Markovian Equilibrium in In�nite HorizonEconomies with Incomplete Markets and

Public Policy1

Manjira DattaArizona State University

Leonard J. MirmanUniversity of Virginia

Olivier F. MorandUniversity of Connecticut

Kevin L. Re¤ettArizona State University

June 2004

1We thank Robert Becker, Alex Citanna, Jeremy Greenwood, Seppo Heikkila,Ken Judd, Tom Krebs, Michael Magill, Jianjun Miao, Martine Quinzii, ManuelSantos, Jean-Marie Viaene as well as seminar participants at the University of Con-necticut, CORE, Erasmus University, Stanford, Warwick and the 2004 NBER/NSFGeneral Equilibrium conference at UC-Davis for helpful discussions. Datta thanksCORE for its gracious support of her research during her stay in the Spring of2002. Re¤ett thanks the Tinbergen Institute at Erasmus University for its gra-cious support of his research during his stay in the Spring of 2002, as well as theDean�s Award for Excellence Summer Research Grant Program at ASU.

Abstract

We develop an order-theoretic geometrical approach to the problem of exis-tence, computation, and characterization of continuous Markov equilibriumfor a class of in�nite horizon multiagent competitive equilibrium models withcapital, aggregate risk, public policy, externalities, single sector production,and incomplete markets. The class of models is large and has been studiedextensively in the applied literature in public economics, macroeconomics,and �nancial economics, and our focus is not limited to symmetric Markovequilibrium.Our methods allow us to provide a sharp characterization of �xed point

constructions based on order continuous operators on compact subsets ofpositive continuous functions endowed with the partial order induced by thecone. Therefore, the paper produces interesting intersections of lattice the-oretic and topological �xed point theory. We construct monotone iterativeprocesses that in principle could be used as the basis for stable computationalalgorithms that construct a positive Markov equilibrium within an class ofmonotone Lipschitzian (and, therefore, Clarke di¤erentiable) Markov equilib-rium via successive approximation. In addition, we develop a new uniquenessproof within this restrictive class of Clarke-smooth Markovian equilibrium.This uniqueness proof weakens substantially previous arguments in the liter-ature. Finally, we are able to prove monotone comparison theorems withinthis restrictive class of Lipschitzian Markov equilibrium in Veinott�s strongset order on the space of public policy parameters.

1 Introduction

Over the last two decades, researchers have proposed a number of new theo-retical frameworks in the macroeconomics and �nancial economics literaturein an e¤ort to assess the quantitative role of behavioral heterogeneity andincomplete markets on both the positive and the normative aspects of equi-librium economic �uctuations. Such models have proven important for re-searchers attempting to study the interplay between public policy, incompletemarkets, income risk, asset pricing, production uncertainty in quantitativewelfare assessments of the costs of business cycles in such diverse �elds aspublic economics, �nancial economies, and macroeconomics. Aside from thefew exceptions that we discuss below, a majority of the characterizations ofMarkovian equilibrium for such economies has been numerical, and relat-ing �xed point constructions pertaining to the existence of equilibrium toarguments related to numerical computation have not been systematicallyaddressed. This paper bridges this gap for an important class of equilibriummodels of economic �uctuations with behavioral heterogeneity, incompletemarkets, public policy, and bounded production non-convexities.It is well known that from both a theoretical and applied perspective, the

characterization of some models with heterogeneous agents and incompletemarkets appear more tractible than others. One issue that is key for relativeto the question of tractibility centers around the number of assets allowedthe model. In models with many assets, little is known about concerningthe characterization of Markovian equilibrium with incomplete markest (es-pecially in situations where the second welfare theorem fails, production isnonconvex, and there is public policy).1 In models with a single asset, a bit

1One important exception to this remark is the work of Kubler and Schmedders [49].In this paper, the authors have addressed the issue of the existence of stationary Markovequilibrium in �nance economies with various asset structures (some including incompletemarkets and collateral constraints). There paper suggests the possibility of developing ageneral isotone iterative method based on "set-to-set" maps for computing Markov equilib-rium. Although such methods appear powerful and show potential for future applications,many details are yet to be explored. The preliminary work of Kubler and Schmedderssuggest possible implementations. Some of these implementations might be extended tomodels with more general production sectors and with public policy.In Re¤ett [59], their results are shown to be a special case of a more general monotone

and/or chain methods for Markovian equilibrium based on condensing maps that mapthe powersets of the space of endogeneous variables and wealth distributions into itself.Numerical methods for computing set-to-set maps could be developed based upon the

1

more is known. In this setting, there have been two central environmentsproposed that introduce behavioral heterogeneity into dynamic general equi-librium models in a tractible manner. On the one hand, researchers havestudied an important class of models commonly referred to as "Bewley mod-els"; models that can be identi�ed in particular with the pioneering work ofBewley [15]. A key feature of a "Bewley model" is that agents in the econ-omy face a continuum of uninsured agent speci�c income risks (that usuallyis assumed via a law of large numbers not to generate aggregate risk). Oftensuch version also introduce aggregate sources of risk to economy wide pro-duction processes, and assume that agents have at their access only a singleasset (e.g., capital or �at money) to self insure. Examples of these economiesin applied macroeconomics include the studies of Aiyagari [1] and Kruselland Smith [48].As Krusell and Smith [48] discuss, this class of Bewley models introduce

signi�cant computational problems that have yet to be resolved (e.g., the ap-proximation of a probability measure and existence of Markovian equilibriumon the so-called "natural state space.") Recently, some existence issues havebeen resolved in a paper by Miao [51] where he proves existence of stationaryMarkovian equilibrium in the Krusell and Smith [48] on an "expanded" statespace. His methods are surprisingly simple, and appeal to a straight-forwardapplication of a local convexity argument to the space of sequences of prob-ability measures. Miao�s construction is unfortunately non-constructive andrequires in general an expanded state space for the Markovian equilibrium.2

methods discussed in Rockafellar and Wets [61].2The approach of Miao [51] is quite simple, and reminiscent of the approach taken

to large anonymous games in the work of Bergin and Bernhardt [14] and Becker andZilcha [10]. The �xed point construction takes place on a space of probability measures,and has the following steps: (i) restrict attention to economies with shocks that satisfystandard Feller properties and there exists a version of the law of large numbers (in acontinuum indexed on the unit interval) where this no-aggregate uncertainty conditionimposed when modelling idiosyncratic ex post risk; (ii) posit a sequence of probabilitymeasures on a compact support that agents used to compute all future prices ; (iii) haveagents best respond to this sequence with appropriately continuous and measurable policyfunctions of aggregate states; (iv) aggregate agent decisions using the LLN showing thatthe aggregation procedure is consistent with the assumption a of compact state spacefor the support of these probability distributions; (v) given the Feller assumptions, showthat one is de�ning a continuous operator on the space of probability measures in a weaktopology on a compact convex set of probability measures, so then existence follows froma standard application of the Schauder-Tychono¤ theorem and (vi) if needed, use thisprocedure to construct payo¤ equivalent Markovian equilibrium expanding the "natural"

2

Because of the former point, Miao�s methods are never tied to computation.The latter issue is also a problem, as Miao�s work suggests problems using ofthe "natural" state space for the Markov equilibrium (where "natural statespace" consist of the state variables such as asset holdings, current shocks(idiosyncratic and/or aggregate), and summary measures of the distributionof states across agents as used in Krusell and Smith [48]). Therefore from acomputational perspective (and from the perspective of comparative analy-sis), theoretical results on the structure of Markovian equilibrium for suchmodels seem of limited use.3

An important second type of heterogeneous agent model (also havingmany of the same features that de�ne the previous class of so-called "Bewleymodels") has also been proposed in the literature over the last two decadesbeginning with Becker [6]. This class of models are often referred to as"Ramsey models", and have the advantage that their state variables canbe easily representated by a collection of functions. This is very appealingfrom a numerical perspective. In this style of model, households still face anuninsured income �uctuation problem, but not with respect to a continuumof random (idiosyncratic) risk. That is, typically these models assume a �nitenumber of households ex post. Recursive versions of these models have beenextended in deterministic settings by Becker [7][8] (see Becker and Boyd [9]for a review of this literature) and Townsend [70], and in stochastic settingsin Scheinkman and Weiss [66], Becker and Zilcha [10], Judd and Gasper [39],Judd, Kubler, and Schmedders [40][41], and Kubler, and Schmedders [49].In closely related work, a class of multisector stochastic incomplete mar-

state space to include current period value functions. The approach has little to say aboutthe structure of equilibrium policy functions.

3We should remark that important progress on order theoretic �xed point methodsin a subclass of economies considered by Miao has recently been provided in Mirman,Re¤ett, and Stachurski [55]. In this paper, under the assumptions of Becker and Zilcha[10] concerning the monotonicity of household income, the authors develop a new mixedmonotone map approach to the problem that computes the Markovian equilibrium in themodels of Aiyagari [1] and Krusell and Smith [48] using two steps: a �rst step that appliesa constructive version of Tarski to obtain extremal �xed points of a �rst step operator thatis isotone; and (ii) uses results on the �xed point structure from Amann [2] for normalcones and partially order Banach spaces to obtain a successive approximation algorithmthat computes a Markovian equilibrium. In that paper, the authors prove a much strongerexistence results in the space of policy functions (not probability measures associated withthe equilibrium policy function), and those functiosn are de�ned on the "natural" statespace as in Krusell and Smith [48].

