Equilibrium with State-Contingent Markets

Magister Ludi


(A) Optimal Risk-Bearing Allocations
(B) Arrow-Debreu Equilibrium
(C) Individual and Social Risks

Selected References



(A) Optimal Risk-Bearing Allocations

The state-preference approach to uncertainty of Kenneth J. Arrow (1953) and Gérard Debreu (1959) lends itself rather easily to Walrasian general equilibrium theory (although see Maurice Allais (1953) for another early attempt at reconciling uncertainty and general equilibrium). In a general equilibrium context with multiple agents and multiple state-contingent commodities, where agents are endowed with state-contingent endowments which they can trade among themselves. Thus, a "competitive equilibrium" is a set of state-contingent prices and state-contingent commodities which satisfy all agents' utility-maximizing choices and clear state-contingent markets.

In order to approach the idea of a general equilibrium with state-contingent markets, it might be useful to remind ourselves of the individual optimum for a single agent. Letting S be the set of states and assuming the existence of state-independent utility functions, then recall from our discussion of the state-preference approach that a particular agent's optimization problem is:

max U = s S psu(xs)


s S psxs s S pses

where xs is a state-contingent commodity bundle, ps a set of state-contingent prices. Recall also that this yields the result that for any commodity i and any two states s, s S:

p su (xis)/pis = p s u (xis )/pis

which was termed the fundamental theorem of risk-bearing.

One of the exogenous components of this are the various vectors of state-contingent prices, ps. In a general equilibrium context, these should be explained. A simple example in the Edgeworth box in Figure 1 might motivate how to go about this. Let there be two states of nature: S = {1, 2} and one physical commodity ("consumption"). Thus, there are two state-contingent commodities, x1 and x2. Let there also be two individuals, A and B, with endowment bundle eA = {e1A, e2A} for agent A and eB = {e1B, e2B} for agent B. Consequently, the total size of the economy is {e1, e2} = {e1A + e1B, e2A + e2B}. This will set the dimensions of the Edgeworth box.

In Figure 1, OA represents the origin of agent A's problem and OB that of agent B. Initial endowment is noted as E in Figure 1. The two 45 lines from the origins are the "certainty lines" of each agent. If there is no aggregate uncertainty, then total endowments are the same in both periods so e1 = e2 and the box is square and thus the 45 lines coincide. If, in contrast, there is aggregate uncertainty, then e1 e2 so that total endowment in one period will be different than in another period. In this case, the Edgeworth Box is not a square and the 45 lines will not overlap but rather be parallel to each other. The situation we see in Figure 1, thus, allows for aggregate uncertainty.

Each agent possesses a utility function, in this case UA and UB, representing preferences over state-contingent commodity bundles x = (x1, x2). If we assume the existence of a state-independent elementary utility functions, then we can decompose them into their expected utility form, specifically:

UA(x) = ・/font> s S p sAuA(xs) = p 1AuA(x1) + p 2AuA(x2)

UB(x) = ・/font> s S p sBuB(xs) = p 1BuB(x1) + p 2BuB(x2)

where uA and uB are elementary utility functions and p sA is agent A's subjective probability of state s and p sB is agent B's subjective probability of state s. Note that we are not assuming they are the same: agents can have different probability assignments to states of nature just as much as they can have different utility functions. In general, the slopes of the indifference curves are:

dx2/dx1|UA = -p 1AuA (x1)/p 2AuA (x2)

dx2/dx1|UB = -p 1BuB (x1)/p 2BuB (x2)

However, along the 45 certainty line stemming from origin OA, where uA(x1) = uA(x2), then agent A's indifference curves will have slope -p 1A/p 2A while along the 45 line stemming from origin OB, where uB(x1) = uB(x2), agent B's indifference curves will have slope -p 1B/p 2B. Once again, there is no requirement that they be the same.

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Figure 1 - Optimal Risk-Bearing Allocation with Aggregate Risk

From the endowment point E, where we have indifference curve UA and UB, notice that there is room for exchanging state-contingent commodities. A movement from allocation E to allocation F would be a mutually-beneficial trade as agent A increases utility to UA and agent B increases utility to UB . Notice that the indifference curves are tangent to each other at point F. This implies that at F:

p 1AuA (x1)/p 2AuA (x2) = p 1BuB (x1)/p 2BuB (x2)

Consequently, we can trace out a "contract curve" from OA to OB representing the series of allocations in the Edgeworth box where the indifference curves of OA and OB are tangent - thus each point on the contract curve is Pareto-optimal or an "optimal risk-bearing allocation" (or ORBA). As the 45 lines do not coincide (i.e. there is "aggregate uncertainty"), then notice that the contract curve will lie between the 45 lines. This is always true if both agents are risk-averse.

