which recursive equilibrium?
TRANSCRIPT
Which Recursive Equilibrium?�
Manjira Datta, Leonard J. Mirman, Olivier F. Morand, Kevin Re¤etty
April, 2010Preliminary and Incomplete
AbstractWe reconsider recent results on the existence and uniqueness of recur-
sive equilibrium in the setting of a simple one-sector nonoptimal growthmodel with a state-contingent tax. We �rst extend the uniqueness resultin Coleman [13] to a much larger class of functions (i.e., spaces of boundedfunctions). Even given this, we then show there exist other �xed point pro-cedures that construct continuous recursive equilibrium outside the set ofcontinous functions considered in Coleman [13]. The result point to the del-icate nature of existing uniqueness results. We then extend Coleman�s pol-icy iteration approach to verify and construct (locally Lipschitz) continuousrecursive equilibrium in economies previously thought not to possess them(e.g., the economy in Santos [51]). We show existing correspondence-basedcontinuation methods (e.g., Miao and Santos [35]) do not, in general, ver-ify the existence of recursive equilibrium (even for simple one-dimensionalcases). We propose a new correspondence-based continuation method (val-ued in function spaces) which does verify the existence of recursive equilib-rium.
�We thank the participants a workshop at Jadavpur University, Kolkata, India during March2009, the Sorbonne Workshop for Economic Theory held in Paris, June 2009, as well as AdrianPeralta-Alva, Rajnesh Mehra, Ed Prescott, Manuel Santos, Yiannis Vailakis, ×ukasz Wozny,and especially, Robert Becker for helpful discussions. Manjira Datta and Kevin Re¤ett thankthe Center d�Economie de la Sorbonne (CES) for arranging their visits during the Spring termsof both 2008 and 2009.
yManjira Datta, Department of Economics, Arizona State University; Leonard J. Mirman,Department of Economics, University of Virginia; Olivier F. Morand, Department of Economics,University of Connecticut; Kevin Re¤ett, Department of Economics, Arizona State University.Do not recirculate, rather write the corresponding author at Kevin.Re¤[email protected] for the mostrecent version.
1
1. Introduction
Since the seminal work of Lucas and Prescott [31] and Prescott and Mehra [44], acentral focus of quantitative work in macroeconomics has been on the computationof recursive equilibrium (RE). For a rigorous implementation of such a numericalresearch program, a great deal of understanding of the set of RE is required. ARE, when it exists, is a special case of a sequential equilibrium, where equilibriumtrajectories can be constructed from a collection of time invariant functions whichare solutions to systems of equilibrium functional equations.1 In the early workon recursive methods for characterizing equilibrium dynamics, the emphasis wasplaced on homogeneous agent economies where the second welfare theorem held.This situation is particularly convenient, as RE allocations and prices could beconstructed by solving a single functional equation, namely a Bellman equationdescribing a pure resource allocation problem faced by a social planner on onlyaggregate states. Then, appealing to arguments of Bewley [8] or Prescott andLucas [43], the social planner�s optimal solutions could be supported as recursivecompetitive equilibrium (RCE) on de�ned along the diagonal of a collection offunctions de�ned on both individual and aggregate states, where these equilibriumdecision rules implied the existence of a suitable in�nite dimensional equilibriumprice system. As well-known tools are available for solving the Bellman equations(both from an theoretical and numerical perspective), the resulting methodologicalapproach was very powerful.2
Unfortunately, things have changed a great deal over the last three decades.
1In the literature, the primary focus has been on stationary Recursive equilibrium stationaryequilibrium de�ned on minimal partitions of the individual and aggregate histories of the econ-omy. Using nonstationary dynamic programming methods (e.g., Hinderer [25]), a more generalclass of nonstationary recursive methods can be developed. Additionally, recursive equilibriumusually refers to as "short-memory" Markovian equilibrium, where the equilibrium is a diagonalof a function de�ned on only current states of dynamical system. Finally, recursive equilibriacan be constructed as compositions of functions (each depending again only on the diagonal ofa function de�ned on current individual and aggregate states), as is the case in "generalized"Markov equilibrium (e.g., Kubler and Schmedders [28] and Miao and Santos [35]) and Feng,Miao, Peralta-Alva, and Santos [21].
2From a theoretical perspective when entertaining questions concerning equilibrium com-parative statics, pricing, and/or existence of stationary Markov equilibrium; from a numericalperspective because of the availability of value and policy iteration techniques for obtaining ac-curate approximation solutions. For motivation from a theoretical perpective, see Prescott andMehra [44]; for motivation from a numerical perspective, see the survey of Rust [50].
In more recent work, a majority of this work that has conducted in the setting ofdynamic economies where the second welfare theorem fails (i.e., so-called "nonop-timal" economies"). For such economies, for even the simplest of cases (e.g., ahomogeneous agent economy with one sector production, a state contingent taxand lump-sum transfers), the social planning procedures of Prescott and Mehra[44] are known generally to fail. As numerous interesting problems in dynamicgeneral equilibrium take place in such nonoptimal settings (e.g., models of opti-mal taxation, monetary economies, heterogeneous agent models with incompletemarkets, among others), a systemic approach to characterizing and constructingdi¤erent classes of recursive equilibrium is needed. The problem is di¢ cult, asin nonoptimal economies, a recursive equilibrium is a solution to a set of equi-librium versions of the household�s decision problems, typically a strictly concaveproblem, whose solutions along equilibrium trajectories for optimal solutions arecharacterized not only by parameterized Bellman equations, but by necessary andsu¢ cient systems of equilibrium Euler equations/inequalities. Therefore, for sucheconomies, the resulting system of functional equations governing the existenceand computation of a recursive equilibrium can become quite complex.Various methods have been proposed in the current literature to resolve some
(or all) these technical issues in the context of various economies. One importantclass of such methods are known as "monotone continuation methods".3 To date,two types of monotone continuation approaches have been proposed: (i) continua-tion methods de�ned in spaces of correspondences (i.e., so-called correspondence-based method �rst proposed in Kydland and Prescott [29], but extended in numer-ous papers including Kubler and Schmedders [28], or Miao and Santos [35]), amongothers), or (ii) continuation methods de�ned in spaces of functions (i.e., so-calledfunction-based continuation method, �rst proposed in Coleman [11][13], Green-
3By a "continuation method", we mean a operator theoretic approach to solving functionalequations that constructs a mapping T : X ! X where X is a collection of functions orcorrespondences, where an element x 2 X is used to parameterized the future structure ofequilibrium of the economy, then one computes the implied decision rules, from when T (x) canbe computed. The approach does bear, in an abstract sense, a resemblence to a continuationmethod in numerical analysis, but the details are still somewhat di¤erent.A classic continuation method, in our terminology, is stationary dynamic programming. Here,
one parameterized the continuation structure of the optimization problem on a state space iss 2 S with V (s) 2 V(S), with V(S) is a complete metric space of function on S). Then, oneconstruct TV in V(S) in the Bellman equation that implied by V (s): A (unique) �xed pointis then constructed to solve the functional equation. Correspondence-based extensions of thesemethods are also available.
wood and Hu¤man [23], Datta, Mirman, and Re¤ett [14] or Mirman, Morand,and Re¤ett [36]).In this paper, we compare these methods in a simple class of nonoptimal
economies, namely, a nonoptimal Solow-Cass growth model with a state-contingenttax under certainty and perfect commitment. In this simple setting, we areable to show some important di¢ culties associated with both function-based andcorrespondence-based continuation methods, all of which carryover to more com-plicated nonempty dynamic economies with production. We �rst focus on thecase of a progressive state-contingent tax �rst studied in Coleman [11]. For theseeconomies, we construct multiple function-based �xed point procedures, each ofwhich construct recursive equilibrium. These procedures di¤ering both in theirde�nition of �xed point operator, as well the domains in which these operatorsare de�ned.. Along these lines, for the well-known policy iteration method �rstproposed in Coleman [11][13], we �rst extend the uniqueness result obtained inColeman [11][13] to much larger class of functions, namely, spaces of bounded func-tions, where implied decision rules are not necessarily continuous or monotone.Still, we are also able construct a second �xed point procedure that veri�es theexistence of continuous recursive equilibrium, whose �xed points exist in classof continuous functions larger than the set of continuous function than our newuniqueness result policy iteration holds. Using these procedures, we show unique-ness claims potentially are only necessarily statements about the �xed points ofparticular �xed point procedures; not claims about the structure of the set ofrecursive equilibriumWe then consider the question of continuous RE in models often thought not
to admit them (e.g., the example presented in Santos ([51], section 3.2)). As theSanto�s example has been a central motivation for adopting correspondence-basedapproaches to characterizing recursive equilibrium (e.g., Miao and Santos [35],Feng et. al. [21], and Peralta-Alva and Santos [46]), reconsidering the question ofcontinuous RE in this class of economies is particularly important. Using a newsimple variation of Coleman�s policy iteration procedure, we construct locallyLipschitz continuous recursive equilibrium for these economies with a regressivetax. Further, contrary to often mentioned limitations of Coleman�s policy iterationmethod (e.g., Miao and Santos [35] and Feng et. al. [21]), these continuousrecursive equilibrium are not monotone necessarily on their state space. We thenconstruct selections in the Miao-Santos procedure equilibrium correspondence thatare not RE. This, in e¤ect, implies both (i) correspondence based methods cannotverify the existence of RE, and (ii) numerical approaches to selecting RE form
approximate equilibrium correspondence (via set approximation methods) mustprovide some systematic guidance for distinguishing approximate RE selectionfrom selections that are not RE.We �nally reconsider the question of when can correspondence-based methods
be used to verify the existence of a RE in our economies with a regressive tax. Wepropose a new correspondence-based approach, based upon "interval iterations"of interval operators valued in function spaces, that integrate the function-basedpolicy-iteration approach of Coleman [11], with the correspondence-based contin-uation method advocated in Kubler and Schmedders [28] and Miao and Santos[35]. For this interval mapping, we prove the existence of continuous recursiveequilibrium for our class of economies, and prove a simple method for computingit approximately. We then develop a formal partial ordering method for makingcomparative dynamics comparisons of our new interval iteration method versusthe existing class of correspondence-based continuation methods.The paper shall proceed as follows: in the next section, we describe the class of
nonoptimal Solow-Cass growth models considered in the paper. In section three,we use function-based continuation methods to study the structure of recursiveequilibrium in situation of progressive taxation as in Coleman [11]. Section fourthen considers the same methods, but for the regressive tax case as in Santos [51].Section �ve reconsiders both models, but only with correspondence-based meth-ods. Here, in addition to applying the Miao-Santo�s procedure, we also construct anew correspondence-based procedure for constructing recursive equilibrium. Sec-tion six, then, makes concluding remarks. In the appendix, will include the proofs,a detailed appendix with all the mathematical terminology used in the paper, andstatements of key �xed point theorems used in the proofs.
2. The Economies
To keep the issues raised in this paper clear, we focus our attention on the caseof recursive equilibrium in simple class of Solow-Cass growth models with a statecontingent income tax.4 What is important is that the class of economies provessu¢ cient compare carefully the di¤erent existing monotone continuation methodsin the literature as for the case of models with a progressive tax, our results relatedto those proved in Coleman [11][13], Greenwood and Hu¤man [23], Phelan and
4Of course, similar issues to those that arise in this simple model also arise in much morecomplicated models (e.g., the stochastic OLG models and asset pricing models with productionstudied in Miao and Santos [35] and Datta, Mirman, Morand, and Re¤ett [15]).
