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Individual and collective risks in large economies
JOURNAL
OF ECONOMIC
Individual
THEORY
15,
279-294 (1911)
and Collective
Risks in Large Economies
SIEGFRIEDBERN~NGHAUS Department
of Economics,
University
of Bonn, 53 Bonn,
Adenauerallee
24-26,
West Germany
ReceivedMarch 17, 1976; revised September 1, I976
INTRODUCTION This paper is concerned with the allocation problem under uncertainty. The classical results in this direction are given by Arrow [2] and Debreu [5, Chap. 71. They extend the well-known theorem of welfare economics (see, for instance, [I]) on the existence of decentralizing Pareto prices to an economy with uncertainty and derive the result that in such economies the decentralization of Pareto-efficient states works by “contingent markets.” In thesemarkets contracts for the conditional delivery of commodity quantities are exchanged where the condition is given by the occurrence of a specified event. Obviously the number of these markets and consequently the number of the corresponding decentralizing “contingent prices” depends on the number of the states of the world and therefore can grow enormously. One way to overcome this difficulty was shown by Malinvaud in his stimulating article, “The Allocation of Individual Risks...“[lO]. Our results are related to a mathematical generalization and conceptual extension of Malinvaud’s paper: the crucial step is the specification of risks in the economy. A large part of risk stems from individual sources, for instance, the state of health of an agent living far away from me will not influence the probability of the event “I catch a cold.” Or the effects of local showers on the farming of a specified region will not influence the agricultural results of farmers living far away. Many other examples of this kind can be found; they express the idea of “individual” events whose occurrence is relevant for a particular individual and has a vanishing effect on the probability of events relevant for any other individual. Intuitively one expects that this kind of risk can be averaged out provided the number of agents in the economy is large enough. This has the important consequencethat decentralization of Paretoefficient states works without contingent markets. Naturally there is another kind of risk called “collective” which cannot be averaged out even in a very large economy. For instance, an event “the nuclear world war takes place” will be called collective. It is characterized by concerning all agents at the sametime. The relation between the concepts 279 Copyright All rights
i^s’ 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0022-0531
280
SIEGERIED BERNINGHAUS
of individual and collective risk is in this paper construed as follows. The probabilistic independence of the individual events of some agents is related to a collective event which can occur for the whole economy; that means we consider the independence of some individual risks under the condition that a specified collective event occurs. The consequence for decentralization in such economies is the existence of contingent markets. But the number of these markets depends on the structure of collective risks. In these markets contracts for the conditional delivery of commodity bundles are exchanged but the conditions are given by the occurrence of collective events.
THE
MODEL
The formal model to be presented here is not very different from an “Arrow-Debreu world” with contingent commodities. The idea is well known [5, Chap. 71: Commodities are characterized by their physical properties, by date and location of their availability, and by the “state of the world” in which they are available. Let the states of the world be represented by an atrbirary measure space (Q, 9); then it is natural to identify the commodity space of the economy with the space of all essentially bounded measurable RIvalued functions (L”(sZ, 9, P))” provided a suitable measure P can be determined.l On the motivation for the choice of this particular commodity space the interested reader is referred to Bewley [3], Radner [14], andPrescott and Lucas [13]. Going beyond the classical Arrow-Debreu frame work we postulate the existence of a probability measure P on (52, 9) governing the realization of the states of the world. It is not necessary that the agents know this measure; they may have quite distinct subjective 1 We establish the following
conventions:
(a) (T(L”, L1))i denotes the product topology on the Z-fold product of the topological space L” endowed with the Mackey topology 7(L”, L’), where L1 is the space of P-integrable real functions. The same is valid for the weak topology (o(Lm, ~5’))~ on (L”)‘. Consequently all convergence arguments in (7(L co, L’))i resp. (u(L”O, L’))i made in the text are understood “coordinatewise” (for any of the I coordinates). (b) The norm topology on (L”)& is induced by the norm II x Ilk, : = maxh--l.....l {II xh llm) where I/xn jlm is the essential supremum of xn E Lm and x = (x1 ,..., xi). An analogous definition is valid for Ij x 11:on (L’)‘. (c) On Ri we define a norm by II x II: = maxn,l,....l {I XIL I} where xn E R and x = (Xl ,..., x2). (d) For fg (J?)~ g E (~2~)~ we compute the value by (f, g) = sf. g dP where the dot “ . ” denotes scalar ‘product on R i. If p E (ba)i, the space of all bounded additive set functions absolutely continous with respect to P, then we have (p, g) = Cl s g, dph (e) E(x) denotes the vector (E(x,),..., E(.Y~)) of expected values of the coordinate functions xn E Lm where x = (x1 ,..., XI).
INDIVIDUAL
AND
COLLECTIVE
RISKS
281
probability evaluations. In the following we only assume that the central planning bureau, responsible for the setting of decentralizing prices, has complete knowledge of this objective probability P. Remark. The concept of commodity space presented above presupposes the existence of the same 1 commodities completely characterized by their physical properties (respectively, local or temporal availability) in any state of the world. Prices (or values) of commodity bundles are elements in the topological dual of (L”)I. Consequently the chosen topology on the commodity space determines the price space.It is well known (e.g., [3]) that the norm topology on (L’)’ with linear functionals in (ba)l gives rise to price systems without economic justification. Problems arise from elements in (ba)l that are not u-additive. Therefore we introduce the Mackey topology (7(L”, L1))z on (L” )‘. 1t has been demonstrated in [3] that this topology is sufficiently strong to admit interesting preference relations; for example, Neumann-Morgenstern utility functions representing preference relations between commodity bundles in (L”)’ are Mackey continuous [3, Appendix IT]. The admissible price systems for this topology consequently are elements of (~5~)“.As the probability space(J2, 9, P) is allowed to be atomlesswe prefer in the sequel to call these elements of (L1)z “price densities” corresponding to the prices which are then understood as u-additive measuresin (ba)l, for we have no economic interpreation for price systemsassigninga certain value to contracts promising delivery in states of the world which all have by definition probability zero. Clearly from a mathematical point of view these two concepts are identified (by the Radon-Nikodym theorem). In this paper we consider only exchange economies: Every consumer i (i == I,..., m) is, as usual, characterized by a consumption set Xi , a subset of (L*)I, a preference relation si , a transitive and complete binary relation on Xi , and an RI-valued function ei E(La)z representing his initial endowment. Remark. Preferences are defined here on contingent consumption plans, respectively, “consumption strategies” in the terminology of Malinvaud [lo]. Consequently the decisions of the consumers involve additional reasoning like risk aversion and subjective probability evaluation (see, for instance, [5, Chap. 71). The central new concept in this economy is the definition of feasibility of allocations. As a natural extension of the feasibility definition in deterministic economies one could propose calling an allocation (x1 ,..., x,) E (L”)I”” (xi E Xi) feasible iff
% (Xi - e,) = 0,
282
SIEGFRIED BERNINGHAUS
which can also be interpreted as the equality of aggregate planned supply and demand in any state of the world. In real economies the risks come mainly from individual sources and under an appropriate assumption of “independence” between the consumersrisks one can expect someregularity in the per capita excessconsumption (l/m) Cy (xi - ei) provided the number of agents is large enough. These short remarks will be made precise in the last section. The reader must accept for the present that under appropriate assumptions on probabilistic dependence between the agents’ risks it is not unreasonable to define feasibility of an allocation (x, ,..., x,,) in large economies by
or by (3) Equation (2) concerns the situation of purely individual risks in the economy, while in (3) one cannot average out a part of risk characterized by ~8, a sub-o-algebra of 9. If 99 is not trivial (that means 99 # SF and #-{Q, D>) one can interpret the events in S? as the collective risks the agents in the economy are bearing.Per capitaplanned excessconsumptionis only influenced by collective events provided the number of agents is sufficiently large. These remarks are plausible consequencesof special probabilistic hypotheses, as will be shown in the last section.
PARETO-EFFICIENT
STATES AND PARETO PRICES
The aim of this section is the analysis of Pareto prices dependent of the risks in the economy. Therefore we must discuss at first the concept of Pareto-efficient states. As we have two new definitions of feasibility, in the sequel denoted by (2) (resp. (3))-feasibility according to Eq. (2) (resp. (3)) is satisfied, we have two concepts of Pareto efficiency. DEF. 1. A (2)-feasible ((3)-feasible) state (x1*,..., x,?&*)E (Lco)z’nLof the economy d = {X, , &, et ; i 7:: I ,..., m} is called (I)-Pareto-efficient (resp. (C)-Pareto-efficient) iff there is no other (2)-feasible (resp. (3)-feasible) allocation (xll,..., x,3 such that
xi1 xi xi*
for every i,
and xi1 >j .xi*
for at least one i.
