{ , 1_~ ))��_ # 7t & 1(_{ , 1_~ )

&p*(_)&)

(12)

for all t�{, since, by the properties of 1_~ , $2(t, 1(_{ , 1_~ ))>0 for all t�{.By condition (i) of the Theorem, the per capita endowment of every

commodity is bounded at each node. Hence, for all t�{, 0<$2(t, 1(_{ ,1_~ ))�2KL �_ # 7t & 1(_{ , 1_~ )

&p*(_)&. Since the inequality holds for everyperiod, for all t�1 we have

\�1+2KL\

�

�\�1+\

�($2(t, 1(_{ , 1_~ ))��_ # 7t & 1(_{ , 1_~ )

&p*(_)&). (13)

Substituting then (13) into (12), and summing the inequalities we obtainfrom t={ to t=T, yields

O1

$2(T+1, 1(_{ , 1_~ ))+ :

T

t={

1�_ # 7t & 1(_{ , 1_~ )

&p*(_)&\�1+2KL\

�

�1

$2({, 1(_{ , 1_~ ))=

1$2(_{ , _{&1)

. (14)

The terms $2(_{ , _{&1) and $2(T+1), 1(_{ , 1_~ )) are, by construction,strictly positive; hence the series on the left-hand side of (14), being positiveand monotonically increasing, must converge.

Since _{ was chosen arbitrarily, the same is true for any other possiblechoice of _{ . Consequently, the above argument shows that the existence ofan improving allocation implies that there exists a sub-tree 1_~ such that forall nodes _{ # 1_~ ,

A(_{ , 1_~ ) := :�

t={

1�_ # 7t & 1(_{ , 1_~ )

&p*(_)&<�.


So if for every sub-tree 1_ , there is a node _� # 1_ for which the seriesA(_� , 1_) diverges, the allocation must be CPO. K

Proof of Corollary 1

For any sub-tree 1_ , and t�t(_),

:_̂ # 7t

&p*(_̂)&� :_̂ # 7t & 1_

&p*(_̂)&

O LimT � �

:T

t=t(_)

1� _̂ # 7t & 1_

&p*(_̂)&

� LimT � �

:T

t=1

1� _̂ # 7t

&p*(_̂)&& :

t(_)&1

t=1

1� _̂ # 7t

&p*(_̂)&. (15)

Since for any sub-tree 1_ , t(_) is finite, the last term in (15) is always afinite number. Consequently, the divergence of the first term on the righthand side of (15) implies the divergence of the left hand side. This in turnimplies the divergence of A(_� , 1_) for some _� # 1_ (since �_̂ # 7t & 1_

&p*(_̂)&��_̂ # 7t & 1(_� , 1_) &p*(_̂)&); we can then apply Theorem 3 to get theresult. K

Proof of Claim in Remark 3

Modifying Peled's notation slightly, let (R(_$; _))_$ # _+ be the vector ofrandom (real) rates of return on money, at node _, and (q(_$; _))_$ # _+ thevector of the supporting one-period contingent claim prices, the vectorbeing uniquely determined by the marginal rates of substitution of the onlyagent in generation _. At a monetary equilibrium, the present value,evaluated at _, of the return on money at node _$ # _+ is *(_$; _) :=R(_$; _) q(_; _)>0; hence, the present value of the one-period return onmoney is �_$ # _+ R(_$; _) q(_$; _)=�_$ # _+ *(_$; _)=1.

Peled [21] has shown that a sufficient condition for CPO, in additionto the requirement that all the numbers R(_$; _) lie in a compact subset ofthe set of positive real numbers, is that there exists =>0 such that, forevery path _� and every t,

`t

{=0

R(_�{+1 ; _�

{ )=> t

{=0 *(_�{+1 ; _�

{ )> t

{=0 q(_�{+1 ; _�

{ )�=.

As we noticed, there is a one-to-one relationship (already exploited inSection 4) between the system of one-period contingent prices [qss$] at allnodes and the system of Arrow�Debreu prices [ p(_)]; in particular,we have here > t

{=0 q(_�{+1 ; _�

{ )= p(_�t+1), so the above sufficient condi-

tion can also be written >t{=0 *(_�

{+1 , _�{ )�=p*(_�

t+1). Summing both


terms across the different first t+1 elements of all paths, we get�_�

t+1 # 7t+1> t

{=0 *(_�{+1 ; _�

{ )�=(�_�t+1 # 7t+1

p*(_�t+1)). The term on the

left hand side can also be written as �_t� # 7t

> t&1{=0 *(_�

{+1 ; _�{ )

[�_$ # _�t+1

*(_$; _�t )]; since �_$ # _+ *(_$; _)=1 for every _, iterating the

argument we get

:_�

t+1 # 7t+1

`t

{=0

*(_�{+1 ; _�

{ )= :_t

� # 7t

`t&1

{=0

*(_�{+1 ; _�

{ )= } } } =1.