Stacchetti [45], Datta, Mirman, and Re¤ett [14], Mirman, Morand, and Re¤ett[36]; for the special case of regressive taxes, our results are directly related to thosein Santos [51], Miao and Santos [35], and Feng, Miao, Peralva-Alta, and Santos[21]).Time is discrete and indexed by t 2 T = f0; 1; 2; :::g. There is a continuum
of in�nitely-lived and identical household/�rm agents. Households get utilityfrom lifetime consumption, solve standard consumption-savings problems, andhave time separable utility with constant discounting. Firms are identical, andendowed with standard constant returns to scale technology (where, again, for thesake of illustration, we shall often study the case of Cobb-Douglas production).There no nonconvexities in production., There are no shocks. The single sourceof nonoptimality in the macroeconomy is a state contingent income tax that isredistributed via lump-sum transfers. To keep our issues focused, we consideronly two types of equilibrium distortion: (i) a progressive income tax, and (ii) aregressive income tax. We assume perfect commitment for the government forall periods, and the proceeds of the taxes are redistributed lump-sum back tohouseholds each period.Households are endowed with a unit of time which they supply inelastically
to competitive �rms, and enters any given period with an individual level of thecapital k 2 K, facing an economy in aggregate stateK 2 R+ (whereK is the per-capita capital stock). Denoting the state of a household by s = (k;K) 2 K�K;and period utility by a function u(c) (where c 2 R+, or, perhaps, R++ in the caseof logarithmic utility), a household�s lifetime preferences are de�ned over in�nitesequences indexed by dates and histories c = (ct) and are given by:5
U(c) =1Xt=0
�iu(ct) (1)
The capital-labor ratio chosen by a typical �rm is denoted by x; where tech-nology is represented by a twice continuously di¤erentiable, concave, constantreturns to scale production function F (x; n) for each �rm in the economy, wheren is the typical �rms labor demand. Anticipating equilibrium, given inelastic la-bor supply, we have n = N = 1; and x = K. 6 From pro�t maximization, this
5Although the original work of Coleman [11] did not cover the unbounded homogeneousreturns case, Morand and Re¤ett [40] show all his results can be extended to this case.
6All of our results can be shown to hold for models with elastic labor supply where leisureenters in separably into preferences (or, more generally, are supermodular). We discuss theseissues at the end of the paper.
implies the market equilibrium prices of capital is r = F1(K; 1), and the price oflabor is w = F2(K; 1). De�ning f(x; 1) = F (x; 1); then F additionally satis�esthe following standard Inada condition:
limx!0
F1(x; 1)!1; and limx!1
F1(x; 1)! 0
To create a simple class of equilibrium distortions, we assume that the governmenttaxes all sources of income using a state-contingent marginal tax rate of �(K) 2[0; 1), and then rebates the tax revenues back to households lump sum in theamount J = �m. In this paper, we only consider two cases of taxation: (i) �(K)increasing in K (e.g., "progressive" taxation"), and (ii) �(K) is decreasing in K(e.g., "regressive" taxation).7
Throughout the paper, we shall make the following assumptions on preferencesand technologies:
Assumption 1: The discount rate is � 2 (0; 1); and period utility functionu : R+ ! R+ where u(c) is C2; strictly concave, strictly increasing, and satis�esthe following Inada condition
limc!0
u0(c)!1; limc!1
u0(c)! 0
Assumption 2: The production function F (x; n) is constant returns to scale,C2; strictly increasing, strictly concave, and satis�es the following Inada condition
limx!0
F1(x; 1)!1
Assumption 3: (i) (Progressive taxation) � : R+ ! [0; 1); is locally Lipschitz,and monotone increasing (i.e., isotone); (ii) (Regressive taxation): � : R+ 
Assumption 1 and 2 imply bounded equilibrium growth (i.e, the existence of amaximal capital stock ku (so the state space can be taken to be bounded andK=[0, ku]); this assumption is only made for convenience (e.g., for the case ofunbounded states spaces, see Morand and Re¤ett [40]). Second, the class ofequilibrium distortions consistent with Assumption 3 have are common in theliterature. The local Lipschitz structure for taxes and transfers is a very mildassumption, and have been used in many papers (e.g., Santos [51]). For theprogressive tax case (e.g., Coleman [11]), if we only seek continuous (but notnecessarily locally Lipschitz continuous) recursive equilibrium, we can relax thelocal Lipschitz properties in Assumption 3 to just continuity. As for the case ofregressive taxation, we do need the Lipschitizian properties for the taxes imposedin Assumption 3.
2.1. Household Decision Problems
We begin by constructing a recursive representation of a typical household�s deci-sion problem that we shall repeatedly appeal to throughout the paper. As �rmssolve static decision problems, the equilibrium prices of capital in the economyare given as
r(K) = F1(K; 1)
and wagesw(K) = F2(K; 1)
These prices are taken as parametric by households when solving their decisionproblems.In this paper, we seek recursive equilibrium on a minimal state space x =
(x1; x2) 2 X= K�K � R2+: Let B(X) denote the space of bounded functions
endowed with (i) the topology of uniform convergence, and (ii) the pointwisepartial order, and Bf (X) be a subset of B(X) that consist of all the sociallyfeasible aggregate laws of motion; i.e.,
Bf (X) = fh(x)j0 � h(x) � r(x2)x1 + w(x2)g � B(X)
Notice, for an equilibrium trajectory, we shall have x = x1 = x2 (i.e., k = K); sor(x)x+w(x) = f(x): Endow Bf (X) with its relative topology and relative partialorder. The collection Bf(X) is a complete sublattice in B(X). 8
8For all needed order theoretic de�nitions used in the paper, as well as useful �xed pointtheorems, see the appendix (also, see Birkho¤ [9] and Topkis [53]).
For simplicity, we assume household�s own the �rms, and rent the factors ofproduction to those �rms in competitive markets. Using the de�nitions of r andw, and appealing to zero pro�ts under constant returns to scale in Assumption 2,the household income process can be written equivalently as either y1 or y2 in thefollowing expression:
y1(k;K) = (1� �(K))ff(K) + (k �K)r(K)g+ J(K) (2)
= (1� �(K))fr(K)k + w(K)g+ J(K)= y2(k;K) (3)
where yi : K�K++!R+ for i = 1; 2:9 Using yi(k; k), the household�s budget
correspondence can be written as either i = 1 or 2 is:
i(k;K) = fc; k0jc+ k0 � yi(k;K); c � 0; k0 � 0g (4)
In equilibrium, where k = K; as the government�s budget constraint is imposed,we require �f = �(rK+w) = J . Therefore, the household�s income processes canbe written, respectively, as
y1(k; k) = f(k)
= y2(k; k) = rk + w
Under Assumptions 2 and 3, as f , r, w; � and J are each at least locally Lipschitz,the household�s feasible correspondence i(k;K) is locally Lipschitz continuouson K�K when K > 0; for each version i = 1; 2.10
Let K� = Kn0, and X�=K�K�: To construct a recursive representationof the household�s decision problem; for a household entering a period in states = (k;K) 2 X� in a candidate recursive equilibrium h 2 Bf (K�K); whenh(K;K) > 0; we can construct a unique value function V � : K�K��Bf (X) thatsatis�es the following parameterized Bellman�s equation:
V �(k;K;h) = supa2i(k;K)
fu(yi(k;K)� a) + �V �(a; h(K; z))g (5)
where the household�s feasible correspondence is simply i(k;K) = [0; yi(k;K)]:Under Assumptions 1-3, appealing to a standard argument in the literature, it
9It will become clear in a moment why keeping track of the two di¤erent equivalent expressionsfor household�s income process is useful.10See Rockafellar [49] for a discussion of Lipschitizian properties of correspondences.
can be shown that the unique real-valued solution to this Bellman equation isa function V �(k;K; h) in the set W = f V : K � K��Bf ! R, v (i) isotonein k, each (K;h), (ii) strictly concave (hence, continuous) in k for each (K;h)}.Further, additionally, by the Mirman-Zilcha lemma, V �(k; k;h) also (iii) has aenvelope theorem in V �1 (k;K;h) = u
0(yia�)r(1� �), where (iv) the optimal policya� = a�(k;K;h) is single valued and continuous in its �rst argument. Allowingfor unbounded returns above in our setting is not a problem. (See Morand andRe¤ett [40] for power utility, or Morand, Re¤ett and Wang [41], more generally).11 Notice, if in addition, h(K) is continuous, a�(k;K;h) is also continuous in K.Therefore, in any recursive equilibrium h� 2 Bf ; when k = K; h� must be suchthat conditions (i)-(iv) hold for equilibrium value function V �(k; k;h�) and uniqueoptimal solution a�(k;K;h�): De�ning the mapping yia� = y
i� a�; for i = 1; 2; wecan construct a necessary and su¢ cient �rst order characterization of the uniqueoptimal solution a� = a�(k;K; h) is for i = 1; 2:
u0(yia�)� �u0(yia�)(a�; h(K)) r(h(K))(1� �(h(K)) = 0 (6)
With these properties of V �(k; k;h�) and a�(k;K;h) now clear, we now readyde�ne an recursive equilibrium for our economies as follows:12
De�nition 1. A recursive equilibrium is any function h�(k; k) 2 Bf (K�K);such that (i) h�(k; k) > 0 and (f � h�)(k; k) > 0 when k 2 K�; (ii) h�(k; k) =a�(k; k;h�(k; k)); and h�(k; k) = 0, else, and (iii) when k > 0; given a law of motionh�(k;K) 2 Bf , when k = K, (a) there is a dynamic program V �(k; k; h�(k; k)) thatsolves (5), strictly concave in its �rst argument, that satis�es the Bellman equation(5) with associated unique optimal solution a�(k; k;h�(k; k)); (b) V �(k; k; h�(k; k))has an envelope theorem V �1 (k; k;h
�) = u0(yia�)r(K)(1��(K)) at a�(k; k;h�(k; k));11That is, the key step to dealing with the unbounded below case is to �rst solve the Euler
equation abstractly, and then use well-known methods for solving dynamic programming prob-lems with unbounded returns in a candidate to show there exists, and has a unique value functionevaluated at a (positive) equilibrium solution that satis�es a necessary and su¢ cient Euler equa-tion. This can be done by a modi�cation of the local contraction arguments in Martins-Da-Rochaand Vailakis [33] in our deterministic model. See Re¤ett [47] for a discussion.12The requirement of interiority of consumption implied in our de�nition is a natural require-
ment for a recursive equilibrium. For example, it is needed to show that the candidate recursiveequilibrium decision rule a�(k; k; h�(k; k)) = h�(k; k) induces sequential equilibrium with a pricesystem in an appropriate dual space to the (in�nite) commodity space. Although this is not thefocus of this paper, proofs for our economies can be built from the recent results presented inMorand, Re¤ett and Wang [41] for this economy.
and (c) a�(k; k;h�(k; k)) can be characterized by the necessary and su¢ cient Eulerequation ( 6 ).
