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Faculty Working Papers 
 
 
 ON THE EFFICIENCY OF COMPETITIVE EQUILIBRIUM 
 IN INFINITE HORIZON ECONOMY AND MONEY 
 
 Masahiro Okuno and Itzhak Zilcha 
 
 #374 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana-Champaign 
 
FACULTY WORKING PAPERS 
 College of Commerce and Business Administration 
 University of Illinois at Urb ana-Champaign 
 
 February 11, 1977 
 
 ON THE EFFICIENCY OF COMPETITIVE EQUILIBRIUM 
 IN INFINITE HORIZON ECONOMY AND MONEY 
 
 Masahiro Okuno and Itzhak Zilcha 
 
 #374 
 
'Vajari 
 
 •': ;•■'(' 
 
 . i'fOK fl'SA ' 
 
ON THE EFFICIENCY OF COMPETITIVE EQUILIBRIUM 
 IN INFINITE HORIZON ECONOMY AND MONEY 
 
 By 
 Masahiro Okuno aad Itzhak Zilcha 
 
 Assistant Professor 
 January 1977 
 
 Revised Version 
 
 Department of Economics 
 University of Illinois 
 Urbana, Illinois 61801 
 
■ \- 
 
1-1 
 
 1 . Introdu ction 
 
 Consider a finite horizon economy with outside money which has 
 no intrinsic (i.e., fiat money). At a self "fulfilling (or rational) 
 expectation equilibrium, not. only the price of money must be zero but also 
 money has no exchange value. For, -at the end of the horizon all the traders 
 attempt to spend away money unless price of money is zero and rational 
 expectation makes price of money to be zero over the horizon. 
 
 If we consider an infinite horizon economy, the story is different. 
 In any period, a trader with positive cash balance may find some other 
 trader who is willing to accept money. In this case, an equilibrium with 
 positive price of money may exist. 
 
 In [7], Samuelson showed that, in a simple balanced growth 
 generation overlapping economy, there are indeed cases where traders are 
 willing to accept money and equilibrium exists with positive price of money. 
 Moreover, it was shown that, to achieve a Pareto efficient allocation through 
 a competitive market, the creation of money is necessary in these cases. 
 
 Cass and Yaari [3] argued that this pecuniary feature arises 
 because of a lack of financial intermediary. In a generation overlapping 
 economy not all traders can trade among each other. Nor there is any 
 central market as in a usual finite economy. Therefore, some type of 
 financial intermediation like money is necessary to establish Pareto 
 efficiency of a competitive equilibrium. Then, one might ask when this type 
 of financial intermediation is called for. 
 
 In terms of Gale [4j, a simple economy of the type of Samuelson 
 possesses two distinct possibilities of stationary competitive equilibrium 
 
1-2 
 
 (golden-rule or no-trade). Moreover, the optimal (or Pareto efficient) 
 equilibrium requires the existence of fiat money if the economy satisfies 
 a condition which Gale called the Samuelson case. Hayashi [5] showed, in 
 more general but still restricted class of economies, that money is necessary 
 for Pareto efficiency of an equilibrium when initial' endowment- is not Pareto 
 efficient. 
 
 However, most of the models above are of very simple nature, 
 i.e., each generation live for two period and/or there is only one good 
 except money, in addition to the property that the economy is either 
 stationary or of balanced growth nature. If these restrictions are removed, 
 a simple result like in [3] or [5] no longer holds. It is worthwhile to 
 mention here that the question of existence of a monetary equilibrium in 
 non-stationary economy was investigated by Okuno [6]; it was shown that, 
 under some very restrictive assumptions, there is an equilibrium where 
 money has a positive exchange value. 
 
 We consider the following model. In each period, a finite 
 number of consumers are born. All the consumers live the same, but finite, 
 number of periods. The number of goods is (finite) constant over time. 
 Those, and only those, consumers who live on the first day of the economy 
 may have positive amount of fiat money. Therefore, the total amount of 
 fiat money is constant over the time. Transactions are costless even for 
 future markets. However, transactions with consumers who are either not 
 born or have died are prohibited. In other words, consumers can issue 
 I.O.U. (inside money) but they must clear their obligations by the time 
 of their death. We treat these I.O.U. and fiat money separately. We 
 
1-3 
 
 call equilibrium monetary if fiat money has a positive value and barter 
 otherwise. 
 
 In view of Cass and Yaari's work the following questions arise; 
 (1) If there is a system of financial intermediaries, notably that of the 
 "social contrivance of money", are . competitive equilibria which utilize 
 these intermediaries necessarily Pareto efficient? Namely, in our set-up, 
 are monetary equilibria always Pareto efficient? (2) If the answer is no, 
 what makes equilibria Pareto inefficient and in what situations equilibria 
 are Pareto efficient? This paper answers these questions. 
 
 In Section 2, we present a model which is slightly different 
 from an economy described above. (The equivalence of the economy above 
 and our model is shown in the Appendix). In Section 3, we prove the existence 
 of a competitive equilibrium, although it can be monetary or barter, Pareto 
 efficient or inefficient. In Section 4, the question (1) of Pareto efficiency 
 of competitive equilibria, especially in regard to the role of money, is 
 answered with the help of a few examples. In Section 5, sufficient con- 
 ditions for efficient equilibrium are discussed by using theories on 
 efficiency prices (especially, by Benveniste [2]). In Section 6, we discuss 
 relationship between our result and that by Cass-Yaari [3], present an open 
 question accompained by an interesting example. 
 
2-1 
 
 2 . Descrip ti on of the Economy 
 
 There are an infinite number of tine periods (t = 1, 2, ...) 
 and in each period there are m identical goods. All the goods are 
 completely perishable. There is also (outside) money which has no in- 
 trinsic value but it is the economy's only store of value. We .assume, 
 for simplicity, that all consumers (except those who were born before 
 period 1) live for the same number of periods, N + 1 (N > 1). Consumers 
 are denoted by i = 1, 2, .... Let O.trd, 2, ...} be the period in which 
 
 i is born. Hence i-th consumer is born in 0. and dies in 9. -+• N . 
 
 1 x 
 
 We also use the notation L. = i9. , 3.+ 1, ... 8.4- N}, a set of periods 
 
 1X1 i 
 
 in which i lives. The set of individuals who are born in the same period t 
 is called t-th generation ana denoted by G , i.e., G - (i|8. ■ t}. We 
 
 assume that for all t = 1, 2, ..., G is a non-empty and finite set. 
 
 00 m 
 Let E = X E where m — m for all t . Let E, be the 
 t=l 
 non-negative cone of E . We assume that the ordering of all the consumers 
 
 in this economy is made according to their age, i.e., for any i , j , 
 
 i 1 j if and only if 8, > 0, , We denote by k. the following type of 
 
 sequence in E ; x. = (x. (1), x.(2), ... ) z: E where x.(t)£E for all t 
 
 and X. (t) = for all t - L. = \ 9 . , - . - , 6 . + N} . We denote bv x. the 
 
 i ill -l 
 
 projection of x. on the coordinates of L . Namely, x = fx (&.)» 
 -i i ' -i v i i 
 
 x.(e i + i), ..., x.(e. -f m)f E m(N+1) . 
 
