General Equilibrium When Economic Growth Exceeds Discountingseaver-faculty.pepperdine.edu/jburke2/articles/General.pdf · Hence, for each positive tolerance =, under price system

Journal of Economic Theory 94, 141�162 (2000)

General Equilibrium When Economic GrowthExceeds Discounting1

Jonathan L. Burke

Department of Economics, IUPUI, Indianapolis, Indiana 46202

jolburke�iupui.edu

Received May 23, 1996; final version received October 9, 1999

After dropping the standard general-equilibrium assumption that preferenceorders discount future consumption faster than the economy grows and droppingcontinuity and weakening utility representation, we establish commodity prices andconsumptions that approach approximate equilibrium to within any practicaltolerance. The Weiza� cker-overtaking criterion defines the best-known non-standard-discounting orders we admit over discrete-time, deterministic consumption paths andover continuous-time, stochastic consumption processes. We also perturb preferences toqualify all approximate equilibrium as full equilibrium, thus showing some well-knownnon-existence examples are singular, and so are inadequate defence of standardassumptions. Journal of Economic Literature Classification Numbers: C60, C62.� 2000 Academic Press

1. INTRODUCTION

We admit new preference orders to general-equilibrium theory by findinga type of limit of approximate equilibrium without assuming orders discountfuture consumption faster than the economy grows. For a discrete-timedeterministic example, if the economy actually grows but consumptionunits are normalized to keep endowments at 1 each period, then we admitorders that negatively discount, or up-count, normalized units in the sensethat 1 normalized unit of future consumption (the future endowment) ispreferred to 1 normalized unit of current consumption (the current endow-ment). Roughly, any discounting that would have been caused byimpatience for consumption in actual units is offset by the growing endow-ment in actual units. The Weiza� cker-overtaking criterion defines the best-known up-counting orders over any infinite-horizon commodity space. For

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1 This paper extends an earlier, unpublished analysis of limit equilibrium for non-discount-ing preferences (Burke [4]) from discrete-time, deterministic commodities to general infinite-dimensional commodity spaces, while offering stronger conclusions. Thanks to an anonymousreferee for careful proofreading and provocative questions.

example, discrete-time, deterministic consumption path (xt)�0 in l� over-takes path (xt) when partial sums �T

t=1 2t(xt&xt)>0 for large T.Likewise, continuous-time, stochastic consumption process (X(t))�0in a suitable L� -space (Subsection 2.2) overtakes process (X� (t)) whenpartial (Lebesgue) integrals �T

0 2t(EX(t)&EX� (t)) dt>0 for large T, whereEX(t) is the expected value from stochastic consumption X(t) at timet # [0, �).

The reason general-equilibrium theorist's assume discounting is faster thangrowth are well-known examples showing the possibility of non-existencewithout discounting, and the failure of standard existence theorems andproofs without discounting. For example, consider a 2-person, discrete-time, deterministic, pure-exchange economy with 1 unit endowment foreach consumer in each period. Despite satisfying all standard assumptionsexcept discounting, the economy has no (exact) equilibrium when one con-sumer has discounted utility u(x)=��

1 2&txt over l +� , and the other has

an up-counting Banach limit v(x)=Lim xt [2]. In fact, for each period,transferring consumption from the up-counting consumer to the discount-ing consumer Pareto-improves utility; hence, the only Pareto optimum andonly candidate for equilibrium allocates the up-counting consumer zeroeach period, which violates standard individual rationality. However, thezero-consumption allocation is a type of limit of approximate equilibriumunder the price system p(x)=��

1 2&txt+Lim xt , where the Banach limit(Lim xt) is a norm-continuous positive linear function over l� that definesa price bubble on certain infinite-lasting consumption and endowmentpaths. Precisely, under p, each consumer's unit endowment has value 2 andthe supremum of each utility over the budget set is also 2. And supremumutility levels are approached by the approximate-equilibrium allocation ofzero to the up-counting consumer for the first 100 years followed by thetotal endowment thereafter (generating utility v(x)=Lim xt=2), with thetotal endowment followed by zero for the discounting consumer (genera-ting utility u(x)=��

1 2&txt=�1001 2&t2=2&2&99).

More generally, for a mixture of up-counting and discounting preferenceorders among a finite number of consumers of continuous-time stochasticconsumption processes, which include discrete-time and deterministic con-sumption as a special case, we will find a price system p and an allocation(xi) of consumptions that approach approximate equilibrium to within anypractical tolerance. For example, partition the first hundred years intoone-second intervals and partition states of nature into one hundredevents E. Hence, for each positive tolerance =, under price system p thereexists an =-approximate equilibrium allocation (x=

i ) for which, whenrestricted to time in any given 1-second interval and states in any givenevent E, each consumer's average consumption in x=

i is within = of averageconsumption in xi . Thus we offer limit-equilibrium price system p and

142 JONATHAN L. BURKE

allocation (xi) for practical normative and positive analysis of acompetitive economy.

Section 2 situates formal analysis with an abbreviated list of standardassumptions for the exchange of commodities among a finite number ofconsumers over an infinite-dimensional Riesz space L. That includesdiscrete-time, deterministic commodity space L=l� ; continuous-time,deterministic space L=L�([0, �)); and continuous-time, stochastic spaceL=L�(0_[0, �)), where the product of the probability state space0 and the time interval [0, �) is endowed with the predictable tribe(sigma-field) of measurable sets, which embodies an exogenous filtration ofcontinuously-evolving knowledge about the actual state of nature (Subsec-tion 2.2). We admit a general class of up-counting orders in each of thoseintertemporal spaces by dropping standard lower and upper semi-con-tinuity assumptions on preferences, weakening the standard representationof preferences by utility functions, and weakening the standard closednessassumption on the utility possibilities set to a type of uniform monotonicity.For example, we admit continuous-time stochastic Weiza� cker orders eventhough they are neither upper- nor lower-semicontinuous in any lineartopology nor representable by any utility function.

Section 3 proves the existence of our limit of approximate equilibria. Ourdefinition of `àpproximate'' is non-standard because some orders likeWeiza� cker are not representable by a utility function. But our definitioncan still be interpreted as satisficing under bounded rationality [13], whichplaces approximation errors on maximization by consumers modeled in theeconomy. Part of our existence proof manages to adapt standard proofs,originally designed for preferences fully represented by utility.