3

kets models in the "Ramsey style" has also been developed recently in a seriesof interesting papers by Krebs [45][47]. In Kreb�s work, he proposes the useof a two sector incomplete markets economy with a �nite number of agentsthat have access to both physical and human capital, but face uninsuredincome risk. To keep his setup tractible, Kreb�s studies a very specializedCES class of economies with no public policy. In this setting, he does providea topological proof of existence of a Markovian equilibrium with no-trade.His results have been extended to more general settings using lattice pro-gramming methods and order theoretic �xed point constructions in Re¤ett[60].We should note researchers have yet to extend Euler equation-type meth-

ods that have been developed for homogeneous agent model within the so-called "monotone-map method" (e.g., the monotone methods of Bizer andJudd [16], Coleman [20], Greenwood and Hu¤man [31], Becker and Foias[11], Datta, Mirman and Re¤ett[24] and Mirman, Morand, and Re¤ett [54])to models with behavioral heterogeneity. Since the papers of Bizer and Judd[16] and Coleman [20], the so-called "monotone map" literature has devel-oped rigorous characterizations of Markovian equilibrium in homogeneousagent economies studying the �xed point structure of a class of monotonenonlinear operators.4 By using order based constructions in the spirit of thework of Tarski [68], this strand of literature has been able to not only es-tablish existence, but provide su¢ cient conditions for global uniqueness ofequilibrium within a very large class of continuous Markovian equilibrium,in addition to provides strong suggestions for computational methods for ac-tual applications.5 These monotone map methods have been extended usingpurely order based �xed point theory and lattice programming methods ap-plied to dynamic equilibrium models found is found in the work of Amir,Mirman, and Perkins [4], Amir [5]), and Mirman, Morand, and Re¤ett [54].6

4For an alternative approach to ours using monotone operators for deterministic Ram-sey equilibrium problems, see Becker and Foias [11]. The latter approach looks promising,although it is yet to be developed for stochastic Ramsey problems where more than oneagent in the economy can make interior savings decisions in all states. The relationshipbetween our methods and Becker and Foias�methods should be explored in future work.

5For other examples, see Coleman [22][21], Greenwood and Hu¤man [31], and Datta,Mirman, and Re¤ett [24].

6In smooth, strictly concave settings, lattice programming and pure lattice based �xedpoint theory are not required to obtain monotone characterizations of equilibrium. Forexample, in Coleman [20] a topological version of the lattice based �xed point theoremof Tarski [68] is available that exploits the continuity of the nonlinear operator and the

4

Unfortunately, none of these methods appear to apply directly to nonsym-metric equilibrium in multiagent settings.In this paper, we provide some important results for heterogeneous agent

economies using an integrated topological and order-theoretic approach. Weprovide a set of su¢ cient conditions under which there exist Markovian equi-librium in an equicontinuous set of functions. Further, as our methods areconstructive, we can provide theoretical algorithms that could prove veryuseful for computing extremal Markovian equilibrium. As the trajectoriesof the operator studied is shown to be monotone in interesting policy para-meters, we can then show how to conduct monotone comparative analysisin the sense of Milgrom and Shannon [53] on the entire equilibrium set forequicontinuous Markovian equilibria. Finally, we can then show that thecomputational methods employed here will converge to the unique equilib-rium.within the set of equicontinuous equilibrium. In this sense, this papercan be viewed as providing an additional level of geometric equivalence fora particular class of heterogeneous agent, single sector, incomplete marketseconomies with capital accumulation and inelastic labor supply. In futurework, we will attack the questions of elastic labor supply in these settings.The remainder of the paper is laid out as follows. In the second section of

the paper, we describe the model. The assumptions are standard in the lit-erature, and are somewhat more general than those in related previous work(e.g., in Becker and Zilcha [10])7. In the third section we construct ouroperator, prove existence of Markovian equilibrium decision processes anda basic monotone comparative analysis result in Veinott�s strong set order.In the fourth section of the paper, we provide a su¢ cient conditions underwhich there are unique Markovian equilibrium decision processes within ourequicontinuous class of Markovian equilibrium, and we discuss how our newapproach to uniqueness generalizes some results obtain in Coleman [22] and

equicontinuity of the equilibrium set of consumption and investment functions. (see forexample Amann [2]). In Mirman, Morand, and Re¤ett [54], no topological constructionsare used.

7One has to be careful when comparing the assumptions used here to those in Beckerand Zilcha [10]. In this latter paper, some of the author�s assumptions are needed toaddress technical issues not discussed here (e.g., measurability issues needed to provestationary Markovian equilibrium), while others are need to prove existence of strategicequilibrium (which is not the focus of this paper).We focus on the class of pure strategy competitive (open-loop) recursive competitive

equilibrium which is emphasized in the computational work of many papers in the macroliterature, e.g., Krusell and Smith [48].

5

Datta et al [24]. In particular, our new su¢ cient condition using results foroperators that are cone compressions. Of independent interest to our environ-ment, we discuss how our method can dispense with the strong monotonicityassumptions that underlies all previous work on uniqueness of Markovianequilibrium. In the last section of the paper we provide a discussion andconclude.

2 A Model Economy with Incomplete Mar-kets and Equilibrium Distortions

The model is similar to the multiagent stochastic version of the in�nite hori-zon growth model originally pioneered in the work of Scheinkman and Weiss[66] and Becker and Zilcha, [10]. In this setting, we amend the classic Ramseyequilibrium setting in Becker�s original work (e.g., [6]) by allowing for Markovshocks to both aggregate technologies and individual labor income streams,equilibrium distortions such as taxes or money, production nonconvexitiesin social returns. Labor income risk is uninsured. Our notion of uninsuredidiosyncratic income risk is related to the notion considered in Aiyagari [1],Krusell and Smith [48], and Mirman, Re¤ett, and Stachurski [55], and it isone example of an important type of income �uctuation problem originallyproposed in continuous time in Scheinkman and Weiss [66].8

Time is discrete and there are J types of in�nitely-lived household/�rmagents who each own capital and an endowment of a unit of labor in eachperiod, each type with a continuum of agents. Agents of the same type areidentical and are identi�ed by the subscript j = f1; 2; :::; Jg. For conve-nience, and without loss of generality, we normalize the mass of agents tobe the unit interval, and there are �j � 0 agents of type j,

PJj=1 �j = 1:

To abstract from the complications introduced by elastic labor supply, weassume that households do not have preferences over leisure, and thereforethey supply labor in �xed supply. Households in addition may face equilib-rim distortions associated with tax policies which following the discussion inGreenwood and Hu¤man [31] or Datta et al [24] we write down in reduced-form (although one could imagine monopolistic competition or valued �at

8A careful reading of the paper reveals that one can incorporate period-by-period unin-sured idiosyncratic shocks as in, for example, Krebs [47] without changing the structureof our arguments at all.

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money via binding cash-in-advance constraints as being the genesis of suchequilibrium distortions).There is a continuum of identical �rms, and each �rm is endowed with the

same production technology. This production technology contains aggregatestochastic technology shocks and aggregate production externalities. The�rms produce a single perishable good for sale in a competitive market, andrent capital and hire labor in competitive factor markets. Households cansmooth consumption by accumulating a single asset. This asset is assumedto be productive and we refer to it as capital.9

Aggregate uncertainty comes in the form of a vector of exogenous shocks� 2 �. The stochastic process of shocks is assumed to follow a �rst-orderMarkov process with stationary transition matrix �(�; d�0): To avoid sometechnical issues associated with measurability, we assume that � has a �nitenumber of states.10 Speci�cally, � = [�ind; �a] where �ind = [�1; :::; �J ] rep-resents the vector of type-speci�c shocks (�j is received by the householdsof type j), and �a is a vector of aggregate shocks to aggregate productiontechnology. For each period and state, the preferences for households of typej are represented by a period utility index uj(c), where the commodity spaceis assumed to be a positive interval in R+. Letting �t = (�1; :::; �t) denotethe history of the shocks until period t, the households lifetime preferencesare assumed to be additively separable and de�ned over in�nite sequencesindexed by dates and histories c = (c�t) and are given as,

Uj(�) = E0

( 1Xt=0

�jtuj(cjt)

)(1)

where the summation in the mathematical expectation in (1) is with respectto the probability structure of future histories of the shocks �t given thetransition matrix �, and �j 2 (0; 1): The assumptions on the period util-

9Although similar constructions can be made for the case of monetary economies andmodels with a bond (e.g., Bewley [15] or Aiyagari [1] and Scheinkman and Weiss [66]).10Our existence results do not depend on whether the state space for the exogenous

shocks are countable, although to deal with the uncountable case, we need to make some-what di¤erent arguments. See Mirman, Re¤ett, and Stachurski [55] for a discussion ofmixed monotone recursive methods. Issues concerning measurable equilibrium decisionrules can be dealt with using the methods discussed in Becker and Zilcha [10] and Hopen-hayn and Prescott [35]. We note, that these issue are critical when discussing the issue ofstationary Markovian equilibrium. As our work is oriented more toward construction ofequilibrium Markovian decision processes, we do not address these issues in this paper.

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ity function uj : R+ 7! R for j = 1; :::; J are standard for models whereinteriority is an issue, and are given as follows.