 Each consumer i has a vector of initial endowments in each 
 
 period t in L. , w.(t) . A sequence of initial endowments is denoted 
 
 by w.£ E + where w.(t) = for all t~L. . The con sumptio n pos sibility 
 
 set for i , X. , is assumed as 
 l 
 
2-2 
 
 X = {c.fE jr. (t) = for all t £ L J 
 
 i - 1 + - ■ i ' i 
 
 We denote by c =' (c.). , - fc, , c„, ... ) the allocation of consumptions 
 
 - i 1=1 -1 • 2 
 
 in the economy. The set of feasible consumption allocations is defined as 
 
 X = c = (c ) . , ic "■ X. for all i and I c.(t) < Z w. (t) for all t • , 
 ~i 1=1 -xi ,i — . - 1 
 
 i=l i=l 
 
 Each consumer i has the preference ordering, ">., , defined over 
 
 i 
 
 E , i.e., over the lifetime bundles of consumption. We assume that 
 
 for each i his preference ordering is represented by a utility function 
 
 u. : E >■ E . Finally, we assume that consumers in the first 
 
 period may have a positive amount of initial money holding, m. •: E, . For 
 
 generality, we denote by m. an amount of initial money holding for i , 
 
 but we assume m. .> for all i , to. = for all i with 9. > 1 and 
 11 i 
 
 m = 2 m. > . 
 
 The following assumptions on initial endowments and the utility 
 functions will be made. 
 
 CO 
 
 Assumption I : For all i w. (t) > for all t L. , £ w.(t) >> and 
 
 i=l 
 £ w.(t)|j < M for all t , for some constant M 
 i=l 
 
 i 
 
 Assumption I I: For all i , u. is continuous, increasing and strictly 
 
 quasi-concave and u.(w.) - . 
 
 Let c* = (c.) in X . e* is Pareto efficient if 
 
 (1) c. -. w. for all i and (2) there is no c = (c). , in X such 
 
 • -x 1 -i ~i i-j. 
 
 -,v * 
 
 that c. >. c for all i and c. >. c. for some j . Note that our 
 
 -l i -i -;} j -j 
 
 definition of Pareto efficiency requires individual rationality as well. 
 
2-3 
 
 Let X* be the set of all Pareto efficient allocations in X . 
 
 A sequence of prices of goods will be denoted by p = (p(l), 
 p(2), ...] E . Given prices p and a consumption allocation c in • X , 
 we have the price of money, p (p, c) . The fact that the price of money 
 can be taken constant over time is shown in the Appendix. 
 
 Let U = (u = (a , a , . ,.)[a. = u.(c.) for all i for some 
 
 c •' XJ and let U* = {u = (a , a , ...)'" U| there is c ■ X* such that 
 
 u.(c.) = a. for all i} . 
 i — z. x 
 
 Defi nition : (p*, c*) is a competitive equilibrium if p* r . E + , p* > 
 c * = (c.'K ' X , p m (p*, c*) > , and for all i 
 
 (a) p*«c. < p**w + p (p*, c*)-m i 
 
 (b) c maximizes u. over {c. " X, jp*«c ± < p*«w ± + p (p*,c*)'m ± j . 
 
 * * * 1 
 
 Definit ion: (p*, u*. c*) where u* = (u , u. , ...) with u i E 1S a 
 
 compensated equilibriu m if p* > , p m (p*, c*) > , c*' X , and for all i 
 
 (a') c. minimizes p*«c. over {c. X.ju.(c) >. u } 
 _ x _ .- 1 . i i ' x — x — x 
 
 (V) p**c. = p**w. + p (p*, c*)»m i . 
 
 Given a competitive equilibrium, (p*, c*) , we call it a monetary 
 equilibrium if p m (p*, c*) > and a bar ter equilibrium if p (p*, c*) - 
 
3-1 
 
 3 . Existence of a Competitive Eq ui libri um 
 
 In this section, we shall prove the existence of a competitive 
 equilibrium in this economy. For this purpose, we shall introduce a few 
 definitions and an assumption. 
 
 For each time period T *. N + 1 let F denote the set of of all 
 
 consumers who will die on or before T , i.e., F = {i ] Q. +N <_ T} . 
 
 T 
 Let. X be the projection of X on F , namely 
 
 J. 
 
 X T = it? = (c). _ I c.(t) < £ w.(t) for all t < T} . 
 - 11 F T i F T X " = i=l X 
 
 Note that the aggregate consumption for F may exceed their aggregate 
 
 T 
 endowment. Let U be the projection of U on F , or 
 
 T T i T T - 
 
 U ={u = (a. )... _ a. =u.(c.) for all i:F„ for some c in X } 
 x x F ' x X -x T 
 
 Next, the sees of (individually rational and) Pareto efficient 
 
 T 
 allocations in X , and the corresponding utility allocations are 
 
 X = ic •■= (c). ^ in X ' c. -. w. for all i- F and 
 
 -T T 
 there is no c • X s.t. c. -. c. for ail i - F„. 
 
 - X ' X - X i 
 
 and c. -. c. for some i F r „} 
 -J J -J 1 
 
 T* T T T T* 
 
 U = {u = (ct.),- ^ in U there exists c X such that 
 
 x *■ - F ' 
 
 1 
 
 a. = u.(c.) for all i- F m } 
 x l -i T 
 
 Let us introduce now the foil owing definitions. 
 
 T T 
 Definition: c = (c). . _ in X is T-efficient if c. -.. w. for 
 ~ -.11 ■■ F„ -l x -l 
 
3-2 
 
 all i F_ and there is no c - (c). ._ , X c. < I c , 
 
 L X F T x F T 
 
 c. -. c. for all i in F m and for some i F m , c . -. c.. 
 -ii-i T T' -j J -j 
 
 c* = (c ) , in X is shor t-run efficien t if for all T , 
 ~i i=l 
 * 
 (c.). , „ is T-efficient. Obviously, a short-run efficient allocation 
 
 is not necessarily Pareto efficient. 
 
 T T 
 Definition: Given c"" - (c . ) . „ in X . let i and i' be any two 
 \.x i . F_ ' 
 
 commodities, t and t' <_ T be any two periods. We say that timed commodities 
 
 T 
 (t, j) and (t ' , j') are resource-rel ated (with respect to c ), if there 
 
 exists i in F , such that c..(t) > and c.,,(t') > . We say 
 
 T 
 that (t, j) and (t* , j') are in directly-r e cource-r elated (w. r. t. c ) 
 
 Q 
 
 if there is a finite sequence ((t, , j, )l lr _ n such that (t , j ) = (t, j) , 
 
 K. K. ix~U O O 
 
 (t £) j y . ) = (t', j') and (t k _ l5 J k-1 ) ls resource-related to (t fc , j fc ) for 
 
 Assumption III: Any (t, j) and (t', j') (1< j, j' <m, 1< t, t' <T) 
 
 T 
 are indirectly resource related with respect to any T-efficient c , for anv 1' 
 
 We shall use Assumption III to prove existence of competitive 
 
 equilibrium. However, we show first that this assumption implies another 
 
 important property of intergenerational substitution of consumption. 
 