This shows that the sufficient condition for CPO given by Peled indeedimplies 1�= �_ # 7t

p*(_), or equivalently 1��_ # 7tp*(_)�=>0.

Proof of Lemma 2

To ease the burden of notation, throughout the proof we write p and Qinstead of p* and Q*.

We first prove claim (ii). Consider the sum of prices over all nodesat a given date t such that the state at t is some prespecified s~ # S,�(_0 , s1 , ..., s~ ) # 7t

p(_0 , s1 , ..., s~ ), and denote this term by p(s~ , t). Recallingthat p(_0 , s1 , ..., st&1 , s~ )= p(_0 , s1 , ..., st&1) qst&1s~ , we get p(s~ , t)=�s # S p(s, t&1) qss~ . But then ( p(1, t), ..., p(S, t))$=( p(1, t&1), ...,p(S, t&1)) } Q, so that, iterating the argument,

( p(1, t), ..., p(S, t))$=( p(1, 1), ..., p(S, 1)) } Qt&1=q_0Qt&1,

where q_0is a row of the matrix Q corresponding to the state at the initial

date.31 But then

:_ # 7t

p(_)= :s # S

p(s, t)=( p(1, t), ..., p(S, t)) } 1S_1=q_0Qt&11S_1 ,

where 1S_1 is an S-dimensional vector whose elements are all equal to 1.Since Q is a strictly positive matrix, by Perron's Theorem (see, e.g., Horn

and Johnson [15, Theorem 8.2.8]) [*(Q)&1 Q] t � L as t � �, where L isalso a strictly positive matrix. So, for any =>0, there exists t(=) such that,for all t>t(=), =>&[*(Q)&1 Q]t&L& for any norm. Hence, the same istrue when we consider the difference of the two matrices componentwise,so that

Lij&=<[*(Q)&1 Q]tij<Lij+= for all (i, j) # S_S and t>t(=)

� L&=1S_S<[*(Q)&1 Q]t<L+=1S_S for all t>t(=),


31 The argument is very similar to the one yielding the conditional probability, for a Markovprocess with stationary transition probabilities, of the realization of a given state s at date t givenan arbitrary initial distribution (see, e.g., Doob [10, pp. 170�185]).

where 1S_S is a matrix with all elements equal to 1. Pre- and post-multi-plying respectively by q_0

and 1S_1 yields

q_0(L&=1S_S) 1S_1

<[q_0*(Q)&t Qt1S_1]<q_0

(L+=1S_S) 1S_1 for all t>t(=)

�1

q_0(L&=1S_S) 1S_1

1*(Q)t

>1

�_ # 7tp(_)

>1

q_0(L+=1S_S) 1S_1

1*(Q)t for all t>t(=)

so that (since q_0is strictly positive)

*(Q)�1 O :�

t=1

1�_ # 7t

p(_)diverges

and

*(Q)>1 O :�

t=1

1�_ # 7t

p(_)converges.

The validity of claim (i) is established next.By Perron's Theorem, since Q is strictly positive, there exists a vector

x # RS++ , &x&=1, such that Qx=*(Q) x (see, e.g., Horn and Johnson [15,

Theorem 8.2.2]). Define the weight function as follows:

*1_0(_0)=1 and *1_0

(_0 , s1 , ..., st)=qst&1st

xst

*(Q) xst&1

.

Note that *1_0( } ) will always take positive values and �st # S *1_0

(_0 , s1 , ..., st)=1 for all nodes (_0 , s1 , ..., st&1) since �s$ # S qss$ xs$=*(Q) xs

for all s # S.The associated map *� 1_0

( } ) satisfies

*� 1_0(_�

t )

p(_�t )

= `{=t

{=1

1qs{&1s{

qs{&1s{xs{

*(Q) xs{&1

=q_0s1

xs1} } } qst&1st

xst

q_0s1} } } qst&1 st

*(Q) x_0} } } *(Q) xst&1

=xst

(*(Q))t x_0

.