Assumption One:(i) Each utility function uj(c) is at least twice continuously di¤erentiable,

strictly increasing, strictly concave.(ii) The derivatives u

0j(c) satisfy the Inada conditions:

limc!0u0

j(c) =1 and limc!1u0

j(c) = 0:

Each household of type j is endowed with a unit of time and an initial stock ofcapital kj0 > 0:We assume that capital and labor are used in the productionof the output good according to a constant returns to scale technology whichallow for spillovers in capital and labor as in Romer [62]. We summarize thedistorted production technology by the continuous function f(k; n; K;N; �a),with f : K�[0; 1] � K�[0; 1] � �a ! R+ and where K �R+ is a compactinterval to be described below, and �a is compact, the total capital stockis K =

PJj=1 �jKj where the vector of capital stocks K = (K1; K2; :::; Kn)

represents the distribution of individual capital holdings across agent. Wemake the following standard assumptions on f :

Assumption Two: The production function is constant returns to scalein its �rst two arguments11 (i.e., homogenous of degree one in (k; n)) and:(i) f(0; 1; K; 1; �a) = 0 for all (K; �a) 2 K��a;(ii) f(k; 1; K; 1; �a) is continuous, increasing, twice continuously di¤er-

entiable, and strictly concave in k; K; and �a;(iii) f(k; 1; K; 1; �a) satis�es the standard Inada conditions in k for all

(K; �a); i.e.,

limk!0

fk(k; 1; K; 1; �a) = 1

limk!1

fk(k; 1; K; 1; �a) = 0;

(iv) there exists k(�a) > 0 such that f(k(�a); 1; k(�a); 1; �a)+(1��)k(�a) =k(�a) and f(k; 1; k; 1; �a) < k for all k > k(�a) for all �a 2 �a.11We could dispense with the constant returns to scale assumption in private inputs, as

in Greenwood and Hu¤man [31] to allow for decreasing returns to scale in (k; n). However,such models allow �rms to make non-zero pro�ts in equilibrium and organizational issuesof the �rm become important. Therefore, we assume there are constant returns to scalein private returns.

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In equilibrium, note that n = N = 1 since there is no preference forleaisure: Assumption Two is standard in the stochastic growth literaturewith representative agents and equilibrium distortions (c.f., Brock and Mir-man [18] and Greenwood and Hu¤man [31], and for a slightly di¤erent ver-sion Becker and Zilcha [10]). Assumption 2. iii is stronger than needed (seeColeman [20], yet we make it for convenience. Under assumption 2.iv, thereexists �k = sup�a k(�a) so that the state space for the aggregate capital stockand output can be taken to be in [0; �k]. Let K = [K1; :::; KJ ] 2 K =�n[0; �k], and denote the aggregate state space for the entire economy as S =[K; �] = [K1;:::;KJ ; �] 2 S = K � � Note that S is compact. Finally, sincethe individual household enters a period with stock kj, the state of an in-dividual household is sj = [kj; S]: Notice that given our assumptions, boththe aggregate and individual state spaces are compact sets.

3 Decision Problems and De�nition of Equi-librium

Firms hire capital and labor in competitive markets, and we let r(S) andw(S) represent the rental rate on capital and the wage rate, respectively.There technologies are allow to be non-convex (e.g., see Romer [62]). Giventhese prices, pro�t maximizing �rms solve the following standard optimiza-tion problem:

�(S) = maxx�0;n2[0;1]

f(x; n; K;N; �a)� rx� wn

Following Becker and Zilcha [10], we assume that each household owns anidentical share of the �rm. With constant returns to scale technologies, an-ticipating an equilibrium with x = K; and n = N = 1; a necessary conditionin equilibrium:

w(S) = f(K; �a)� f1(K; �a)K (2)

r(S) = f1(K; �a) (3)

where f(K; �a) = f(K; 1; K; 1; �a) is the output along an equilibrium path,f1(K; �a) = f1(K; 1; K; 1; �a); and w is the value of the household endowmentof a unit of labor.Households face an idiosyncratic income risk. Speci�cally, a household of

type j does not necessarily receive a single unit of e¤ective labor time, but,

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rather, the amount of �j > 0 units. To control the size of the state space, weassume that these e¢ ciency unit shocks do not generate aggregate risk forany state, i.e.,

PJj=1 �j�j = 1 in each state of the world, (so that, necessarily,

�j � 1 for all j). We denote C(S) to be the space of bounded, positivecontinuous functions de�ned on the compact S and with range RJ+ and weequip C(S) with the standard C0 uniform topology and the standard partialorder (component wise). C(S) is many things including (i) a completemetric space, (ii) a normal and minihedral cone of continuous functions,(iii) a Banach lattice under the sup metric topology (and endowed with thepointwise partial order induced by the cone). Denote the cross-section ofaggregate per capita capital stocks denoted K = [K1; :::; KJ ] can be used tocharacterize prices in the aggregate economy. We assume this distributionhave the following recursive representation:

K 0 = h(S);h 2 C(S)

We also allow the government to tax both capital and wage income. As-sume that these policy interventions have the following standard separableform:

�r(K; �a) = [1� �k(S)]f1(K; �a) and �w(K; �a) = [1� �n(S)]f2(K; �a)

where � = [�k; �n] 2 � is a continuous mapping S![0; 1] � [0; 1]. Weassume the dual partial order for a function lattice is placed on �; i.e, �0 �� if �0(S) � �(S) for all S: In addition we restrict attention to � in thede�nition of � which satisfy the following regularity conditions on the spaceof admissible distorted prices under consideration:Assumption Three: The vector of distortions � = [�k; �n] and f are such

that the distorted wage �w and the distorted rental rate �r satisfy the followingconditions:(i) �r is strictly decreasing in the vector K and such that:

limK!0

�r(K; �a)!1:

(ii) The distorted income for the household �w(K; �a) + K �r(K; �a) +J(K; �) is increasing in K for each �:

Here, J(K; �) is the lump-sum transfer that equals �n(K; �a)w(K; �a) +�k(K; �a)Kf1(K; �a): The assumption in (i) is similar to an assumption made

10

in both Becker and Zilcha [10] except that we do not require the distortedwage rate to be concave in the mean capital-labor ratio K as they do. Thisassumption is needed to get the best responses of agent of each type j toexhibit a particular form of monotonicity in the state variables. The as-sumption in (ii) is standard in work using monotone methods, and is foundin particular in the work on representative agent models in Coleman [20],Greenwood and Hu¤man [31], and Datta, Mirman, and Re¤ett [24]. No-tice that without policy induced distortions, assumption (ii) amounts tothe identical restriction on the size of the externality used in representativeagent models with production externalities as in Greenwood and Hu¤man[31]. Finally the Inada condition on �r is merely a convenience, and a weakerversion of this condition only imposing that there exist a state for capitalis always su¢ ciently marginally productive relative to the discount factor isalso available (see for example Coleman [20]).Each household of type j faces the following ex ante individual feasibility

constraint �(sj) :

cj + k0j = gj(kj; K; �); c; k

0 � 0

where gj(kj; K; �) = �j �w(K; �a)+(kj�K)�r(K; �a)+J(S); and J(S) is a con-tinuous function that describes the lump sum transfers from distortionarytaxes collected by the government: Notice that �(sj) is well behaved, andexpanding in (kj; K). In particular since gj is continuous in each of itsarguments, is a non-empty, compact and convex-valued, continuous corre-spondence.We can now state the decision problem for each household of type j under

assumptions One-Three. Given the state variable sj at the beginning of anyperiod, the decision problem for the individual of type j can be representedas the solution of the dynamic programming problem summarized by thefollowing Bellman equation:

v(sj) = sup(cj ;k0j)2�(sj)fuj(cj) + �jZ�

v(s0j)�(�; d�0)g (4)

Standard arguments appealing to the contraction mapping theorem showthe existence of a function v� 2 G that satis�es this functional equation,and it can also be established that the function v is increasing in (kj; K) andstrictly concave in its �rst argument (see as we have unbounded returns, seeDuran [29]). Further appealing to the arguments in Mirman and Zilcha [52],

11

the strict concavity of v� in its �rst argument also implies that the envelopetheorem is available, and v� is at least once di¤erentiable in kj for each j.We are now prepared to de�ne equilibrium.

De�nition 1 A (recursive) competitive equilibrium consists of functions �and C; a value function for the household v(sj) and the associated individualdecisions c�j(sj) and k

�0j (sj) such that: (i) given � and C, v

�j (sj) for each

household type j that satis�es the household�s Bellman equation for eachtype of agent j = f1; 2; :::; Jg; (ii) c�j(sj) and k�0j (sj) solves the right-handside optimization in the Bellman�s equation; (iii) all markets clear, that is,k�0(s) = �(S) = K 0; c�(s) = C(S) and c�j(sj) + k

�0j (sj) = gj(Kj; K; �) for

all j; (iv) the government budget is balanced, that is J(S) = �n(S)w(S) +�k(S)Kf1(K; �a)

4 Existence of Equilibrium

Prior to proving the existence of a continuous recursive competitive equilib-rium, we begin by de�ning a few terms to be used in the sequel.