 Definition : Let c X' be T-efficient (c*' X be short-run efficient, 
 
 respectively) and c > . Period 1 is said to be e-resource relate d to 
 
 period t <, T with respect to c (c* , resp.) if there exists c = (c ) . „ 
 
 ~ rj. 
 
 such that 
 
 (a) c.(l) = Z c*<l) + (1, 1 , 1) , 
 
 1 ■ r ? 1 - F T 
 
3-3 
 
 (b) _ c.(t) = E c.(t) - _(1, 1, ..., 1) 
 
 (c) c (T) - c*(t) for all t £ 1 , X ± I, t, 
 
 i .- F T x i _ F_ X 
 
 (d) c. X. and c. >:. c. for all i£ F m . 
 -i . x -l x -x T 
 
 T T T 
 I.emna 1 : F or each T , there exist positive scal ers {_ ,■ _ , •-. e } 
 
 X x* T 
 
 such that for all T-efficient allocations, c X , period 1 is c'-resource 
 
 _ — __ ______ __ _ 1 — _ 
 
 related to period t . 
 
 T T* 
 
 Proof : For each c - X , by Assumption III and monotonicity of a. , 1 
 
 is e-resource related to t for some e > . Let e (c ) be the maximum 
 
 of such e > (one can take maximum, for _ is defined over a compact set 
 
 X* . X* 
 as is easily seen by the definition) . For any T-efficient c v X , and for 
 
 T* "T 
 
 any 6,0<6<_(c")by Assumption III, there exists c satisfying 
 t ~ 
 
 condition (a), (c) and 
 
 (b r ) Z c.(t) = _ c?(t) - 6(1, 1, ..., 1) 
 i ■■■ F_ i - F T 
 
 * ■ 
 
 (d') c • X. and c. ■-. c. for all i " F r „ . 
 ~x x -x j -x T 
 
 Since u. is continuous, there is a (relatively) open neighborhood of 
 
 X* x T* T T T* 
 
 c in X , N(c 6)7 X , such that for all c - N(c , 6) , peirod 1 is 
 
 T 
 6-resource related to period t . Therefore e (c ) is continuous. Since 
 
 T 
 by assumption III, the set of all T-efflcient c in X is compact, there 
 
 T T T I 
 
 is e = min {e (c )|c X Lent) > . 
 
 t c _ 
 
 Given Assumptions I-II [ , we can prove 
 Theorem 1 : There exists a c ompetit Ive equilibrium . 
 
3-4 
 
 Let us prove first the following Lemma. 
 Lemma 2 : For each T , T > N + 1 , there exists c = ( e . ) . in X , 
 p E v;ith p (t) = for all t_JL_T , and p > such that for all i : F 
 
 T 
 
 T* * T* , mT* - 
 (a) p "C < p »w. + p -ra. 
 
 « v * . . r r- v ± i T* T* mT* - , 
 
 (.b; c. maximizes u, over lc." X i p *c. < p *w.H-p 'm.} 
 -l i i ' C ,.i — i i 
 
 Proof: The proof is essentially similar to Arrow-Rahn's [1, ch. 5]; However, 
 due to our special structure and existence of money, some parts are subs- 
 tantially different. 
 
 Given T , T > N + 1, define 
 
 S T - (p* E + | |ip T (l)|; = 1 , !|p T {t)|[ = ~ for all 2 < t < T and p X (t) = ( 
 
 £ 
 
 r r TtT 
 
 for all t > T}, where the sequence of positive scalars (el -, is given by 
 
 Lemma 1. Define a mapping p' : S > E, by 
 
 p (p ) = - •■ - P (t) w (t) 
 
 m i : p m t=T-N 
 
 ra T - 
 In other words, p (p )m, the value of the money, is made so that 
 
 PTi 
 
 it is exactly equal to the value of endowments in X , which does not 
 
 T T 
 properly belong to the consumers in F . Given prices p ■■ S and allocation 
 
 T .. T T 
 
 c X , we define budget surplus tor all i in F , 
 
 / T T. i . m,. T - 
 
 s.Cp s c ) - p (w. ■ + p (p )a. . 
 
 T T 1 
 
 Let Z = {v = (v ) v , ! v.£ ,L' and v. = 1} . Define 
 
 1:F I 
 
 T T T" T T 
 
 ijj : I *■ U in the following way: For v c £ , define 
 
3-5 
 
 rr, rp rp rp rrl m T TA 
 
 ^ (v~) - max {a > | Av ~U }*v . *J> is well defined because (1) u f U 
 
 if u. = for all i £ F , since E w. (t) - E w.(t) > for all t 
 
 1 T i=l 2 i ■ F * 
 
 T t 
 I-B+l < t <T (note the definition of X ) and (2) any Pareto efficient 
 
 T T* 
 allocation c X is also weakly Pareto efficient because of Assumption III 
 
 (i.e., if c Pareto-dominates c then u.(c„) > u.(c.) for all i- F_ 
 
 s T ,T . - 
 
 for some c in X ) . V is a continuous function. 
 
 T i T* 
 
 For each u = (a ) . in IT , define corresponding consump- 
 
 T T* 
 
 tions c In X by, 
 
 T* HP T" Tit 
 
 W (u ) = {c ■- X I u.Cc.) =a. for all i£ F_} . 
 
 i _ i i j- 
 
 T 
 W is single-valued and continuous by Assumption II. 
 
 For each c in X , define 
 
 B x (c ) = {(x(l), ...» x(7)j E m j there exist 
 
 -X 
 
 c = (c.). _ such that c. > c, 
 _i'i F T -l 'x -i 
 
 for all i F_ and 
 
 E c (t) < E c.(t) + x(t) for all t < T} 
 
 i F_ ~~~ i - F m x 
 
 T T 
 
 T T 
 
 B (c ) is strictly covex and contains a non-empty interior. Furhtermore, 
 
 rj rj, Vr. 
 
 (0, ..., 0) -f Int 3 (c ). Define P : X — * S, as follows: 
 
 P T (c 1 ) = fp 1 S T | (t)x(t) > for all (x(l) , . . . , x(T)J in B (c T )} 
 
 t=3 
 
3-6 
 
 By the well-known separation theorem, there exists a non-trivial prices 
 
 fp(l),..., p(T)J E™ satisfying the condition of V (c ) except that 
 
 T T 
 p S . By the definition of e-resource relatedness and Lemma 1, 
 
 fx(l), . .., x(T)) B T (c T ) when x(l) = (1, -.., 1), x(t) = -£*«(1, ..., 1) 
 
 'T 
 
 and x(t) = for all t > T , r.J 1, t . Therefore || p(l)|| -r e- c ||p(t) j| > 
 
 rp rp -j™ T T 
 
 or |jp(t)[|< i|p(D|| /e • Hence p S • P (c ) is non-empty, convex, 
 
 compact-valued, and upper semi-continuous. 
 
 T T T T 
 
 Finally, define V : S x X — > I , 
 
 V T (p T , c T ) = {v T L T j Vj . = if s i (p T , c T ) < 0} . 
 
 T 1 T 
 
 V (p , c ) f <i , convex, compact valued, and upper-semi-continuous. Now, 
 
 consider the following correspondence. 
 
 TTTTTTTT T T T T 
 
 P °W °»K XV * W °<i> : S x Z * X > S x 2 x X 
 
 This correspondence satisfies all the conditions of Kakutani's fixed point 
 
 theorem . 
 