INDIVIDUAL
AND
COLLECTIVE
RISKS
283
It is easy to see that the set of (2)- or (3)-feasible allocations is larger than the set of allocations which are feasible in the sense of Eq. (1). Therefore the existence of Pareto-efficient states to be proved first cannot easily be deduced from well-known results. To deal with the existence problem we need the followjng property of consumption sets. (A.l) the sets
Every X, is a convex and weakly
are weakly compact for arbitrary c E R,‘. We can give an example of a consumption PROPOSITION
.satiTjies property
I.
closed subset of (L+“)‘such
that
set Xi fulfilling this property.
The set Xi dejned bl*
(A. 1).
Proof. Clearly Xi is convex and weakly closed. Consider the set Ajt, = {xi - E(xi) j xi 6 Xi , 0 < E(xf) < cl for a certain vector c E R, I. For an arbitrary element xi -- E(xi) in AiC we have
As a (norm-) bounded and weakly closed set in the dual of the Banach space (L’)’ the set a,C is weakly compact [6, p. 4241. Q.E.D. Remark. The property of consumption sets expressed in Proposition 1 does not imply any compactness of consumption sets. It restricts the consumption plans to those nonnegative (I,“)‘-functions whose deviation from its expected value is of the sameorder of magnitude. In the proof of the existence theorem we need an important result in mathematical economics which can be found, for example, in [I 5, p. 5811. For our purpose we need the most general version. LEMMA 1. Let A be CIcompact set on l~~hieh a transitire binary refution s is defined such that the sets {x E A 1x 2 x>
are closedin Af%r ewry ,YE A, then there exists n maximal elementfor ,< ik .4_ THEOREM 1. The economy 6 7 (X-,) <’ -17 ei ; i = I,..., m) hns an (I)Pareto-ejkient (rcsp. (Cl-Pareto-eflrient) state iffor euery i
284
SIEGFRIED BERNINGHAUS
(a) Xi satisjies (A. 1), (b)
the sets {xi E A’, / xi kjixij
are (a(L”,
L1))Wosed,for
erety
.Ti E Xi .
(1) (I)-Pareto-efficient states. We will show the existence of (I)-Pareto-efficient states by applying Lemma 1 to the following situation. On the set of (2)-feasible allocations defined by Proof.
we define the Pareto ordering sr by (Xi’)
k&i”)
-3 xi1 5; xi2
for every i.
By assumption (b) the set
{(xi) E F I C-r;) ZAK)l is weakly closed in F for all (Xi) E F. To apply the lemma it remains to show weak compactness of F: Consider the sets Aib = {xi E Xi 10 < E(xJ < b], At = (.Y; - E(xJ j xi E Ai”>, I2 = {X E (L+“)z / 0 < x < b, Lx is constant), b E R+2(one can verify that I7 is weakly compact). By definition of F we have F C n proji F2 C n Atb C n(d,” + Z7). By (A.l) the set Aib is weakly compact. As F is weakly closed the weak compactnessof F follows. Consequently by Lemma 1 there exists a maximal element for sp in F that meansan (I)-Pareto-efficient state. (2) (C)-Pareto-efficient state. We define the set of (3)-feasible allocations by
Clearly F C F and F is weakly closed as ES(.) is a weakly continuous operator on (IY,“)~ [ll, p. 301. Then Lemma I gives us the existence of a (C)-Paretoefficient state. Q.E.D. Remark. To make the continuity assumption of Theorem 1 compatible with the model introduced in the preceding section we remark here that the convex (T(L”, L1))z-closed sets in (L”)2 are (o(L”, L1))z-closed. As we need in the sequel the convexity of preferences we can regard the weak upper semicontinuity of preferencesasequivalent with Mackey upper semicontinuity. 2 projd A denotes the projection projiA CX,.
of A C (Lm)l.m onto its ith coordinate.
Consequently
INDIVIDUAL
AND COLLECTIVE RISKS
285
As a preparation for the proofs of the main theorems on Pareto prices we introduce the following notation: To the economy 8 we associatean economy 8 whose commodity space is R1 where d need not have any economic meaning in our context
where g{ Ui(a)} denotesa greatest element of the set (ii(a) := {xi EX< /E(xJ = a}. To the economy 6’ we associate furthermore the economy 2 whose commodity space is (Lm(!2, d, P))” by a similar method -
Remark. The preference relations T7i and xii are constructed such that the following conclusions hold: If (x1*:., x,,*yis an (I) (resp. (C))-Paretoefficient state in 8, then (X1*,..., X,, *) is a Pareto-efficient state in 8, respectively, ( Z1*,. .., Z,12 *) is a Pareto-efficient state in Z.3 By the following lemma it is demonstrated that the definition of the preference relations s’i , si makessenseprovided an appropriate assumption on Li is made. LEMMA 2. Let Xi and si fuljill the snme assumptionsas in Theorem 1; then there exists a greatest element for sj in U,(a) (resp., Ui(f)) inhere u E Rz,f~ (L=(Q, .SY,P))“.
Proof.
By (A.1) the sets U&z) -
a = {xi -
E(x<) / E(xi)
= a, xi E Xi , a E R+z}
are weakly compact, consequently Ui(a) is weakly compact. By Lemma I there exists a greatest element in Ui(a). From the relation U,(f) C Ui(f) the same is valid for Ui( f) provided we can show that U2(f) is weakly closed: Let {x,} be a generalized weakly convergent sequencein Ui( f) where x, + x0 . 3 Pareto effkiency in 6(8)iis based on the usual concepts of feasibility. That means for 2, an allocation (aJ E R”.m is feasible iff Xy ai = Cl: e, (ei E Rz). For 3, an allocation (fJ E (Lm)z.n is feasible iff Cyyi = CT ei (ei E (~5”)~).
286
SIEGFRIED
BERNINGHAUS
Consequently JB x, + JB x0 for every B E g’. As JB x’, == jB E,*(x,) = jsj for all ,?:we have se I?‘(xJ = jBf= SDEa(x,). Therefore E*(x,) -.r Q.E.D. For the proof of the next theorem we need a property of preference relations which has a well-known analog in deterministic economies: A preference relation -& on Xi is called “locally nonsatiated” (with respect to (o(L”, f.r))’ iff for any xi E Xi (xi is no satiation consumption) there exists in every weak relative neighborhood 17,~n Xi of x, a consumption plan .I-;’ such that s,’ ‘- _, .I-, .
2.
THEOREM
that,for
ererjf
Let
6’ = {Xi , 5, , ej ; i -: I ,..., m)
be an economic
such
i
(a) X, = (L,“)l, (b) Giwn satiation in (ba)‘)
sI
is conrex
and locally
nonsatiated
(ti+th
req.
to (n(L’
, Ll))‘).
an (I)-Pareto-ejficient state (xl*,..., x,*) where some x: is no consumption then there exists a price system TT* (o-additive measure with constant density function p* E Rz (p* + 0) such that for ever)’ i xi xi
xi*
=c- E(p*
. xi) > E(p*
. xi*).
Proof. The proof utilizes a theorem on the existence of Pareto prices in deterministic economies [5, 6.4 (l)]. Suppose we can apply this theorem to the economy 8 = {xi , si , FL; i == l,..., m> where (XI* ,..., .Ynl*) is the Pareto-efficient state in d constructed from (xl*,..., x7,,*): Then there exists a price system p* E RL(p* f 0) such that for every i: ai &
Xi*
3 p* . a, > p* . 2,“.
(1)
As .Y[* is a consumption strategy in an (I)-Pareto-efficient state having the sameexpected value as the elementsin Ui(Xj*) we have the relation g{Ui(aJl
2Ji
8{~f(%*)l
-i
(2)
Xi*
for all ai satisfying ai xzi Xi*. Assume there exists in contrast to the statement of the theorem an -yi’ E X7 such that x,’ >,j xi*
2 E(p*
. xi’)
< E(p*
’ xi*).