In this de�nition above, we emphasize the requirement that any recursive equi-librium must satisfy conditions (iii.a-iii.c). This continuity requirement says thata recursive equilibrium holds along the diagonal of a function a�(k;K; h�(k; k)),namely, we have a�(k; k; h�(k; k)) = h�(k; k): It also bears mentioning that for thecorrespondence-based continuations methods in the literature, the requirement ofcontinuity in the individual state variables along the diagonal is demanding. Forexample, it implies that for G�(k; k) the equilibrium correspondence generatedby the "APS" type operator, one must guarantee the existence of a selection,say g�(k; k) 2 G�(k; k), that is continuous in its �rst argument (hence, guaran-teeing the resulting Generalized Markov equilibrium decision rule for investmentk0 = x�(k; k; g�(k; k)) 2 X�(k; k; g�(k; k)) is a continuous selection in its �rstargument). This condition is very di¢ cult to check even in very simple one di-mensional problems as we shall argue in Section 5 of the paper, as the typicalequilibrium correspondence G�(k; k) is simply a nonempty upper semicontinuouscorrespondence in its arguments.
3. The Progressive Tax Case
We �rst consider function-based continuation methods for our economies underAssumptions 1, 2, and 3(i) and 3(iii) (that is, nonoptimal Cass growth with a state-contingent progressive tax, and lump-sum transfers.) For these economies, weprove two new results. First, we extend the uniqueness result obtained in Coleman[13] for his policy-iteration procedure to a much larger class of domains (namely,a class of bounded functions with bounded consumption functions, not necessarilyeven monotone). We then construct a second new �xed point procedure that isnot policy iteration, but admits a complete lattice of (locally Lipschitz) continuous�xed points, where at least its least �xed point (for equilibrium investment) is arecursive equilibrium. This recursive equilibrium cannot be guaranteed to be inColeman�s �xed point set obtained using policy iteration.
3.1. Some Useful Function Spaces
Any discussion of solution methods for functional equations begins with a discus-sion function spaces that serve as domains for �xed points of operators used to
solve the equations. At this stage, we de�ne a number of function spaces that weuse in the paper. We begin with subsets of the bounded socially feasible decisionrules Bf (X). As will be clear in our subsequent discussion, to guarantee our solu-tions are recursive equilibrium satisfy conditions (iii.a)-(iii.c) along their diagonala function (where x1 = x2), it is important keep track of individual vs. aggregatestates separately. Therefore, partition the components of the state space x 2 Xas x = (x1; x2) 2 X1�X2 = X � R2
+, where x1 can be viewed as a individualsholds of capital, while x2 is the aggregate state of the economy�s per capita capitalstock, X1 = X2 = K: Consider a subset of Bf (X) consisting of a set of uppersemicontinuous, monotone functions on X:
UCS(X)= fh(x) 2 Bf j h(x) monotone increasing (isotone)
and upper semicontinuous in xg
For an element h 2USC(X), if we interpret a typical element h(x) as an candidateequilibrium investment decision rule, notice the implied equilibrium consumptionc�(x) = y1(x)� h�(x) is lower semicontinuous in x (and not necessarily isotone).An important subset of USC(X) occurs when consumption decisions rules
c = y1 � h are also isotone on X, namely, the space:
C(X) = fh(x) 2 USC(X)jh(x) continuous, s.t.y1(x)� h(x) is isotone in xg
The space C(X) is the domain for policy iteration methods studied in Coleman[11][13] (as well as many subsequent related papers on policy iteration methodsbased upon equilibrium versions of the household�s Euler equations). Therefore,for h 2 C; as both h(x) and c(x) = y1(x) � h(x) are isotone in x, h(x) and itsimplied consumption c(x) are necessarily continuous (as with both c(x) and h(x)are increasing, hence, locally Lipschitzian with modulus y0(x) = f 0(x) near x 2 X;x > 0). Give the subcollections USC(X) and C(X) their relative topologies andpartial orders to the space Bf (X).13
In our �rst lemma, mention the order completeness properties of subsets ofBf under pointwise partial orders:
13In our subsequent discussion, when the context for X is obvious, we shall refer to thesespaces spaces Bf ; USC, and C, respectively, where the domains of the functions de�ning eachspace is understood.
Lemma 2. Bf is a complete lattice; (ii) USC is subcomplete in Bf ; (iii) C issubcomplete USC(X):
Proof: To see the completeness claims, let B � Bf : As the pointwise inf andsup of B satis�es the pointwise bounds, i.e., 0� infxB � m; and 0� supxB � m;we have ^B 2 Bf and _B 2 Bf : Hence, Bf is a complete lattice.Further, as monotonicity in x (resp, equicontinuity at x) are preserved also
under pointwise sup and inf operations in X, if B � C, ^B 2 C and _B 2 C:Hence, C is a complete lattice.Finally, if B �USC, then the pointwise inf of any arbitrary B is upper semi-
continuous (e.g., Aliprantis and Border [5], lemma 2.41)). As _USC=y1(x) iscontinuous, by the characterization of a complete lattice in Davey and Priestley([17], Theorem 2.31), USC is complete lattice.Finally, noting obvious sublattice and subchain inclusions, the subcomplete-
ness and subchain completeness claims in the lemma follow. �
We next consider subsets of Bf (X) where the restrictions on h 2 Bf (X)are stated in terms of their implied properties on the inverse of marginal utility ofconsumption. To do this, we �rst construct an analog to space of bounded feasibledecision rules Bf (X) in terms of inverse marginal utilities. We can let the inversemarginal utility implied for any element h 2 Bf (X) be denoted by:
mh(x) =1
u0(y1(x)� h(x)) ; for h 2 Bf (X), u0(y1 � h) > 0
= 0; else
Under Assumption 1, the function mh(x) is well-de�ned: Recalling Assumptions1, 2, and 3(i) and 3(iii), it is known that for our economies, there exists a maxi-mal sustainable capital stock, say ku > 0: Therefore, we can de�ne the maximalsustainable inverse marginal utility of consumption as mu
0 =1
u0(y1(ku;ku)) : Noticingmu(0; 0) = 0;de�ne the set of socially feasible inverse marginal utilities is given asfollows:
Mf (X) = fmj0 � m � mu(k; k)gNotice, as promised, the spaceMf (X) is simply a restatement of the space Bf (X)in the previous section. That is, we have h 2 Bf i¤m = 1
u0(h) 2Mf :
One important subset of Mf (X) occurs when continuous m 2Mf (X), and isan element of the following subcollection:
MA(X)= fh(x) 2 Bf (X)jh(x) continuous, 0 � mh � mu0 ; s. t.
0 � jmh(x0)�mh(x)j �
1
u00(y1(ku; ku))g
For an investment function h(x) to be consistent with the inverse marginal utilitylevel m(x) 2 MA(X); we only require the equilibrium decision rules to have animplied consumption function that has an implied variation of its inverse mar-ginal utility an element of a collection of functions that each exhibit (uniform)equicontinuity near each x bounded above by m0
f : Therefore, for h 2MA, as u(c)is C2; c(x) = y1 � h is also locally Lipschitz continuous in x. Therefore, as y1is also locally Lipschitz, h(x) is locally Lipschitz (as Lipschitz structure is closedunder scaler multiplication and addition).Finally, consider the following subset of Mf (X) that also prove useful in our
subsequent arguments:
M(X)= fm 2Mf (X)jm s.t.R� (k)
m(k; k)strictly decreasing k for k > 0g (7)
The subsetM(X) �Mf (X) is closely related to the domain of functions studiedon Coleman [13] for his uniqueness argument (the di¤erence being that form(x) 2M(X), we do not required m(x) to be continuous).EndowMf (X) with the pointwise partial order, and give the subsetsMd and
M each their relative partial orders and topologies. First, note, that M is notorder closed; hence, is not a suitable domain for existence arguments via ordertheoretic �xed point methods (e.g�Tarski�s theorem or its variants). M(X) willprove very useful for uniqueness arguments. In the Lemma 3, we discuss the ordercompleteness properties of the remaining function spacesMf andMA:
Lemma 3. (i)Mf (X) is a complete lattice; (ii)MA(X) is chain complete.
Proof: That Mf (X) is a complete lattice follows directly from Bf (X) a com-plete lattice (noting, the one-to-one lattice morphism de�ned in (7) between ele-ments of Mf (X) and Bf (X):
Further, as MA is equicontinuous and pointwise compact, it is a compactsubset of the space of continuous functions on X in the topology of uniformconvergence. Hence, in the pointwise partial order, by a theorem in Amann (e.g.,Amann ([4], Theorem 10),MA(X) is chain complete.�
In the subsequent discussion, when the context is clear, for all function spaces,we will delete the reference to the state-space X (e.g., C(X) is denoted as C).We are now ready to discuss function-based approaches to equilibrium in oureconomies with a progressive tax.
3.2. The Coleman Procedure
The �rst function-based method we consider is the policy iteration method pro-posed in Coleman [11] [13]).14 Coleman�s procedure can be de�ned as follows: forh 2 C; rewrite the equilibrium version of the household�s Euler equation in (6) asthe mapping ZA : K�X�C! R:
ZA(x; k; k; h) = u0(x)� �u0(y1h(y1x))R� (y1x) (8)
where, the function R� (K) = r(K) � (1 � �(K)) denotes the distorted returnon capital, and, y1h = y1 � h: 15 Then, a nonlinear operator A(h)(k; k) can beconstructed implicitly using ZA as follows:
A(h)(k; k) = x� s.t. ZA(x�(k; k; h); k; k; h) = 0; k > 0; h > 0, 8k= 0 else.
It is important to remember the operator equation A(h)(k; k) = h is only anabstract operator equation (with solutions that are not necessarily recursive equi-librium). Therefore, to make its �xed points of A(h) recursive equilibrium, furtherargument is typically required.The properties of iterative methods based upon the Coleman procedure have
been studied extensively in the literature. For the sake of completeness, we sum-
14We shall referred to this procedure as the "Coleman�s procedure". See also Bizer and Judd[10]. This procedure has been studied in a number of other papers. See Mirman, Morand, andRe¤ett ([36], Section 4) for a detailed set of references.15We should note, we consider the mapping ZA (and all similar mappings in the paper) to
be a real-valued function. We are careful to only use ZA to de�ne our operators when it isreal-valued. This makes the need for the extended reals unnecessary.
marize what is known about the solutions to the operator equation A(h) = h inC:16
Proposition 4. (Coleman [11][13] and Mirman, Morand, and Re¤ett [36]). LetCA be the set of �xed points associated with A(h)(k; k): Then, under Assumptions1-3(i) and 3(iii), CA = f0; h�(k; k)g � C, with h�(k; k) = _A > 0 when k > 0:Further, the iterations limnA
n(f) = supnAn(f) ! h� (where the convergence is
both in topology and order, respectively). Finally, h�(k; k) is C1 when k > 0.
We now extend the uniqueness result for the policy iteration methods in Propo-sition 4 to a more general setting. Following Coleman [13], we construct a secondoperator whose �xed points can be shown to be isomorphic to those of A(h)(x)using the domainM(X). To do this, de�ne the function H(m) implicitly by:
u0(H(m)) =1
mfor m > 0, 0 elsewhere.