 T»fc T^ T^» 
 Let (p , v , c ) be the fixed point. By our definition, 
 
 T* T* T* T T* / T* T* „T*. 
 
 c X . Let us show that when u = ■i 1 (v . ) , tP > L l » £ ' 
 
 satisfies (a') and (b ' ) of the definition of compensated equilibrium for 
 
 all i- F . Assume s.(p , c ) < for some i . By definition of 
 
 V , v. = and u.(c.) =0 . Since u.(w.) = we come to 
 
 X* * * 
 
 0>s.(p , c.) >_ p (w.-c.) which contradicts the fact that 
 l „ _ i — „i ~ i 
 
 T* T T* T* T* , r- . 
 
 p P (c ) . Therefore s . (p , c ) > for all i . By the defxnition 
 
 m T'ft — — 
 
 of p (o ) and notin n. , we see that 
 
 i 
 I 
 
 Z s.(p T , c 1 ) = (w - c )]- p m Cp L )-m=-0 
 
 if F J " " i t F 
 
 T T 
 
3-7 
 
 X" ■£* 
 Hence s . (p , c ) = for all i in F • 
 
 Finally, because of Assumption III and monotonicity of u. "s, 
 
 conditions for compensated equilibrium are equivalent to those of competitive 
 
 X* * • I* 
 
 equilibrium (i.e., p *c. > for all i F » since p (t) >> for 
 
 all t < T) . 
 
 7* T* mT* v 
 Proof of Theo rem 1 : Let (c , p , p .) be the fixed point 
 
 X* 
 
 we obtained in Lemma 1. Let u be the corresponding utility values, 
 
 * -T -T -T -ffiT, °° -T T* -T , 
 
 which is in U . Let (p , u , c , p ) T _-i satisfy; p = p ; u • li 
 
 -T T* -T -T T* -mT mT* 
 
 such that Proj u = u ; c X such that Proj c = c ; p = p 
 
 p p 
 
 T T 
 
 ,_X _X _'£. oo 
 
 Then there is a subsequence of [p (1) , u", c J which converges 
 
 f -* *1 il II * 
 
 coordinate-wise to [p*(l), u , c J such that || p A (l) || = 1 , u U , and 
 c* X . 
 
 To show that c* is short run efficient, notice that by 
 Assumption III it can be proved, in a straightforward manner, that a limit 
 
 of k-efficient allocations (for some finite k) is a k-efficient allocation. 
 
 k k 
 This follows from the fact that if c = ( is Pareto-dominated, 
 
 _i i ■ F. 
 
 - k *• k k 
 
 then there exists some k-efficient (c). ._, s.t. u . (c . ) > u.(c.) for 
 
 X X X -1 1 -X 
 
 K 
 
 all i in F. . 
 
 V:'; 
 
 Given anv T for all T > T , (c. ) . . is T -efficient 
 
 o o . i i fc t o 
 
 o 
 
 -t x* T* 
 
 (otherwise, for some (c.). _ with I c. < '[. c. we have 
 
 T o a ^T * " F T 
 
 ~X x* T * ~T *T 
 
 u.(c.) > u.(c. ) for all i in F r „ . This implies I p~ (c. - c. ) > 
 x-ii-x 1 . .. _ _ ^x ~x 
 
 o x F T 
 
 o 
 
 * 
 
 which is impossible.) This implies that (e.) is T -efficient for 
 
 -i'icF„ ~ 
 
 any T , hence c* ia short-run efficient, 
 o - 
 
3-8 
 
 Since c* is short-run efficient, by Assumption III, there 
 
 exists a sequence of positive scalars {e , z , ..., ) such that 1 is 
 
 c - resource related to t for ail t with respect to c*. Therefore, 
 t 
 
 for given t , there exist T and (c.). , which satisfy the con- 
 
 o „ A 
 
 ditions of 6-resource relatecness with 6 (0 < o < e )' and with c. > c 
 
 L. — X 1 1 
 
 for all i6F (by Assumption III again). Because of the coordinatewise 
 
 o T * 
 
 convergence, period 1 must be 6-resource related to t for all c for 
 
 T sufficiently large. Therefore {p (t) }__. must be bounded and we 
 
 can find a converging subsequence. Finally, for such a subsequence, 
 
 oo N-M 
 mT* mT* — i ml T* *■ T* 
 
 {p } is bounded, for . Z„ p -m = P -m <_ £ p -c i I z p (t)w (t) , 
 
 1 G l . i€G~ - 1 " i=l t=l ' * 
 
 which is unformly bounded. Let (p*, p ) be the limit of such a converging 
 
 subsequence. Then p™ > , p* > since ||p*(l)|] = 1 , and 
 
 m* 
 s.(p*, c*) = for all i . It can be shown easily that (p*, u*, c*, p ) 
 
 is a compensated equilibrium, which, in turn, is a competitive equilibrium 
 
 by Assumption III, 
 
 Q.E.D. 
 
4 . Efficiency of an Equilibrium 
 
 In their paper [3], Cass and Yaari, in effect, claim as follows. 
 In an economy with overlapping generations, there is no "Walras market", 
 for some consumers in different generations may not be allowed to trade' 
 each other. To achieve an efficient equilibrium in a limited market 
 structure, some device like a financial intermediary is called for. 
 Therefore, there may exist an equilibrium with money actively traded. 
 Example 1 
 
 Consider the following economy, which has the same structure as 
 
 our model. Let m (the number of goods) be one. Let each consumer live 
 
 for two periods. We index them as 0,1,2,... . 6 = i for all i > 1 . 
 
 For i = , he lives only at t = 1 . Their initial endowment is as in 
 
 the Table 1 while we assume m„ = 1 and m. = for all i 5 1. 
 
 x 
 
 i n^ 
 
 1 
 
 2 
 
 3 
 
 
 
 ' 1 
 
 
 
 1 
 
 •> 
 
 1 • 
 
 
 2 
 
 
 3 
 
 1 
 
 • 
 
 
 
 "" . 
 
 Table 1 
 
 Finally, let utility u. = Ha c.(e.) + £n c ± (Q ± + 1) for all i £ 1 
 
 and u = In c„ (1) . 
 
 There are two equilibria, one is barter and the other monetary. 
 Barter equilibrium is such that consumption is exactly the same as initial 
 
4-2 
 
 t-1 
 
 allocation and prices satisfy ?(t) = 3 p(l; for all t. Monetary 
 
 equilibrium is such that consumption is as in Table 2, prices satisfy 
 p(t) = p(l) = p m for all t . The Barter equilibrium is Pareto inefficient, 
 for it is dominated by the monetary one and the monetary equilibrium is 
 Pareto efficient. Therefore, the example not only seems to confirm 
 Cass-Yarri contention but also indicate importance of distinguishing a 
 Pareto efficient equilibrium from an inefficient one. 
 
 i \v 
 
 X 
 
 2 
 
 3 
 
 
 
 2 
 
 
 
 1 
 
 2 
 
 2 
 
 
 2 
 
 
 •i 
 
 £ 
 
 2 
 
 Table 2 
 
 Example 2 
 
 However, it is not too difficult to construct an inefficient 
 monetary equilibrium. For example, let us keep essentially the same 
 structure of the first example but change their initial endowment slightly 
 as in Table 3 with e > 0. Then the barter equilibrium (of consumption 
 and prices) in the first example becomes a monetary equilibrium in the 
 new situation with p™ = ep(l). Since the aggregate endowments are the 
 same as before, our monetary equilil Ls Pareto inefficient. 
 