Consequently g(Ui(Xi’)> kj g{Ui(Xi*)} by (2). That implies E(p* . xi’) 2 . xi*) by (l), a contradiction. Therefore it remains to show that 8 fulfills the assumptions required for an application of the theorem: Clearly xi and 3i is convex. A more detailed argument is given for local nonsatiatedness of si : Assume there is an an E’ > 0 such that for all a FX< with /I N ~ Si* 11< c’ we have a sj .Yi* where xi* E X, is no satiation consumption. E(P*
INDIVIDUAL
AND COLLECTIVE
RISKS
287
Consequently g{Ui(a)} si g{U&*)) := zi and xi si zi for all xi E U(a) where a has the property described above. But for zi we take the weak neighborhood {x E (L”)I 1(X - zi , 1) < E’) denoted by r/,,,, . By assumption (b) of the theorem there is a -yi’ E Xi n lJ1,,, such that xi’ >i zi. As 1 ?cj’ - xi* 1~< E’ we have a contradiction. Q.E.D. This theorem has an important economic interpretation: It assuresthe existence of decentralizing Pareto prices of the form r*(E) = p* * P(E) for a random event E E 9 where p* is a constant density function and P is the probability measure introduced in the preceding section. Therefore in an economy with individual risks the decentralization problem is solved in the following way. The central planning bureau announces only one price for each of the 1 commodities. Together with communication of P the agents evaluate their consumption strategies by J x &r* = E(p* * x). Contingent markets are not neededin the specialcaseof economieswith purely individual risks. For the analysis of price implications of (C)-Pareto-efficiency we will first prove a proposition on the decentralization of “usual” Pareto-efficient states, that means, states whose feasibility definition is based on Eq. (1). A serious difficulty in proving results of this kind in an infinite-dimensional setting is the application of separation theorems: First we must ensure that the convex set to be separated has a nonempty interior [6, p. 4171. Second, even if we can apply an appropriate separation theorem we must not hope without further assumptions that the separating Pareto prices are o-additive measureshaving their density function in (L1)z. To overcome thesedifficulties we introduce further assumptions on preference orderings (for further discussion see [4]). As a generalization of the munotanicity property in deterministic economies we call a preference ordering Li on Xi = (L+“)L monotone iff xi -1 k . lA > i xi for arbitrary vectors k > 0 and P(A) > 0. The second assumption which is weaker than Mackey continuity of preferencesis taken from [13]: (A.2) Assume xi1 >i xi2 (xil, xi2 EXi) and let {F,} be a sequence of measurablesetssuch that P(F,) - I, then 1F,xil > i xi2 for sufficiently large n. An economic interpretation of (A.2) is given in [131:change a consumption strategy on events with diminishing probability then the agent will consider these plans similar to the original one. PROPOSITION
2. Let 8 = {Xi , &;i , ei ; i = I,..., m} be an econom.ysuch
that for erery i (a) Xi == (L-=-)l, (b) & is conrex, monotone, and.fuIfills (A.2) (c) xy ei > 0 a.e.
288
SIEGFRIED
BERNINGHAUS
Given a Pareto-eficient state (x1*,..., ,y,,,*) where no xi* is a satiation consumption, there exists a price T* with density fbtction p* E (L1,jl (p* # 0) such that for every i xi & xi* 3 E(p* . xi) 3 E(p* +xi*). Proof. Consider the convex set Z = Cy &(xi*) where &(xi*) := {xi E Xi j xi >i xi*} By definition of Pareto efficiency we have Cy ei $ Z. By the separation theorem for convex sets in topological vector spaces [6, p. 4171 there is a IT* E (ba)l where r* # 0 such that
CT*,2) 3 ( n-*,f
for all z E2,
ei
1
1
0)
as by the monotonicity assumption Z has a nonempty (norm) interior. Take, for example, xi* + k (k > 0) that lies in the norm interior of 2i(xi*). Next we will prove the inequality
( 1
T*, $ ei < (7~~*,z)
for all z E 2,
where 7~~*is the countable additive part of T*.~. Assume to the contrary (r*, Cy ei) > (niTc*,z’) for at least one z’ EZ. We construct the functions 2, ’ := (rrC*, zn’) = (n*, z,‘) for these n, in contradiction to (I). From (2) then follows, together with assumption (c), (rc*,teI)
<(rc*,z)
forallzEZ.
Furthermore inequality (2) together with (c) implies (rc*, z) 3 (n*, Cy ei) > 0. Therefore 7~~*# 0 and is nonnegative. Let Z be the weak closure of Z. Clearly (nc*, rrae rc*, z) for all z E Z. The relations Cyzi(xi*) C Z, yn ,=zm> iwn and x1 xi* E Z imply Cy xi* minimizes (rc*, Cy xi) on 1 x,*2 l ez Cyzi(xi*) as xi* E~~(x~*) because of local nonsatiatedness. Consequently c’irc*, xi*> minimizes (7~~*,xi) on {xi E Xi j xi & xi*} that gives us the (Q.E.D. result by takingp* := density function of TV*. 4 By a theorem of Yosida and Hewitt [16, p. 521 every nonnegative T E (LKz)~is representable as the sum ?r = nTc+ q (T, , up > 0) where T, is a-additive and up is purely finitely additive (that means up > p > 0 and p u-additive * v = 0).
INDIVIDUAL
AND
COLLECTIVE
RISKS
289
Remark. We need the result of Proposition 2 as an important step in the proof of the next theorem. Nevertheless one could regard the result as interesting in itself. But then one either had to verify that (A.2) does not exclude interesting preference relations or prove Proposition 2 with the Mackey continuity assumption of preferences, which is easy to do. 3. Let 6’ = (Xi, si,
THEOREM
pi ; i = I,..., m) be an economy such that
for er)ery i (4 (b)
Xi = (L,“)‘, si is convex, monotone, and (T(L”,
(c)
x11:”e, > 0 a.e.
L1))z-continuoz~s,
Given a (C)-Pareto-eficient state (x1*,..., x7,*) where no xi* is a satiation consumption, there exists a price system with %‘-measurable density function p* E (Ll)l (p* # 0) such thatfor every i xi 2; xi* 3 E(p*
. xi) 2 E(p”
. xi*).
Proof. The main step of the proof is the application of Proposition 2 to the economy 2 whose commodity space is (L”(Q, Z?I,P))“: By means of this proposition we have a 5%measurable price density p* # 0 such that fi & fi” * E(p”
.fj) 3 qp*
. xi*).
Consequently xj & jc,* * E(p* * Xi) 2 E(p”
. xi*),
by the same argument given in Theorem 2. Therefore it only remains to show that the hypotheses of Proposition 2 are satisfied in the artificial economy 2 provided assumptions (a)-(c) are fulfilled in b: (a’)
We have from (a) xi
= (L+-(Q,
9, P))“.
(b’) zi is convex, and the monotonicity can be demonstrated as follows. Assume to the contrary fi + lAkz;ifi for some k>O, A~92 and P(A) > 0. By definition of 3% we have g{ U,(h + 1Ak)} & g{ U,(f,)) : = zi. Because of (.zzi+ l,k) E U& + 1Ak) we have zi + 1Ak ki Zi , a contradiction to assumption (b). The relation si fulfills (A.2): AssumeA yi hi and there exists a sequence of g-measurable sets (F,} such that P(F,) + 1 and a subsequence{F,k} such that lFnbJ; zi hi . By definition of zi this implies
290
SlEGFRlED
BERNINGHAUS
I Fnb~ilis an element of U( 1Fn,Ji) but 1F,,kzil converges in the Mackey topology against zil (see [3, Appendix I (24)J). Therefore lFnlczil Lpi zi2 for sufficiently large IZ as & is Mackey continuous, a contradiction to (b). (c’)
We have C:” ci > 0 a.e., from (c).
Q.E.D.
The solution of the decentralization problem in economies with individual and collective risks has the following interpretation in the light of Theorem 3. The central planning bureau announces contingent prices m*(E) = JEp* dP to the agents. Contingent markets are necessary for decentralization but their number depends on the structure of collective risks in the economy. For example, if 9I is finite it suffices to announce a finite number of contingent prices p*. Together with the knowledge of P which is communicated to the agents by the planning bureau the agents are able to compute the value of their consumption strategies by J x dn* = E(p* . x).