4.1 De�nitions

Lattice. Consider a setX ordered with a re�exive, transitive, antisymmetricrelation denoted \ � ": We refer to X as a partially ordered set. An upper(lower) bound for a set B � X is an element xu(xl) 2 B such that 8x 2 B;x � xu (xl � x): The set X is a lattice if any two elements x and x0 in Xhave a least upper bound and a greatest lower bound (denoted respectivelyby x ^ x0 and x _ x0): A lattice is complete if for any subset of X, say B;there is a least upper bound and a greatest lower bound. The supremum ofB, if it exists, is a least upper bound. A chain C � X is a subset of X forwhich all pairs of elements are ordered. A subset B � X is a sublattice of Xif it contains the sup and the inf (with respect to X) of any pair of points inB: Notice that the product of a arbitrary collection of lattices equipped withthe product (coordinatewise) order is a lattice. An order interval is a subset[a; b] of an ordered topological space X with a < b that is closed in X.Let X be a partially ordered set and P (X) the power sets of X (i.e., the

collection of all subsets of X). Consider the following set orders introducedin Veinott [71]: (i) the strong set ordering on P (X)n;, denoted by �sis such

12

that B �s B0 if for each x 2 B and each x0 2 B0, x0 ^ x 2 B0 and x0 _ x 2 B;(ii) the pointwise strong set order on P (X)n;; denoted by �ss; is such thatB �ss B0 if for all x 2 B and x0 2 B0; x � x0: Notice that if X is a lattice, andP (X)n? ordered with the strong set order on X; (P (X)n?;�s) is a partiallyordered set.Vector Lattices and Banach Lattices. A partially ordered vector

space X is a real vector space equipped with a partial order� that is compat-ible with the following algebraic structure: (i) if x � x0; then x+z � x0+z; forall z 2 X; (ii) if x � x0; then �x � �x0 for all � � 0: The set X+ = fx 2 X,x � 0g is the cone ofX: A coneX+ is solid if the interior ofX+ is nonempty.Any partially ordered vector space which is a lattice is called a vector lattice.If the space has a norm k x kX which satis�es whenever j x j�j x0 j in X;k x k�k x0 k, we say X has a lattice norm: A complete normed vector spaceis a Banach space. A normed vector lattice is a vector lattice equipped witha lattice norm. A normed vector lattice X that is complete in the Cauchysense, and is endowed with a lattice norm is referred to as a Banach lattice.Mappings on a Lattice: We say a mapping A : X ! Y where X and

Y are partially ordered sets is isotone if for a; b in X, we have Aa � Ab whena � b: We say a mapping is antitone if Aa � Ab when b � a: A sequencefhn ! hg in H is order convergent if the exists two monotonic sequencesof elements from H, one decreasing fh#ng; and one increasing fh"ng, suchthat h = inf h#n = suph"n and h"n � hn � h#n: A necessary and su¢ cientcondition for a directed increasing sequence hn ! h to be order convergentis then for h = suphn: An operator Ah is order continuous on H if wheneverhn ! h in order, Ahn ! Ah in order. For example, if an operator A alongdirected subsets satis�es satis�es _A(hn) = A(_hn) (see Davey and Priestley[23], chapters 2 and 3 for an excellent discussion).With this investment in terminology, we are now ready to study the

equilibrium problem.

4.2 Construction of Equilibrium

Appealing to now standard arguments, the Euler equation associated withthe right side of the Bellman equation (3) above can be rewritten for each

13

agent j as:

u0

j(cj(sj)) � �j

Z�

u0

j(cj(s0

j))�r(K0; �0a)�(�; d�

0);

= if k0

j > 0 (5)

where here the 0 notation on a variable refers to next period value of theparticular variable, and using constant returns to scale we have k

0j = �j(1�

�n(S)[f(K; �a)�Kf1(K; �a)]+kj(1��k(S))f1(K; �a)+J(S)�cj = gj(kj; K; �)�cj. In a recursive competitive equilibrium, a candidate consumption functionfor the household of type j necessarily satis�es cj(sj) = Cj(S) pointwise, sothat K

0j = gj(K; �) � Cj(S) = h(K; �) 2 C(S) where in gj now we impose

the government budget condition that J(S) = �n(S)w(S)+�k(S)Kf1(K; �a).As this is true for all j = 1; :::; J , we can re-write the J Euler inequalities asa system of integral equations, that must be satis�ed in equilibrium by thevector of investment functions i = i(K; �), as follows:

u0

j(gj � ij) � �

Z�

u0

j(gj(i; �0)� ij(i; �0))�r(i; �0a))�(�; d�0);

= if ij > 0 for j = 1; :::; J (6)

where the notation g � i = [g1 � i1; ::::; gJ � iJ ] where g = g(K;K; �) =g(K; �) is equilibrium income for the household (and we implicitly imposethe government budget constraint on each agent in the de�nition of g.)We use this system of equations to implicitly de�ne a nonlinear operator

in the space of equilibrium investment decisions, denoted by A, for which thepositive �xed point such that the vector of consumptions c = [c1; :::; cJ ]� 0is an equilibrium. The operator A is de�ned on a space H of candidateinvestment equilibrium functions h : S ! RJ+ satisfying the following condi-tions:(i) h = [h1; :::; hJ ] continuous;(ii) hj 2 [0; gj] for all j; h(0; �) = 0(iii) 0 � hj(K 0; �)�hj(K; �) � gj(K 0; �)�gj(K; �) for all j when K 0 � K

for each � 2 �:That is, we shall seek recursive equilibrium on a very simple set of func-

tions: namely a space consisting of continuous savings functions such thatboth consumption and savings are increasing in distribution of wealth forthe aggregate economy. Of course, in this environment we suspect there are

14

many other Markovian equilibrium decisions processes. We seek to provide astable monotone algorithm for computing a subclass of this potentially verylarge set. Also, notice that as gj is a continuous and di¤erentiable functionde�ned on a compact set, gj is therefore Lipschitzian for each agent j (andtherefore Clarke di¤erentiable). We extend a result in Montrucchio [56] toan environment with many agents, production nonconvexities, and incom-plete markets. We also remark that, by a standard argument, H is a closedequicontinuous subset of functions in the positive cone of continuous func-tions ordered by the pointwise Euclidean order. It is compact in an topology�ner than the interval topology, and therefore, by Frink�s theorem, H can beshown to be subcomplete in the relative order on C(S).More speci�cally, letC(S) denote the cone of positive continuous maps h :

S ! RJ endowed with the standard C0 uniform topology and the pointwisepartial order induced by the cone.

Lemma 1 H is a convex, complete sublattice of the Banach lattice of con-tinuous functions C(S). H is also a compact order interval of Lipschitzcontinuous functions in C(S).

Proof: Convexity and the fact thatH is a closed sublattice of the Banachlattice of continuous functions when equipped with the standard pointwisepartial order on C(S) is obvious. H is also clearly an order interval.For compactness, simply note thatH is an equicontinuous set of functions

de�ned on a compact space under the restriction that g�h is increasing inK,� is discrete, and g�h a continuous function on a compact set (and thereforethe variation in h is uniformily bounded by a uniformily continuous functiong). Therefore, and by a standard application of the Arzela-Ascoli theorem,His relatively compact in C(S). As H is already closed in C(S), H is compactin C(S). As g is Lipschitz in K we therefore conclude that H consists ofLipschitz continuous functions.Finally as H is a closed sublattice in C(S) that is also equicontinuous

in the C0 uniform topology, H is compact in the interval topology, andtherefore H is a complete sublattice by the complete characterization ofcomplete lattices in the order topology by Frink [30] and Birkho¤ [17], so forany subset of H, say H � H; supK;�H = hu(K; �) and infK;�H = hl(K; �)are in H.�

15

Letting u0(g � i) = [u01(g1 � i1); ::::; u

0J(gJ � iJ)]T denote the vector of

marginal utilities, the system of J Euler equations can be written more com-pactly as follows: u

0(g� i) = �

R�u0(g(i; �0)� i(i; �0))�r(i; �0))�(�; d�0): De�ne

in the obvious vector notation the mapping Z : H�X�K��! �RJ whereX � K � RJ

+

Z(x;K; �; h) = 2(x)�1(x;K; �; h) (7)

1 = �

Z�

u0(g(x; �0)� h(x; �0))�r(x; �0)�(�; d�0);

2 = u0(g � x);

De�ne then the following nonlinear operator A : H!H0 as follows:

Ah(K; �) = fx(K; �; h) 2 x�(K; �; h)j 0 < h < g; x isotone in K given �; hg= f0 elsewhereg:

Here,

x�(K; �; h) = fxjZj(x;K; �; h) � 0; and Zj(x;K; �; h) = 0 if (gj � xj) > 0g:

Recalling that a compact operator is an operator that maps boundedsubsets in the domain into relatively compact subsets in the range, and acompletely continuous operator is a compact operator that is also continuous,we can now prove a key proposition useful in the subsequent proofs of themain theorems of the paper.

Proposition 2 Under Assumptions One-Three, (i) for any h 2 H and(K; �) 2 S, Ah(K; �) is well-de�ned;(ii) Ah(K; �) � H; (iii) Ah and g�Ahare strictly increasing in K, each �; (iv) Ah is a continuous and compactoperator (e.g., completely continuous) on H in the uniform topology on H;(v) Ah is isotone on H; (vi) Ah is order-continuous on H when H equippedwith the standard pointwise partial order.