4-3 
 
 H 
 
 
 1 
 2 
 
 1-e 
 
 3-fe 1-3 e 
 
 3+3" 1 , 
 
 -2 
 
 1-3 "~£ 
 
 3+3 e 1-3 e 
 
 Table 3 
 
 As a matter of fact, a monetary equilibrium and a barter equili- 
 brium are not too different in some cases. Suppose we obtain a barter 
 
 equilibrium with equilibrium prices p which satisfy lim inf jj p(t) [J > 0. 
 
 t-x» 
 Then, in general, we can redistribute initial endowments in each period 
 
 from a young generation to a old generation, and make the original barter 
 
 equilibrium a monetary equilibrium in a new economy. Moreover, we can 
 
 make this redistribution as small as possible, as is the case of £ in 
 
 our example. Namely, any barter equilibrium with lim inf |j p(t) j| > 
 
 t-*» 
 is considered as the limit of a sequence of monetary equilibria where each 
 
 monetary equilibrium belongs to a different economy but these economies 
 
 converge to the economy in question. 
 
 More disturbingly, if at any equilibrium, lim sup j| p(t) [j = , 
 
 then the equilibrium can never be monetary because the value of money 
 
4-4 
 
 m — 
 p .m must sooner or later exceed the value of economy's total initial 
 
 endowments. But as we show in Theorem 2 below, such a ("barter) equilibrium 
 
 is always Pareto efficient. 
 
 Some readers may observe that the Example 2 is of very absurd 
 nature. If p is unbounded, one unit of good in period 1 has the same 
 value as a very small units of goods in a distant future. Hence one can 
 make a sequence of transfers of goods, of the same value, from young 
 generations to older generations in each period, and eventually make this 
 transfer disappear in the limit, thus creating a Pareto-dominating 
 allocation. A. seemingly natural conclusion, in view of strict-quasi- 
 concavity of the utilities (Assumption II), seems to be the following one. 
 If lim sup i| p(t)jj < °° , then a (monetary) equilibrium is Pareto efficient- 
 Unfortunately, this contention is, again, incorrect as the following 
 example shows. 
 Example 3 
 
 Let the generational structure be the same as in Example 1. 
 Let the initial endowments be as in the Table 4. Namely, ^nC^) = 1 - £"3 , 
 
 M.(l) = 3 + £-3 q , 
 
 2 
 io. (6.) = 3 + c *3 for all ii2, 
 
 -q+~ 
 (i).(S.+l) = 1 - e«3 for all i > I , 
 
 2 
 where c is an arbit smaJ ositive number and q = tt /6 . Let 
 
 utility functions be 
 
4-5 
 
 l-e*3 
 
 3+e • 3" q 
 
 l-e*3 
 
 3+e • 3 
 
 -al-1 
 
 q+1 
 
 -q+- 
 
 1-e -3 
 
 -q+- 
 
 3+e«3 
 
 -q+- 
 
 l-e-3 
 
 Table 4 
 
 i Z -l 
 
 u.(c(!3 i ),c(6 i +l)) = 
 
 .2 .2 
 
 c(8.) x + c(6.+l) 1 
 
 L x 
 
 i -1-] .2-1 
 i 
 
 for all i>2, 
 
 ^(0(1)^(2)) = in c{l) + £n c(2) and u (c(l)) - In. c(l) 
 
 Then there is a monetary equilibrium with consumption allocation as in 
 Table 1 with prices 
 
 t-1 .2 
 
 x 
 
 p(t> = ?(!)• TT 3 
 i=l 
 
 t-1 
 
 ■p(l) for all t > 2 
 
 To see this, observe that for any ii 1, marginal rate of substitution of 
 
 i/i 2 
 c(0.) for c(e.+l) is [c(8.}/c(8.+l)j . Therefore, at the 
 i x i x 
 
 equilibrium, MRS = Z ' . Moreover, since )_ — =■ - ff /6 = q , 
 
 i=l i 
 
 lim p(t) is finite. However, a, this equilibrium allocation is Pareto 
 
4-6 
 
 dominated by the allocation in Table 2. Because for any i > 1, the equilibrium 
 consumption (c*(0 .) ,c*(9 .+1)) = (3,1) is indifferent to (1,3) . 
 Since consumption in Table 2 is a convex combination of these two, strict 
 qua'si-concavity of utility functions asserts that Table 2 consumption 
 dominates the equilibrium allocation. 
 
5-1 
 
 5 . Chara c terizati on of Efficient Equilibi i . 
 
 In this section, we shall ask the following question; under 
 what conditions there is a (monetary or barter) Pareto efficient equilibrium? 
 Let us start with a relatively simple case. 
 
 HI" 
 
 Theorem 2: Let (p*, c*, p ) be a com petitive equ ilib rium . If . 
 
 lira sup ||p*(t)|j = , then c* is an efficient equilib rium . 
 t -*- °° 
 
 Pro of: Suppose that c* is not efficient. Then there exists an allocation 
 
 * 
 
 c X which dominates c*, i.e., c. r . . c. for ail i and for some j , 
 
 ~ -i x -i J 
 
 * ft * 
 
 c ■- . c. . Therefore, P**c. — p A «c for all i and p*«c. > p*-c. . 
 - J -J ~ -x ~ - -i ~ ~3 ~ ~J 
 
 Let T = h . + K , the last period in which j lives. Then for any T >* T , 
 o j o 
 
 F contains j and 
 
 < X p*.(c. - c.) 
 
 i ; F ~ " a " X 
 
 T 
 
 T-N+l ^ T A 
 
 Z I P*(t)[c.(t)-c.(t)l+ Z Z p*(t)fc.(t) -c.(t) 
 
 ifF T t=l X i F t=T-N+2 a X 
 
 < Z Z p*(t)[c.(t) - c At)] . 
 
 if F t=T-N+2 
 
 * 
 However the right hand side goes to zero as T — > » , for I [c.(t) -c. (t) 
 
 i : F X L 
 T 
 
 is uniformly bounded by Assumption I, a contradiction. 
 
 Q.E.D . 
 
5-2 
 
 If lim sup j!p(t)]{ > , we need some conditons for Pareto 
 t -> <» 
 efficiency. The following approach makes direct use of the result by 
 
 Benveniste [2] on efficiency prices. In the proof of Theorem 1, we show 
 
 that a competitive equilibrium is always short-run efficient. 
 
 Let K, = !i|T< 8 < T •+ N - 1} , i.e., the set of all the 
 
 consumers who live in both periods T + N - 1 and T + N . Given a 
 
 short-run efficient allocation c -. X , define 
 
 D T (c) = ((xT; X*) = (x T (T), ..., x T (T + N-l); x T (T + N), ..., 
 
 -T ^\ - ?mN , -i 
 
 x (T+2N-l)JrE ■ There exists (c"> . „ such that 
 
 - ± ' \ 
 
 c. S. c. for all i: H and 
 -1 l -x I 
 
 c,(t) - 1 c.(t) + x (t) for all t , 
 i H i H 
 
 T < t < T + 2N - 1) 
 
 Since c is short-run efficient, we can define a social (aggregate) 
 
 T 
 indifference curve at c . D (c) is (the translate of) upper contour 
 
 set for this indifference curve. In the examples in the previous section 
 
 where there is only one good, there is only one consumer in each generation 
 
 T 
 
 and each consumer lives only for two periods, D"(c) is exactly equal to 
 upper contour set for the consumption. In Example 3, we showed that if 
 the indifference curve loses "strict" curvature over the time, we might 
 find a Pareto inefficient equilibrium. fore, we shall define the 
 "coefficient of strictness" o rence curve similar to 
 
 Benveniste [2]. 
 