THE
CONCEPTS
OF
INDIVIDUAL
AND
COLLECTIVE
RISK
The results of the last section can be interpreted as follows. Change the definition of feasibility of allocations as described by Eqs. (2) and (3); then the implications for the classical decentralization theorem of welfare economic are given by Theorems 2 and 3. But why should one regard these special feasibility conditions as plausible? In this section we will particularize the probabilistic framework to deduce the feasibility conditions as a consequence of some “Laws of Large Numbers” utilizing the following method. Results are deduced for a limit economy with infinitely many agents; these results are then applied to economies with a large but finite number of agents. When talking about individual risks in an economy one has the idea of random events concerning one person or a small group of individuals only. For example, the breakdown of a tractor of a farmer in a foreign country does not influence the probability of a breakdown of my car. A large part of risk concerning the agent’s environment is based on these events. Let us assume first that all risks in the economy are of this kind. That is an idealized situation studied first systematically by Malinvaud [lo]. We will in the following give a possible formalization of this idea. Clearly a suitable mathematical model behind this concept is that of “stochastic independence.” Therefore we propose the definition DEF. 2. An economy d = (Xi , 5 i , e, ; i == I,..., m> is called an economy with purely individual risks iff any allocation (x1 ,..., x,,,) F (15,“)~‘~ (xi E X,) consists of independent random variables, and furthermore (e,,..., e,,) is a collection of independent random variables.
INDIVIDUAL
AND
COLLECTIVE
RISKS
291
Remark. This definition is restrictive in the sense that “interaction effects” between agents in a small group are excluded. We prefer to present the simplest caseto clear the basic ideas. For a discussionof generalizations see[ 10, Appendix]. In economies with purely individual risks one can expect that the risks are averaged out if the number of agents is sufficiently large, especially such that the vector of per capita excessconsumption (l/m) 1.:” (xi - ci) does not differ “too much” from (l/m) zy E(.uj - ei) as N + co by the law of large numbers for independent random variables. Therefore the feasibility condition in such economies should be formulated in expected values of excessconsumption (see Eq. (2)). In the following we give a more rigorous version of that and give an example of exchange economies which satisfy Definition 2. Consider a sequenceof sub-u-algebras {Sn} of 9 such that the u-algebras Sr ,.... .F, are independent for arbitrary N (the existence of such a sequence can always be assured by constructing (Q, 9, P) as an appropriate product space). We consider a sequenceof exchange economies8’” = {iu, , 5n , e, ; II 7~~I,.... IV) which is associated to the sequence{SI,!,,3in the following way.
(a) XT, = (L+a(Q, 35 , P)Y; (b) P,, is &-measurable; (c) C‘l -l differs from Q” by the one consumer whose consumption plans and initial endowments are &,+1- measurable random variables and any allocation (-x1,..., x,“) in 8” is regarded as a finite subsequenceof an allocation is,)- for the limit economy P. Let us make the assumption. (A.3) The variances of any sequence of random variables {.u, - e,} taken from {8iVj are uniformly bounded. Then we have as a straightforward application of the strong law of large numbers for independent random variables [7, p. 2041
This motivates our definition of feasibility of allocations in economies with purely individual risks: If N is sufficiently large we can replace in 8.” the usual definition of feasibility as given in Eq. (1) by the new definition given by (2) without making a great mistake. Although a large part of risk stems from individual sources, we must admit that there are events whose occurrence influences the “individual events” of all agents. To take up the example, above imagine the economies of the world are in the collective state “the great nuclear world war takes
292
SIEGFRIED
BERNINGHAUS
place”; then the breakdown of a car in a foreign country caused by a nuclear explosion is accompanied by a breakdown of my car (provided we regard that war as the simultaneous explosion of the nuclear potential in the world). Collective risks are characterized by events whose occurrence concerns all agents in the same way. The individual events are conditioned to the collective framework in which they occur. As a possible mathematical model for economies with collective and individual risks we propose the concept of “conditional independence.” DEF. 3. An economy 6 = {X, , si, ei ; i = l,..., m} is called an economy with individual and collective risks iff any allocation (x1 ,..., x,,) E (L+“)z’” consists of conditional independent random variables, and furthermore 6% ,..., e,,) is a collection of conditional independent random variables where the condition is given by the collective risks in b. To give an example of economies which fulfill this definition we take a sequence {gn} of sub-a-algebras of 9 which are conditional independent with respect to g[, , where J%‘, denotes the o-algebra of terminal events generated by the sequence {Z%}. Obviously the economies dN defined above satisfy Definition 3 and g’, can be interpreted as the u-algebra of collective risks associated with the sequence {bN}. Intuitively one expects that in the sequence (l/N) Cy (x, - e,) the collective risks cannot be averaged out and consequently the limit must somehow be conditioned to a &YC gA, . By means of the following assumption we can make this precise. (A.4) Allocations (x1 ,..., x,,,) in bN are admissible iff they give rise to equally distributed individual excess consumption (x1 - e, ,..., xN - eN). Naturally this is a restriction for the set of allocations in an Remark. economy. But we do not see another way to derive the results below without this assumption. One can weaken the restrictive character of the situation by making (A.4) only for allocations in the types of a specified type economy (see [IO]). But this minor modification can not avoid the artificial character of (A.4). The trouble arises from the indeterminacy of allocation problems. In deterministic economies one need not specify the origin of an allocation; the consumption plans may be allocated by a central planner or they may arise as the result of an individual choice of an agent. But when one wants to discuss some kind of probabilistic dependence between the agents an assumption on the formation of allocations is necessary. This difficulty vanishes in the analysis of demand functions where the origin of consumption is the individual choice and probabilistic dependence of demand functions follows from probabilistic dependence of the characteristics of the agents [S]. Now consider a sequence {x, - e,) of individual excess consumption plans taken from the sequence {a.V}. By the construction of the sequence of economies these random variables are exchangeable [9, p. 3641. Then we
INDIVIDUAL
AND
COLLECTIVE
RISKS
293
have from the law of large numbers for exchangeable random variables [12, p. 1441
where E&u’(x, - e,) is the conditional expectation of an arbitrary member of the sequenceand 9X” is the a-algebra of terminal events generated by the sequence{x, - enj. This motivates the formuiation of the feasibility condition in conditional expected values as described in Eq. (3) provided we consider an economy F”Nwith sufficiently large N. But there is a discrepancy between (3) and the result derived above. In (3) we have a fixed u-algebra ~8 for the construction of conditional expectations but by the reasoning above we see that the u-algebras gK,” resulting from the law of large nubers depend on the allocation under consideration. As we have described S”, as the u-algebra of all collective events in the economy we can interpret gX” as the collective risks concerning the special allocation. But by construction of our model we have a’,,” C g’,, for any $2X generated by an admissible allocation. Consequently an allocation feasible (in the senseof (3)) with respect to Ezu(.) is feasible with respect to ,!?7-‘(e). To fix a g-algebra as required in Eq. (3) we propose to take gm . This avoids the problem of the choice of a special s’,” and is the best way to express the effects of all collective risks on feasibility. Inserting a’, for gzx where &=,” g $a only means that we dispensewith further “smoothing” of the special allocation in the feasibility condition. For we get a better “smoothing effect” in the limit by the operator Es(.) when we deal with functions which are “more constant” (meaning, they generate a proper sub-o-algebra of the o-algebras cFn) than functions generating the F,, .
REFERENCES ARROW, An extensionof the basictheoremsof classicalwelfareeconomics,in “Proceedingsof the SecondBerkeleySymposiumon MathematicalStatisticsand Probability” (J. Neyman,Ed.), pp. 507-532,Univ. of California Press,Berkeley,
I.
K.
2.
K. ARROW,
1951. Stud.
3.
T.
The role of securities in the optimalallocationof risk-bearing,Rev.
Econ.
31 (1964), 91-96.
BEWLEY,
J. Econ.
Existenceof equilibriain economies with infinitely many commodites, 4 (1972),514-540. “Theory of Value,” Wiley, New York, 1959. ValuationequilibriumandParetooptimum,Proc. Nat. Acau’. Sri. U.S.A.
Theory
4. G. DEBREW, 5. G. DEBREU,
40 (1954), 588-592. 6.
N.
DUNFORD
AND
J. SCHWARZ,
“Linear Operators,Part I,” Interscience, New York,
1966. 7.
B. GNEDENKO,
“The Theory of Probability,” Chelsea, New York, 1968.
294
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HILDENBRAND, Random preferences and equilibrium analysis, J. Ero~r. Theory 3 (1971) 414-429. M. LOEVE, “Probability Theory,” Van Nostrand Reinhold, New York, 1963. E. MALINVAUD, The allocation of individual risks in large markets, J. EWU. Theory 4 (1972), 312-328. P. MEYER, “Probability and Potentials,” Blaisdell, Toronto/London, 1966. J. NEVEU, “Mathematical Foundations of the Calculus of Probability,” Holden-Day, San Francisco, 1965. E. PRESCOTT AND R. LUCAS, A Note on price systems in infinite dimensional space, Internat. Econ. Rev. 13 (1972), 416422. R. RADNER, Efficiency prices for infinite horizon production programs, Rec. Econ. Stud. (1967), 51-66. D. SCHMEIDLER, Competitive equilibria in markets with a continuum of traders and incomplete preferences, Econometrica 37 (1969), 578-585. K. YOSIDA AND E. HEWITT, Finitely additive measures, Trans. Amer. Math. Sot. 72 (1956), 46-66.