So we have �Tt=1 (*� 1_0

(_�t )�p(_�

t ))=�Tt=1 (xst

�(*(Q))t } x_0). Since

&x&=1, it follows that if *(Q)>1 then there exist a weight function, *1s0,

and a scalar A<� such that, for every path _�(1_0),

A(_�(1_0); (1_0

, *1_0)) := :

�

t=1

*� 1_0(_�

t )

p*(_�t )

�A.

But then the result in claim (i) follows easily by invoking the result in claim(ii). K

Proof of Theorem 4

The main part of the result follows almost immediately from Lemma 2,using Corollary 1 and Theorem 1.

If *(Q)�1, by Lemma 2(ii) we get in fact that ��t=1 (1��_ # 7t

p(_))diverges, hence CPO follows by Corollary 1.

On the other hand, if *(Q)>1, applying results (ii) and (i) of Lemma 2and then Theorem 2 we find that CPO necessarily fails.

What remains to be shown is that in the latter case (i.e., if *(Q)>1)there exists an alternative stationary allocation which is Pareto improving.A proof of this was already in Aiyagari and Peled [1] and Gottardi [14].We provide here a partly different argument which more closely parallelsthe ones of the results of the earlier sections.

Since Q is strictly positive, by Perron's Theorem we know there exists avector x # RS

++ , &x&=1, such that Qx=*(Q) x. Since *(Q)>1, we havethen Qx>x, i.e., �s$ # S qss$xs$&xs>0 for all s # S.

Define \� :=maxs # S, h # H \� s, h(2KH(1+S)), where \� s, h(2KH(1+S)) iswell defined and positive by Assumption 1 and Lemma 1; : :=mins # S[�s$ # S qss$xs$&xs], ; :=:�(\� maxs # S x2

s ). Note that :>0 and:;=\� ;2 maxs # S x2

s . Considering next the vector ;x we see that

:s$ # S

qss$(;xs$)&(;xs)

�; mins # S {:

s$

qss$xs$&xs==:;=\� ;2 maxs # S

x2s

O :s$ # S

qss$(;xs$)&(;xs)�\� (;xs)2 for all s # S

O :s$ # S

p*(_) qss$ (;xs$)&( p*(_) ;xs)�\�( p*(_) ;xs)

2

p*(_)for all s # S

(16)


Define x̂(s$, s; s, h� ) :=x*(s$, s; s, h� )+#;xs$ , x̂(s; s, h� ) :=x*(s; s, h� )&#;xs

for some h� and for s, s$ # S, where # # (0, 1] is suitably chosen32 so thatx̂(s, h� ) # R1+s

++ and &x̂(s, h� )&x*(s, h� )&�2KH(1+S) for all s # S. Setx̂(s, h)=x*(s, h) for all h{h� . So, we have constructed a feasible stationaryallocation for which, recalling again that ( p*(_0 , s1 , ..., st&1 , s)s # S)=( p*(_0 , s1 , ..., st&1) q*st&1s)s # S) and using (16), we see that

:s # S

p*(_0 , s1 , ..., st&1 , s)[x̂(s, st&1 ; st&1 , h� )&x*(s, st&1 ; st&1 , h� )]

+ p*(_0 , s1 , ..., st&1)[x̂(st&1 ; st&1 , h� )&x*(st&1 ; st&1 , h� )]

�\�( p*(_0 , s1 , ..., st&1)[x̂(st&1 ; st&1 , h� )&x*(st&1 ; st&1 , h� )])2

p*(_0 , s1 , ..., st&1)

for all (_0 , s1 , ..., st&1), t;

i.e., the bounded Gaussian curvature condition is satisfied for the membersof generations born at any t affected by the transfer. Noting that the initialold (of type h� ) receive a positive transfer, we conclude that the alternativestationary allocation x̂ is indeed CPO improving. K

Proof of Theorem 5

The result follows by a fairly immediate reformulation of the argumentof the proofs of Theorems 1 and 2 where the reformulation reflects the newform of the Gaussian curvature conditions for the economy with completespot markets and bonds. There is only one extra piece of argument in theproof of sufficiency which we need to add.

Note that if the non-vanishing Gaussian curvature condition holds forall agents at a given node, by taking averages across agents we get

:h _

1H

+� (_$, _, h)(�_$ # _+ +� (_$, _, h))

$2(_$, _, h)&�&$1(_)+\�

($1(_))2

&p� (_)&. (17)

Since both +� (_$, _, h) and $2(_$, _, h) are nonnegative, for all (_$, _, h), wealso have

H _:h

1H

+� (_$, _, h)(�_$ # _+ +� (_$, _, h))&_:

h

1H

$2(_$, _, h)&�:

h _1H

+� (_$, _, h)(�_$ # _+ +� (_$, _, h))

$2(_$, _, h)& .