The function H(m) = c(m) is the consumption level required to obtain the inversemarginal utility level of 1
mwhenm > 0: Under Assumption 1, as u0(c) is strictly de-
creasing, for eachm > 0, the mappingH is well-de�ned, bounded, strictly increas-ing, and it has the following important boundary properties: (i) limm!0H(m) = 0,and (ii) limm!f H(m) = f = m
u:
Using the function H, for each m 2 M; next consider the mapping ZA:K�K�K�M based, again, upon an on equation (6) as follows:
ZA(x; k; k;m) = �1
x+ �
R� (y1 �H(x))
m(y1(k; k)�H(x); y1 �H(x))
De�ne a new nonlinear operator bAm(k; k) implicitly in ZA(x; k; k;m) as follows:bAm(k; k) = fx�(k; k; ~m) j bZA(x; k; k;m) = 0 for ~m > 0, 0 elsewhereg:
In Lemma 5, we show the operator bAm(k; k) is well-de�ned, transforms thespace M into itself, and has strong geometric properties when restricted to thedomain M.
16Although the focus here is on economies with bounded state spaces, Morand and Re¤ett[40] extend Coleman [11][13] to the case of unbounded state spaces, and power utility. Furthergeneralizations are also available. See Morand, Re¤ett and Wang [41]
Lemma 5. Under Assumptions 1, 2, 3(i) and 3(iii), bAm(k; k) is well-de�ned inM(X), with A(m)(k; k) 2 M, and H( bAm)(k; k) 2 C(X): Finally, bAm(k; k) 2M(X); bAm(k; k) isotone, pseudo concave, and k0-monotone.Proof: As ZA is (i) strictly increasing in x; each (k; k;m); m > 0; bAm(k; k) is
well-de�ned. For k1 � k2 > 0; the second term in ZA falls. Therefore, for such(m; k); have A(m)(k1; k1) � A(m)(k2; k2): Noting the de�nition of bAm(k; k); underAssumption 2 and 3(i), A(m)(k; k) is isotone in (k; k): Therefore, A(m)(k; k) 2M:To see A(m)(k; k) is such that H(A(m))(k; k) 2 C; simply note that as
A(m)(k; k) is increasing in k; when k1 � k2 > 0; m > 0; then as the �rst term ofZA must fall, A(m)(k; k) must be such that the second term of ZA falls; hence, asm 2 M; A(m)(k; k) is such that y1(k; k) � H(A(m))(k; k) is increasing in k: AsH(A(m))(k; k) is also increasing in k; A(m)(k; k) is such that H(A(m))(k; k) 2 C:Let m0 � m;m > 0 and k > 0: As ZA is strictly decreasing in m, for such
(m; k), A(m0)(k; k) � A(m)(k; k): Again, noting the de�nition of bAm(k; k) else-where, we have A(m)(k; k) isotone onM.Finally, that A(m)(k; k) is pseudo-concave and k0�monotone follows from
Coleman ([13], Lemma 3 and 4, respectively). �
We now are ready to prove the following important extension of the mainuniqueness theorem in Coleman [13]:17
Theorem 6. Let h� be the unique positive �xed point in Proposition 4. Then,under Assumptions 1, 2, 3(i) and 3(iii), the set of �xed points of bAm(k; k) is M
A
={0, m�g � M(X); with m� > 0 when k > 0. The iterations infn bAn(mu)!m�
where the convergence is in order and topology, where H(m�)(k; k) = h� 2 C(X):
Proof: First, consider m 2 M(X); m > 0, k > 0: Under Assumptions 1,2, and 3(i), as cZA is strictly in x, x 2 K 2 R. Therefore, cZA (x; k; k;m) isupper-semicontinuous from the left, and lower semicontinuous from the rightin x. By Assumptions 1 and 2, we have additionally limx!0 cZA = +1 and
17Also, it is important to note that Coleman [11] and others have studied economies withstochastic Markov shocks de�ned in a discrete shock space. Therefore, obviously, our uniquenessresult extends with a trivial modi�cation to such economies.
limx!f bZA = �1: Hence, at all such points (k;m); by Guillerme�s coincidence the-orem (Guillerme [24], Theorem 3), there exists a root bAm(k; k) = x�(k; k; ~m) suchthat cZA (x�; k; k;m) = 0. Further, as cZA is strictly increasing in x, bAm(k; k) =x�(k; k; ~m) is unique. Noting bAm(k; k) = 0 else, bAm(k; k) is well-de�ned inM(X).Next, note that the minimal �xed point of A(m)(k; k) is by de�nition 0: To
establish the only other �xed point of cAm(k; k) is m�(k; k) with m� > 0 whenk > 0; using the de�nition of m, we have:
H(m0(k; k)) =1
u0(c0(k; k)):
By the de�nition of the operator A(m)(k; k) in equation (3.2), we have
1bAm0(k; k)= �
(R� (y
1(k; k)�H( bAm0(k; k))
m0(y1(k; k)�H( bAm0(k; k))
)or, equivalently (from the de�nition of c0):
1bAm0(K; z)= �fR� (y1(k; k)�H( bAm0(k; k)))
� u0(c0(f(k; k)�H( bAm0(k; k))g:
Therefore, by construction, Ac0 satis�es:
u0((Ac0)(k; k) = �fR� (y1(k; k)� Ac0(k; k))�u0(c0(y1(k; k)� Ac0(k; k))g:
By the uniqueness of bAm0; it must be that 1= bAm0 = u0(Ac0) (or, equivalently,H( bAm0) = Ac0)). By induction, for all n = 1; 2; :::; Anc0 = H( bAnm0). Hence, a�xed point of cAm(k; k) corresponds with a �xed point of A(h)(k; k):Next, we prove a �xed point ofA(h)(k; k) corresponds to �xed point of bAm(k; k):
To see this, consider an x such that Ax = x, and de�ne z = 1=u0(x) (or, equiva-lently H(z) = x). By de�nition, x satis�es:
u0(x(k; k)) = �fR� (y1(k; k)� x(k; k))� u0(x(y1(k; k)� x(k; k)g for all (k; k):
Substituting the de�nition of y into this expression, we have:
1
y= �
R� (y1(k; k)�H(z(k; k))
z(y1(k; k)�H(z(k; k)) ;
hence, z(k; k) is a �xed point of bA. Therefore, h�(k; k) 2 A , m� 2 A:As bAm(k; k) is pseudo-concave and k0-monotone, it has at most two �xed
points, one non-zero (e.g., Coleman ( [13], Theorem 5). Therefore, the �xedpoint set of A(m)(k; k) is M
A=f0;m�g; with m� = 1
u0(c�(k;k))) for c�(k; k) > 0
when k > 0: Hence, H(m�(k; k)) = c�(k; k) 2 C:Finally, to show uniform convergence of the iterations infn bAn(mu) = limn
bAn(mu)! m�; �rst note that for any m 2 M; as A(m)(k; k) is increasing in k; we havethe following inequality when k1 � k2 > 0:
� 1
A(m)(k1; k1)+
�R� (y
1(k2; k2 �H(A(m)(k2; k2)))m(y1(k2; k2)�H(A(m)(k2; k2)); y1(k2; k2 �H(A(m)(k2; k2)))� 0
Hence, A(m)(k; k) is such that y1(k; k)�H(A(m)(k; k) is increasing in k: Hence,A(m)(k; k) implies H(A(m)) 2 C as f� H(A(m)) increasing in k: Hence, each el-ement of a {An(mu)(k; k) equicontinuous set such thatH(A(m)) and f�H(A(m))has maximal variation f 0(k) at each k > 0: Therefore, by Dini�s theorem, for eachk > 0; as the limiting functionm� is continuous, and limn
bAn(mu)!m� uniformlywhen k > 0: Noting the de�nition of A(m) when k = 0, this convergence is uni-form on X. Finally, convergence in order is implied by the fact that pointwiseand uniform convergence coincide in C, and {An(mu)(k; k)g forms a subchainin C with sup and inf operations for {An(mu)(k; k)g equal to pointwise/uniformlimits. �
Theorem 6 extends the uniqueness result in Coleman [13] In particular, ele-mentsm 2M(X) do not require monotonicity of either investment, nor continuityof either consumption or investment. All that is required is that consumption bemonotone (jointly) in (k; k) continuous in its �rst argument, bound in its secondargument.
3.3. More Continuous Recursive Equilibrium
We next propose a second �xed point procedure that veri�es the existence of re-cursive equilibrium potentially outside the class of continuous functions consideredin the Coleman procedure. The result is not inconsistent with Theorem 6, as thenew existence argument is proven relative to a domain of continuous functions notincluded in the class of functions where the uniqueness argument of Theorem 6is considered (namely,M). As our new operator does not have the requisite con-cavity and monotonicity properties needed to guarantee the existence of unique�xed points in its domain; we cannot, therefore, guarantee that �xed points ob-tained via policy iteration will be the only recursive equilibrium for economieswith progressive taxes..The new procedure is a "two-step" continuation method. To de�ne the method,
�x a pair of functions h 2 USC and h = h(K;K) 2 Bf ; and consider the mappingZB(x; k; k; h; h) :
ZB(x; k; k; h; h) = �u0(y2(k; k)� x) (9)
+ �u0(r(h(k; k))x+ w(x)� h(h; h) �R� (x)
where, in the de�nition of ZB; we have used the fact that that y2(k; k) = r(k)k+w(k) = f(k): For �xed h 2Bf ; de�ne a "�rst-step" nonlinear operatorB1(h)(k; k; h)in the space USC discussed in Section 3 in Lemma 2 implicitly in the household�sequilibrium Euler equation as follows:
B1(h)(k; k; h) = f if @ an x st r(h(K))x+ w(x)� h(h(K;K); h(K;K)) > 0;when k > 0; h(k) > 0; h 2 Bf
= x�j for ZB(x�; k; k; h; h) = 0; else, when k > 0; h(k) > 0; h 2 Bf
= 0 , elsewhere.