5-3 
 
 Let (p*, c*, p' ) be a competitive equilibrium, then c* is 
 
 short-run efficient as was shown In the proof of Theorem 1. 
 
 Define 
 
 T+N-l 
 
 Q T (c*, p*) » {(X* X 1 ) D T (c*) | Z p*'(t)-x T (t) < 0} 
 
 ~ t=T 
 
 The following definition of jj-strictness guarantees that if 
 
 c* is short-run efficient with support p* , then given T we can find 
 
 * T 
 
 a tangent "paraboloid" at (c.) containing D (c*) . 
 
 -x 1 " H 
 
 m* 
 
 Definition: Given a competitive equilibrium (p*, c*, p ) , c* is 
 
 — T -T T 
 jj -strict at T if for all (X., , X„) in Q (c*, p*) , we have 
 
 T+2N-1 T+N-l u (T+N-l V 
 
 P*(t)-,x i (t)>- Z p*(t)-x i (t) + ^q ¥rr - Z p*(t)-x(t)| 
 
 t=T+N t=T /' t/ ,, /t ,| t=T 
 
 L 
 
 t=T 
 
 p*(t)-w(t) 1 - 
 
 where w(t) = Z w,(t) , i.e., the aggregate endowment of the economy at 
 
 i=l x 
 t . Denote by IL = sup {y ! c* -strict at T'l . 
 
 Remark : Our definition of u -strictness generalizes that of Benveniste. 
 The reader is referred to [2! for detailed tnation of this condition. 
 
 OS 
 
 Given c* in X , I e ! , cients of resource related- 
 
 t t=i 
 
 ness if f° r all t e - sup 1 is £ resource related to 
 
 period t (w.r.t. c*)}. 
 
 m* 
 Theorem 3 A co mpetitive equil ibrium p ) i s Pareto efficient 
 
 if, at c* , 
 
 00 
 
 (a) the c of resource relatedness (c ) , satisfy 
 
 t t= a — _ — _ 
 
 E e = «> , and 
 t=l 
 
5- 
 
 (b) t here exists \i > such th at for al l. T > 
 u >_ u , where c* is 
 
 Remark : Examples 1 and 2 can be considered as counter examples for the 
 
 theorem without (a) and example 3 as a counter example for the theorem 
 
 without (b). By assuming uniform smoothness in the analogous manner 
 
 (see [2]), we can prove that condition - . cf icient for Pareto 
 
 efficiency of a competitive equilibrium. 
 
 Proof : Because of (a), an argument similar to the one in the proof of 
 
 T 
 
 Lemma 1 shows that E l/|{p(t)|| — > °° as T ► °" . 
 
 t-1 
 
 * 
 
 Suppose there, exists c X such that c. .-. c. for all i 
 
 * 
 
 and c. ■-. c. for some j . Define a sequence t. = (k - 1)N + 1 for 
 
 2 3 J k t, t,+N-l 
 
 kik where k is such that j •" H . Define p = ip*(t)l 
 
 o o t, v K * J t=t, 
 
 k k 
 
 o 
 
 t. t t 
 
 and (L , JL ) such that X (t) = I . (t) - c.(t)] for all t 
 
 J. JL _, 3 X 
 
 i H 
 
 \ 
 
 _t _t 
 
 with t < t < t + 2N - 1 . Defd , - p -X n for all k > 1 and 
 
 _t. _t 
 
 & 2 ? K * X ? ' for a11 k - 2 ■ 
 
 p*[c. - c.J = p -X + p -X = 6 + o > 
 
 * C k 
 for all k with strict inequality with k = k . Since c* is an 
 
 . t + N~l° 
 
 K .A 
 
 equilibrium allocation and ccX , X. E p*(t)[c.(t) - c.(t)j 
 
 t=t 
 k 
 
 t t t 
 = P k lX k + X 2 k_1 J = 6 k + d k < for all k > 2 and 
 
5-5 
 
 - N_1 * - c i - c i 1 
 
 £ I p*(t)[c.(t) - cAc)] = p -X - 6., <_ ■ 
 
 i=l t=I 
 
 .Therefore, o" (6 k , respectively) - a in-decreasing ' (non- increasing) 
 
 ' k k 
 
 sequence of non-negative (n numbers and 6 2 > (& 1 < 0) 
 
 for all k > k + 1 . 
 o 
 
 t t t 
 Hence, (x\ X^) C Q fc (c* s p*) for all k > k Q + 1 and using (b), 
 
 there exists u, > V > such tl 
 
 5 k+1 > _ 5 k + -^(6 k ) 2 > -i k + — (S k ) 2 > for all k > k + 1 (1) 
 2 = 1 n. l = J- n k i ~ ° 
 
 t +N-1 
 
 where n. = S p*(t)-w(t) > . Since -«J >-« 2 > ° for a11 
 t=t. 
 
 K. 
 
 k > k + 1 , (1) is rewritten i -. 
 — o 
 
 ,k, 2 
 
 6 7 +1 i 6 2 ■ 
 
 Define V, = 5^/rj, . For all k > k + 1 , V > , and since 
 k 2 k ok 
 
 t 
 
 x (t) < w(t) for all t and k , V . Then, (2) becomes 
 
 ViVi i Vk + ^\ ( V > ° £or ali k « k o + l (3) 
 
 Taking the inverse of the both sides, 
 
 1 < 1 K._ X 
 
 VlVl ^ \\ 1 + u \ 
 
5-6 
 
 or 
 
 JL < 1 + V 
 
 — — •.'— - ! for all k > k + 1 
 
 k V k \+l V ] . ~ ° 
 
 (4) 
 
 Adding (A) from k = k + 3 to K - 1 and cancelling terms, 
 
 K-l 
 
 i_ < i + p r i_ 
 
 k=k +1 "k ]i \\ +l\ 
 
 i_ i < i±Ji l__J: 
 l, V, i = - 
 
 o o 
 
 k+1 k+i 
 
 '. I Lt ft. ' -1- R. ^ 
 
 L ° ° 
 
 and £ ~ < « 
 
 k=k +1 it 
 o 
 
 However , 
 
 t, +N-1 
 k 
 
 t, +N-1 
 
 00 > Z — : 
 
 k=k +1 'k k „ 
 
 o ok 
 
 k+i t-V, p*<t>- W (t) - k . k+1 t . t 
 
 E l/j|p*(t)jl-i|w(t)i| 
 
 Since v(t) is uniformly bounded by Assumption I, it implies 
 
 £ l/|jp*(t)jj < °° , which is a contradiction. 
 
 t=t. 
 
 k +1 
 o 
 
 Q.E.D. 
 