Section 4 perturbs preference orders to qualify all approximate equi-librium as exact equilibrium. Our topology measuring the perturbation isnon-standard because some orders like Weiza� cker are not representable byany utility function and are not continuous in any linear topology over thecommodity space. Qualifying approximate equilibrium as exact equilibriumgives an alternative interpretation to approximate equilibrium that placesapproximation errors on the modeler of the economy. Roughly, our topologyon preferences will imply the original and perturbed orders cannot bedistinguished by observing actual preferences if measuring the scale ofactual consumption is imprecise. Thus, any observations of actual preferen-ces that are consistent with our assumptions are also consistent withperturbed preferences that have exact equilibrium. Furthermore, qualifyingapproximate equilibrium as exact equilibrium shows the well-known non-existence examples without discounting are singular, which with thegeneral existence of limit equilibrium implies the presence of such examplesis not a logical reason to assume discounting is faster than economicgrowth.

143NON-CONTINUOUS PREFERENCES

2. ASSUMPTIONS

2.1. Standard Assumptions. To admit discrete-time deterministic con-sumption or continuous-time stochastic consumption and to admit futuregeneralization, the commodity space can be any infinite-dimensional Rieszspace L.2 There are only a finite number of consumers, i=1, ..., I. Eachconsumer has a non-zero commodity endowment ei in the positive orthantL+ . To admit the most general preference orders below, assume eachindividual endowment is at least a fraction of the total endowmente :=�i ei ; that is, ei�:e for some positive scaler :. An allocationx=(xi) # LI

+ specifies consumption that balances commodity materials,�i xi=e. An individual consumption is feasible when it is part of someallocation (when xi�e).

Each consumer has a strict preference order oi over L+ satisfying freedisposal (x$i�xi and xi oi x i and xi�x$i imply x$i oi x$i) and the convexityconditions that each preferred set [xi : xi oi xi] is convex, and that x i oi xi

implies :xi+(1&:) x i oi xi for each scaler : in (0, 1].

2.2. Overtaking Examples. Over any infinite-horizon commodity space,the Weiza� cker overtaking criterion defines the best-known orders satisfyingour abbreviated assumptions but violating standard assumptions. Fix anyconcave, non-decreasing, locally insatiable felicity (cardinal utility) functionv: RN

+ � [&�, +�) and an up-counting factor ;>1.For the commodity space L :=lN

� of bounded paths of N-tuples, discrete-time deterministic consumption path xi=(x t

i)�0 overtakes path xi=(x ti )

when partial sums �Tt=1 ;tv(x t

i)>�Tt=1 ;tv(x t

i ) for large T.Define the continuous-time stochastic commodity space L :=LN

� tocontain the bounded, measurable RN-valued stochastic processes over asuitable measure space (0_[0, �), P, +). Each state of nature s # 0 is onepossible realization of all exogenous uncertainty over the entire timehorizon [0, �). + is the product of an exogenous probability measure over0 and Lebesgue measure over [0, �). Finally, P is the predictable tribe(sigma-field) on 0_[0, �) defined by an exogenous filtration of con-tinuously-evolving knowledge about the actual state of nature. Roughly,3 aP-measurable stochastic processes is a function xi=(xi (s, t)) that, at eachtime t, specifies consumption on the basis of knowledge attained up to, butnot including, time t.


2 The standard and non-standard assumptions below extend those in an earlier,unpublished analysis of non-discounting preferences for discrete-time, deterministic consump-tion (Burke [4]).

3 Precisely, the predictable tribe equals the tribe generated by the left-continuous processesadapted to the filtration of knowledge [5, Sect. 3.2]. Duffie and Zame use the predictable tribefor stochastic consumption processes over a finite horizon [6].

(For example, each state s is a particular sample path for the canonicalBrownian motion [11] drawn, under the Wiener probability measure,from the space 0 :=C[0, �) of continuous functions s=s( } ) on [0, �).Up to time t, the (ex-post) actual state s=s( } ) is known over time interval[0, t). Hence, at each time t, a P-measurable stochastic processes specifies,for almost every4 pair of sample paths s=s( } ) and s= s( } ) in 0, consump-tions that agree xi (s, t)=xi (s, t) whenever the sample paths s=s( } ) ands= s( } ) agree over [0, t).)

Using the previous felicity function v and up-counting factor ;, for anystochastic consumption process (xi (s, t))�0 in LN

�(0_[0, �), P, +),

Ev(Xi (t))=|0

v(xi (s, t)) dP(s)

is expected felicity from stochastic consumption variable Xi (t) :=xi ( } , t)over 0 at time t. Hence, utility accumulated up through time T is thepartial integral

|T

0;tEv(Xi (t)) dt=|

0_[0, T];tv(xi (s, t)) d+(s, t).

Finally, stochastic consumption process (xi (s, t))=(Xi (t))�0 overtakesprocess (xi (s, t))=(X� i (t)) when partial integrals �T

0 ;tEv(Xi (t)) dt>�T

0 ;tEv(X� i (t)) dt for large T.

2.3. Non-standard Assumptions. All up-counting Weiza� cker orders violateseveral assumptions of standard existence theorems [9]. Consider the deter-ministic example from the Introduction, with linear felicity and up-countingfactor 2. Although the discrete-time consumption path xi :=(2, 1, 1, 1, ...)overtakes the path xi=(1, 1, 1, 1, ...), for each positive scaler $ the reducedpath (1&$) xi does not overtake x i , and the original path xi does notovertake the expanded path (1+$) x i . Thus, the order is neither lower norupper semi-continuous in any linear topology. And the third pathx� i :=(2, 0, 2, 0, ...) neither overtakes x i nor is overtaken by xi . Thus, theorder is not negatively transitive5 and not representable by utility.

Weiza� cker orders have weaker properties, however, which are shared byother up-counting orders, and which we will adopt as assumptions to find


4 Precisely, there is an almost-sure (Wiener-probability-one) event 0$/0 such that thestated property holds for every pair s and s in 0$.