The following lemma characterizes the properties of the "�rst step" operatorB1(h)(k; k; h) = B1(h; h) :
Lemma 7. For h = h(K;K) 2 Bf ; for each h 2 USC; B1(h; h) 2 USC: Further,B1(h; h) jointly increasing on in (h; h(K;K)) 2 USC�Bf ; for each (k; k) 2K�K:
Proof : Under Assumptions 1 and 2, ZB is strictly decreasing in x 2 K � Rwith ZB ! 1 for x ! 0 and ZB ! �1 as x ! m2: Noting the de�ni-tion of B1(h; h) elsewhere, we conclude B1(h; h) is well-de�ned. Further, notingthat ZB is upper-semicontinuous and increasing in (k; k) (hence, right continuousin k), and ZB is continuous in x; the root x�(k; k; h; h) is right continuous andmonotone (hence, upper semicontinuous) in k Finally, for each (k; k); k > 0; wehave ZB(x; k; k; h1; h1 )� ZB(x; k; k; h2; h2 ) when (h1; h1) � (h2; h2): Therefore,B1(h1; h1) � B1(h2; h2) for such k: Noting the de�nition of B1(h; h); elsewhere,B1(h; h) is jointly increasing in (h; h) on USC�Bf for each (k; k):�
Let USCB1 (h) � USC be the �xed point correspondence for the operatorB1(h; h) at h 2 Bf . We now prove an important lemma concerning the �xedpoint set of the "�rst-step" of our modi�ed policy iteration procedure18 :
Lemma 8. The �xed point set USCB1 : Bf ! 2USCn? is a nonempty completelattice-valued correspondence, with USCB1 (h) ascending in Veinott strong set or-der on Bf . Further, for each �xed point h� 2 B1(h); h� 2 C(X): Addition-ally, for �xed h 2 Bf ; the iterations limnB
1n(0; h) ! infnB1n(0; h) = h�(h) =
^USCB1 (h) 2 C, such that h�(h) > 0 and f(k)� h�(h) > 0 when k > 0; where theconvergence is uniform. Finally, the selection B2^(h) = ^USCB1 (h) is an increasingselection of USCB1 (h):
Proof: As B1(h; h) 2 USC(X); isotone in h, each (k; k; h); and USC(X) ascomplete lattice, by Tarski�s theorem, USCB1 (h) is a nonempty complete lattice.Further, that the �xed point correspondence USCB1 (h) is ascending in Veinott�sstrong set order follows from a theorem in Veinott ([55], Theorem 14, Chapter4).19
Let h�(h) 2 USCB1 (h): By the de�nition of B1(h; h), when k > 0;
h�(h) = B1(h�; h) =
inffx�(k; k; h�; h); f)19See also Topkis ([53], Theorem 2.5.2).
where ZB(x�(k; k; h; h); k; k; h; h) = 0 . Therefore, when k1 � k2 > 0; using thenotation h�k(h) = h
�(k; k; h), and x�k(h) = x�(k; k; h�; h); the following:
u0(y2(k; k)� x�k(h))��u0(r(h�k1(h)h
�k1(h) + w(h�k1(h))� h(h; h) �R� (h
�k1(h)) �
u0(y2(k; k)� h�k2(h))��u0(f(h�k2(h))� h(h; h) �R� (h
�k2(h)
= 0
as r(h�)h� + w(h�) = f(h�) by the de�nition of income process y2, and h�k(h) isincreasing in k:Therefore, if B1(h�(h); h) = x�k(h), we have B
1(h�(h); h) such that u0(y2(k; k)�h�k(h)) is decreasing in k. Therefore, y
2(k; k) � h�k(h) must be increasing in k:Hence, B1(h�(h); h) 2 C: Further, if k > 0; then B1(h�(h); h)=f , hence wetrivially have y2 � B1(h�(h); h) increasing in k: Therefore, when k > 0; forh� 2USC(X)) B1(h�(h); h) 2 C(X): Elsewhere, B1(h�(h); h) = 0 2 C: There-fore, for all k 2 K; h�k(h) 2 USC) h�k(h) = B
1(h�(h); h) 2 C(X). Finally, asthe subset C(X) is compact in USC(X));we have any �xed point h�k(h) 2 C.To conclude the proof, as for each h = h (K;K); the iterations limnB
1n(0; h)!h�k(h) = ^USCB1 (h) form a monotone sequence of continuous function with thelimit h� continuous, hence, by Dini�s theorem, the convergence is uniform.�
We next construct a new operator from the �xed point correspondence of our"�rst-step" operator, say B3(h)(k; k); as follows
B3(h)(k; k) = B2^(h)(k; k); h 2MA; h < f; k > 0
= f; h = f; k > 0
= 0; else
That is, B3(h)(k; k) is the greatest selection of the �xed point correspondence forthe �rst stage operator, B2^(h) = h
�(k; k; h(K;K)); when this greatest �xed point
is evaluated at k; i.e., the function h = h(k; k): Therefore, the operator B3(h) isnot necessarily an element of C.In Theorem 9, we now show the existence of additional recursive equilibrium
inMA(X). Further, we show that when B3(h)(k; k) is restricted to C, B3(h)(k; k)has a (unique) positive �xed point in C (hence, by Theorem 6, Coleman�s policyiteration procedure is robust to alternative �xed point procedures.Let M
A
B3 be the set of �xed points of B3(h)(x): We have the following:
Theorem 9. The operator B3 : MA ! MA is isotone in MA: Therefore, its�xed points M
A
B3 form a chain complete set, with ^MA
B3 2 MA a continuousrecursive equilibrium. Further, when the operator B3(h)(k; k) is restricted to C,M
A
B3 = f0; h�g � C:
Proof: Let h 2MA; with h < f; k > 0: Let k1 � k2 > 0: As h 2MA; and B3
is increasing in k; we have
j�u0(f(B3(k1; k1)� h(h(k1; k1); h(k1; k1)) �R� (B3(k1; k1)��u0(f(B3(k2; k2)� h(h(k2; k2); h(k2; k2)) �R� (B3(k2; k2)j � 0
Hence, B3(k; k) must be such that
ju0(y2(k1; k1)�B3(k1; k1))� u0(y2(k2; k2)�B3(k2; k2))j
Using the de�nition of the inverse marginal utility in (3.2), using cm = y1 � B3;de�ning mB3 to be the implied inverse marginal utility at cm; this implies
jmB3(k1; k1)�mB3(k2; k2)j �1
u00(mu)
Noting the de�nition of B3 elsewhere, B3(k; k) 2MA:To see B3(h) is isotone, note that when h0 � h; the second term of ZB rises
(noting that the least �xed of B1(h; h); B2_(h) , rises by Veinott�s �xed pointcomparatives statics result (also, Topkis ([53], Theorem 2.5.2). Therefore, notingthe de�nition of B3(h) elsewhere, B3(h) is isotone inMA:AsMA is chain complete, by Markowsky�s �xed point theorem ([32], Theorem
9), the �xed point set for B3; MA
B3 ; is chain complete.
To complete the proof of the �rst part of the Theorem, appealing to the Inadaconditions in Assumptions 1 and 2, the greatest �xed point has ^MA
B3 > 0 whenm > 0; and k > 0. By the local Lipschitz continuity ^MA
B3 near each k, theimplied �xed points for consumption and investment at ^MA
B3 near each k areLocally Lipschitz continuous. Under Assumptions 1, 2, 3(i), all the primitive datathat de�nes ZB is locally Lipschitz continuous. Further as m 2MA, m is locallyLipschitz. As it is known in our setting, local Lipschitz structure is closed undercomposition,20 we have ZB3 locally Lipschitz in k at _M
A
B3 : As under Assumption1, for any element of Clarke partial @mZ , the element does not vanish (as, forexample, the Clarke gradient of the �rst term does not vanish), by Clarke�s implicitfunction theorem, as ^MA
B3 = B3(h�) = B2^(h�; h�)(k; k) is just the root of ZB3 ;
when ^MA
B3 > 0; k > 0; ^MA
B3 )(k; k) is locally Lipschitz near any such k: Notingthe de�nition of ^MA
B3 elsewhere, ^MA
B3 (k; k) is continuous.Finally, the continuity of equilibrium decision rules guarantee we evaluated
the pair of conditions( 5) and (6) in the de�nition of a recursive equilibrium,and verify they are satis�ed with a�(k; k;^B3) being the optimal solution atV �(k; k;^MA
B3 ).To see that when h 2 C; ^MA
B3 = h� 2 C; notice �rst that for such h; B3(h)
in ZB3 in
u0(y2(k1; k1)�B3(k1; k1))��u0(f(B3(h)� h(h(k1; k1); h(k1; k1)) �R� (B3(k1; k1)
is now increasing in (k; k) such that f � B3 is increasing in (k; k):Therefore,B3 2 C; additionally. Therefore, noting that in Lemma 2, C is a complete lattice,as B3 is isotone, the �xed point set B3 � C: Therefore, ^M
A
B3 2 C. It can easilybe veri�ed that appealing to obvious modi�cations of the arguments in Theorem 6,the �xed points of B3(h) for h 2 C can be related one-to-one with the �xed pointsof A(m)(k; k) de�ned in theorem in M: Hence, MA
B3 = f0; h�g as in Proposition4 �
4. Recursive Equilibrium with Regressive Taxes
We now consider the nonexistence of continuous recursive equilibrium in the ex-ample studied in Santos [51] for case of regressive income taxes. For this case,20If f : I1 ! I2; and g : I2 ! R; f and g Lipschitz any I1 and I2 compact in (0; ku], then
g � f is Lipschitz on I1.
we develop a two-step modi�cation of Coleman�s policy iteration procedure toestablish the existence of (locally Lipschitz) continuous recursive equilibrium inBf : To do this, consider the following modi�cation of Coleman�s procedure: forh 2 C(X); and h(K;K) 2 C(X); de�ne the mapping ZAS(x; k; k; h; h) as follows
ZAS(x; k; k; h; h) = u0(y1(k; k)� x)� �u0(y1h)(x; x))r(x)(1� �(h(K;K)))`
where, under Assumption 3(ii), the income tax �(K) is now assumed to be de-creasing and Lipschitzian in its argument. Fixing both h and h, we can de�nea nonlinear operator AS(h)(k; k; h) = AS(h; h) implicitly in ZAS(x; k; k; h(K)) ath = h(K;K) as follows:
AS(h; h) = x� s.t. ZAS(x�(k; k; h; h(K)); k; k; h; h) = 0;
h > 0, k > 0; all K
= 0 else.
For �xed (h; h) 2 C�C; we �rst prove some basic properties of the operatorAS(h; h) :
Lemma 10. For (h; h) 2 C�C; at h = h(K;K); AS(h; h) 2 C. Further, when
k > 0; k = K; As(h)(k; k; h(k; k)) is locally Lipschitz:
Proof: For k = 0; and (h; h) 2 C�C;the operator is well-de�ne. Fixh(K) 2 C, k > 0: For such points, the fact that the operator AS(h)(k; k; h(K))is well-de�ned, isotone, and continuous in the uniform topology in h follows fromColeman ([11], Proposition 4). Further, if K� = (0; ku]; under Assumptions 1,2, 3(ii)-(iii), noting local Lipschitz continuity is pointwise closed under compo-sition in this context, for (h; h) 2 C�C; the mapping ZA(x; k; k; h; h) is (x; k)jointly locally Lipschitz on K� �K�. Further, under Assumptions 1 and 2, not-ing y1(k; k) = f(k), as each partial Clarke gradient for ZA in x does not vanish,i.e., the Clarke generalized gradient @xZA is of full rank, by Clarke�s Implicit
Function Theorem, x�(k; k; h; h(k; k)); k; k; h; h(k; k)) = AS(h)(k; k; h(k)) is lo-cally Lipschitz near each k > 0.21�
We now study the monotonicity of the operator AS(h)(k; k; h(k)) . In partic-ular, we prove the mapping AS(h)(k; k; h(K)) is jointly isotone on C�C :
Lemma 11. AS(h)(k; k; h(k)) increasing jointly in (h; h).
Proof: When h > 0; k > 0; as ZAS(x; k; k; h; h) is strictly increasing in x;and decreasing jointly in (h; h) 2 C�C; AS(h)(k; k; h) is jointly isotone forsuch (h; h); when k > 0: Noting the de�nition of AS(h)(k; k; h(K)) elsewhere,AS(h)(k; k; h(K)) is jointly isotone on C�C:�
Let CAS (h)(k; k) be the set of �xed points of AS(h)(k; k; h(K)) at h(K) 2
C(X) when k = K: We have the following result:
Lemma 12. CAS (h)(k; k) is a nonempty complete lattice for each h 2 C(X).Furthermore, BS(h) = ^hCAS(h)(k; k) is an increasing selection.