6-1 
 
 6 . Conclusions and Remarks 
 
 In an economy with a generational structure, one would not 
 expect to have the existence, of Kalras market. Hence, unless there is 
 some "double coincidence of wants" between generations, an equilibrium 
 may be Pareto inefficient in a sequence of markets generated by a genera- 
 tional structure. This is precisely the reason why Cass-Yaari looked 
 into a "closed-loop" model, which is a finite analogue of a stationary 
 generational economy. It seems that they conclude that if money is 
 introduced to the market, a central clearing house, which was lacking 
 without money, was effectively restored because money plays a role of 
 intermediation. As we saw in Section 4, it is incorrect if one extends 
 his attention beyond a stationary economy. It is true, in some cases, 
 that the introduction of money makes a new equilibrium Pareto efficient. 
 But not all monetary equilibrium are Pareto efficient, namely financial 
 intermediation does not always remedy the lack of Walras markets. 
 
 As we observed in Section 5, the question of Pareto efficiency 
 of a competitive (barter or monetary) equilibrium is closely related to 
 the question of efficiency prices. , r , Lems seem to be inherent to 
 the infinity of horizon more . . - else. 
 
 However, it should be noted that our f: igs do not refute the 
 claim by Cass-Yaari altoge • pie where there is 
 
 neither monetary nor bart^ i ent eq i ura. We believe 
 
 that the following conjecture eco] tot find a Pareto 
 
 efficient equilibria inter:.. (money), there is a Pareto 
 
 efficient equilibrium in t. r.y with money. To this end, one 
 
6-2 
 
 must show the existence of a Pareto efficient (barter or monetary) 
 equilibrium. One might hope that a direct application of Arrow-Hahn. [1] 
 approach may do just this. Unfortunately, there is some technical 
 obstacle, that is, the utility frontier U* is not closed in Lhe 
 topology of convergence in coordinates {the most natural topology in 
 this economy) . 
 
 So, it is an open question if the non-existence of a Pareto 
 efficient barter equilibrium is a sufficient condition for the existence 
 of a Pareto efficient monetary equilibrium. Let us conclude this paper by 
 presenting yet another example, which shows that it is not a necessary 
 condition. Namely, an example which contains both Pareto efficient barter 
 and Pareto efficient monetary equilibria. 
 
 There are two goods in the economy in addition to money. In 
 each period, consumers 2t and 2t + 1 are born. T A'e refer the first 
 consumer as an "even" consumer and the second as an "odd" consumer. All 
 the even consumers and all the odd consumers have the same characteristics, 
 
 respectively, except the date of birr . for an even consumer, 2t , 
 
 r , i 711711, 
 
 initial endowment is (w.(.t), w.(t -f l)j = (— , — , — , — ) , i - 2t , his 
 
 utility function 
 
 u. = In {min fc, (t), c.„(t)]} + In {min [c- (t-KL) , c .„(t+l)]}, i = 2t . 
 
 1 11 1Z li 12 
 
 An odd consumer 2t + 1 has initial endowment 
 
 it), w i (t + 1)) = (4, 2- 0, 12), i = 2t + 1 , 
 
 u - lMc,.,(t)] 1 +^[c (t)] 1 }~ h + Uc.,(t+1)] I + [c. (t+l)l 1 ~ h , i = 
 
 1 11 1 Z I. J. li 
 
 2t + l 
 
Consumers and 1 live only in perio* with 
 
 w Q (l) = (~, |) and ».,(!) = (0, 12) 
 
 u = An {min [c l),llc Q (l)]} 
 
 6-3 
 
 • i .-•< 
 
 u n = in C[c_,(l)] "■ -r [c 19 (l) x } ' 2 - 
 
 12 
 Finally, let m. = ror all i f 1 and m_ =•' — . The initial endow- 
 ■ i. i 5 
 
 merit allocation is summed up in Table 5. Then there exist at 1 eat two 
 distinct equilibria, one monetary and the other barter. 
 
 2*4- 
 
 { 2 ' ?/ 
 
 (0, 12) 
 
 ( 71 1 } ,71 1 
 
 C4 S 2) (0, 12) 
 
 ,71 1, ,71 1_, 
 2 ' 2 ^ 2 ' 2 
 
 ,12) 
 
 Le 5 
 
 A monetary equi [ 1 Location is shown in 
 
6-4 
 
 Table 6 while prices are p(t) = (1, 1) for all t and p = 1 . A 
 barter equilibrium consumption allocation is shown in Table 7 while prices 
 are p(t) = (1 , 4) for all t and p "" = C . Since both allocations 
 satisfy conditions for Theorem 3. Pareto . efficient . 
 
 t i 
 
 
 
 (33, 3) 
 
 
 1 
 
 ,36 36 
 <. 5 , 5 ) 
 
 
 2 
 
 (33, 3) 
 
 (33, 3) 
 
 3 
 
 
 .36 36 
 
 { 5 ' 5 
 
 4 
 
 
 (33, 3) 
 
 ,9 9. 
 (t. t) 
 
 ,36 36. 
 5 3 
 
 Table 6 
 
6-5 
 
 t I 
 
 
 1 
 2 
 3 
 
 4 
 5 
 
 V 2 ' 2 ; 
 
 (16, 3) 
 
 ,55 
 2 
 
 
 / 5 _1 5. 
 ( '2 ' 2' 
 
 (4, 2) (16, 3) 
 
 ,55 5. 
 ( T' 2° 
 
 (4, 2) 
 
 ,55 5 
 
 ) 
 
 ( 2 » 2 
 
 (16, 8) 
 
 Table 7 
 
7-1 
 
 APPE 
 
 We consider the following economy . - before there are m goods 
 and a finite number of consumers in each • od t . Let H be a set of 
 individuals living at t , i.e., S. - U|t - N < f - . < t} - Each consumer 
 
 i is characterized by so that he lives from period 9. up to period 
 
 i i 
 
 + N . *\]j the transactions are costless but we assume that transactions 
 i 
 
 can be made only among living generations for the periods they live. Hence 
 in each period t , there are m spot markets for each commodity, m futures 
 markets each for delivery date t = t+1, .-., t+N and a bond market each 
 for maturization date T = t+1, ..., t+N . Let x (t) be a vector of net 
 purchases that consumer i makes in period t to be delivered at 
 x = t, t+1, ... t+N . Each x' T (t) :E and we write 
 
 x 1 = (x X (t), x 1 (t), ... x* (t)} which is the list of spot and future 
 
 i . . „m(N+l) 
 transactions l made at time t . x is zn E 
 
 A unit of bond is a contract to deliver one unit of money at the 
 
 ib , K , 1 
 
 ma 
 
 turization date. Let x (t) h E be a net purchase of bond in period t 
 
 for delivery at T, T > t and x* = [x"\, , (t), ..., x .. w (t)J is the bond 
 
 — t t+1 t+N 
 
 transactions at t . Let us use infinite sequences of vectors in E 
 
 'L 
 
 is defined as x 1 = [O, 0, . .., x^(t) , ..., x ...(t), 0, ...J and x__ is 
 is defined as x* b = (O, 0, ..., x^Ct) , ..., x^^t), 0, ...). Let 
 
 m 1 ?. E be the cash balance of consumer i at the end of period t and 
 
 t 
 
 m 1 = (0, 0, . .., in* . .., mZ ,„, 0, . ..). Let us write the commodity price 
 
 6 c 1 .+M 
 
 i i , 
 
 in period t as p - (O, 0, ..., 0, ? Jt) , .... P t+N (t), 0, ...} where 
 
 p (t)~ E™ is a vector of prices at ■: in the future market for delivery 
 
 at t , t < t < t+N . Also be od t , 
 
 b /-„ „ b , ^ b 
 
 , t = (0, ..., o, P t+1 (t), ... 
 