8. W.
9. 10. 11. 12. 13. 14. 15. 16.
BERNINGHAUS
OF ECONOMIC
Individual
THEORY
15,
279-294 (1911)
and Collective
Risks in Large Economies
SIEGFRIEDBERN~NGHAUS Department
of Economics,
University
of Bonn, 53 Bonn,
Adenauerallee
24-26,
West Germany
ReceivedMarch 17, 1976; revised September 1, I976
INTRODUCTION This paper is concerned with the allocation problem under uncertainty. The classical results in this direction are given by Arrow [2] and Debreu [5, Chap. 71. They extend the well-known theorem of welfare economics (see, for instance, [I]) on the existence of decentralizing Pareto prices to an economy with uncertainty and derive the result that in such economies the decentralization of Pareto-efficient states works by “contingent markets.” In thesemarkets contracts for the conditional delivery of commodity quantities are exchanged where the condition is given by the occurrence of a specified event. Obviously the number of these markets and consequently the number of the corresponding decentralizing “contingent prices” depends on the number of the states of the world and therefore can grow enormously. One way to overcome this difficulty was shown by Malinvaud in his stimulating article, “The Allocation of Individual Risks...“[lO]. Our results are related to a mathematical generalization and conceptual extension of Malinvaud’s paper: the crucial step is the specification of risks in the economy. A large part of risk stems from individual sources, for instance, the state of health of an agent living far away from me will not influence the probability of the event “I catch a cold.” Or the effects of local showers on the farming of a specified region will not influence the agricultural results of farmers living far away. Many other examples of this kind can be found; they express the idea of “individual” events whose occurrence is relevant for a particular individual and has a vanishing effect on the probability of events relevant for any other individual. Intuitively one expects that this kind of risk can be averaged out provided the number of agents in the economy is large enough. This has the important consequencethat decentralization of Paretoefficient states works without contingent markets. Naturally there is another kind of risk called “collective” which cannot be averaged out even in a very large economy. For instance, an event “the nuclear world war takes place” will be called collective. It is characterized by concerning all agents at the sametime. The relation between the concepts 279 Copyright All rights
i^s’ 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0022-0531
280
SIEGERIED BERNINGHAUS
of individual and collective risk is in this paper construed as follows. The probabilistic independence of the individual events of some agents is related to a collective event which can occur for the whole economy; that means we consider the independence of some individual risks under the condition that a specified collective event occurs. The consequence for decentralization in such economies is the existence of contingent markets. But the number of these markets depends on the structure of collective risks. In these markets contracts for the conditional delivery of commodity bundles are exchanged but the conditions are given by the occurrence of collective events.
THE
MODEL
The formal model to be presented here is not very different from an “Arrow-Debreu world” with contingent commodities. The idea is well known [5, Chap. 71: Commodities are characterized by their physical properties, by date and location of their availability, and by the “state of the world” in which they are available. Let the states of the world be represented by an atrbirary measure space (Q, 9); then it is natural to identify the commodity space of the economy with the space of all essentially bounded measurable RIvalued functions (L”(sZ, 9, P))” provided a suitable measure P can be determined.l On the motivation for the choice of this particular commodity space the interested reader is referred to Bewley [3], Radner [14], andPrescott and Lucas [13]. Going beyond the classical Arrow-Debreu frame work we postulate the existence of a probability measure P on (52, 9) governing the realization of the states of the world. It is not necessary that the agents know this measure; they may have quite distinct subjective 1 We establish the following
conventions:
(a) (T(L”, L1))i denotes the product topology on the Z-fold product of the topological space L” endowed with the Mackey topology 7(L”, L’), where L1 is the space of P-integrable real functions. The same is valid for the weak topology (o(Lm, ~5’))~ on (L”)‘. Consequently all convergence arguments in (7(L co, L’))i resp. (u(L”O, L’))i made in the text are understood “coordinatewise” (for any of the I coordinates). (b) The norm topology on (L”)& is induced by the norm II x Ilk, : = maxh--l.....l {II xh llm) where I/xn jlm is the essential supremum of xn E Lm and x = (x1 ,..., xi). An analogous definition is valid for Ij x 11:on (L’)‘. (c) On Ri we define a norm by II x II: = maxn,l,....l {I XIL I} where xn E R and x = (Xl ,..., x2). (d) For fg (J?)~ g E (~2~)~ we compute the value by (f, g) = sf. g dP where the dot “ . ” denotes scalar ‘product on R i. If p E (ba)i, the space of all bounded additive set functions absolutely continous with respect to P, then we have (p, g) = Cl s g, dph (e) E(x) denotes the vector (E(x,),..., E(.Y~)) of expected values of the coordinate functions xn E Lm where x = (x1 ,..., XI).
INDIVIDUAL
AND
COLLECTIVE
RISKS
281
probability evaluations. In the following we only assume that the central planning bureau, responsible for the setting of decentralizing prices, has complete knowledge of this objective probability P. Remark. The concept of commodity space presented above presupposes the existence of the same 1 commodities completely characterized by their physical properties (respectively, local or temporal availability) in any state of the world. Prices (or values) of commodity bundles are elements in the topological dual of (L”)I. Consequently the chosen topology on the commodity space determines the price space.It is well known (e.g., [3]) that the norm topology on (L’)’ with linear functionals in (ba)l gives rise to price systems without economic justification. Problems arise from elements in (ba)l that are not u-additive. Therefore we introduce the Mackey topology (7(L”, L1))z on (L” )‘. 1t has been demonstrated in [3] that this topology is sufficiently strong to admit interesting preference relations; for example, Neumann-Morgenstern utility functions representing preference relations between commodity bundles in (L”)’ are Mackey continuous [3, Appendix IT]. The admissible price systems for this topology consequently are elements of (~5~)“.As the probability space(J2, 9, P) is allowed to be atomlesswe prefer in the sequel to call these elements of (L1)z “price densities” corresponding to the prices which are then understood as u-additive measuresin (ba)l, for we have no economic interpreation for price systemsassigninga certain value to contracts promising delivery in states of the world which all have by definition probability zero. Clearly from a mathematical point of view these two concepts are identified (by the Radon-Nikodym theorem). In this paper we consider only exchange economies: Every consumer i (i == I,..., m) is, as usual, characterized by a consumption set Xi , a subset of (L*)I, a preference relation si , a transitive and complete binary relation on Xi , and an RI-valued function ei E(La)z representing his initial endowment. Remark. Preferences are defined here on contingent consumption plans, respectively, “consumption strategies” in the terminology of Malinvaud [lo]. Consequently the decisions of the consumers involve additional reasoning like risk aversion and subjective probability evaluation (see, for instance, [5, Chap. 71). The central new concept in this economy is the definition of feasibility of allocations. As a natural extension of the feasibility definition in deterministic economies one could propose calling an allocation (x1 ,..., x,) E (L”)I”” (xi E Xi) feasible iff
% (Xi - e,) = 0,
282
SIEGFRIED BERNINGHAUS
which can also be interpreted as the equality of aggregate planned supply and demand in any state of the world. In real economies the risks come mainly from individual sources and under an appropriate assumption of “independence” between the consumersrisks one can expect someregularity in the per capita excessconsumption (l/m) Cy (xi - ei) provided the number of agents is large enough. These short remarks will be made precise in the last section. The reader must accept for the present that under appropriate assumptions on probabilistic dependence between the agents’ risks it is not unreasonable to define feasibility of an allocation (x, ,..., x,,) in large economies by
or by (3) Equation (2) concerns the situation of purely individual risks in the economy, while in (3) one cannot average out a part of risk characterized by ~8, a sub-o-algebra of 9. If 99 is not trivial (that means 99 # SF and #-{Q, D>) one can interpret the events in S? as the collective risks the agents in the economy are bearing.Per capitaplanned excessconsumptionis only influenced by collective events provided the number of agents is sufficiently large. These remarks are plausible consequencesof special probabilistic hypotheses, as will be shown in the last section.