Proof: As AS(h)(k; k; h(K)) is isotone in h on C, each 2 C; the �xed pointcorrespondence CAS (h)(k; k) is a nonempty complete lattice follows by Tarski�stheorem ([52], Theorem 1). By Veinott�s �xed point comparative statics result inthe appendix (e.g., Veinott [55], Theorem 14, Chapter 4), CAS (h)(k; k) is strongset order isotone jointly in the parameter h(k; k) with all isotone selections forminga complete lattice. That ^hCAS (h)(k; k) is an increasing selection follows fromthe fact that as CAS (h)(k; k) is also complete lattice-valued that is ascending in
21By a theorem in Matou�kova (e.g, [34], Theorem 2.4), as AS(h)(k; k; h(k; k)) is Lipschitzon each closed I � (0; ku]; there exists a Lipschitz extension of AS onto [0; ku] with the sameLipschitz module. As we really only need this property in equilibrium, we defer this issue to theexistence of equilibrium result in Theorem 10 below.
Veinott�s strong set order, hence, ^hCAS (h)(k; k) is well-de�ned an is the leastincreasing selection for investment. �
Let CBS be the �xed point set of the operator BS : C! C de�ne in Lemma
12:We now prove the main theorem of this section, namely, that our modi�cation ofColeman�s policy iteration algorithm converges to a greatest continuous recursiveequilibrium:
Theorem 13. Under Assumptions 1, 2, and 3(ii)-(iii), there exists complete lat-ticeCBS of �xed points ofB
S(h) inC(X): Furthermore, the iterations limnBSn(0)!
h� = ^CBS : Finally, h� a recursive equilibrium that is continuous, locally Lipschitzwhen k > 0, and convergence of limn ^BSn(f)! h�is uniform on X.
Proof: As BS(h) is isotone in C, C a complete lattice, the set BS is anonempty complete lattice by Tarski�s theorem. Further, by the isotonicity ofB(h); the iterations from the maximal element of C, namely ^C =0 is the minimalinvestment (or, maximal consumption associated with a 1 period economy): Then,limnB
Sn(0)gn ! h� = ^BS .As the limiting �xed point AS(h�(k; k); k; k; h�(k; k)) is de�ned implicitly in
ZAS , when k > 0; from Lemma 10, the �xed point h�(k; k) =AS(h�(k; k); k; k; h�(k; k))is locally Lipschitz when k > 0.Finally, for k 2 [0; ku]; let d1(k) = 0; and d2(k) = f: Notice, d1 � h� � d2
for all k 2 (0; ku]: Hence, this holds for all closed subsets I � (0; ku]: UnderAssumption 2, both d1 and d2 are pointwise bounded, such that 9c > 0; such thatfor all k1; k2; d1�d2 � cjk1�k2j for all (k1; k2) 2 [0; ku]� [0; ku]: Therefore, by atheorem in Matou�kova ([34], Theorem 2.2), h�(k; k) is a continuous extension onto[0; kU ], with h�(k; k) at k = 0. Further, as {BSn(0)gn is an increasing sequence ofcontinuous functions converging pointwise to a continuous function h� = ^BS .Hence, by Dini theorem, this convergence is uniform (also, see Amann [3], Theorem6.1).�
We make a few remarks on this theorem. First, there is no claim to uniquenessof RE. Second, and perhaps most importantly, in the next section, we shall showsolutions to the equation (6) for this economy with regressive taxes that are notrecursive equilibrium (e.g., see Lemma 15 in the next section). So approximatinga continuous recursive equilibrium for this economy could prove very di¢ cult as
the set of selection k0 = �(k) that satisfy the equilibrium functional equation (15)could be very large. This would prove particularly di¢ cult for correspondence-based methods not valued in function spaces.Finally, it important to note that Santos ( [51], Section 3.2) never establishes
the existence recursive equilibrium in his example; rather, he only considers thenonexistence of continuous recursive equilibrium on a state space of K. As weshall show in the next section, discontinuous solutions to the equilibrium Eulerequation of the household�s can be constructed. They are not RE. Further, Itbears mentioning that in Santo�s proof, to character the local properties of acontinuous RE near a particular unstable steady state, it appears he calculateseigenvalues needed for this characterization applying the Grobmann-Hartmanntheorem. Indeed, it is precisely this application that obtains the necessary con-tradiction with the continuity of RE on an open manifold near this "spiral sink".It is important to remember that the Grobmann-Hartman theorem in his contextwould assume the local RE decision rule is a smooth dynamical systems mappingin C1�manifold.
5. Correspondence-based Continuation Methods
We �nally consider a correspondence-based continuation methods for constructingrecursive equilibrium. Our main focus in this section will be on economies witha state contingent regressive taxes (as this case has been studied extensively us-ing correspondence-based continuation methods in the work of Santos [51], Miaoand Santos [35] and Feng, et. al. [21]). In the Miao-Santos correspondence-based continuation method, instead of parameterizing the continuation structureof the economy in the previous section with functions, we use correspondences,say G(x) 2 G(X), where G(X) is a complete lattice of correspondences underthe set inclusion partial order. We will de�ne a mapping in G essentially as fol-lows: (i) given an element G(x)2 G, we can solve the �rst order conditions forall solutions to the household�s Euler equation that are consistent with this im-plied continuation structure for the economy; then, (ii) we use these solutions tode�ne mapping, say T (G)(x) 2 G(X); that returns the updated values of thesecontinuation variables today: We then compute �xed points of this "set to set"mapping.It will turn out that for the regressive tax case, the critical complication for the
correspondence-based approach is the fact that under Assumptions 1 and 3(ii),
the function
�(x; k) =u0(y1(k; k)� x)(1� �)(x)
does not exhibit any particular pattern of monotone comparative statics in thepair (x; k): Hence, for each (k; k); the set of solutions to a modi�ed version ofour equation ZAS in equation (4) continuing to an element g 2 G(k0; k0) will becorrespondence T (G)(x) that is simply nonempty, upper semicontinuous corre-spondence and preserves compactness. Unfortunately, such a correspondence willnot admit continuous selection in its �rst argument, in general, and, hence gener-alized Markovian decision rules, say k0 = x�(k; k; g�(k; k)); cannot be guaranteedto be optimal solutions to a strictly concave dynamic programming problem forthe household in its individual state k in equation (5). This will be core issuewith generating situation where the Miao-Santos procedure fails guarantee theexistence recursive equilibrium selections without additional arguments.
5.1. The Miao-Santos Procedure
To de�ne the Miao-Santos operator T (G)(x); consider for a subset D � R; thecollection of subsets D0 = 2D, where D is compact. It is known that the pair(D0;�) is a continuous lattice (hence, a complete lattice) under the set inclusionpartial order. Endow (D0;�) with the Hausdor¤metric. Under this metric, D0 isalso a complete metric space. De�ne the following set of correspondencesG � D0;de�ned as follows:
G= G(X) = fG(x)jG : X! D0; G(x) � Gu(x) = D 8 x = (k; k) 2 X;G(x) a nonempty, compact-valued, and upper-semicontinuous in xg
When de�ning G, we require a top element, say Gu. When applying the Miao-Santos procedure to our economies with Inada conditions, this element can oftenbe challenging to construct without prior knowledge of the actual recursive equi-librium. For the moment, we shall assume that for our economy with a regressivetax, such a greatest element Gu = D can be speci�ed for all (k; k) such that k > 0.22.22It is not clear often how to do this (for example, in our economy). In Miao and Santos [35]
and Feng et. al. [21], the avoid this question by endowing agents with an interior income point.For even simple models like ours, no such element exists.
Consider an operator T (G)(k; k) � G mapping in spaces of correspondencesde�ned implicitly in the equilibrium version of the household�s Euler equation (6)as follows:
T (G)(k; k) = fg0j9x� st u0(y1 � x�)(1� �)(x�) � �g; g 2 G(k
0; k0) 2 G (10)
k0 = x�; g0 = u0(y1 � x�)f 0(k); x�(k; k; g) 2 X�(k; k; g); y1 = y1(k; k)g
where x� = x�(k; k; g) 2 X�(k; k; g) is an implied selection for investment decisionin equilibrium de�ned on the expanded state space that includes the "shadowvalue" of household capital holdings in equilibrium, namely g. That is, when g 2G(k0; k0) is an equilibrium envelope in a recursive equilibrium tomorrow, today�sdecision rules for investment in equilibrium will be selections from X�(k; k; g).Therefore, a continuation g will induce an auxiliary state variable in the decisionrules for equilibrium policies k0 = x�(k; k; g). This is precisely the "generalizedMarkov equilibrium" structure that is studied in the literature (e.g., see addition-ally for example Phelan and Stacchetti [45] and Kubler and Schmedders [28]).As shall be mentioned in a moment (e.g., in Proposition 14 below), it is known
that the operator T (G)(x) maps the space of correspondences G into itself, andis isotone under the set inclusion order. Hence, T (G)(x) has a complete latticeof �xed points by Tarski�s theorem (ordered under set inclusion). Further, underAssumptions 1 and 3(ii) on the Lipschitz structure of u0 and � ; as the mappingZT de�ned as
ZT (x; k; k; g) =u0(y1 � x)(1� �)(x) � �g
is jointly continuous in all its arguments, by a standard argument, T (G)(k; k) is(pointwise) Hausdor¤ continuous. This implies T (G)(k; k) is order continuous onG.23 Therefore, by the Tarski-Kantorovich theorem (e.g, Dugundji and Granas[19], Theorem 4.2), T (G)(k; k) will have a greatest �xed point in the down-set{GjG 2 G; G � Gu = Dg under set inclusion that is computed as
infnT n(Gu)(k; k)! G�(k; k)
Finally, given "self-generation" arguments �rst discussed in Abreu, Pearce, andStacchetti (e.g., [1][2]), it turns out that our interest is only on this greatest �xedpoint G�(x):
23 See Re¤ett [47] for discussion.
Given the existence of such an upper bound Gu = D 2 G; Miao and Santos[35] and Re¤ett [47] have proven a number of results concerning iterative methodsbased upon the correspondence-based operator T (G)(k; k):We state the key factsin the next proposition proved in Miao and Santos [35]:
Proposition 14. (Miao and Santos [35]; Feng et. al. [21]). For each (k; k) 2 X;k > 0; T (G(k; k)) 2 G is isotone on G under set inclusion. Further, if 9 Gu 2 Gsuch that Gu(k; k) � T (Gu)(k; k) under set inclusion, for all (k; k); then, theiterations limn T
n(Gu)(k; k) ! infn Tn(Gu)(k; k)G�(k; k) (where the inf is with
respect to set inclusion); and G�(k; k)=G� the greatest �xed point of T (G)(k; k);where the convergence both in the Hausdor¤.
