A note on price systems in infinite dimensional space.pdf

Economics Department of the University of Pennsylvania Institute of Social and Economic Research Osaka University A Note on Price Systems in Infinite Dimensional Space Author s Edward C Prescott and Robert E Lucas Jr Reviewed work s Source International Economic Review Vol 13 No 2 Jun 1972 pp 416 422 Published by Wiley for the Economics Department of the University of Pennsylvania and Institute of Social and Economic Research Osaka University Stable URL http www jstor org stable 2526035 Accessed 21 12 2012 22 27 Your use of the JSTOR archive indicates your acceptance of the Terms these social totals are constrained to satisfy the equality y x An m n tuple xi yj one xi for each i one yj for each j is called a Manuscript received January 14 1971 revised May 4 1971 1 Since completing an earlier version of this note we have discovered 1 which approaches this problem and others from a similar point of view 416 This content downloaded on Fri 21 Dec 2012 22 27 38 PM All use subject to JSTOR Terms and Conditions PRICES IN INFINITE SPACE 417 state of the economy A state xi y is called obtainable if xi E Xi for all i yje Yj for all j and y x 2 VALUATION EQUILIBRIUM v z will denote a real valued linear form on L which gives the value of any commodity point A state x y is a valuation equilibrium with respect to v z if 2 1 x y is obtainable 2 2 For every i xi C Xi and v xi v x implies xi i x Each consumer s satisfaction is maximized subject to a budget constraint 2 3 For every j yj C Yj implies v y x then xi tx 1 t x x for 0 t 1 Convexity of preferences These two conditions insure that any valuation equilibrium is a Pareto Optimum Theorem 1 of 2 2 3 INFINITE TIME HORIZON Let z zo z be an infinite sequence of elements such that zt C Lt where Lt is a normed linear space Denote the norm of Lt by Ilt Defining scalar multiplication and addition of sequences to be termwise the space of sequences z such that 3 1 Izil sup llztllt n Let zn denote the projection of z on Ln The following additional assumptions will be made ASSUMPTION III If xi E Xi then xl E Xi if yj C Yj then yJ E Yj ASSUMPTION IV If xi x4 C Xi and xi i x then there is an N which may depend on the choice of i xi and x such that n N implies xl i x i Assumption III implies both consumption and production can be truncated while IV imposes a continuity requirement on preferences to the effect that sufficiently distant consumption is discounted 2 To prove the converse of Theorem 1 of 2 Theorem 2 three additional assumptions are needed as stated in 2 This content downloaded on Fri 21 Dec 2012 22 27 38 PM All use subject to JSTOR Terms and Conditions 418 EDWARD C PRESCOTT AND ROBERT E LUCAS JR Let v z be a continuous linear functional on L Then we have LEMMA 1 The limit p z lim v z n oo exists PROOF Let Z i be the sequence with the i th term equal to zi and all other terms 0 Then v z v z j Let Z zi if v z i 0 and z z otherwise Then n v zfll E v z v zj Since v z is continuous llvll is finite which proves that p z exists and there fore that p z exists It is clear that p z is a continuous linear form on L equal to v z on Ln for all n although not in general equal to v z on L We now establish THEOREM 1 Suppose that x y is a valuation equilibrium with respect to a continuous linear form v z and that x is not a saturation point for any i Then under Assumptions I IV x y satisfies 3 2 for every i xi E Xi and xi 1 x implies p x1 p x 3 3 for every j yj C Yj implies p yj i x implies v xi v x 3 5 for every i xi C Xi and xi 1 x implies p xi v x 3 6 for every j yj C Yj implies p y1 j x For xiEXi and xi 2ix set xi t txi 1 t x for 0 t i x4 Suppose xi 2i x and v xi v x Then for t sufficiently near zero v xi i x This contradiction to 2 2 establishes 3 4 Suppose for xi E Xi that xi xQ and p xi v x For t sufficiently near zero p xi t v x For n sufficiently large then V Xi t n p xi t i x and X t n C XI given III and IV This con tradiction to 3 4 proves 3 5 Suppose 3 6 is false Then p yj v y for some yj X Y1 for some j By III one can choose n so that yJ C Yj and v yJ7 p yJ v y contradicting 2 2 This proves 