An Arrow-Debreu equilibrium in this scenario would be an allocation of state-contingent commodities (x1A, x2A, x1B, x2B) and a vector state-contingent prices (p1, p2) where markets clear and all agents are at their individual optimum, in this case where:

(1) (x1A, x2A) argmax p 1AuA(x1) + p 2A uA(x2) s.t. p1x1A + p2x2A p1e1A + p2e2A (A's optimum)

(x1B, x2B) argmax p 1BuB(x1) + p 2BuB(x2) s.t. p1x1B + p2x2B p1e1B + p2e2B (B's optimum)

(2) x1A + x1B = e2A + e1B (market for good 1 clears)

x2A + x2B = e2A + e2B (market for good 2 clears)

Such a situation is shown in Figure 1 at allocation F. It is easy to notice that as, we know, forming a Lagrangian for each agent, the first order conditions for a maximum imply that:

dL/dx1 = p 1AuA (x1) - l Ap1 = 0

dL/dx2 = p 2AuA (x2) - l Ap2 = 0

from the first agent's problem (l A is the Lagrangian multiplier) and:

dL/dx1 = p 1BuB (x1) - l Bp1 = 0

dL/dx2 = p 2BuB (x2) - l Bp2 = 0

for the second agent. Consequently, we see that these imply that:

p 1AuA (x1)/ p 2AuA (x2) = p1/p2 = p 1BuB (x1)/ p 2BuB (x2)

thus the indifference curves are tangent to each other and to the equilibrium price line (shown in Figure F with slope -p1/p2). That an Arrow-Debreu equilibrium is an optimal risk-bearing allocation is clear enough from the tangencies and standard convexity proofs will confirm this more generally. Notice that if we assume agents have the same subjective probabilities, i.e. p 1A/p 2A = p 1B/p 2B, then the tangency condition on the contract curve reduces simply to uA (x1)/uA (x2) = uB (x1)/uB (x2), or cross-multiplying:

uA (x1)/uB (x1) = uA (x2)/uB (x2)

so that the ratio of marginal utilities of both agents in a given state is the same across states. This is also known as the "optimal risk-bearing ratio" in the case of identical beliefs.

It might be worthwhile to concentrate on the impact of different degrees of belief, risk and risk-aversion on the final equilibrium. Consider Figure 2 where we again have our two agents, one good and two states. Suppose that in one state, agent A has all the endowment and B none. In the second state, B has all the endowment, and A none. Let us say, then, that eA = (1, 0) and eB = (0, 1), thus we are at the southeast corner of an Edgeworth box at point E. Notice that in this particular case we do not have aggregate uncertainty as {e1, e2} = {e1A + e1B, e2A + e2B} = (1, 1) so aggregate e1 = e2. Thus, the 45 certainty lines of the agents coincide. Thus, the only uncertainty is distributive uncertainty, i.e. only distribution of total endowment is uncertain, but the total is not as aggregate endowment is state-independent.

In the absence of aggregate uncertainty, then if subjective probability estimates are the same for both agents, then the contract curve will be the 45 line connecting OA and OB. This implies that at any optimal risk-bearing allocation, both agents will "insure" completely, i.e. both lie on the 45 certainty line. Assuming, further, that both have the same attitudes towards risk (i.e. same utility functions), then agent A promises to pay agent B half his endowment if state 1 occurs on the condition that B pays half his endowment to A when state 2 occurs. They both ensure completely as consumption in both states are {1/2, 1/2} for both - shown as point F in Figure 2 where A achieves utility UA(F) and B achieves utility UB(F).

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Figure 2 - ORBA with no aggregate risk and different beliefs.

However, if subjective probability assignments are different, say p 1A/p 2A > p 1B/p 2B, then the tangency conditions for ORBA remain [p 1A/p 2A][uA (x1)/uA (x2)] = [p 1B/p 2B][uB (x1)/uB (x2)] - which implies that uA (x1)/uA (x2) < [uB (x1)/uB (x2)], or:

uA (x1)/uB (x1) < uA (x2)/uB (x2)

i.e. the ratio of marginal utilities across states are no longer equal at an optimal risk-bearing allocation. Going back to utility levels, this implies that:

uA(x1)/uB(x1) > uA(x2)/uB(x2)

so at any ORBA, A has a comparatively greater utility in state 1 than 2 whereas B has a comparatively greater utility in state 2 than state 1. In short, both have purchased relatively more in the state they believe to be more likely.