Therefore, the question of existence of recursive equilibrium is now reduced toguaranteeing the existence of selection g�(k; k) 2 G�(x) such that the generalizedMarkov equilibrium decision rule k0 = x�(k; k; g�) 2 X�(k; k; g�) is a recursiveequilibrium per our de�nition in Section 2.A few remarks on Proposition 14. First, and most importantly, Proposition
14 is not su¢ cient to establish the existence of a recursive equilibrium. That is,although T (G)(k; k) maps G into itself, G� 2 G does not guarantee the existenceof selection g�(k; k) 2 G�(k; k), such that the implied decision rule for investment,for example, namely k0 = a�g� = a
�g�(k; k; g
�(k; k)) = x�(k; k; g�(k; k)) satis�es thefollowing two necessary and su¢ cient conditions for the existence of a recursiveequilibrium when k > 0:
V �(k; k; g�) = supy2i(k;k)
fu(y1(k; k)� a�g�) + �V �(a�g� ; g�)g
u0(y1(k; k)� a�g�)� �u0(y1g�)(a�g� ; a�g�)) r(a�g�)(1� �(a�g�) = 0Unfortunately, the nonexistence of a selection g�(k; k) 2 G�(k; k) continuous its�rst argument (e.g., Aubin and Frankowska [6], Example, p. 358) implies thatProposition 14 does not verify the existence of a recursive equilibrium.Second, it should also be mentioned that in studying the structure of the house-
hold�s �rst order necessary and su¢ cient conditions in (6), for a�g�(k; k; g�(k; k))
to be continuous in its �rst argument, y1g�(k0; k0) must be continuous in its second
argument. One su¢ cient condition guarantee this continuity condition of y1g� in its
second argument is to have the recursive equilibrium a�g�(k; k; g�(k; k)) be jointly
continuous in (k; k): Hence, it is not clear what class of discontinuous recursiveequilibrium are possible given the restrictions imposed by equations (5) and (6)along equilibrium trajectories.Third, the topological convergence result as proved in Miao and Santos [35]
can be shown to imply the convergence of T (G)(k; k) in an important order topol-ogy (namely, the Scott topology) in the context of the set inclusion partial. Inparticular, it implies the operator T (G)(k; k) is Scott-continuous on G. In Ref-fett [47], this fact is important when addressing the question of the existence ofcontinuous models of computation for recursive equilibrium in dynamic economieslike the one considered in this paper. One problem that is noted in that paperis that operators such T (G)(k; k) that map in domains of correspondences thatare valued in simple Euclidean spaces, as opposed to function spaces, are notScott continuous relative to the set of recursive equilibrium selections in G�(k; k);rather, they are only Scott-continuous relative to greatest �xed point of T (G)(x)inG: This implies in our context, the Miao-Santos procedure (even in our contextof Cass growth with a regressive tax) is not constructive. As a key motivation forcorrespondence-based methods is numerical, such a loss of continuity per the setof recursive equilibrium poses a signi�cant challenge to rigorously operationalizingthe Miao-Santos procedure for the construction of both approximate and actualrecursive equilibrium solutions.24
5.2. Not all Selections in G� are Recursive Equilibrium
We now construct selections in the Miao-Santos equilibrium correspondenceG�(k; k)in Proposition 14 that do not induce recursive equilibrium decision rules for in-vestment. To do this, we construct a new operator TB(h)(k) that studies solutionsto the household�s Euler equation in the regressive tax case that are (i) a sub-set of the bounded feasible decision rules Bf studied in Lemma 2, but are (ii)not su¢ ciently continuous to be consistent with being an optimal solution to thedynamic program in equation (5). Such solutions that are not RE can easily beconstructed.24For example, in Feng et. al. [21], a discretized version of the step function approxima-
tion scheme for correspondences based upon Beer [7], has been proposed. That approximationscheme, although novel for constructing set approximations for upper semicontinuous correspon-dences, is silent on rigorously constructing approximate solutions for appropriately continuousselections g�(k; k) 2 G�(k; k) that are actually recursive equilibrium.
To do this, let h 2 Bf : Then when h > 0; k > 0; consider the mappingZTB(x; k; h) =
ZTB(x; k; h) = u0(f(k)� x)� �u0(r(h)x+ w(x)� h(h(k))r(x)(1� �(h(k))
De�ne the nonlinear operator TB(h)(k) mapping in Bf as follows
TB(h)(k) = f if @ an x st r(h(k))x+ w(x)� h(h(k)) > 0;when k > 0; h(k) > 0; h 2 Bf
= x�(k; h) for ZB(x�(k; h); k; h) = 0; else when k > 0; h(k) > 0; h 2 Bf
= 0 , else.
We mention the following Lemma that characterizes the operator TB(h)(k) :
Lemma 15. The operator TB(k) is well-de�ned with TB(h)(k) 2 Bf : Further,TB(h)(k) is isotone on Bf : Therefore, the �xed point set B
f
TB of TB(h) is anonempty complete lattice.
We can now use the operator BS(h) to construct an iterative procedure thatgenerates the existence of (i) a limiting subset subintervals of functions of G�(k; k)de�ned Proposition 14, and (ii) elements of this subset are not necessarily RE.To do this, �rst, for k > 0; de�ne a lower bound for stationary investment by thecontinuous function to be simply hL(k) = f � x�(k; k; f ;h) in the following:
u0(x�)� �u0(f(f � x�))r(x�)(1� �(f)) = 0
Similarly, under the Inada conditions in A1 and A2, 9 a continuous hu < f; suchthat TB(hu) � hu < f: Further, both hL(0) = hu(0) = 0: This is the level ofinvestment in a two-period economy with a tax rate parameterized at h(k) = f:For such an hL; we have hL < TB(hL):Then, de�ne the set of functions (viewing h as investment) as follows:
�Bf= fh 2 Bf jhL � h � hu]}
For all k > 0; under Assumptions 1, 2, and 3(ii), have 0 < hL � TB(h) � hu < f .Next, for h 2 �Bf ; recalling the de�nition of the the inverse marginal utility mh
level constructed in section 3 in equation (7), de�ne an continuation envelopegh(k; k) for an investment function h 2 �Bf as follows:
gh(k) =r(k))(1� �(h(k)))
mh(k)
which is well-de�ned and increasing in h on �Bf : Then, the function space ofcontinuation envelopes in a candidate equilibrium h 2 �Bf for the individualscapital holdings k is given follows:
GB = fgh(k; k)jghu � gh � ghL ; h 2 �Bfg
Let I(GB) be the collection of all sub-order intervals in the space GB; namely,
I(GB) = fIGjIG = [g1; g2]; g1 � g2; g1 2 GB; g2 2 GBg [?
Endow I(G �Bf ) with the Hausdor¤ metric, so I(G �Bf ) is a complete metric space.We refer to I(G �Bf ) as the interval power domain of GB. Let I(�Bf) denote asimilar interval power domain for the set �Bf with typical (nonempty) elementIB = [h1; h2]; h1 2 �Bf ; h2 2 �Bf : Endow both I(G �Bf ) and I(�Bf ) with the reverseset inclusion order. As we shall argue in the sequel, when endowing an intervalpower domain with the reverse set inclusion order, the resulting partial order hasa nice interpretation from the perspective of best approximation. Further, bothI(G
�Bf ) and I(�Bf ) are continuous lattices (hence, necessarily complete lattices).We can now de�ne a particular iteration of a correspondence-based proce-
dure using the interval operator T (I)(x) operating in the space of subintervalsI(G
�Bf ).25 Let IGu = [0; gu] � GB; and de�ne a new operator T (IG)(x) that maps
subintervals I(G �Bf ) into themselves as follows:
T (IG) = fghjh 2 IB; IB 2 I(�Bf )
Consider interval iterations on T (IG)(x) from the top element of I(G �Bf ), namely,IGu = GB. This element be view formally as the "worst" approximation of arecursive equilibrium selection inGB in the reverse set inclusion order.26 As IGu �Gu = D (as IGu consists of selections with additional properties in G
u); IGu � Guunder reverse set inclusion, and IGu a "better" approximation than the initial guess
25See Moore [38] [39] for a discussion of interval operators.26This lower bound is, of course, the upper bound for our procedure under set inclusion.
Further, in Proposition 14, we are taking the correspondence Gu = IGu :
Gu relative to any recursive equilibrium in the space of correspondencesG orderedunder reverse set inclusion.Using equation (5.2), consider the iterations of these two di¤erent operators
from their respective top elements. That is, generate the trajectories of the in-terval operator T (IG)(x); namely {T n(IGu )(x)g1n=1; recursively from initial inter-val IGu using equation (5.2); and the trajectories of the Miao-Santos operator{T n(Gu)(x)g1n=1 de�ned using equation (10). For any orbit n; respective opera-tors are partially ordered under reverse set inclusion as follows
T n(IGu )(x) � T n(Gu)(x)
The iterations T n(IGu )(x) are just selections in Tn(Gu)(x))) by construction . Ap-
pealing to Lemma 15, and noting the order continuity of T (IG)(x), we have
infnT n(IGu )(x) = lim
nT n(IGu )(x)
= I(G
�Bf )
T
It can be shown, in particular, that the interval limit �Bf
TB induces selection g�
that can be used to de�ne "generalize Markovian equilibrium" policies for in-vestment, namely a�g�=a
�(k; k; g�(k; k)) 2 �Bf
TB de�ned in Lemma 15. Further,as both I(G �Bf ) and GB are metric spaces under the Hausdor¤ metric, relativeto any recursive equilibrium selection a�g� 2
�Bf
TB ; one can show that our intervalapproximations minorize in uniform distance a RE selection induced by a candi-date generalized Markov equilibrium selection constructed from the Miao-Santoslimiting correspondence using the topology of uniform convergence on G�Bf .Unfortunately, even with all this structure, as our interval operator only con-
structs selections a�g� 2 �Bf
TB , these selections are not necessarily RE. Therefore,this particular interval iteration scheme using T (IG)(x) de�ned in I(G �Bf ) doesnot provide a method for verifying even the existence of recursive equilibrium. Ofcourse, what the iterations T n(IGu )(x) do allow us to conclude is that withoutfurther argument, Proposition 14 also does not verify the existence of recursiveequilibrium decision rules a�g� induced for some selection g
�(k; k) 2 G�(k; k) (aswe have I(G
�Bf )
T� G�(k; k) under reverse set inclusion, and the set of recursive
equilibrium induced by selection a�g� 2 Bf
TB could be empty).
5.3. A New Correspondence-Based Method
We can now provide a very simple �x for this situation. To do this, restrict ourinterval operator T (IG)(x) to a smaller domain, say subintervals of the set �C �Bf : Further, de�ne this restriction of T (IG)(x) by using the mapping BS(h) =_hAS(h)(k; k) de�ned Lemma 12. To see the details, let the interval powerdo-main of C be de�ned by:
I(C) = fIC jIC = [h1; h2]; h1 2 C;h2 2 Cg [?
Modifying the de�nition of gh, replacing �Bf in the de�nition of GB; and restrictingthe set of continuation envelopes to be those de�ned using the operator Bs(h)de�ned in C; we can de�ne an "APS" method valued in function spaces. That is,using BS(0), recompute the candidate continuation envelopes GC as follows:
geh(k; k) = gh(k; k) for h 2 �Cwith
�C = fh 2 Cj Bs(0) � 0 � hu < fgThen, de�ne GC to be the collection of continuation envelopes:
GC = fgeh(k; k)jgehu � geh � geBS(0); h 2 �Cg
We now study the operator T (I)(x) de�ne in equation (5.2) in the smaller intervalpower domain I(GC) given by
I(GC) = fIC jIC = [g1; g2]; g1 � g2; g1 2 GC ; g2 2 GCg [?