PARETO-EFFICIENT
STATES AND PARETO PRICES
The aim of this section is the analysis of Pareto prices dependent of the risks in the economy. Therefore we must discuss at first the concept of Pareto-efficient states. As we have two new definitions of feasibility, in the sequel denoted by (2) (resp. (3))-feasibility according to Eq. (2) (resp. (3)) is satisfied, we have two concepts of Pareto efficiency. DEF. 1. A (2)-feasible ((3)-feasible) state (x1*,..., x,?&*)E (Lco)z’nLof the economy d = {X, , &, et ; i 7:: I ,..., m} is called (I)-Pareto-efficient (resp. (C)-Pareto-efficient) iff there is no other (2)-feasible (resp. (3)-feasible) allocation (xll,..., x,3 such that
xi1 xi xi*
for every i,
and xi1 >j .xi*
for at least one i.
INDIVIDUAL
AND
COLLECTIVE
RISKS
283
It is easy to see that the set of (2)- or (3)-feasible allocations is larger than the set of allocations which are feasible in the sense of Eq. (1). Therefore the existence of Pareto-efficient states to be proved first cannot easily be deduced from well-known results. To deal with the existence problem we need the followjng property of consumption sets. (A.l) the sets
Every X, is a convex and weakly
are weakly compact for arbitrary c E R,‘. We can give an example of a consumption PROPOSITION
.satiTjies property
I.
closed subset of (L+“)‘such
that
set Xi fulfilling this property.
The set Xi dejned bl*
(A. 1).
Proof. Clearly Xi is convex and weakly closed. Consider the set Ajt, = {xi - E(xi) j xi 6 Xi , 0 < E(xf) < cl for a certain vector c E R, I. For an arbitrary element xi -- E(xi) in AiC we have
As a (norm-) bounded and weakly closed set in the dual of the Banach space (L’)’ the set a,C is weakly compact [6, p. 4241. Q.E.D. Remark. The property of consumption sets expressed in Proposition 1 does not imply any compactness of consumption sets. It restricts the consumption plans to those nonnegative (I,“)‘-functions whose deviation from its expected value is of the sameorder of magnitude. In the proof of the existence theorem we need an important result in mathematical economics which can be found, for example, in [I 5, p. 5811. For our purpose we need the most general version. LEMMA 1. Let A be CIcompact set on l~~hieh a transitire binary refution s is defined such that the sets {x E A 1x 2 x>
are closedin Af%r ewry ,YE A, then there exists n maximal elementfor ,< ik .4_ THEOREM 1. The economy 6 7 (X-,) <’ -17 ei ; i = I,..., m) hns an (I)Pareto-ejkient (rcsp. (Cl-Pareto-eflrient) state iffor euery i
284
SIEGFRIED BERNINGHAUS
(a) Xi satisjies (A. 1), (b)
the sets {xi E A’, / xi kjixij
are (a(L”,
L1))Wosed,for
erety
.Ti E Xi .
(1) (I)-Pareto-efficient states. We will show the existence of (I)-Pareto-efficient states by applying Lemma 1 to the following situation. On the set of (2)-feasible allocations defined by Proof.
we define the Pareto ordering sr by (Xi’)
k&i”)
-3 xi1 5; xi2
for every i.
By assumption (b) the set
{(xi) E F I C-r;) ZAK)l is weakly closed in F for all (Xi) E F. To apply the lemma it remains to show weak compactness of F: Consider the sets Aib = {xi E Xi 10 < E(xJ < b], At = (.Y; - E(xJ j xi E Ai”>, I2 = {X E (L+“)z / 0 < x < b, Lx is constant), b E R+2(one can verify that I7 is weakly compact). By definition of F we have F C n proji F2 C n Atb C n(d,” + Z7). By (A.l) the set Aib is weakly compact. As F is weakly closed the weak compactnessof F follows. Consequently by Lemma 1 there exists a maximal element for sp in F that meansan (I)-Pareto-efficient state. (2) (C)-Pareto-efficient state. We define the set of (3)-feasible allocations by
Clearly F C F and F is weakly closed as ES(.) is a weakly continuous operator on (IY,“)~ [ll, p. 301. Then Lemma I gives us the existence of a (C)-Paretoefficient state. Q.E.D. Remark. To make the continuity assumption of Theorem 1 compatible with the model introduced in the preceding section we remark here that the convex (T(L”, L1))z-closed sets in (L”)2 are (o(L”, L1))z-closed. As we need in the sequel the convexity of preferences we can regard the weak upper semicontinuity of preferencesasequivalent with Mackey upper semicontinuity. 2 projd A denotes the projection projiA CX,.
of A C (Lm)l.m onto its ith coordinate.
Consequently
INDIVIDUAL
AND COLLECTIVE RISKS
285
As a preparation for the proofs of the main theorems on Pareto prices we introduce the following notation: To the economy 8 we associatean economy 8 whose commodity space is R1 where d need not have any economic meaning in our context
where g{ Ui(a)} denotesa greatest element of the set (ii(a) := {xi EX< /E(xJ = a}. To the economy 6’ we associate furthermore the economy 2 whose commodity space is (Lm(!2, d, P))” by a similar method -
Remark. The preference relations T7i and xii are constructed such that the following conclusions hold: If (x1*:., x,,*yis an (I) (resp. (C))-Paretoefficient state in 8, then (X1*,..., X,, *) is a Pareto-efficient state in 8, respectively, ( Z1*,. .., Z,12 *) is a Pareto-efficient state in Z.3 By the following lemma it is demonstrated that the definition of the preference relations s’i , si makessenseprovided an appropriate assumption on Li is made. LEMMA 2. Let Xi and si fuljill the snme assumptionsas in Theorem 1; then there exists a greatest element for sj in U,(a) (resp., Ui(f)) inhere u E Rz,f~ (L=(Q, .SY,P))“.
Proof.
By (A.1) the sets U&z) -
a = {xi -
E(x<) / E(xi)
= a, xi E Xi , a E R+z}
are weakly compact, consequently Ui(a) is weakly compact. By Lemma I there exists a greatest element in Ui(a). From the relation U,(f) C Ui(f) the same is valid for Ui( f) provided we can show that U2(f) is weakly closed: Let {x,} be a generalized weakly convergent sequencein Ui( f) where x, + x0 . 3 Pareto effkiency in 6(8)iis based on the usual concepts of feasibility. That means for 2, an allocation (aJ E R”.m is feasible iff Xy ai = Cl: e, (ei E Rz). For 3, an allocation (fJ E (Lm)z.n is feasible iff Cyyi = CT ei (ei E (~5”)~).
286
SIEGFRIED
BERNINGHAUS
Consequently JB x, + JB x0 for every B E g’. As JB x’, == jB E,*(x,) = jsj for all ,?:we have se I?‘(xJ = jBf= SDEa(x,). Therefore E*(x,) -.r Q.E.D. For the proof of the next theorem we need a property of preference relations which has a well-known analog in deterministic economies: A preference relation -& on Xi is called “locally nonsatiated” (with respect to (o(L”, f.r))’ iff for any xi E Xi (xi is no satiation consumption) there exists in every weak relative neighborhood 17,~n Xi of x, a consumption plan .I-;’ such that s,’ ‘- _, .I-, .
2.
THEOREM
that,for
ererjf
Let
6’ = {Xi , 5, , ej ; i -: I ,..., m)
be an economic
such
i
(a) X, = (L,“)l, (b) Giwn satiation in (ba)‘)
sI
is conrex
and locally
nonsatiated
(ti+th
req.
to (n(L’
, Ll))‘).
an (I)-Pareto-ejficient state (xl*,..., x,*) where some x: is no consumption then there exists a price system TT* (o-additive measure with constant density function p* E Rz (p* + 0) such that for ever)’ i xi xi
xi*
=c- E(p*
. xi) > E(p*
. xi*).
Proof. The proof utilizes a theorem on the existence of Pareto prices in deterministic economies [5, 6.4 (l)]. Suppose we can apply this theorem to the economy 8 = {xi , si , FL; i == l,..., m> where (XI* ,..., .Ynl*) is the Pareto-efficient state in d constructed from (xl*,..., x7,,*): Then there exists a price system p* E RL(p* f 0) such that for every i: ai &
Xi*
3 p* . a, > p* . 2,“.
(1)
As .Y[* is a consumption strategy in an (I)-Pareto-efficient state having the sameexpected value as the elementsin Ui(Xj*) we have the relation g{Ui(aJl
2Ji
8{~f(%*)l
-i
(2)
Xi*
for all ai satisfying ai xzi Xi*. Assume there exists in contrast to the statement of the theorem an -yi’ E X7 such that x,’ >,j xi*
2 E(p*
. xi’)
< E(p*
’ xi*).