 P* = (0, ..., 0, " p* (t), ..., ]* * -..} where P ^(t)?E^ is the 
 
7-. 
 
 m 1 
 
 price of bond in t which natures at t , T > t . Finally p - E, is thi 
 
 price of money in t 
 
 . e.+N ., e.+N 
 
 , r i , Xv 1 , ib. i i- 
 
 A consumption plan for i, [c. , (x ) _ , (x ) , m 
 
 _ l. t~y . _. t t — o . _ 
 i i 
 
 is budget feasibl e if 
 
 x i i . „ i ^ n 
 
 a) c = w + Ex > 
 
 t~ 
 
 , . mi mi i b ib ra .. ib , . . - 
 
 b) pm _, = p ra - p x - p x + p i x (t) -- 
 F t t+1 *t t _t_t ~t_t *t t = 
 
 7 < t 
 
 where m. , = m"*" . 
 
 3 -1 
 1 
 
 A consumption plan for consumer i is optimal if it is budget 
 feasible and it is a u maximizer (over the set of all budget feasible 
 consumption plan of i). 
 
 An equilibrium is a pair of prices and feasible consumtion plans 
 
 (i.e., budget feasible for each i) , namely 
 
 . 6,+N ., 8 +N 
 
 UP,,' P,.* PJ f _, » (c , (x ) fl , (x ) , m ) J sucn that 
 
 ~c _r l t- 1 ~ -, t t— <v , _ i_ t— o „ ~ l—r 
 
 i l 
 
 a) The consumption plan of each consumer i is optimal, 
 
 b) I x = for all i. , 
 i 
 
 ) Z x =0 for all t , 
 . .. t 
 
 i 
 
 d) Z m = m for all t (m i amount of money) . 
 
 t 
 
 i 
 
 Proposition : Jj}£_ equil ;ribed in the Appendix .is 
 
 equivalent to the equilibr xt. 
 
 P roof : ► : An equilibr , in the ■': in this Appendix 
 
 requires either p = i- t or . - ail t. For if 
 
/-J 
 
 D =0 and p'' , > for some t , it would contradict to the mo no tonicity 
 F t t-1 
 
 i i — j m m 
 
 of u and the fact that " m = m ' , Also if p = and p > 
 
 i 
 
 the above conditions are violated. mdly, the set of feasible plans is 
 
 homogeneous of degree zero (for p' , p and p for any t) . Therefore 
 
 we can normalize prices in each period. We can do it by taking .p = p v. E 
 
 t ~r 
 
 for all t . Next, we shall see that P T (t) = p.( T ) and P (t) = p" for 
 ail r , t < t - 
 
 p (t) < p (x) and p (t) < p are obvious, for otherwise consumer 
 
 X — T 1 — 
 
 i with 6. < t < t < 6.+N can sell the good (bond) at t and obtain the 
 
 b b 
 
 arbitrage profit. Also p (t) ' p (t+1) and p (t) > p (t+1) and 
 
 T — 1 t — T 
 
 b IP 
 
 p_(t-l) > p for all t otherwise by the equilibrium conditions, there 
 
 T — 
 
 exist- i with 9 < t < t+1 < 6 +N ar 'd m > who can make arbitrage 
 i = - - i t 
 
 profit by buying the good (or bond). Proceeding by induction, p (t) = p (t) 
 
 and p_(t) = p" for all t and r . 
 
 Given the equilibrium in the Appendix define x = l x (t) for all 
 
 *- t < t 
 
 j n 
 
 i , the consumption plan c with c = (c )._■,• Then this consumption allocation 
 
 m 
 
 with normalized prices and p are a competitive equilibrium in the economy 
 
 described in the text. 
 
 •* : Given the equilibrium in the text (p, c) , let us find an equilibrium 
 
 « ~q in Tii m 
 of the economy in the Appendix. Define p (t) = p , p (t) = p , p = p 
 
 XT t 
 
 for all t and t . Let c = c for all i and t . 
 
 t t 
 
 Assume p > , for the case p =0 can be treated analogously. 
 By the monotonicity of u , p >> for all t . Hence for any arbitrary 
 
 index k for good, p, > for all t . Fix the index k . Let x (t) = 
 
 kt ~ 
 
 for i , t , t , and let x. (t) = i and t , and all x > t+1 
 
 if j ^ k and all T ; t+2 if j = k 
 
7-4 
 
 Let x (1) = x for . . .et P.C . be the set of households 
 
 with p (Dxj^l) - p'm = -; < be those with 
 
 p^CDx^CL) - iyn 1 - 5* > . E e* . a is well defined 
 
 [ ' N i ~ P, 
 
 ~ itt~ ~ rr — i -j 
 
 for p m = p Z m > Lon. :h i./ e x* (1) = 6' /p 
 
 and for each i € F ' . Let c /:,, . By construction, 
 
 1 2k 1 1 k 
 
 p x* - p^m 1 < ana = . Define in* = (p^ 1 - p.O/ P m . 
 
 ~ -L ~ L 1 - J. ill 
 
 -i i 
 Given m , x for all i and t < T - 1 , define 
 
 ^ i- i "1 
 
 x (T) = x + x,L,(r -1) . Define. P„, as the set of households with 
 
 L L i. I 
 
 1 * T*l - 
 
 ,-< T (T) - p T K. T 
 
 p T (T)x,_,(T) - p"ra v __. = t^ < and N„, as the sec of households with 
 
 p m (T)x_(T) - p„cC . = 5_ > . Proceeding exactly the sarae way as before, 
 T T T T— 1 t — 
 
 we can find x.„ and m^ which satisfies both budget condition and market 
 
 "a. X 
 
 clearing conditio . 
 
 Ev induction, we car. construct x and m for all i and t . 
 
 ■- - 
 
 This is an equilibrium in the Appendix. 
 
erences 
 
 [1] Arrow, K. J., and F. Hahn, Ltd \ An aly sis , Hoiden-Day, 
 
 San Francisco, 197i. 
 
 [2] Benvenistt M., "A lomplete Characterization of Efficiency for a 
 General Capita] Ac. ion Model," Journal or Economic 
 ... 325-337. 
 
 [3] Cass, D., and M. E. Yaari, "A Re-Examination of the Pure Consumption 
 Loans Model," J.P.E. (1966) pp. 353-367. 
 
 [4] Gale, D. s "Pure Exchange Equilibrium of Dynamic Economic Models," 
 J ournal of Economic T heory 6, (1973) pp. 12-36. 
 
 [5] Hayashi, T., "Monetary Equilibrium in Two Classes of Stationary 
 
 Economies," Review o f E conomic Studies 3& (1976) pp. 269-284. 
 
 [6] Okuno, M. s "On A Monetary Equilibrium in a Generation Overlapping 
 
 Economy," (1975) Department of Economics, University of Illinois, 
 Urbana, Illinois. 
 
 [7] Samueison, ? . , "An Exact Consumption-Loan Model of Interest with or 
 Without the Social Contrivance of Money," JPE, (1958). 
 
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