Proof: See appendix.�Since H is a non-empty, compact and convex order interval in the space of

continuous functions onK��, andA is a continuous and compact operator, a

16

simple topological argument for the existence of Markovian equilibrum mightappear to be available. For example, Schauder�s theorem guarantees thatexistence of a �xed point for such an A (see, Hutson and Pym [37], p 208,Theorem 8.2.3.). In our case, this approach is not useful from the point ofestablishing existence of equilibrium for we already know A has a trivial �xedpoint (namely the zero consumption plan), and it proves di¢ cult to associatea (strictly concave) value function. An alternative approach would be tofollow Becker and Zilcha [10] and Miao [51] and de�ne an operator associatedin the space of probability measures associated with the on the optimal policyfunctions of agents (aggregated), and then set up an application of a localconvexity �xed point (e.g. Schauder�s theorem, or Fan-Glicksberg) on suchan operator. This approach is very powerful for existence, but unfortunatelyis di¢ cult to relate to applied work using numerical methods. Further, itdoes not allow one to conduct monotone comparative analysis on the spaceof economies, and in particular the space of admissible policy parameters.(e.g., monotone comparative analysis in the tax rate �k(K; �)):We we do notfollow this approach either.We follow an alternative approach. Rather than trying to rede�ne the

domain of A so as to exclude the trivial �xed point, we follow the suggestion�rst made in Coleman [20]. We exploit the monotonicity of A combined withstrong order continuity properties of our operator to apply a strong versionof Tarski�s theorem which both (i) proves existence of a complete latticeof �xed pints, and (ii) allows use to computing the extremal �xed pointsby successive approximation on countable indexations of iterations of ouroperatorAh o¤the endpoints of the space. Tarski�s theorem (e.g., Tarski [68],Theorem 1) guarantees that the set of �xed points of our increasing mappingA from a complete lattice into itself has a non-empty complete lattice H of�xed points:Using the underlying order continuity of the problem, we canthen apply a version of Tarski�s theorem due to Kantorovich to establishthe success of a successive approximation algorithm in converging to theminimal and maximal �xed points. Further, we show that in the �xedpoint correspondence for this operator, we must have a "positive" Markovianequilibrium. In a multiagent setting, this requires two things (i) all agentsto have strictly positive (�uctuating) consumption; (ii) at least one agenthaving strictly positive capital holdings.Finally, because of the constructive approach taken here, we can also

consider the issue of monotone comparative analysis on a set of deep policyparameters. In particular, using the strong set order suggested in Veinott [71]

17

for the equilibrium set, we can compare the equilibrium set in a parameterprior to establishing uniqueness within the class of Markovian equilibriumwe consider. But actually, we do more. Using an argument new argumentfor uniqueness, we weaken the su¢ cient conditions in the existing literature(e.g. Coleman [22] and Datta, Mirman, and Re¤ett [24]) for uniqueness ofMarkovian equilibrium within our equicontinuous subset of equilibrium map-pings. The su¢ cient conditions to apply our new method in the multiagentcontext are strong, but can be shown in the homogeneous agent setting to beweak relative to existing su¢ cient conditions. We should also note that in amultiagent incomplete markets setting, this uniqueness argument should beinterpreted as a global stability argument for monotone iterative methods,and does not estiblish uniqueness relative to a broad class of discontinuousMarkovian equilibrium (as in the case of the homogeneous agent economy).Therefore it could prove very useful in numerical implementations of our �xedpoint operator relative to the class of continuous Markovian equilibrium.In the rest of the paper we will consider the standard partial order on

the lattice C(S) de�ned as follows: h0 � h if h0(S) � h(S) for all S 2 S.In this partial order, it is straightforward to verify that H is a sublattice inC(S):We now consider the application Knaster-Tarski�s �xed-point theoremto the operator A relative to the space H: We denote the �xed points ofthe operator Ah by E(�), and as we will be interested eventually in howthis correspondence changes with perturbations of �; we make explicit thedependence of E on the policy distortions �.The application of the Knaster-Tarski theorem we apply is due to Kan-

torovich (see for proof for example, Vulikh [72], Lemma XXII.2.1, p 337).The theorem states that for a isotone operator A that is order continuouson sequences, mapping a non-empty complete lattice H into itself for whichthere exists some element �h 2 H which has A�h � �h has nonempty set of �xedpoints. In addition, the sequence fAn�hg converges by successive approxi-mation to the maximal �xed point h�u = supE(�):

12 Using a dual argument,you we can obtain similar results for the minimal �xed point h�l : In ourproblem, h�u = g is a (trivial) maximal �xed point, and is not decentralizableas a competitive equilibrium. So in our work, we will focus on the minimal�xed point (of which as long as h�l 2 E(�); h�l such that g � h�l >> 0; h�l is12Note that there are other �xed point arguments would deliver the same successive

approximation results. For example, see Amann [3] and Heikkila and Lakshmikanthan[34], chapter 1.

18

an Markovian equilibrium decision process).

Theorem 3 Under Assumptions One-Three for any given � 2 �, (i) the setof �xed points of Ah, namely E(�);is nonempty complete lattice; (ii) amongthe set of �xed points of A, there exists a maximal (resp, minimal) �xed pointh�u(�) = supE(�) (resp; h

�l (�) = inf E(�)) such that limnA

n(g) = h�u(�) = g(resp;limn!1An(0) = h�l (�) � 0) and the convergence is uniform. (iii) theminimal �xed point h�l (�) = supE(�) is such that at least for one index j 2 J;gj � h�jl(�) > 0; (iv) E(�k) is isotone on ( �k;�d) in the strong set order�sin �k where �k ={�kj(�k; �n) 2 �g and �d is the dual standard pointwisepartial order on �k; and supE(�) and inf E(�) form isotone selections.

Proof: (i) The set H is a nonempty complete lattice by the �rst lemmain section 4, and the operator A is a isotone self map on H: Therefore bythe main theorem in Tarski ( [68], Theorem 1), E(�) is nonempty completelattice for each �.(ii) First we note that Ah by Proposition 2(ii), A(H1) = fAhjh 2 H1 �

[0; g]g is relatively compact as Ah is a compact operator (e.g., A mapsbounded subsets of [0; g] into relatively compact sets of the order interval[0; g]); by Proposition 2(iii) Ah is completely continuous; and by Propostion2(iv) Ah is order continuous. That A(g) = g is by de�nition the maximal�xed point. By the lemma above in Vulikh [72], successive approximationslimnA

n0 converges to a minimal �xed point h�l = inf E(�) � 0. The uni-form convergence follow to the minimal �xed point follows from Amann ([2],Corollary 6.2) noting the operator is completely continuous, fAn(0)g is anequicontinuous sequence and compact and S and H are compact, and there-fore convergence to h�l is uniform.(iii) The minimal �xed point h�l 2 E(�) is such that g � h�l > 0 follows

from an obvious modi�cation of the main theorem in Greenwood and Hu¤-man [31], where An0 is the optimal plan for the economic agents n periodsfrom her terminal date (along an equilibrium trajectory where k = K), T nJi0is the value function associated with the Bellman operator for agent i, withJi0 = 0, and the Inada condition in assumption 3 implying that A(0) 6= 0:(iv): First we note that E(�) is complete lattice valued for each �: Let

�0 � � with �0k � �k; �0n = �n: Notice that Ag = g for all � 2 �: From the

de�nition of Z in (7), when �0k � �k; Z(�

0k;h;Ah(�k;K; �); K; �) � Z(�k;h;

Ah(�k;K; �); K; �) = 0 where now we make the dependence of Ah on �kexplicit by adding it as an argument to Z in (7). Then Ah(�

0k;K; �) �

19

Ah(�k;K; �): By an argument Topkis ([69], Theorem 2.5.2) E(�) must beincreasing in the strong set order, i.e., E(�

0k) �s E(�k) with supE(�) and

inf E(�) forming monotone selections�Our work can be related to the important �ndings on determinancy of

equilibrium of Kehoe, Romer, and Woodford [43] and Santos [63]. A so calledNegishi problem is constructed, which is a modi�ed planning problem thatcan be used to characterize the equilibrium manifold, and a su¢ cient condi-tion for establishing comparative analysis results is that the value functionbe C2:13 In contrast, we take an order based approach to comparative analy-sis, and now consider the question of global uniqueness within the class ofequicontinuous Markovian equilibrium where both investment and consump-tion decisions are monotone.

5 Computation of Equilibrium via a Succes-sive Approximation Algorithm

When applying numerical methods to this problem, we assume that the re-searcher applies the approximation methods to a set of basis functions thatare known to deliver asymptotically consistent approximations for an un-known element of a equicontinuous collection functions on a compact subsetof Rn (in particular, for any unknown element h 2 H(S)). That is, we as-sume that the basis elements are chosen such that there exists an sequenceof approximations hn 2 Hn such that hn ! h 2 H:uniformly as n ! 1,n 2 N;N the set of natural numbers. In our case, our nonlinear �xedpoint operator A de�ned in (7) can be shown to be both topologically andorder continuous. Relative to topological continuity, in particular we canprove that the trajectory fAn(0)g forms an equicontinuous seequence in H;it can therefore it can be shown that an iterative projection methods basedupon Ahn ! Ah for suitable Hn (e.g., piece-wise linear as in Coleman [19])will have the desired convergence properties from an initial function hn0 = 0):Therefore asymptotic convergence issues for successive approximations of ex-tremal �xed points are straightforward, and can be addressed with variousapproaches as in the work of Judd (e.g, [38]).An issue we address in this section concerns the characterization of the

limits arbitrary trajectories of iterative procedures based upon the operator

13Santos [63] provides a set of su¢ cient conditions.