Consequently g(Ui(Xi’)> kj g{Ui(Xi*)} by (2). That implies E(p* . xi’) 2 . xi*) by (l), a contradiction. Therefore it remains to show that 8 fulfills the assumptions required for an application of the theorem: Clearly xi and 3i is convex. A more detailed argument is given for local nonsatiatedness of si : Assume there is an an E’ > 0 such that for all a FX< with /I N ~ Si* 11< c’ we have a sj .Yi* where xi* E X, is no satiation consumption. E(P*
INDIVIDUAL
AND COLLECTIVE
RISKS
287
Consequently g{Ui(a)} si g{U&*)) := zi and xi si zi for all xi E U(a) where a has the property described above. But for zi we take the weak neighborhood {x E (L”)I 1(X - zi , 1) < E’) denoted by r/,,,, . By assumption (b) of the theorem there is a -yi’ E Xi n lJ1,,, such that xi’ >i zi. As 1 ?cj’ - xi* 1~< E’ we have a contradiction. Q.E.D. This theorem has an important economic interpretation: It assuresthe existence of decentralizing Pareto prices of the form r*(E) = p* * P(E) for a random event E E 9 where p* is a constant density function and P is the probability measure introduced in the preceding section. Therefore in an economy with individual risks the decentralization problem is solved in the following way. The central planning bureau announces only one price for each of the 1 commodities. Together with communication of P the agents evaluate their consumption strategies by J x &r* = E(p* * x). Contingent markets are not neededin the specialcaseof economieswith purely individual risks. For the analysis of price implications of (C)-Pareto-efficiency we will first prove a proposition on the decentralization of “usual” Pareto-efficient states, that means, states whose feasibility definition is based on Eq. (1). A serious difficulty in proving results of this kind in an infinite-dimensional setting is the application of separation theorems: First we must ensure that the convex set to be separated has a nonempty interior [6, p. 4171. Second, even if we can apply an appropriate separation theorem we must not hope without further assumptions that the separating Pareto prices are o-additive measureshaving their density function in (L1)z. To overcome thesedifficulties we introduce further assumptions on preference orderings (for further discussion see [4]). As a generalization of the munotanicity property in deterministic economies we call a preference ordering Li on Xi = (L+“)L monotone iff xi -1 k . lA > i xi for arbitrary vectors k > 0 and P(A) > 0. The second assumption which is weaker than Mackey continuity of preferencesis taken from [13]: (A.2) Assume xi1 >i xi2 (xil, xi2 EXi) and let {F,} be a sequence of measurablesetssuch that P(F,) - I, then 1F,xil > i xi2 for sufficiently large n. An economic interpretation of (A.2) is given in [131:change a consumption strategy on events with diminishing probability then the agent will consider these plans similar to the original one. PROPOSITION
2. Let 8 = {Xi , &;i , ei ; i = I,..., m} be an econom.ysuch
that for erery i (a) Xi == (L-=-)l, (b) & is conrex, monotone, and.fuIfills (A.2) (c) xy ei > 0 a.e.
288
SIEGFRIED
BERNINGHAUS
Given a Pareto-eficient state (x1*,..., ,y,,,*) where no xi* is a satiation consumption, there exists a price T* with density fbtction p* E (L1,jl (p* # 0) such that for every i xi & xi* 3 E(p* . xi) 3 E(p* +xi*). Proof. Consider the convex set Z = Cy &(xi*) where &(xi*) := {xi E Xi j xi >i xi*} By definition of Pareto efficiency we have Cy ei $ Z. By the separation theorem for convex sets in topological vector spaces [6, p. 4171 there is a IT* E (ba)l where r* # 0 such that
CT*,2) 3 ( n-*,f
for all z E2,
ei
1
1
0)
as by the monotonicity assumption Z has a nonempty (norm) interior. Take, for example, xi* + k (k > 0) that lies in the norm interior of 2i(xi*). Next we will prove the inequality
( 1
T*, $ ei < (7~~*,z)
for all z E 2,
where 7~~*is the countable additive part of T*.~. Assume to the contrary (r*, Cy ei) > (niTc*,z’) for at least one z’ EZ. We construct the functions 2, ’ :=
<(rc*,z)
forallzEZ.
Furthermore inequality (2) together with (c) implies (rc*, z) 3 (n*, Cy ei) > 0. Therefore 7~~*# 0 and is nonnegative. Let Z be the weak closure of Z. Clearly (nc*, rrae rc*, z) for all z E Z. The relations Cyzi(xi*) C Z, yn ,=zm> iwn and x1 xi* E Z imply Cy xi* minimizes (rc*, Cy xi) on 1 x,*2 l ez Cyzi(xi*) as xi* E~~(x~*) because of local nonsatiatedness. Consequently c’irc*, xi*> minimizes (7~~*,xi) on {xi E Xi j xi & xi*} that gives us the (Q.E.D. result by takingp* := density function of TV*. 4 By a theorem of Yosida and Hewitt [16, p. 521 every nonnegative T E (LKz)~is representable as the sum ?r = nTc+ q (T, , up > 0) where T, is a-additive and up is purely finitely additive (that means up > p > 0 and p u-additive * v = 0).
INDIVIDUAL
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289
Remark. We need the result of Proposition 2 as an important step in the proof of the next theorem. Nevertheless one could regard the result as interesting in itself. But then one either had to verify that (A.2) does not exclude interesting preference relations or prove Proposition 2 with the Mackey continuity assumption of preferences, which is easy to do. 3. Let 6’ = (Xi, si,
THEOREM
pi ; i = I,..., m) be an economy such that
for er)ery i (4 (b)
Xi = (L,“)‘, si is convex, monotone, and (T(L”,
(c)
x11:”e, > 0 a.e.
L1))z-continuoz~s,
Given a (C)-Pareto-eficient state (x1*,..., x7,*) where no xi* is a satiation consumption, there exists a price system with %‘-measurable density function p* E (Ll)l (p* # 0) such thatfor every i xi 2; xi* 3 E(p*
. xi) 2 E(p”
. xi*).
Proof. The main step of the proof is the application of Proposition 2 to the economy 2 whose commodity space is (L”(Q, Z?I,P))“: By means of this proposition we have a 5%measurable price density p* # 0 such that fi & fi” * E(p”
.fj) 3 qp*
. xi*).
Consequently xj & jc,* * E(p* * Xi) 2 E(p”
. xi*),
by the same argument given in Theorem 2. Therefore it only remains to show that the hypotheses of Proposition 2 are satisfied in the artificial economy 2 provided assumptions (a)-(c) are fulfilled in b: (a’)
We have from (a) xi
= (L+-(Q,
9, P))“.
(b’) zi is convex, and the monotonicity can be demonstrated as follows. Assume to the contrary fi + lAkz;ifi for some k>O, A~92 and P(A) > 0. By definition of 3% we have g{ U,(h + 1Ak)} & g{ U,(f,)) : = zi. Because of (.zzi+ l,k) E U& + 1Ak) we have zi + 1Ak ki Zi , a contradiction to assumption (b). The relation si fulfills (A.2): AssumeA yi hi and there exists a sequence of g-measurable sets (F,} such that P(F,) + 1 and a subsequence{F,k} such that lFnbJ; zi hi . By definition of zi this implies
290
SlEGFRlED
BERNINGHAUS
I Fnb~ilis an element of U( 1Fn,Ji) but 1F,,kzil converges in the Mackey topology against zil (see [3, Appendix I (24)J). Therefore lFnlczil Lpi zi2 for sufficiently large IZ as & is Mackey continuous, a contradiction to (b). (c’)
We have C:” ci > 0 a.e., from (c).
Q.E.D.
The solution of the decentralization problem in economies with individual and collective risks has the following interpretation in the light of Theorem 3. The central planning bureau announces contingent prices m*(E) = JEp* dP to the agents. Contingent markets are necessary for decentralization but their number depends on the structure of collective risks in the economy. For example, if 9I is finite it suffices to announce a finite number of contingent prices p*. Together with the knowledge of P which is communicated to the agents by the planning bureau the agents are able to compute the value of their consumption strategies by J x dn* = E(p* . x).