Letting ICu = _GC ; denoting the restriction T (IB)(x) to I(GC) by Te(I)(x), wenow prove a stronger version of Proposition 14 relative to interval mapping Te(IC):Further, we can give a very simple explicit operator on the set of �xed points ofTe(I)(x); say CTe, that computes a jointly continuous selection g
�(k; k) 2 G�(k; k),such that g�(k; k) induces an recursive equilibrium decision rule a�g�(k; k; g
�(k; k))that is a generalized Markov equilibrium in the sense of Miao and Santos [35].
Theorem 16. For every n; and all x 2 X; the orbits for the operators Te(I)(x);T (I)(x); and T (G)(x) are ordered, respectively, under reverse set inclusion fromtop elements ICu ; I
Bu ; and G
u, as follows: T ne (ICu )(x) � T n(IBc )(x) � T n(Gu)(x):
Further, limn Te(ICu ) = supn Te(I
Cu ) ! ^ Te
= [0; g�] � TB � G�(k; k); wherethe sup is taken with respect to the reverse set inclusion order. Finally, g�(x)corresponds to a recursive equilibrium h� = ^BS in Theorem 13:
Proof: First, for n = 1; that we have
Te(ICu )(x) � T (IBu )(x) � T (Gu)(x)
follows from ICu � IBu � Gu; and the fact that T ne (ICu )(x), by de�nition, con-
sists of selections in C from T n(IBu )(x); and Tn(IBu )(x) consists of selections B
f inT n(Gu)(x): As reverse set inclusion is a closed on the powersets D0; this compar-ative dynamics result per iterations is preserved in the limit: i.e,
limnT ne (I)(x) � lim
nT n(I)(x) � lim
nT (Gu)(x)
Finally, as by construction, T ne (ICu )(x)=[0,
r(f�Bs(0))m(f�Bs(0)) ]; we have g
�(k; k) =r(f�^
BS)
m(f�^BS):�
6. Appendix: De�nitions and Results
To keep the paper self-contained, many de�nitions needed in the paper are nowprovided. For a more complete accounting of the material in this section, seeBirkho¤[9], Gierz, et. al. [22], and Veinott [55].
6.1. Spaces
An arbitrary set (P;�) is partially ordered set (or Poset) if P is equipped withan order relation �: P � P ! P that is re�exive, antisymmetric and transitive.If every element of a poset P is comparable, then P is chain. If P is a chainand countable, P is a countable chain. The space P op shall denote the poset Pequipped with its dual partial order �op. An upper ( respectively, lower) boundfor a set B � P is an element xu(respectively, xl) 2 P such that for any otherelement x 2 B; x � xu (respectively, xl � x) for all x 2 B: If there is a point xu(respectively, xl) such that xu is the least element in the subset of upper boundsof B � X (respectively, the greatest element in the subset of lower bounds ofB � P ), we say xu (respectively, xl) is the supremum (respectively, in�mum) ofB: Clearly if the supremum or in�mum of a set P exists, it must be unique.We say a set L is a lattice if for any two elements, say x and x0 in L; L
is closed under the operation of in�mum (denoted by x ^ x0); and supremum(denoted x _ x0):The former is referred to as �the meet�of the two points, whilethe latter is �the join�: A subset L1 of L is a sublattice of L if it contains the supand the inf (with respect to L) of any pair of points in L1: A lattice is completeif any L1 � L, upper bound (denoted _L1) and a greatest lower bound (denoted
^L1) are both in L. If this completeness property only holds for countable subsetsLc, the lattice is ��complete. In a poset P , if every subchain C � P is complete,then P is referred to as a chain complete poset (or equivalent, a complete partiallyordered set or CPO). A set C is countable if it is either �nite or there is a bijectionfrom the natural numbers onto C: If every chain C � P is countable and complete,then P is referred to as a countably chain complete poset. An order interval isde�ned to be [a; b] = [a) \ (b], a � b.
6.2. Mappings in Posets
Let (P1;�P1) and (P2;�P2) be Posets. A function (or, equivalently, operator)f : P1 ! P2 is isotone (or order-preserving) if f(x0) �P2 f(x); when x0 �P1 x; forx; x0 2 P1. A function f(x) is antitone (or order-reversing) if f(x) �p1 f(x0) whenx0 �P1 x; for x; x0 2 P1: A function that is isotone or antitone is monotone. If P1and P2 be Posets, X a set, g : X ! P2 a function, we say a function g(x) admitsan isotone decomposition f(p1; p2) if there exists a function f : P1�P1 ! P2 suchthat f(p1; p2) is isotone on P1 � P1: If X and Y are three sets, f : X �X ! Y;the diagonal of a function f(x; y) is a function g = f(x; x):Finally, a sequence fhn ! hg in H is order convergent if there exists two
monotonic sequences of elements fromH, one decreasing fh#ng; and one increasingfh"ng, such that h = inf h#n = suph"n and h"n � hn � h#n: A necessary andsu¢ cient condition for an increasing sequence hn ! h to be order convergent ish = suphn: An operator Ah is order continuous on H if for all countable chainsC 0 = fhng, _A(C 0) = A(_C 0).For a set X; de�ne by 2X the powersets of X, and L(X) the nonempty sub-
lattices of L, and L1 and L2 be two arbitrary sublattices. Let RX2 be an orderrelation on 2X :We say a correspondence F : P �! 2X2is ascending in the relationRX2 from a poset (P;�) to 2X2 if F (x0)RX2F (x); when x0 � x: If this set relationRX2 induces a partial order on the a subclass of the powersets 2
P2 ; say P (X2);and if F (x) : P �! P (X2), we refer to F (x) is a isotone correspondence. Dually,we can de�ne a descending and antitone correspondence.In this paper, we shall focus primarily on a few order relations on the powersets
2X of a set X. For an arbitrary set X, the Set inclusion Partial order �SI is thefollowing:A �si B ifB � A: Set inclusion induces a continuous lattice structure on2X with ^ = \; _ = [: If X = L is additionally a lattice, de�ne L(X) = fL1 � XjL1 a nonempty sublattice}, and let L1; L2 2 L(X): Then, de�ne Veinott�s StrongSet Order �s on L(X) : L1 �s L2; if for any a 2 L1; b 2 L2, a ^ b 2 L2 and
a _ b 2 L1: Veinott�s strong set order will be used extensively in this paper.Finally, let F : X ! 2Xn? be a non-empty valued correspondence for each
x 2 X: The correspondence F is said to have a �xed point if there exists an x�
such that x� 2 F (x�): If this correspondence is actually a function, say f(x);then a �xed point x� has x� = f(x�): A correspondence F : X � T ! 2Xn? isreferred to as parameterized correspondence. For F (x; t); denote the �xed pointset at t 2 T by XF (t) : T ! 2X : A �xed point x� 2 XF (t) is minimal (resp,maximal) in X if there does not exist another �xed point, say y� 2 XF (t), suchthat y� � x� (resp, x� � y�): If a �xed point is either minimal or maximal, we sayit is extremal. If x�L(t) = ^XF (t) exists relative to the order structure in X (resp,x�G(t) = _XF (t) relative to X), then x� is the least �xed point (resp, greatest �xedpoint) of F relative to X:
6.3. Some Useful Fixed Point Theorems
In our constructions, we shall apply various versions of Tarski�s �xed point the-orem. We begin with an interesting version of Tarski�s �xed point theorem dueto Veinott. Let X be a complete lattice, 2X the powersets of X; T a partiallyordered set, and F : X � T ! L(X) be a parameterized correspondence, whereL(X) � 2X is the collection of nonempty sublattices of X endowed with Veinott�sstrong set order. The set (L(X);�) is the largest partially ordered set in 2X :27Fixing t 2 T; let XF (t) be the set of �xed points of F (x; t) in X: We have thefollowing version of Tarski�s theorem. 28
Proposition 17. Veinott [54][55]. Let X be a complete lattice, F (x; t) 2 L(X)is nonempty, subcomplete-valued correspondence that is jointly strong set orderascending. Then (i) XF (t) is a nonempty complete lattice, (ii)
XF (t) is strong set
order ascending, and (iii) _XF (t) and ^XF (t) are isotone selections.
Tarski�s original theorem (Tarski [52], Theorem 1) occurs as a special caseof Proposition 17 where F (x; t) = f(x); and f : X ! X is a operator (i.e., afunction).
27Lc(X) is not generally a lattice (e.g., for X not distributive). If X is completely distributivecomplete (i.e., X � RI; where I is any arbitrary set, RI given the product order, and X=[0,1]I
with its relative partial order), then L(X) is a complete lattice (actually, completely distributivecomplete lattce, hence, a continuous lattice).28See Topkis ([53], Theorem 2.5.2) for proof.
There are many useful constructive versions of Proposition 17 in the literaturefor the special case that the parameterized correspondence F (x; t) is a parame-terized operator f : X � T ! X. In the �rst case, we assume for each t 2 T;the partial map ft(x) : X ! X is order continuous. For this case, we have thefollowing version of Tarski-Kantorvich-Markowsky theorem (e.g., Dugundji andGranas [19], Theorem 4.2, and Markowsky [32], Theorem 9).
Proposition 18. Let (X;�) be a CPO, f : X�T ! X isotone, 9 a xL 2 X suchthat xL � f(xL; t); andXf (t) : T ! 2X a �xed point correspondence for f; and forall t 2 T;Then, (i) XF (t) � fx 2 XjxL � xg is nonempty CPO. Further, if (X,�)is countably chain complete, and f(x; t) is order continuous on X; each t 2 T; theiterations supn f
n(xL) = �(t) 2 Xf (t); where �(t) =infx2fx2Xjx�xLgXf (t):
An important special case of the above corollary occurs when was add addi-tional structure to both (X;�) and f(x; t): We state the theorem in the contextwe shall apply it in the paper, namely when (X;�) is a compact order interval inthe space of bounded real-valued continuous functions C(Y ) de�ned on compactY � Rnendowed with (i) the topology of uniform convergence, and (ii) pointwisepartial order . This version of the theorem is proven in Amann ([3], Theorem6.1), with the continuous comparative �xed point comparative static result citedproven in Re¤ett [16]. The existence result holds in more general topologicalspaces, while the continuous �xed point comparative statics result holds in moregeneral ordered metric spaces.
Proposition 19. In Corollary 18, if additional, (a) (X;�) is a compact orderinterval in C(Y ), (b) f(x; t) continuous in x; each t2 T , then, (i) the iterationssupn f
n(xL; t) = limn f(xL; t) = �(t) 2 XF (t); and �(t) =infx2fx2Xjx�xLgXF (t):Further, if, additionally, (c) T � C(Y ), (d) f(x; t) is jointly continuous, (e) andrelative to the set X1 = fx 2 Xjx � xLg � X; for all x0 2 X1; limn f
n(x0; t) !�(t) 2 Xf (t), then (ii) �(t) is continuous on T.
Proposition 19 is a particularly useful result for many of our arguments. TheTarski-Kantorvich theorem tells us that the upper envelope (resp, dually, lower
envelope) from some least point in X (resp, any greatest point in X); order con-verges to least (resp, greatest) �xed points in X:We can make this convergence intopology by introducing the Scott topology (an order topology where the bases ofthe order topology is form from directed sets in X): This topology is not alwayseasily related to approximate solution methods. On the other hand, in Proposi-tion 19, the topology is standard in the numerical approximation literature (i.e.,uniform approximation). So the constructive result is particularly appealing inthis context.
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