20

A. In Theorem 3, we provide a strong characterization concerning the com-putation of extremal �xed points. That is, the limit of monotone iterationslimnA

n(0) is stable for the minimal �xed point: In this next section, we iden-tify additional conditions that sharpen the characterization of the stabilityof Ah from any initial h0 2 H; h0 < g: That is, we characterize the limitlimnA

n(h0) for any h0 2 H; h0 < g . For this, we require an additionalinteriority condition, namely A(0) > 0: We will refer to this as the"strong"interiority case.In theorem 4, we �rst provide a complete characterization of strictly in-

terior Markovian equilibrium within the class H. We then address the issueof uniqueness and stability of monotone iterative procedures based uponAh within the class H. It is especially interesting that in the homogeneousagent (nonoptimal) case, we can sharpen the uniqueness result reported inthe monotone-map literature (e.g., Coleman [20][22], Datta et al [24], andMorand and Re¤ett [58]). We discuss this after the proof of the theorem. Weshould mention that even in the multiagent context, this uniqueness/stabilityresult is not vacuous, as one simple example of such an economy where thesu¢ cient conditions are apparently met is the two agent example studied inBecker and Zilcha [10]. We develop a new proof of uniqueness within theclass H using the following results in Guo and Lakshminkantham [33]: (i)a strongly sublinear operator on a solid cone is a e-concave operator ([33],Theorem 2.2.1); (ii) an isotone e-concave operator on a solid cone has atmost one strictly positive �xed point ([33], Theorem 2.2.2), and (iii) a in-creasing completely continuous e-concave mapping A has a strictly positive�xed point if and only if A is a cone compression ([33], p65).14

Theorem 4 Under Assumptions One-Three, for � 2 �; h0 < g, h0 2 H; (i)A is a completely continuous, isotone, and e-concave operator with a �xedpoint correspondence E(�) with a strictly positive element if and only if A:is acone compression. Further, (ii) if A(0) > 0; then minE(�) = h�l = limnA

nh0> 0 is the unique strictly interior Markovian equilibrium within the class of

14Let P be a solid cone. An operator A : P! P is strongly sublinear if for a strictlyinterior m 2 P; and for any t 2 (0; 1) we have Atm > tAm for all (K; �):An operator Ah : P! P is a cone compression on the cone P if their exists an R; r > 0

such that

Ah � h; for h 2 P;khk < r; h 6= 0;Ah � (1 + �)h; for h 2 P; khk > R; � > 0:

21

equicontinuous Markovian equilibrium in H; i.e., h�l is a globally stable �xedpoint of Ah relative to the set Hng.

Proof of Theorem: We �rst prove (ii), i.e., that Ah has at most one singlestrictly interior �xed point in H. A direct implication of this (noting thatA(g) = g by de�nition) is that we show is that if h0 < g; h0 2 H; the minimal�xed point h�l = limnA

nh0 > 0: The strategy of our proof is somewhatindirect, and it shows that the �xed point of A coincide with the �xed pointsof another operator, denoted A; which is shown to have at most one �xedpoint.The operator bA will be de�ned on a space M of functions m : S ! RJ+

satisfying the following conditions:(i) m(K; �) = [m1(K; �); :::;mJ(K; �)] withmj : K

J��!KJ continuousfor all j;(ii) for K > 0; 0 � mj(K; �) � 1

u0j(gi(K;K;�))=mu for all j;

(iii) mj(0; �) = 0 for all j and all �;(iv) �r(K0;�)

mj(K0�) <�r(K;�)mj(K;�)

when K 0 > K; with weak inequality when K 0 � K:For anym 2M, we de�ne the functionH(m;K; �) = [H1(m;K; �); :::; HJ(m;K; �)]

implicitly in the following system of n equations:

u0(Hj(m(K; �))) =1

mj(K; �)for m > 0

Hj(m;K; �) = 0 for mj = 0: (8)

which under our concavity, continuity, and monotonicity assumptions em-bodied in Assumptions One-Three can be solved globally for a unique inverse(using the inverse function theorem along with the fact that all mappings areproper). Notice, this mapping Hj can be de�ned if a particular agent j isborrowing constrained, i.e., Hj = gj: If we substitute for Ah and h into(7) using (8), we can obtain an equivalent representation of the �xed pointcorrespondence for Ah in terms of a new operator Am: We can then ex-ploit the geometric properties of Am to characterize E(�); the �xed pointcorrespondence for Ah:To begin our argument, �rst note that if A(0) > 0; then recalling mu =

supM; we have Amu < mu: Therefore, the indicator variables in (7) are all 0.Under the strict concavity and boundary assumptions on ui(c) in assumptionOne, by a standard inverse function argument relative to the de�nition of Hin (8), we ha ve H well de�ned for each m(K; �) 2 M. In particular, using

22

the de�nition of Ah (which is a monotone selection in the correspondencex�(t) as discussed in the proof of Proposition 2 in the appendix), noting that"today�s" consumption in (7) is de�ned using H = g�Ah; and "tomorrow�s"consumption H 0 = g � h; we can rewrite 1 and 2 in (7) using (8) and thede�nitions of (Ah; h) as follows:

1 = �

Z�

u0(g(x; �0)� h(x; �0))�r(x; �0)�(�; d�0)

= �

Z�

r(g �H( ~m(K; �)); �0)m(g �H( ~m(K; �)); �0)�(�; d�

0);

2 =1

~m:

Z(x;K; �; h) in (7) then becomes

Z(m; ~m;K; �) =1

~m� �

Z�

r(g �H( ~m(K; �)); �0)m(g �H( ~m(K; �)); �0)�(�; d�

0); (9)

where we now have a new operator bA de�ned implicitly as follows:A(m) = f ~m j Z(m; ~m;K; �) = 0 for m > 0; 0 elsewhereg:

In (8), we observe m = 1u0(g�Ah) when h < g: Notice m de�ned above by

de�nition satis�es u0(g(m(K; �))� Ah(m(K; �))) = 1mj(K;�)

where h > 0.

Under assumptions One-Three and notingm 2M, Z is strictly increasingin m, and strictly decreasing in ~m: In addition, for �xed m > 0; K > 0 and�; ~m = 1

u0(g(K;�)�Ah(K;�)) ! 0 implies that Z ! +1, and ~m ! 1u0(g(K;�)) ;

implies that Z ! 1: Consequently, by following a Am for each m > 0;K > 0; and �: Note that Am = 0; elsewhere.The operator bA is related to Ac = g�Ah in the following manner (where

for the moment, we write the operator constructed in the Proposition in termsof consumption.15 To any orbit of the operator Ac (i.e., any sequence Acnh015In this proof, it is clearer, from a notational perspective, to write the operator in

the proof in terms of consumption, not investment. That is why we use the notaton ofAc = g � A; where A is the operator in our Proposition. We are looking for Markovianequilibrium where investment is such that consumption is also monotone (in this notation,we seek Ah such that Ach is increasing in K).

23

for h 2 H in the proposition) corresponds an orbit of the operator bA; whichcan be obtained through the following construction. Given any h0 2 H ,we have m0 =

1u0(h0)

2 M. Equivalently, m0 satis�es H(m0) = h0: By

construction, there exists a unique bAm0 that satis�es bZ(m0; bAm0; K; �) = 0;that is,

1bAm0(K; �)= �E�

[r(g(K; �)�H( bAm0(K; �)); �0)

m0(g(K; �)�H( bAm0(K; �)); �0);

or, equivalently (from the de�nition of h0),

1bAm0(K; �)= �E� u

0(h0(g(K; �)�H( bAm0(K; �)); �0))�r(g(K; �)�H( bAm0(K; �)); �

0):

By construction, Ah0 satis�es,

u0((Ach0)(K; �)) = �E�u0(h0(g(K; z)�Ach0(K; �); �0))�r(g(K; �)�Ach0(K; �); �0):

Therefore, by uniqueness of bAm0 it must be that 1= bAm0 = u0(Ach0) (or,

equivalently, that H( bAm0) = Ach0 = g � Ah0). By induction, we have for

all n = 1; 2; :::; Anh0 = H( bAnm0). It is also true that any �xed point of Ac

(and therefore A) corresponds a �xed point of bA. Indeed, for any h� �xedpoint of A, de�ne m� = 1

u0(h�) (or, equivalently, H(m�) = h�). By de�nition

h� satis�es (using the de�nition that Ac = g � A,

u0(g � h�(K; �)) = �E�u0(g(h�(K; �); �0))� h�(h�(K; �); �0))r(h�(K; �); �0);

or, equivalently,

1=m�(K; �) = �E�[r(g(K; �)�H(m�(K; �)); �0)]

[m�(g(K; z)�H(m�(K; �)); �0)];

using the fact that Ac = g � A; with m� a �xed point of bA.First we note that by an obvious modi�cation of the arguments in Propo-

sition 2, Am is is (i) well-de�ned (as in Proposition 2), (ii) isotone on M(since Z is increasing in m, and decreasing in ~m; Am0 � Am so that bA ismonotone), and (iii) Am � M (from a similar proof to that in Proposition2): We also note that A is strongly sublinear operator.To establish the strong sublinearity of A, notice as bZ is decreasing in

its second argument. Therefore, a su¢ cient condition for Atm > tAm fort 2 (0; 1) and m strictly positive, is

Z(tm; tAm;K; �) > Z(tm; Atm;K; �) = 0;

24

which is true since m 2M; m > 0; m such that rmis decreasing in K

Z(tm; tAm;K; �) =1

~m� �

Z�

�r(g �H(t ~m); �0)m(g �H(t ~m); �0)�(�; d�

0) > 0:

As Am is a strongly sublinear operator on a compact subset of solid cone ofpositive continuous functions, Am is an e-concave operator ([33], Theorem2.2.1). As further Am is an isotone operator (in addition to being e-concaveoperator on a solid cone), therefore Am has at most one strictly positive �xedpoint ([33], Theorem 2.2.2). That proves (ii).