32 Since x*(s, h� ) # R1+S++ , this is always possible.

Hence the inequality in (17) can also be rewritten as

H+� (_$) $2(_$, _)�&$1(_)+\�

($1(_))2

&p� (_)&.

The above expression is the same, except for the fact that H+� (_$) appearsinstead of 1�*1_~ (_$) , as condition (4) obtained in the proof of Theorem 1.The rest of the argument can then proceed as in the proof of Theorem 1,simply by replacing 1�*1_~ (_$) with H+� (_$). K

REFERENCES

1. S. R. Aiyagari and D. Peled, Dominant root characterization of Pareto optimality and theexistence of optimal monetary equilibria in stochastic OLG models, J. Econ. Theory 54(1991), 69�83.

2. Y. Balasko and K. Shell, The overlapping generations model. I. The case of pure exchangewithout money, J. Econ. Theory 23 (1980), 281�306.

3. L. M. Benveniste, A complete characterization of efficiency for a general accumulationmodel, J. Econ. Theory 12 (1976), 325�337.

4. A. Bose, ``Pareto Optimality and Efficient Capital Accumulation,'' Discussion Paper 74-4,University of Rochester.

5. J. L. Burke, Inactive transfer policies and efficiency in general overlapping generationseconomies, J. Math. Econ. 16 (1987), 201�222.

6. D. Cass, On capital overaccumulation in the aggregative neoclassical model of economicgrowth: A complete characterization, J. Econ. Theory 4 (1972), 200�223.

7. D. Cass, R. Green, and S. Spear, Stationary equilibria with incomplete markets andoverlapping generations, Int. Econ. Rev. 33 (1992), 495�512.

8. S. Chattopadhyay, Optimality in stochastic OLG economies with a continuum of states,mimeo, University of Alicante, 1999.

9. G. Debreu, ``The Theory of Value,'' Wiley, New York, 1959.10. J. L. Doob, ``Stochastic Processes,'' Wiley, New York, 1953.11. J. Dutta and H. M. Polemarchakis, Asset markets and equilibrium processes, Rev. Econ.

Stud. 57 (1990), 229�254.12. J. D. Geanakoplos and H. M. Polemarchakis, Overlapping generations, in ``Handbook

of Mathematical Economics'' (W. Hildenbrand and H. Sonnenschein, Eds.), Vol. IV,pp. 1898�1960, North-Holland, New York, 1991.

13. P. Gottardi, Stationary monetary equilibria in overlapping generations models withincomplete markets, J. Econ. Theory 71 (1996), 75�89.

14. P. Gottardi, Constrained efficiency of stationary monetary equilibria in overlappinggenerations models with incomplete markets, mimeo 1996.

15. R. A. Horn and C. R. Johnson, ``Matrix Analysis,'' Cambridge University Press, Cambridge,1985.

16. R. E. Lucas, Jr., Expectations and the neutrality of money, J. Econ. Theory 4 (1972),103�124.

17. R. Manuelli, Existence and optimality of currency equilibrium in stochastic overlappinggenerations models: The pure endowment case, J. Econ. Theory 51 (1990), 268�294.

18. A. Mas-Collel, ``The Theory of General Economic Equilibrium: A Differentiable Approach,''Cambridge Univ. Press, Cambridge, UK, 1985.


19. T. J. Muench, Optimality, the interaction of spot and futures markets, and the non-neutrality of money in the Lucas model, J. Econ. Theory 15 (1977), 325�344.

20. M. Okuno and I. Zilcha, On the efficiency of a competitive equilibrium in infinite horizonmonetary economies, Rev. Econ. Stud. 42 (1980), 797�807.

21. D. Peled, Informational diversity over time and the optimality of monetary equilibria,J. Econ. Theory 28 (1982), 255�274.

22. R. Radner, Existence of equilibrium of plans, prices, and price expectations in a sequenceof markets, Econometrica 40 (1972), 289�303.

23. S. Spear, Rational expectations in the overlapping generations model, J. Econ. Theory 35(1985), 251�275.

24. I. Zilcha, Characterizing efficiency in stochastic overlapping generations models, J. Econ.Theory 55 (1991), 1�16.


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