3 6 From 3 5 p x v x for all i Adding over all i gives p x 2 v x Similarly using 3 6 and adding over j gives p y x4 By IV n may be chosen so that xl 1 x If p is trivial one has v xq p x7 0 v x which contradicts 2 2 This proves that p is non trivial This completes the proof of Theorem 1 Letting Pt Zt P Z t one has a sequence Pt of linear forms on Lt t 0 1 29 In terms of this sequence the value p z of any element of L can be written as the present value expression 3 10 Z Pt zt t O In particular if Lt is Euclidean space each term in the sum 3 10 can be represented as an inner product P zt where Pt E Lt 4 UNCERTAINTY In this section we consider a particular specification of the spaces Lt appli cable to economic systems in which the future is uncertain Let Ut t Pt be a probability space where Ut has the interpretation as the set of possible states of the world up to and including period t and Pt is a probability measure governing the occurrence of these states The period t commodity space Lt is the space of Kt tuples ztl u ztK u zt u of real valued Pt measurable functions ztk u on Ut that 4 1 JIztfjt essential sup Iztk U t Zsk otherwise For a sequence Aj An EC st the notation An 4 0 will mean A D An2 and limnp Pt An 0 Then to Assumptions I IV are added ASSUMPTION V If An 0 xi C Xi and yj C Yj there is an N such that n N implies xi An E Xi and yj A C Y3 ASSUMPTION VI If An 4 0 x x4 E Xi and xi j x there is an N such that n N implies xi An i X Roughly speaking V supplements III by requiring that conditionally truncated elements of consumption or production sets also belong to consumption production sets provided the truncation probability is sufficiently small Assumption VI supplements IV by requiring that a consumption sequence with a sufficiently low probability of truncation is nearly as good as the same sequence not truncated We first prove LEMMA 2 For each fixed t k there exists a sequence An 4 0 and a measurable function qtk such that 4 5 qtk Ztk lim Ptk Ztk An 4tk U ztk u dPt PROOF Let IA be the indicator function of A CE t By Theorem 1 24 of 3 52 Ptk IA m A mf A where mc is completely additive and mf is purely finitely additive Write mf mf mf where m m 0 By Theorem 1 22 of 2 52 there is a sequence of sets A 4 0 with mi A m Ut for all n Then for any A C Pt m A A O Similarly there is a sequence An 40 such that m A A 0 for all A CESt and all n Let An An U A Then An 4 0 and for all ACE 1 and all n mf A An m A A UA m A A UAA mf A A A m A A Aj 0 Thus pk IA A 1 Ptk IA A mc A An This content downloaded on Fri 21 Dec 2012 22 27 38 PM All use subject to JSTOR Terms and Conditions PRICES IN INFINITE SPACE 421 If pt A 0 IlIAII 0 so by the continuity of Ptk Ptk IA 0 Thus Ptk iS absolutely continuous with respect to Pt Then applying the Radon Nikodym Theorem and using the fact that nc is countably additive we have Pik Ztk A qtk U Ztk u dPt Ut A 1 where qt is unique almost surely on U AI Now letting n tend to infinity gives 4 5 Observe that llqtkll lIPtkll so that the convergence of the right side of 4 3 follows from the convergence of the expression 3 10 Thus repeating the argument used to prove Theorem 1 for each fixed t and applying Lemma 2 one proves THEOREM 2 Suppose that x y is a valuation equilibrium with respect to a continuous linear form v z on L where L is as specified at the beginning of this section Suppose further that x4 is not a saturation point for any i Then under Assumption I VI xQ yQ satisfies 4 6 for every i xi C Xi and xi i x implies q xi q x 4 7 for every j yj C Yj implies q yj q yO where q is given by 4 3 where q is non trivial 5 AN EXAMPLE The function of Assumptions IV and VI was to impose a continuity require ment on preferences beyond continuity in the norm of the space In this section we provide a simple example illustrating the need for some continuity require ment of this sort Consider a single period where trading is completed prior to the realization of a random variable u assumed to be uniformly distributed on the unit interval The production set is Y y 0 y u essential inf x u One can readily verify that all assumptions of 2 are satisfied as is our Assump