By a theorem in Guo and Lakshmikantham [33], p65), Am is an increas-ing completely continuous e-concave mapping A has a strictly positive �xedpoint if and only if A is a cone compression ([33], p65). An operator is com-pletely continuous if it is continuous and compact. By a simple modi�cationof a result in Coleman ([20], Theorem 5), as K�� is compact, noting Mis equicontinuous and closed (and therefore compact in the uniform metrictopology on continuous functions), as we have Amn ! Am pointwise for all(K; �), this convergence is uniform on K��, and therefore A is a contin-uous operator on M. That A is a compact operator follows from the factthe AM = fAmjm 2 Mg � M; M compact. therefore Am is completelycontinuous. As we have already proven Am is isotone and e-concave, by theaforementioned result above, Am is necessarily a cone compression. Thisproves (i). �We have several remarks. First in the case of CES preferences, we never

need to work with Am: That is, if A(0) > 0; as u0(tm) = u0(t)u0(m); then asin Coleman [20] and Datta, Mirman, and Re¤ett [24], one can directly showAh is a strongly sublinear operator. In these papers, one can show that thenonlinear �xed point operators have unique strictly positive Markov equilib-rium in a compact subset of a solid cone of positive continuous functions usinga nonlinear �xed point operator that can be shown to be completely contin-uous, e-concave, and isotone. Our result below say this result is equivalentto their operator being a cone compression. As isotonicity and e-concavity issu¢ cient for uniqueness of strictly positive �xed points, therefore the strongmonotonicity assumption in the hypotheses of their uniqueness result (e.g.,k0�monotonicity) is not necessary.16 Coleman ([20], Theorem 8) holds un-der more general conditions than our uniqueness result, based on strongly

16For the de�nition of a pseudo-concave operator A, see the de�nition of a stronglysublinear operator in footnote 14, but eliminate the requirement that domain of the op-

25

sublinear mappings on a solid cone, but it is su¢ cient for uniqueness to onlycheck the strongly sublinear structure of the nonlinear �xed point operator(as the domain/range for the �xed point problem is a compact order intervalin a solid cone.)Notice also that because of the cone compression characterization of the

existing of strictly positive �xed points, we can provide a geometric intre-pretation for the necessity of the hypothesis A(0) > 0 for the existence of astrictly positive �xed point (let alone a unique strictly positive �xed point).If A(0) � 0; then Ah is not e-concave. That is, consider Am when A(0) � 0(and therefore Amu � mu with strict inequality for necessarily only for oneof equations j 2 J). In this case, we modify the de�nition of Z to includethe borrowing constaint. This implies that Am is not strongly sublinear onall of M; i.e.,

Z(m; Am;K; �) =1

Am� �

Z�

�r(g �H(Am); �0)m(g �H(Am); �0)

�(�; d�0)� �I(Am = mu)

=1

tAm� �

Z�

�r(g �H(Am); �0)tm(g �H(Am); �0)

�(�; d�0)� �I(Am = mu)

t

= 0

� 1

tAm� �

Z�

�r(g �H(Atm); �0)tm(g �H(Atm); �0)

�(�; d�0) + �I(Atm = mu);

where the last (weak) inequality follows from having Atm � Am eventhough tm < m. This lack of strict isotonicity is the result of the occasion-ally binding inequality constraints that a¤ect the shape of A over the entirespace M. In previous work on uniqueness, without borrowing constraints( e.g., the homogeneous agent economy in Coleman [22]), interiority condi-tions are strong enough to guarantee the needed strict form of isotonicity.The borrowing constraint, therefore, implies that when A(0) � 0 (and notA(0) > 0), the operator Am is only sublinear, not strongly sublinear.From a computational perspective, Theorem 4 has additional implica-

tions. It points out that, in general, any numerical solution procedure for

erator, namely P, be a solid cone. We say an operator A is said to be K0-monotone ifit is isotone and if there exists for any strictly positive �xed point m1 of A some K0 suchthat the following is true: for any K1 2 [0;K0] and any m2 such that m1 � m2 impliesm1(K; �) � m2(K; �) for all K � K1.

26

the approximation of Markovian equilibrium in this economy must deal withoccasionally binding constraints globally unless A(0) > 0 (this because ingeneral Markovian equilibrium are not strictly positive): In addition, thiscondition provides a su¢ cient condition for strictly interior equilibrium tobe unique in the spaceM.We make a few �nal remarks on the limitations of these methods. First, to

operationalize the numerical implementation of our operator Âm or Ah, oneneeds to develop algorithms for constructing monotone selections. Second,Theorem 4 only provides the basis for pointwise strong set order comparativestatics for all possible Markovian equilibrium in H: That is, it is importantto realized that our uniqueness argument above does not rule out existenceof Markov equilibrium outside the class H (orM). Given the way selectionsare formed in the de�nition of the operator Ah, we believe, for example, therecould be other Markovian equilibrium with in the class of bounded functions.Finally, the assumptions on the boundedeness of the state space can be

relaxed somewhat, allowing for models with unbounded growth, for example,in the two sector incomplete markets economy studied in Krebs [47]. Usingthe methods in Morand and Re¤ett [58], one can use pure lattice theoretic�xed point constructions to study the operator Ah, and all the results of themain theorems will hold for such economies. In later work, we will addressthe issues of long-run stationary Markovian equilibrium. The order theoreticlimit theory of Hopenhayn and Prescott [35] for characterizing the long-runergodic measure for the capital stocks and idiosyncratic shocks is needed andit requires joint mononicity of equilibrium in endogeneous states and randomshocks.

6 Conclusion

This paper pursues an extension of monotone mapping techniques developedin Coleman [20] for representative agent economies, to multiagent incompletemarkets models with production. The methods are based on the existence ofEuler equations (which are needed to provide sharp geometric characteriza-tions of the nonlinear operators used to construct Markovian equilibrium),and concavity, therefore, plays a key role in the analysis in the paper. Morethan concavity, it is actually strong-concavity which is required to charac-terize the operator on the boundary of the space of candidate equilibriumconsumption functions. One advantage of this method is thus to develop a

27

single valued continuous operator that maps an equicontinuous set of func-tions into itself. This fact is critical when evaluating (and proposing) nu-merical approaches to computing the equilibrium for this type of models. Inparticular, versions of the Stone-Weirstrass theorem can be used to provethat projection methods (for example piecewise linear methods based uponColeman [19] or Judd [38]) will be asymptotically consistent for the actual�xed point. That is, as the dimension of the basis set used to approximateboth the elements of the domain as well as the operator tend toward in�nity,the projection method will converge to the actual equilibrium �xed point.The policies in our equilibrium set are Lipschitz continuous (since they

belong to an equicontinuous set), which is a key property to achieve theerror bounds constructed in Santos and Vigo [65] and Santos [64]. Sincetrajectories of operators can be tied to value functions, in principle it wouldthen be possible to construct asymptotic error bounds to numerical methodsbased upon our monotone operator A along the lines of Santos work. Thiswe leave for future work (See for example Morand and Re¤ett [58].Additionally, the issue of existence of long-run stationary Markovian equi-

librium remains. One can change the assumptions on the primitive data ofthe model to obtain su¢ cient conditions under which Markovian equilibriumwill be jointly monotone in (K; �): For example, adapting a condition usedstudied in Mirman, Morand, and Re¤ett [54] to our setting, one might as-sume u and f are such that u00(c)f1f2 + u0(c)f12 � 0: In our setting, this issu¢ cient to show that the Markovian equilibrium in Theorem 3 are actuallyjointly monotone in (K; �): In this case, the limiting distribution theory ofHopenhayn and Prescott [35] and Danthine and Donaldson [26], and one canconstruct a stationary Markovian equilibrium in the sense of Du¢ e et al [27].We leave further work along this line to future research.Finally, it might be possible to develop a method of monotone compara-

tive analysis for the Markovian equilibrium set for this economy under muchmore general assumptions on the primitive data of taste, technology, and theclass of equilibrium distortions considered. In many situations concavityof the value function is often lost, and therefore single valued operators onspaces of continuous functions are no longer available. However, it is oftenpossible to apply generalizations of Tarski�s theorem to such problems by ex-ploiting the fact that equilibrium best responses of agents (under weak super-modularity assumptions along an equilibrium path) are uniformly bounded,monotone functions. Spaces of such functions form complete lattices, andtherefore existence and characterization using order based �xed point theory

28

is often available. Mirman, Morand, and Re¤ett [54] show how such meth-ods can be used to construct monotone Markovian equilibrium for a verylarge class of non optimal representative agent models.Further, using thesemethods, it should be possible to study with monotone methods multisector,multiagent economies with incomplete markets such as those presented in aseries of papers by Krebs [47][45]. His results concern a class of symmetricclass with identical CES preferences, constant returns to scale technologies,and no public policy. Our methods should be able to be used to extendthese results. One might ask if one could begin to develop a set of tools thatwould make very general versions of his environments tractible for appliedresearchers interested in incorporating human capital and physical capitalinto a richer model of economic �uctuations and growth that also allow forincomplete markets and trade equilibrium. Perhaps such methods can beused to obtain similar analytic results.

7 Appendix

In this appendix, we prove the proposition two in the paper concerning theproperties of the operator Ah: Prior to proving this proposition, we provesome lemmata needed in the proof of the proposition. We �rst now notesome properties of the mappings Z as de�ned in (7).

Lemma 5 Under assumptions One-Three, for each i, (i) Z(x;K; �; h) isstrictly antitone in hi and K; (ii) Z strictly isotone in x.