5 Representation by utility requires negative transitivity, meaning if path xi is better than x i ,than any third path is necessarily either better than the lesser path xi or worse than thegreater xi .

equilibrium. Although no utility function fully represents any up-countingWeiza� cker order over any infinite-horizon commodity space, the limit

ui (x i) :=lim infT � �

;&T :T

t=1

;tv(x ti) (1)

quasi-represents the discrete-time, deterministic order in the sense that ifpath xi overtakes path xi , then ui (xi)�ui (xi). To see that, xi=(x t

i) over-taking xi=(x t

i ) implies the partial sums of utility are eventually ordered,;&T �T

1 ;tv(x ti)>;&T �T

1 ;tv(x ti), so their limits (1) are also ordered,

ui (x i)�ui (xi). Likewise, the limit

ui (x i) :=lim infT � �

;&T |T

0;tEv(X i (t)) dt (2)

quasi-represents the continuous-time, stochastic order.Limit utility, over either discrete-time deterministic (1) or continuous-

time stochastic (2) consumption, is evidently concave, non-decreasing, andinsatiable. And assuming the total endowment is strictly positive (com-ponents et or e(s, t) are uniformly bounded away from 0), utility ismonotone in the total endowment, meaning ui (x i+:e)>ui (xi) for eachfeasible consumption xi and positive scaler :. Combining those propertiesimplies a useful type of uniform monotonicity.

Define the I-tuple u(x) :=(ui (xi)) of utility generated by each allocationor specification x=(xi) of consumption. And define the set u(X) of feasibleutilities as the image of u over the set X of all allocations.

Uniform Monotonicity. For each positive scaler =, there exists a positiveI-tuple $=($i)>>0 such that

u(x)+[$]�u((1+=) X ). (3)

That is, for each allocation x=(xi) in X there exists some allocationx=(xi) in X for which u(x)+$�u((1+=) x), meaning ui (x i)+$i�ui ((1+=) xi) for each consumer.

To verify uniform monotonicity when utility functions are concave, non-decreasing, and monotone in the total endowment, for each positive scaler=, define the I-tuple $=($i) by

$i := infxi�e _u i \xi+

=I

e+&ui (x i)& . (4)


To prove $i>0, for each feasible consumption xi�e, concavity implies

ui \xi+=I

e+�\1&=2I+ ui (xi)+

=2I

u(xi+2e).

Hence, xi+2e�2e, xi�e, and utility being non-decreasing and monotoneyield the required lower bound

ui \x i+=I

e+&ui (x i)�=2I

[u i (x i+2e)&ui (xi)]

�=2I

[ui (2e)&ui (e)]>0.

That is, $i�=2I [ui (2e)&ui (e)]>0. And to prove the utility inequality (3),

for each allocation x=(x i), the required allocation x=(xi) is a convexcombination of (xi) and the ad-hoc constant allocation ( 1

I e). Specifically,xi := 1

1+= xi+=

1+=1I e. Hence, (1+=) x i=x i+

=I e, so the definition (4) of $i

implies

ui (x i)+$i�u i \xi+=I

e+=u i ((1+=) x i) (5)

which yields the required utility inequality (3) for uniform monotonicity.Uniform monotonicity is also implied by standard assumptions, which

include the full representation of each preference order by utility ui that isnon-decreasing and monotone in the total endowment, and for which thelower cone of the utility set u(X) is closed. For proof, truncate the lowercone of the utility set into

U :=[v=(vi) : ui (0)�vi�ui (xi) for some x=(xi) # X]. (6)

Monotonicity implies that each utility vector v�(ui (xi)) in U is dominatedby the vector v :=(ui (xi+

=I e)) in u((1+=) X). Thus, U is covered by the

open sets O(v) :=[v: v>>v] parametrized by v # u((1+=) X). But since thelower cone of the utility set u(X) is closed the truncation U is also closed,and since utility is non-decreasing the truncation is also bounded. Hence,the truncation is compact and is covered by a finite list O(v1), ..., O(vm) ofthe open sets. Compactness also implies the sup-norm distance between thevectors v1, ..., vm and U is bounded below by some scaler $>0. That is, foreach vector v=(vi) in U, there exists some vector v=(vi) among the finitelist for which vi+$�vi for each consumer. In particular, the definition ofU and u((1+=) X) imply uniform monotonicity (3): for each allocationx=(xi) in X there exists some allocation x=(x i) in X for which ui (xi)+$�ui ((1+=) xi) for each consumer.


Hence, in addition to the abbreviated list of standard assumptions (Sub-section 2.1), the forthcoming existence theorem assumes all preferenceorders can be quasi-represented by quasi-concave, non-decreasing, mono-tone, uniformly monotone (3) utility functions.

Since preferences are only quasi-represented by utility, however, theassumed utility properties need not translate into interpretable propertiesof the underlying preference orders. The main problem is utility propertiesmay depend on the choice of functions that represent preferences. Forexample, Leontief-type preferences (xi oi xi if xi�xi+=e for some positivescaler =) are quasi-represented by all non-decreasing utility functions,regardless of whether those functions are quasi-concave, monotone, oruniformly monotone.

We can, at least, bound the restrictiveness of the assumed utility proper-ties by deriving them from a list of interpretable properties of preferenceorders.

Lemma 1. Consider any economy whose preference orders satisfy theabbreviated list of standard assumptions (Subsection 2.1) and the weaktransitivity property that xi oi x$ i whenever x i oi x i and x i oi x$ i .

Then, orders can be quasi-represented by quasi-concave, non-decreasing,monotone, uniformly monotone (3) utility functions if each order embodies atleast minimal substitution in the sense that, for each positive scaler =, thereexists a positive $ such that xi+=eoi xi+$e whenever xi oi xi for feasibleconsumptions.

The literature on the full representation of preferences by utility requiresstrict preference orders to be negatively transitive. But such transitivitydisqualifies all up-counting Weiza� cker orders. Hence, Lemma 1 weakenstransitivity.6

``Minimal substitution'' above places a lower bound on how goods maysubstitute for one another. To see ``minimal substitution'' fail, consider anup-counting discrete-time, deterministic Weiza� cker order with felicityv(0)=&�. At the path xi=(0, 2, 2, 2, ...), all partial sums of utility are�T

t=1 ;tv(x ti)=&�, which implies that path is overtaken by path

xi :=(1, 1, 1, 1, ...) generating finite utility. The first path xi was overtakenbecause the greater quantities of goods 2, 3, ... do not compensate or sub-stitute for the lesser quantity of good 1. Thus, goods are poor substitutes.Precisely, ``minimal substitution'' fails because xi oi xi holds but x i+=eoixi+$e fails for all positive scalers =, $ in which =e<<(1, 1, ...).


6 It may be possible to accommodate more intransitive preferences and still find therequired utility functions, but it may be easier to accommodate intransitivity through analternative analysis that does not use utility at all.