THE
CONCEPTS
OF
INDIVIDUAL
AND
COLLECTIVE
RISK
The results of the last section can be interpreted as follows. Change the definition of feasibility of allocations as described by Eqs. (2) and (3); then the implications for the classical decentralization theorem of welfare economic are given by Theorems 2 and 3. But why should one regard these special feasibility conditions as plausible? In this section we will particularize the probabilistic framework to deduce the feasibility conditions as a consequence of some “Laws of Large Numbers” utilizing the following method. Results are deduced for a limit economy with infinitely many agents; these results are then applied to economies with a large but finite number of agents. When talking about individual risks in an economy one has the idea of random events concerning one person or a small group of individuals only. For example, the breakdown of a tractor of a farmer in a foreign country does not influence the probability of a breakdown of my car. A large part of risk concerning the agent’s environment is based on these events. Let us assume first that all risks in the economy are of this kind. That is an idealized situation studied first systematically by Malinvaud [lo]. We will in the following give a possible formalization of this idea. Clearly a suitable mathematical model behind this concept is that of “stochastic independence.” Therefore we propose the definition DEF. 2. An economy d = (Xi , 5 i , e, ; i == I,..., m> is called an economy with purely individual risks iff any allocation (x1 ,..., x,,,) F (15,“)~‘~ (xi E X,) consists of independent random variables, and furthermore (e,,..., e,,) is a collection of independent random variables.
INDIVIDUAL
AND
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RISKS
291
Remark. This definition is restrictive in the sense that “interaction effects” between agents in a small group are excluded. We prefer to present the simplest caseto clear the basic ideas. For a discussionof generalizations see[ 10, Appendix]. In economies with purely individual risks one can expect that the risks are averaged out if the number of agents is sufficiently large, especially such that the vector of per capita excessconsumption (l/m) 1.:” (xi - ci) does not differ “too much” from (l/m) zy E(.uj - ei) as N + co by the law of large numbers for independent random variables. Therefore the feasibility condition in such economies should be formulated in expected values of excessconsumption (see Eq. (2)). In the following we give a more rigorous version of that and give an example of exchange economies which satisfy Definition 2. Consider a sequenceof sub-u-algebras {Sn} of 9 such that the u-algebras Sr ,.... .F, are independent for arbitrary N (the existence of such a sequence can always be assured by constructing (Q, 9, P) as an appropriate product space). We consider a sequenceof exchange economies8’” = {iu, , 5n , e, ; II 7~~I,.... IV) which is associated to the sequence{SI,!,,3in the following way.
(a) XT, = (L+a(Q, 35 , P)Y; (b) P,, is &-measurable; (c) C‘l -l differs from Q” by the one consumer whose consumption plans and initial endowments are &,+1- measurable random variables and any allocation (-x1,..., x,“) in 8” is regarded as a finite subsequenceof an allocation is,)- for the limit economy P. Let us make the assumption. (A.3) The variances of any sequence of random variables {.u, - e,} taken from {8iVj are uniformly bounded. Then we have as a straightforward application of the strong law of large numbers for independent random variables [7, p. 2041
This motivates our definition of feasibility of allocations in economies with purely individual risks: If N is sufficiently large we can replace in 8.” the usual definition of feasibility as given in Eq. (1) by the new definition given by (2) without making a great mistake. Although a large part of risk stems from individual sources, we must admit that there are events whose occurrence influences the “individual events” of all agents. To take up the example, above imagine the economies of the world are in the collective state “the great nuclear world war takes
292
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BERNINGHAUS
place”; then the breakdown of a car in a foreign country caused by a nuclear explosion is accompanied by a breakdown of my car (provided we regard that war as the simultaneous explosion of the nuclear potential in the world). Collective risks are characterized by events whose occurrence concerns all agents in the same way. The individual events are conditioned to the collective framework in which they occur. As a possible mathematical model for economies with collective and individual risks we propose the concept of “conditional independence.” DEF. 3. An economy 6 = {X, , si, ei ; i = l,..., m} is called an economy with individual and collective risks iff any allocation (x1 ,..., x,,) E (L+“)z’” consists of conditional independent random variables, and furthermore 6% ,..., e,,) is a collection of conditional independent random variables where the condition is given by the collective risks in b. To give an example of economies which fulfill this definition we take a sequence {gn} of sub-a-algebras of 9 which are conditional independent with respect to g[, , where J%‘, denotes the o-algebra of terminal events generated by the sequence {Z%}. Obviously the economies dN defined above satisfy Definition 3 and g’, can be interpreted as the u-algebra of collective risks associated with the sequence {bN}. Intuitively one expects that in the sequence (l/N) Cy (x, - e,) the collective risks cannot be averaged out and consequently the limit must somehow be conditioned to a &YC gA, . By means of the following assumption we can make this precise. (A.4) Allocations (x1 ,..., x,,,) in bN are admissible iff they give rise to equally distributed individual excess consumption (x1 - e, ,..., xN - eN). Naturally this is a restriction for the set of allocations in an Remark. economy. But we do not see another way to derive the results below without this assumption. One can weaken the restrictive character of the situation by making (A.4) only for allocations in the types of a specified type economy (see [IO]). But this minor modification can not avoid the artificial character of (A.4). The trouble arises from the indeterminacy of allocation problems. In deterministic economies one need not specify the origin of an allocation; the consumption plans may be allocated by a central planner or they may arise as the result of an individual choice of an agent. But when one wants to discuss some kind of probabilistic dependence between the agents an assumption on the formation of allocations is necessary. This difficulty vanishes in the analysis of demand functions where the origin of consumption is the individual choice and probabilistic dependence of demand functions follows from probabilistic dependence of the characteristics of the agents [S]. Now consider a sequence {x, - e,) of individual excess consumption plans taken from the sequence {a.V}. By the construction of the sequence of economies these random variables are exchangeable [9, p. 3641. Then we
INDIVIDUAL
AND
COLLECTIVE
RISKS
293
have from the law of large numbers for exchangeable random variables [12, p. 1441
where E&u’(x, - e,) is the conditional expectation of an arbitrary member of the sequenceand 9X” is the a-algebra of terminal events generated by the sequence{x, - enj. This motivates the formuiation of the feasibility condition in conditional expected values as described in Eq. (3) provided we consider an economy F”Nwith sufficiently large N. But there is a discrepancy between (3) and the result derived above. In (3) we have a fixed u-algebra ~8 for the construction of conditional expectations but by the reasoning above we see that the u-algebras gK,” resulting from the law of large nubers depend on the allocation under consideration. As we have described S”, as the u-algebra of all collective events in the economy we can interpret gX” as the collective risks concerning the special allocation. But by construction of our model we have a’,,” C g’,, for any $2X generated by an admissible allocation. Consequently an allocation feasible (in the senseof (3)) with respect to Ezu(.) is feasible with respect to ,!?7-‘(e). To fix a g-algebra as required in Eq. (3) we propose to take gm . This avoids the problem of the choice of a special s’,” and is the best way to express the effects of all collective risks on feasibility. Inserting a’, for gzx where &=,” g $a only means that we dispensewith further “smoothing” of the special allocation in the feasibility condition. For we get a better “smoothing effect” in the limit by the operator Es(.) when we deal with functions which are “more constant” (meaning, they generate a proper sub-o-algebra of the o-algebras cFn) than functions generating the F,, .
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Existenceof equilibriain economies with infinitely many commodites, 4 (1972),514-540. “Theory of Value,” Wiley, New York, 1959. ValuationequilibriumandParetooptimum,Proc. Nat. Acau’. Sri. U.S.A.
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HILDENBRAND, Random preferences and equilibrium analysis, J. Ero~r. Theory 3 (1971) 414-429. M. LOEVE, “Probability Theory,” Van Nostrand Reinhold, New York, 1963. E. MALINVAUD, The allocation of individual risks in large markets, J. EWU. Theory 4 (1972), 312-328. P. MEYER, “Probability and Potentials,” Blaisdell, Toronto/London, 1966. J. NEVEU, “Mathematical Foundations of the Calculus of Probability,” Holden-Day, San Francisco, 1965. E. PRESCOTT AND R. LUCAS, A Note on price systems in infinite dimensional space, Internat. Econ. Rev. 13 (1972), 416422. R. RADNER, Efficiency prices for infinite horizon production programs, Rec. Econ. Stud. (1967), 51-66. D. SCHMEIDLER, Competitive equilibria in markets with a continuum of traders and incomplete preferences, Econometrica 37 (1969), 578-585. K. YOSIDA AND E. HEWITT, Finitely additive measures, Trans. Amer. Math. Sot. 72 (1956), 46-66.
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