Proof: (i) Let h0 � h, for h; h0 in H: Then for each Zj(x;K; �; h0j) �Zj(x;K; �; hj): If h0j � hj; and h0j 6= hj; then the inequality is strict by thestrict concavity of the vector of period utility functions u(c). Therefore Z isstrictly antitone in h. Now �x h 2 H; and take K 0 � K: A similar argumentshows shows that Zj is strictly antitone in K: Again, if K 0 6= K; and K 0 � K;all the inequalities are strict.(ii) For each (h;K; �);when x0 � x; 2 decreases and 1 increases, and

therefore Z(x0; K; �; h) � Z(x;K; �; h). If x0 � x; x0 6= x then the inequalityis strict.�

Proof of Proposition Two:(i) Fix t = (K;h):

29

Recall the mapping Z : X�K���H! �RJ where X � K � RJ+

Z(x; t; �) = 2(x)�1(x; t; �);

1 = �

Z�

u0(g(x; �0)� h(x; �0))�r(x; �0)�(�; d�0);

2 = u0(g � x):

This is used to de�ne the nonlinear operator A : H!H0.De�ne,

Ah(K; �) = fx(t; �) 2 x�(t; �)j 0 < h < g; x isotone in K given �; hg= f0 elsewhere g;

where,

x�(t; �) = fxjZj(x; t; �) � 0; and Zj(x; t; �) = 0 if (gj � xj) > 0g:

De�ne the mapping �(x; t; �) componentwise as

�j(x; t; �) = j2(x)�j1(x; t; �) when j2(x)�j1(x; t; �) > 0;= 0 elsewhere for j = 1; 2; :::; J:

Notice that, for �xed t; �(x; t; �) is well-de�ned; it is continuous in x; andsuch that i2(x)� �(x; t; �) is strictly decreasing in x for each t:De�ne (x; t) : X�K���H! �RJ where X � K � RJ

+ as follows,

(x; t; �) = 2(x)� 1(x; t; �) and

1 = �

Z�

u0(g(x; �0)� h(x; �0)))�r(x; �0)�(�; d�0) + �(x; t; �)Ix(x = 0);

2 = u0(g � x);

where Ix(xi = 0) is the indicator for vector xj when xj = 0. We use tode�ne the operator Ah(K; �): That is, when h > 0; a solution x�(t; �) to(x�(t; �); t; �) = 0 can be used to de�ne Ah(K; �):We �rst prove x�(t; �) is nonempty for �xed (t; �): Observe that under

assumptions One-Three as xj ! gj; ! 1 for each j. If x ! 0; given

30

the Inada condition on �r and the de�nition of �; then (x; t; �) � 0: Giventhat is strictly increasing on the interior of [0; g], there exists a pair ofvectors (xl(t; �); xu(t; �)) 2 [0; g] � [0; g] such that (i) (xl(t; �); t; �) < 0;(ii) xu(t; �) such that (xu(t; �); t; �) � 0; and (iii) xl(t; �) � xu(t; �) in thestandard component product order on RJ+: Finally note that as is strictlyincreasing jointly in x in the interior of [0,g] and nondecreasing on all of [0,g],it is is nondecreasing in xj on [0; gj] for �xed xj = (x1; x2; :::xj�1; xj+1;:::; xJ),x = (xj; x

j) each j 2 J . The mapping (x; t; �) satis�es the semi-continuityconditions in Guillerme ([32], p2118, Theorem 1). And, from Guillerme ([32],p2118-2119, Theorem 1 and 3), we conclude that correspondence x�(t; �) isnonempty on the order interval [xl; xu] � [0; g] � RJ+.17 Further, as strictlyincreasing in x on the interior of [0; g]; and neither endpoint is a solution,we conclude x�(t; �) is an antichain in [0; g] for each (t; �) relative to thecomponentwise order on RJ .We next consider the question of existence of a isotone selection in x�(t; �).

First, we make a few additional observations. Suppress the notation � forthe moment. Fix x(t) 2 x�(t): By the de�nition of x(t), we have (x(t); t) =0: Consider t0 � t; t 6= t0: By lemma 5, as is strictly antitone in t inProposition 2; we have �1 < z = (x(t); t0) < 0: De�ne the directed setG(x(t)) = fxjx � x(t);(x; t) = z � 0 > z; z �niteg. Compute xT = supG(x(t)) 2 [0; g], which exists in [0; g] as [0; g] is a complete sublattice ofRJ+: Compute z

T = (xT ; t0) � z � 0 where the inequality zT � z followsfrom the de�nition of join and the fact that is strictly increasing in x: Bythe continuity of is x, de�ne G = fxjzT � Z(x; t) � 0 > zg where, byconstruction, xT = sup G � [0; g]: Also, note that, G(x(t)) is nonempty (ascertainly we have x(t) 2 G(x(t)), and therefore, so is G(x(t)):We can now prove the existence of an isotone selection in x�(t) using a

variation of a theorem on Veinott [71] ([71], Chapter 4, Theorem 5). Weprove that x�(t) is ascending in the weak induced set order (see Topkis, [69],p 38 and Smithson ([67], conditions 1 and 2, p307). That is, let x�(t) :T ! 2[0;g]; x�(t) such that for any t1 � t2; t1; t2 2 T; then (C1) for allx1 2 x�(t1) there exists a x2 2 x�(t2) such that x1 � x2; and (C2) for anyx02 2 x�(t2) there exists a x

01 2 x(t1) such that x1 � x2: Additionally, if x�(t)

is antichained valued, then by a corollary to Veinott�s Theorem, x�(t) admitsan isotone selection. Notice that, in the last claim (relative to Veinott�soriginal statement of the theorem), as x�(t) is antichain valued, x�(t) is chain

17Using Guillerme�s notation, take 1 = g;2 = f; to apply the theorem.

31

subcomplete in 2[0;g]: So we replace the hypothesis of chain subcompletenessin Veinott�s theorem with the stronger condition that x�(t) is antichainedvalued. We weaken the notion of "ascending" used in Veinott�s proof (whichis increasing in the weak set order, i.e., for any x1 and x2 as above in (i);x1 ^ x2 2 x�(t1) or x1 _ x2 2 x�(t2)) to either (i) or (ii) in Smithson. Inour case, we have both (i) and (ii). Then the existence of least increasingmajorant of any minimal selection in x�(t) under (i) (or greatest increasingminorant of any maximal selection in x�(t) under (ii)) is guaranteed by a keyresult in Veinott ([71], Chapter 2, Lemma 8).To prove x�(t) is ascending in the weak induced set order, as is (i)

strictly increasing and continuous in x on int([0; g]) and nondecreasing every-where, (ii) is continuous in x on the interior of [0; g] and upper-semi continu-ous everywhere in x, and (iii) the order interval [0; g] convex (and, therefore,connected), by we can again apply of Guillerme�s ([32], Theorem 3) interme-diate value theorem to (x; t0; �) for semi continuous functions in x relativeto the order interval [x(t); xT ] � [0; g] � RJ+ to conclude for any x(t) 2 x�(t);there exists a x(t0) 2 x�(t0) such that x(t) � x(t0): This proves (C1). A simi-lar argument shows that for any x(t0) 2 x�(t0); there exists and x(t) 2 x�(t)that minorizes x(t0): This proves (C2). Then by the version of Veinott�s iso-tone selection theorem cited above, we conclude because x(t) satis�es (C1)and is antichained valued (respectively, (C2) and is antichained value), thereexists is an least increasing majorant of any minimal selection in x�(t) (angreatest increasing minorant isotone selection x(t) 2 x�(t)). Therefore, anisotone selection exists, and Ah(K; �) is well de�ned.18

(ii) We note that as Ah(K; �) isotone in K; we have,

�1(Ah(K 0; K 0; �; h))+2(Ah(K; �)) � �1(Ah(K;K; �; h))+2(Ah) = 0

As each j2(x) is only a function of gj�Ahi(K; �) for each j; by the concavityof the vector of marginal utilities u(c) we have Ah(K; �) such that for eachj; gj � Ahj is isotone in K; each �: As � is �nite, and both Ah and g � Ah18We de�ne the operator Ah as any monotone selection from x�(t), so Ah is a func-

tion, and we apply a version of either the main theorems in Amann [2] or Dugundji andGranas [28] to construct extremal Markovian equilibrium within the set H by successiveapproximation. We later show that, in this set H, Markovian equilibrium is unique.Now, alternatively we could de�ne a correspondence, say Ch , as any monotone selection

in x�(t); h > 0; 0 else. We could obtain the existence of Markov equilibrium via the �xedpoint theorems for multifunctions in Smithson [67]. See the discussion after Theorem 4.

32

are isotone in K; and g is uniformly continuous in K; Ah is an element of anequicontinuous set of functions on the compactum S). Therefore Ah � H:(iii) Follows from the fact that �r and u0(c) are strictly decreasing in K:(iv) Continuity follows from an argument in Coleman [20], Proposition

4). As A(H1) = fAhjh 2 H1 � Hg � H; the closure of A(H1) is compact.Therefore Ah is continuous and compact in h, i.e., completely continuous inh.(v) Consider any h0 � h; with h; h0 2 H: Since is increasing in h and

(Ah;K; �; h) = 0 by de�nition of the operator A, then Z(Ah;K; �; h) �0. Consequently, with is strictly increasing in its �rst argument, by anargument similar to (i) for the existence of monotone selection, it must bethat Ah0(K; �) � Ah(K; �) for all (K; �):(vi) Let H � H; hu = sup(H). The point hu is well-de�ned as H is a

complete lattice.Consider any increasing chain hi2I in H such that hi ! hu in order (and

I is not necessarily countable). As Ah is topologically continuous in theuniform topology A(hi2I) ! Ahu uniformly in H. Since Ah isotone, as hi2Iis a chain, we have _A(hn) = A(hu) = A(limhi2I): But by de�nition hu =_(hi) = _H: Therefore A(_H) = _(A(hi)) = _A(H): As similar argumentworks for a decreasing chain hi2I ! hu for A(^hi )=^A(hi2I) = ^A(hu):Therefore Ah is order continuous.�

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