Proof of Lemma 1. The standard choice ui (xi) :=sup [: # R+ :xi oi :e] for utility quasi-represents the order, where ui=0 when the set isempty. For proof, preference xi oi x i and transitivity imply the inferiorinterval [: # R+ : x i oi :e] of scalers includes the inferior interval[: # R+ : xi oi :e]; hence, their supremum are ordered, ui (xi)�ui (xi).Free disposal of preference orders evidently implies utility is non-decreasing.

To prove utility is quasi-concave, consider any convex combination:xi+(1&:) xi of a pair of vectors xi and xi such that utility is ordered,ui (x i)�ui (xi). If utility ui (xi)>0, then for any positive tolerance =<ui (x i),the utility order and the definition of utility imply xi oi (ui (x i)&=) e andxi oi (ui (x i)&=) e. Hence, the convexity of preferred sets implies the con-vex combination is also preferred, (:xi+(1&:) xi)oi (ui (x i)&=) e, whichimplies u i (:xi+(1&:) xi)�ui (xi)&=, which holding for any positivetolerance implies ui (:xi+(1&:) xi)�ui (xi). That inequality also holds ifutility ui (xi)=0. Hence, utility is quasi-concave.

To prove the remaining utility properties, for each positive scaler =, let$ be the ``minimal substitution'' parameter. For each feasible consumptionxi , consider any : # R+ such that xi oi :e. Minimal substitution impliesxi+=eoi :e+$e, so ui (xi+=e)�:+$, which holding for every such :implies ui (xi+=e)�ui (x i)+$, which holding for every feasible consump-tion implies utility is both monotonic in the total endowment anduniformly monotonic. K

3. APPROXIMATE AND LIMIT EQUILIBRIUM

3.1. Statements. Consider any economy whose preference orders satisfythe abbreviated list of standard assumptions (Subsection 2.1) and arequasi-represented (Subsection 2.3) by quasi-concave, non-decreasing,monotone utility functions satisfying uniform monotonicity (3).

To admit the most non-standard preference orders, follow Aliprantis�Brown�Burkinshaw and only require price systems p to be finite-valuedlinear functions over the order ideal [1]

L(e) :=[x # L : |x|�*e for some *>0]

generated by the total endowment (where vector |x| # L is the absolutevalue of vector x). L(e) contains all feasible consumptions, the total endow-ment, and each individual endowment. Restricting finite-value and linearityto the order ideal, rather than requiring it over the entire commodity space,is innocuous in intertemporal L� spaces because the entire L� space equalsthe order ideal generated by any strictly positive total endowment (whosecomponents are uniformly bounded away from 0). The purpose of the ideal


is to admit the most non-standard orders to non-intertemporal and non-L� commodity spaces. In those spaces, we have prices restricted to theorder ideal since expanding them to the entire commodity space is routine:Either (1) allow infinite prices for infeasible commodity vectors [1,Theorem 4.15]; or (2) deduce a finite-valued extension after assuming sometype of properness [9, Sect. 10]. (It remains to be seen what non-standard,non-L� orders satisfy the standard properness assumption.)

For discrete-time, deterministic commodity space l� with a strictly-positive total endowment, all price systems are of the form px=��

1 pt } xt+bx over paths x=(xt) in l� , where ( pt) is a summablesequence and b is a norm-continuous positive linear function over l� thatis zero over every finite-lasting path (every path such that xt=0 for larget). We follow Gilles and interpret b as an asset price bubble [7]. However,for continuous-time L� spaces, with or without uncertainty, norm-con-tinuous positive linear functions can be pure charges that place significantvalue on short-lived consumption, like the function p over L�([0, �))with value p/(1&=, 1)=1 over the characteristic vector of every non-emptyinterval (1&=, 1). While Gilles further calls such systems ``price bubbles''[7], Bewley asserts such systems ``have no economic interpretation'' [3,p. 516]. With further work, one could avoid such controversial continuous-time systems by restricting ``bubbles'' to functions that are zero over everyfinite-lasting deterministic path or stochastic process.

The following definition of approximate equilibrium is non-standard(and complex) because it is defined directly on preference orders, ratherthan utility, and so allows Weiza� cker-type orders not (fully) represented byutility.

Theorem 1. There exists a price system p consistent with approximateequilibrium for arbitrarily small approximation tolerances. Precisely, for eachpositive tolerance =, there exists an allocation x=(xi) of consumption eachsatisfying the budget constraint pxi�pei and the approximate maximizationcondition that expanded consumption (1+=) xi is not oi -worse than thecontraction (1&=) xi of any consumption xi satisfying pxi�pei .

To interpret such =-approximate equilibrium, since an =-increase in thescale of approximate-equilibrium consumption, from xi to (1+=) x i , yieldsconsumption that is not worse that the =-contraction of any affordablealternative, approximate-equilibrium consumption xi comes within thefraction = of being maximal, and may therefore be satisficing [13]. Wecould add an innocuous continuity assumption (quasi-lower-semicontinuity[4]) to strengthen the approximate maximization conclusion so thatexpanded consumption is not oi -worse than any consumption satisfying thebudget constraint, rather than the contraction of any such consumption.


But we avoid extra assumptions since we will give an alternative interpretation(Theorem 2) to all the approximate equilibrium of Theorem 1.

The benefit of finding a limit of approximate equilibrium, rather thanjust individual approximate equilibrium, is that limit-equilibrium prices areindependent of the approximation tolerances. Thus a positive analysis ofprices is robust to approximation tolerances. And since prices determinebudget sets and the budget-constrained supremum of utility (when utilityfunctions represent preferences), the limit equilibrium is sufficient forrobust normative analysis.

Positive analysis of consumption is, likewise, robust when the set offeasible consumptions is suitably compact. For example, here is a corollaryfor continuous-time, stochastic consumption, where the set of feasibleconsumptions is weakly compact:

Corollary 1. Consider any commodity space L�(0_[0, �), P, +)of real-valued, predictable stochastic processes, constructed from a filteredprobability space (0, 7, P). For each economy, there exists a price system p,an allocation x=(xi), and an I-tuple v=(vi) of set functions over event tribe7 that approach approximate equilibrium in the following sense.

For each positive tolerance =, each finite partition 0=T0<T1< } } } TM<� of the time horizon, and each finite partition 00 , ..., 0N of the state spaceinto positive-probability events in 7, there exists an allocation x=(xi) forwhich ( p, x) is an =-approximate equilibrium with average consumption closeto the average specified by x=(xi)

} 1Tm&Tm&1

|Tm

Tm&1

1P(0n) |0n

[x i (s, t)&xi (s, t)] dP(s) dt }<= (7)

over short-run intervals [0, T1), ..., [Tm&1 , TM) and events 00 , ..., 0N , andclose to the average7 specified by v=(vi)

} vi (0n)&LimT � �1

T&TM|

T

TM

1P(0n) |0n

x i (s, t) dP(s) dt }<= (8)

over the long-run interval [TM , �) and events 00 , ..., 0N .

We offer limit-equilibrium allocation x=(xi) and I-tuple v=(vi) for thepositive analysis of consumption in a competitive economy. For example,partition the first hundred years into one-second intervals and partitionstates of nature into one hundred events 0n of similar states. Hence,for each positive tolerance =, under price system p there exists an


7 Since Banach limits (Lim) are restricted to sequences, restrict truncation times T tonatural numbers.

=-approximate equilibrium allocation (x=i ) for which, when restricted to

time in any given 1-second interval and states in any given event 0n , eachconsumer's average consumption in x=

i is within = of average consumptionin xi (7). Likewise, long-run averages in each event 0n are within = ofvi (0n) (8).

The special case of deterministic consumption simplifies closenessbounds (7), (8) to

} 1Tm&Tm&1

|Tm

Tm&1

[x i (t)&x i (t)] dt }<=,

} v i&LimT � �1

T&TM|

T

TM

xi (t) dt }<=.

(It remains to be seen what new non-L� economies have suitably compactfeasible sets to prove robustness like Corollary 1).

3.2. Proofs.

Proof of Theorem 1. Extending an earlier, unpublished proof (Burke[4]) from discrete-time to general infinite-dimensional commodity spaces,fix the quasi-concave, non-decreasing, monotone, uniformly monotone (3)utility functions that are assumed to quasi-represent the preference orders.Our forthcoming Appendix adapts Bewley's original existence proof toestablish a non-standard type of quasi-equilibrium, consisting of a pricesystem p normalized by pe=1 and a utility vector u� in the closure cl u(X)of the utility set so that each utility is price supported in the sense thatui (x i)>u� i implies pxi� pei . We will prove p is the required limit-equilibrium price system.

To find consumption, fix =>0. Since each individual endowment is atleast a fraction of the total endowment, fix any positive scaler :<= smallenough so that :e�(=�2) ei for each consumer, and so that 1+:< 1+=

1+=�2 .Containment u� # cl u(X) of quasi-equilibrium utility and uniform mono-tonicity (3) (for positive tolerance :) imply ui ((1+:) xi)>u� i for someallocation x=(xi). Other than material balance, we will prove thespecification of consumption 1

1+=�2 xi for each consumer satisfies both therequired properties of approximate equilibrium.

First, prove the budget constraint, p 11+=�2 xi� pei . Since each of other

consumer's consumptions generate utility u @((1+:) x @)>u� @ , the pricesupport of utility u� @ implies (1+:) px @� pe @ . But, the material balance ofallocation x=(xi) and price normalization imply �i (1+:) pxi=(1+:) pe=1+: and �i pei=1. Hence

(1+:) pxi=1+:& :@{i

(1+:) px @�1+:& :@{i

pe @= pei+:.


In particular, pxi�pei+:pe. But since : is small enough so that :pe�(=�2) pei , we have pxi�(1+=�2) pei and budget constraint p 1

1+=�2 xi� pei .Second, prove expanded consumption (1+=) 1

1+=�2 xi=1+=

1+=�2 xi is notoi -worse than the contraction (1&=) xi of any consumption xi satisfyingpxi�pei . To prove the contra-positive, for each preferred consumption(1&=) xi oi

1+=1+=�2 x i , quasi-representation by utility implies ui ((1&=) xi)�

ui (1+=

1+=�2 x i), and so ui ((1&=) xi)�ui ((1+:) xi), since : is small enough sothat 1+:< 1+=

1+=�2 . Hence, the utility inequality ui ((1+:) xi)>u� i (from thedefinition of allocation x=(xi)) implies ui ((1&=) x i)>u� i . Hence, the pricesupport of utility u� i implies (1&=) pxi�pei , and so pei>0 implies thebudget violation pxi> pei . Thus, =-expanded consumption is not O i -worsethan the =-contraction of any consumption satisfying the budget constraint.

Finally, define the required approximate-equilibrium allocation ofconsumptions as 1

1+=�2 xi+;ie by choosing the scaler ;i�0 so thatconsumption satisfies the budget constraint with equality, 1

1+=�2 pxi+;i pe= pei . Free disposal implies such increased consumption preserves therequired =-approximate maximization condition. As for material balancefor the approximate-equilibrium, material balance for allocation x=(x i)implies total consumption is proportional to the total endowment

:i \

11+=�2

xi+;ie+=1

1+=�2:i

xi+\:i

; i+ e

=\ 11+=�2

+:i

;i+ e.

Hence, budget-constraint equality implies 11+=�2+�i ; i=1 and material

balance for the approximate equilibrium. K

Proof of Corollary 1. Applying Theorem 1 to a sequence of positivetolerances ==1�:, for :=1, 2, ..., yields a price system p and a sequence of(1�:)-approximate-equilibrium allocations x:=(x:

i ). Alaoglu's theorem [9]implies the order interval [0, e] of feasible consumptions, containing eachconsumer's consumption sequence [x:

i ], is (weak) _(L� , L1)-compact.Also, for each event 0$ in tribe 7, each consumer's sequence of long-runaverage consumptions

v:i (0$) :=LimT � �

1T |

T

0

1P(0$) |0$

x:i (s, t) dP(s) dt (9)

is bounded between 0 and the long-run average of e, and so is also con-tained in a compact set. Hence, the Tychonoff theorem [10] yields a subsetunder which each consumer's consumption x:

i weak-converges to some


consumption xi in L+ , and each long-run average v:i (0$) converges to

some scaler vi (0$).The list x=(xi) of limit consumptions evidently preserves material

balance from allocations x:=(x:i ), and so is an allocation. To see that

limits x=(xi) and v=(vi) fulfill the other requirements of Corollary 1, con-sider each positive tolerance =, each finite partition 0=T0<T1< } } } TM<� of the time horizon, and each finite partition 00 , ..., 0N of the statespace into positive-probability events in 7. Weak convergence x:

i � xi

implies

} 1Tm&Tm&1

|Tm

Tm&1

1P(0n) |0n

[xi (s, t)&x:i (s, t)] dP(s) dt }� 0. (10)

The definition (9) of long-run average consumption implies

v:i (0n) :=LimT � �

1T&TM

|T

TM

1P(0n) |0n

x:i (s, t) dP(s) dt

which with scaler convergence v:i (0n) � vi (0n) implies

} vi (0n)&LimT � �1

T&TM|

T

TM

1P(0n) |0n

x:i (s, t) dP(s) dt }� 0. (11)

Hence, for sufficiently large :, ( p, x:) is an =-approximate equilibrium withaverage consumption in x:=(x:

i ) close (10) to the average specified byx=(xi) over the short-run intervals and events (7) and close (11) to theaverage specified by v=(vi) over the long-run interval and events (8). K

4. EXACT EQUILIBRIUM

This section perturbs preference orders to qualify approximate equi-librium as exact equilibrium (=-approximate equilibrium with tolerance==0). Our topology measuring the perturbation is different from varioustopologies over utility functions and is different from the standard topologyof closed convergence over continuous preference orders [8] becauseorders like Weiza� cker are not (fully) representable by any utility functionand are not continuous in any linear topology over the commodity space.

Precisely, fix any infinite-dimensional Riesz space for the commodityspace and fix individual non-zero endowments in the positive orthant thatare each at least a fraction of the total endowment. Hence define the spaceof economies E=(oi) spanned by all preference orders oi that satisfy the


abbreviated list of standard assumptions (Subsection 2.1) and are quasi-represented (Subsection 2.3) by quasi-concave, non-decreasing, monotone,uniformly monotone (3) utility functions.

Theorem 2. For each economy E=(oi) and each positive tolerance=<1, there exists a perturbed economy E$=(o$i) for which each=-approximate equilibrium ( p, x) of E is an exact equilibrium of E$.Preference orders in the two economies are within = for each consumer, in thesense

xi oi xi if (1&=) xi o$i (1+=) xi ; (12)

xi o$i xi if (1&=) xi oi (1+=) x i (13)

for every pair of consumption vectors.

Qualifying approximate equilibrium as exact equilibrium gives an alter-native interpretation to `àpproximate'' equilibrium that places approxima-tion errors on the modeler of the economy, rather than on the maximiza-tion by satisficing consumers modeled in the economy. Roughly, theoriginal oi and perturbed o$i orders cannot be distinguished by observingactual preferences if measuring the scale of actual consumption is soimprecise that the vector xi cannot be distinguished from the scaler multi-ples (1&=) xi or (1+=) xi . Thus, any observations of actual preferencesthat are consistent with an order oi satisfying our assumptions are alsoconsistent with the perturbed order o$i that has exact equilibrium.

Qualifying approximate equilibrium as exact equilibrium also combineswith the general existence of approximate equilibrium (Theorem 1) to implythe density of economies with equilibria. To be precise, measure the distanced(o$i , o$i ) between two preference orders as the supremum of the positivescalers =<1 for which both =-closeness relations (12) and (13) hold. If thereare no such =<1, then set d(oi , o$i )=1. Hence, dense existence in thetopology generated by d-open balls. Thus the well-known non-existenceexamples without discounting are singular, which with the general existenceof limit equilibrium (Theorem 1) implies the presence of such examples is nota logical reason to assume discounting is faster than economic growth.

The significance of approximate equilibrium and dense existence dependson the fineness of our topology. To evaluate fineness, our topology is finerthan the Euclidean topology on parameters of Cobb�Douglas orders. Forexample, d(o:, o0.5)=1 for any parameter 0.5<:<1 of orders o:

represented by utility u:(x1 , x2)=x:1 x1&:

2 . For proof, if d(o:, o0.5) wereless then 1, then both =-closeness relations (12) and (13) for orders o: and


o0.5 would be satisfied by some =<1. Hence, fix any positive scaler $ smallenough so that $2:&1<(1&=)�(1+=), and so

u:((1&=)(1, $))=(1&=) $1&:>(1+=) $:=u:((1+=)($, 1)).

Hence, =-closeness (12) and (13) for orders o: and o0.5 imply utilityu0.5(1, $)>u0.5($, 1), which contradicts the symmetry of utility at para-meter :=0.5. Therefore, d(o:, o0.5)=1 after all.

More generally, consider the preference orders that are ray continuous,meaning xi o$i xi implies (1&=) xi o$i (1+=) xi for some positive =. Thatincludes all standard orders (which are continuous in some linear topology)and any non-standard order (fully) represented by finite-valued concaveutility, including the up-counting order represented by the Banach limitv(x)=Lim xt from the Introduction and the up-counting orders (variationsof Weiza� cker) that are fully represented by limit utilities (1) or (2). We willprove our topology is separated (Hausdorff) over ray-continuous preferences.In contrast, the standard topology of closed convergence over continuouspreferences is not separated for any infinite-dimensional L� commodity space[8].

Most generally, where preference orders may not be ray continuous andstandard topologies may not be defined, our topology is not separated.However, the extent of non-separability is limited. Perturb any non-ray-continuous preference order oi by defining xi o$i xi if, and only if, (1&=) xi oi

(1+=) xi for some positive =. One can show that perturbed order o$ipreserves all our assumptions, and gains ray-continuity. Although anynon-ray-continuous order like Weiza� cker cannot be separated from its ray-continuous perturbation, d(oi , o$i)=0, we will prove the order can beseparated from any other ray-continuous order and be separated from anynon-ray-continuous order whose ray-continuous perturbation is differentfrom o$i .

Finally, we will prove our topology is fine enough that the Weiza� ckerorders, and other up-counting orders, are topologically separable frompreferences satisfying standard continuity. Thus finding equilibrium for up-counting preferences is a distinct problem from finding equilibrium forstandard preferences. In particular, dense existence cannot be proved bysimply approximating up-counting preferences with standard preferences.

Putting it all together:

Theorem 3. (a) The subset of economies with equilibria is dense under thetopology generated by d(oi , o$i ).

(b) d(oi , o$i)>0 for any pair oi {o$i of ray-continuous orders. Infact, the topology defined by d(oi , o$i ) is pseudo-metric over the entire spaceof orders [12], and is metric over the subspace of ray-continuous orders.


(c) For each linear topology { over the commodity space, each ray-con-tinuous preference order oi that is not {-upper-semicontinuous cannot beapproximated by {-upper-semicontinuous orders o$i ; that is, inf o$i

d(oi , o$i )>0.

To apply the separation in Part (c) to the Weiza� cker order, one canprove its ray-continuous perturbation violates standard upper-semicon-tinuity in the Mackey {(L� , L1) topology. Hence, Part (c) implies the per-turbation is separated from Mackey upper-semicontinuous preferences.Hence, the topology being pseudo-metric (Part (b)) implies the originalWeiza� cker order is likewise separated.

Proof of Theorem 2. Fix tolerance = and economy E=(oi), and definethe new preference orders to be xi o$i xi whenever

x i=*xi+(1&*) x� i for some * # [0, 1)

and x� i s.t. (1&=) x� i oi (1+=) xi . (14)

To prove the perturbation o$i preserves standard assumptions, if xi o$i x i ,then the o$i -definition (14) implies xi=*xi+(1&*) x� i for some * # [0, 1)and some x� i such that (1&=) x� i oi (1+=) x i . Hence, oi -free-disposalimplies x� i oi xi , and so oi -convexity implies *xi+(1&*) x� i oi x i , and soxi=*xi+(1&*) x� i implies xi oi xi . The derived implication xi oi x i ifxi o$i xi means that the same utility function that quasi-represents oi alsoquasi-represents o$i . In particular, o$i inherits all non-standard assump-tions from oi . Next, prove o$i also inherits the abbreviated list of threestandard assumptions:

First, prove free disposal: x$i�xi and xi o$i xi and x i�x$i imply x$i o$i x$i .For proof, the o$i -definition (14) and xi o$i xi imply xi=*xi+(1&*) x� i forsome * # [0, 1) and some x� i such that (1&=) x� i oi (1+=) xi . Hence, x$i�xi

and xi�x$i imply x$i=*x$i+(1&*) x� $i for some vector x� $i�x� i . Hence,oi -free-disposal and x� $i�x� i and (1&=) x� i oi (1+=) xi and xi�x$i imply(1&=) x� $i oi (1+=) x$i . Hence, the o$i -definition (14) and x$i=*x$i+(1&*) x� $i imply the required order x$i o$i x$i .

Second, prove the convexity condition that xi o$i xi implies :xi+(1&:) xi o$i xi for : # (0, 1]. For proof, the o$i -definition (14) implies x i=*xi+(1&*) x� i for some * # [0, 1) and some x� i such that (1&=) x� i

oi (1+=) xi . Hence, for any convex combination,

:xi+(1&:) xi=:(*xi+(1&*) x� i)+(1&:) xi=*� x i+(1&*� ) x� i

for scaler *� :=:*+(1&:) # [0, 1).8 Hence, the o$i -definition (14) and(1&=) x� i oi (1+=) xi imply the convex combination :x i+(1&:) xi o$i xi .


8 The second equality follows from the algebraic identity 1&[:*+(1&:)]=:(1&*).

For the final standard assumption, prove each o$i -preferred set [xi :xi o$i xi] is convex. For proof, fix any two vectors xi and x$i in the set. Theo$i -definition (14) implies the two vectors are convex combinationsxi=*xi+(1&*) x� i and x$i=*$x i+(1&*$) x� $i for some scalers * and *$ in[0, 1) and some vectors x� i and x� $i such that (1&=) x� i oi (1+=) xi and(1&=) x� $i oi (1+=) xi . Hence, any convex combination :x i+(1&:) x$i ,: # [0, 1], of the two original vectors is also a convex combination of thethree vectors xi , x� i , and x� $i . Precisely, after collecting terms, :xi+(1&:) x$i=(:*+(1&:) *$) xi+:(1&*) x� i+(1&:)(1&*$) x� $i , which reads

:xi+(1&:) x$i=*� x i+(1&*� )(:x� i+(1&:) x� $i)

for scalers *� in [0, 1) and : in [0, 1] defined by the three equations :*+(1&:) *$=*� , :(1&*)=(1&*� ) :, and (1&:)(1&*$)=(1&*� )(1&:).9 Butthe assumed convexity of each oi -preferred set implies the convex com-bination :x� i+(1&:) x� $i satisfies

(1&=)(:x� i+(1&:) x� $i)oi (1+=) xi .

Hence, the o$i-definition (14) implies :xi+(1&:) x$i o$i xi ; that is, the convexcombination :xi+(1&:) x$i is in the o$i-preferred set [xi : xi o$i xi].

Next, prove the required =-closeness. For the first relation (12), theimplication x i oi xi if xi o$i xi (from the first paragraph) and o$i -free dis-posal imply the required relation xi oi xi if (1&=) xi o$i (1+=) xi . Theother relation (13) is just the o$i -definition (14) applied to scaler *=0,xi o$i xi if (1&=) xi oi (1+=) xi .

Finally, to prove each =-approximate equilibrium ( p, x) of E=(oi) is anexact equilibrium of E$=(o$i), consider any better consumption, xi o$i xi .Hence, the o$i -definition (14) implies xi=*x i+(1&*) x� i for some* # [0, 1) and (1&=) x� i oi (1+=) xi . But =-approximate maximization byxi and (1&=) x� i oi (1+=) xi imply px� i> pei . Hence, budget constraintpxi= pei (which holds with equality because of material balance), x i=*xi+(1&*) x� i , and *<1 imply pxi> pei . K

Proof of Theorem 3. Part (a) merely combines Theorem 1 andTheorem 2. For Part (b), we will just prove the essential metric propertyd(oi , o$i)>0 for any pair oi { o$i of ray-continuous orders. That is, proved(oi , o$i)=0 implies the orders equal. For any pair of consumptions, ifxi oi xi , then oi -ray-continuity implies (1&=) xi oi (1+=) xi for somepositive scaler =. But d(oi , o$i)=0 implies =-closeness (12) and (13), which


9 For explicit definitions, solve the first equation for *� , the second equation for :, and notethe third equation is then also satisfied since the sum of the left sides of all three equationsis identically 1[:*+(1&:) *$+:(1&*)+(1&:)(1&*$)=1] and the sum of the right sidesis also 1.

implies xi o$i xi . Summing up, xi o$i x i if x i oi x i . Likewise, xi oi x i ifxi o$i xi , so the orders equal.

For Part (c), since oi is not {-upper-semicontinuous, there exists anorder xi oi xi between two vectors for which there exists a {-converging netxn

i � xi such that xi o� i xni for all n. oi -ray-continuity and xi oi xi imply

(1&=)2 x i oi (1+=)2 xi for some positive =. Hence, unless inf o$id(oi , o$i)

>0 over all {-upper-semicontinuous orders o$i , then there exists some{-upper-semicontinuous order o$i satisfying the =-closeness relations (12)and (13), which with (1&=)2 xi oi (1+=)2 xi implies (1&=) xi o$i (1+=) xi .Hence, since convergence xn

i � xi implies (1+=) xni � (1+=) x i , upper-

semicontinuity of o$i implies (1&=) xi o$i (1+=) xni for large n. But for each

such n, =-closeness (12) and (13) implies xi oi xni , which contradicts

xi o� i xni . Therefore, inf o$i

d(oi , o$i)>0 over all {-upper-semicontinuousorders o$i . K

APPENDIX

Throughout, consider any economy whose preference orders satisfy theabbreviated list of standard assumptions (Subsection 2.1) and are quasi-represented by quasi-concave, non-decreasing, monotone, uniformlymonotone (3) utility functions.

For the proof of Theorem 1 (Subsection 3.2), it remains to find a pricesystem p over the order ideal L(e) normalized by pe=1 and a utility vectoru� in the closure cl u(X) of the utility set so that each utility is price suppor-ted in the sense that ui (xi)>u� i implies pxi� pei . To that end, equip idealL(e) :=[x # L : |x|�*e for some *>0] with the norm

&x&e :=inf[*>0 : |x|�*e].

Evidently, norm & }&e makes L(e) a topological vector lattice, and e is inthe norm-interior of the positive orthant.

Step 1. Define a certain correspondence 8, sending the closed (I&1)-dimensional simplex 2 into non-empty sets of I-tuples that has the propertythat any of its zeroes yields the required price system and utility vector.

First, monotonicity implies each utility ui (ei)>&�; hence, add a con-stant to utility (if necessary) so that ui (ei)=0. Second, truncate the lowercone of the utility set into10

U :=[v=(vi) : 0�vi�ui (xi) for some x=(x i) # X].


10 The truncation below the lower bound ui (ei)=0 while the truncation (6) used in theprevious discussion of the literature uses the lower bound ui (0).

For any s in 2, let v(s) be the point in the set cl U that is farthest from 0on the ray from 0 through s. Since all components of vectors in U are non-negative and bounded from above by the utility of the total endowment e,the set cl U is compact, which implies such a farthest point exists, and themap s [ v(s) is upper semi-continuous.

For each v(s), sum preferred sets

Z=:i

[xi # L+(e) : ui (x i)�v i (s)].

Since each utility is non-decreasing, ui (e)�vi (s) and each individualpreferred set is non-empty, so the sum Z is non-empty. Since each utilityis quasi-concave, each individual preferred set is convex, so the sum Z isconvex. And the total endowment e is not in the & }&e -norm interior of thesum because, if it were, then (1&$) e, for some $>0, is also in the sum;hence, utility monotonicity implies some allocation has utility u(x)>>v(s);hence, u(x)�*v(s) for some *>1, which implies *v(s) # U and so con-tradicts the definition of v(s).

Hence, the Separation Theorem yields some non-zero linear function pover L(e) separating Z from e=� i ei . Hence, utility non-decreasing impliesthe function p is positive, and so is a price system and can be normalizedby pe=1. The separation implies

:i

Ei ( p, vi (s))�:i

pei (15)

for consumers' expenditure functions Ei ( p, vi (s)) :=inf[ pxi : xi # L(e), ui (xi)>vi (s)]. (The strict utility inequality is for later convenience.)

To define the desired correspondence, for each s in 2, let 8(s) denote theset of (income-transfer) I-tuples T=(Ti) such that

T�( pe1&E1( p, v1(s)), ..., peI&EI ( p, vI (s))); :i

Ti=0 (16)

for some normalized price system p satisfying (15). Thus the corres-pondence 8 is well-defined.

Finally, if 0 # 8(s), then the definition of 8(s) implies, for some pricesystem, Ei ( p, vi (s))� pei . Vector v(s) being in the closure of the lower one(6) of the utility set implies u� �v(s) for some utility vector in the closurecl u(X) of the utility set. In particular, according to the definition of theexpenditure function, Ei ( p, u� i)� pei , which holding for each consumerdescribes p and u� as the required price system and utility vector.

Step 2. Show that 8 is convex-valued, has a closed graph, has a boundedrange, and is inward pointing at the boundary, meaning if si=0 and T # 8(s),then Ti�0.


The convexity of 8(s) follows from the convexity of the set of normalizedprice systems and the concavity of expenditure Ei ( } , v(s)) in price, which inturn follows from the quasi-concavity of utility. The boundedness of 8(2)follows because the normalization of price implies pei and Ei ( p, vi (s)) arealways between 0 and 1. 8(s) pointing inward at the boundary followsbecause, if si=0, then the definition of v implies vi (s)=0; hence, the nor-malization ui (ei)=0 of utility implies pei&Ei ( p, vi (s))�0. Hence, thedefinition (16) of 8 implies Ti (s)�0.

Finally, to prove that the graph of 8 is closed, consider sequences sk � sin 2 and T(sk) � T� . Thus we must show T� # 8(s). Specifically, we must finda normalized price system p� such that

T� �( p� e1&E1( p� , v1(s)), ..., p� eI&Ei ( p� , vI (s))). (17)

To that end, the definition of each T(sk) implies that, for some normalizedprice system pk,

T(sk)�( pke1&E1( pk, v1(sk)), ..., pkeI&Ei ( pk, vI (sk))). (18)

But Alaoglu's theorem [9] implies the set of normalized price systems isweak-compact. Hence, some subnet weak-converges to some system p� ; thatis, pkx � p� x for each x in the order ideal L(e). In particular, pkei �p� ei . Since we also have T(sk) � T� , the problem of proving (17) reducesto showing that expenditure is upper semi-continuous: Ei ( p� , vi (s))�lim supk Ei ( pk, vi (sk)).

To that end, consider any consumption xi satisfying ui (xi)>vi (s) andshow p� xi�lim supk Ei ( pk, vi (sk)). The upper semi-continuity of v implies,for all large k, ui (xi)>vi (sk); hence, pkx i�Ei ( pk, vi (sk)). Hence, taking thelimit as pk � p� yields p� xi�lim supk Ei ( pk, vi (sk)).

Step 3. Conclusion.Concluding that 8 has a zero from its properties listed above (Step 2) is

a routine application of Kakutani's fixed-point theorem. Thus the requiredprice system and utility vector exist (Step 1). K

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General Equilibrium When Economic Growth Exceeds Discountingseaver-faculty.pepperdine.edu/jburke2/articles/General.pdf · Hence, for each positive tolerance =, under price system