The important thing to note when there are differing subjective probability assignments is that, controlling for the same degree of risk-aversion, agents will not insure completely. The agents' final allocations would be off the 45 certainty line and allow for different allocations in different states, e.g. xA = (3/4, 1/4) and xB = (1/4, 3/4), thus indicating that A thinks state 1 is more likely and B thinks state 2 is more likely. This is shown as point G in Figure 2. Note that the indifference curve of A, UA(G), is much steeper at the 45 line than the indifference curve of B, UB(G), an illustration of the different probability assessments of both agents. As a consequence, the contract curve denoting ORBA points will not be the 45 line in this case but will be below it, shown in Figure 2 as the curve connecting OA and OB lying below the 45 line.

How might this be different if there is aggregate risk, as we had in Figure 1? There are several implications. The first is that, unless one of the agents is completely risk-neutral, the final allocation will always be within the parallel 45 lines so that neither agent ensures completely. This is true regardless of whether probability assignments are different or not. Now, at point F in Figure 1, we know that the slopes of the indifference curves are equal to each other and the price line, i.e.

[p 1A/p 2A][uA (x1)/uA (x2)] = p1/p2 = [p 1B/p 2B][uB (x1)/uB (x2)]

As the agents at point F are off their respective 45 lines, then by the diminishing marginal rates of substitution, this implies that:

[p 1A/p 2A][uA (x1)/uA (x2)] < p 1A/p 2A


[p 1B/p 2B][uB (x1)/uB (x2)] < p 1B/p 2B

which implies that p 1A/p 2A, p 1B/p 2B > p1/p2.

Suppose for the moment that both agents share the same probability assignments. Then, in equilibrium, p 1/p 2 > p1/p2. Now suppose further that p 1 = p 2 = 1/2 (i.e. equal probability), then this condition implies that p2 > p1. In other words, commodity in state 2 has a greater price than commodity in state 1. Now, as Figure 1 shows, by its shape, that e1 > e2 (so aggregate endowment of good in state 1 is greater than aggregate endowment of good in state 2), then p2 > p1 implies that the price for a contingent unit of a commodity is greater for the state for which the good is scarcer.

There is a further implication. Suppose the aggregate endowment of good 1 is 20 and the aggregate endowment of good 2 is 10, i.e. e1 = 20, e2 = 10. Yet, note that the amount of consumption that a unit of the second commodity gives (0 in state 1 and 1 in state 2) is negatively correlated with the aggregate initial endowment (20 in state 1 and 10 in state 2), whereas the amount of consumption the first commodity gives (1 in state 1 and 0 in state 2) is positively correlated to aggregate initial endowment. Yet we also know, following our example, that p2 > p1, thus, contingent consumption is comparatively more valuable if the amount of consumption it gives in different states are negatively correlated with the aggregate initial endowment. This relationship underlies a fundamental theorem in financial economics: a financial instrument (i.e. state-contingent god) is comparatively more valuable if its return is negatively correlated with the "market return" (i.e. aggregate endowment).

Notice that the degree of risk-aversion will also affect agent's utilities in the end. Suppose we are in Figure 3, where there is aggregate risk but the agents share the same probability assignments. If A is risk-neutral (linear indifference curves) and B is risk-averse (convex indifference curves), then there will always be a movement to B's certainty line, i.e. the risk-neutral agent (A) takes on all risk. This is shown in Figure 3 at point H where agent A has (linear) indifference curve UA(H) and agent B has (convex) indifference curve UB(H). As H is on B's certainty line, then B is being insured completely while A is taking a massive amount of risk.

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Figure 3 - ORBA and Risk Aversion

How will increasing the degree of risk-aversion affect equilibrium? This is in principle hard to tell, but can be visualized as follows. Consider an initial ORBA position F in Figure 3 where agent A has utility UA and agent B has utility UB. If, all of a sudden, B becomes more risk-averse, his indifference curve will become more convex. This is shown in Figure 3 by the change from UB to VB. Note that now with UA and VB, F is no longer ORBA - implying that there are further trades that can be conducted which would make everybody better off. An example is shown in Figure 3 by point G, where B remains at the same utility level VB as he had at F but now agent A's utility has increased to UA . Notice that A is gaining a tremendous amount of good 1 and losing only a little bit of good 2 in the movement from F to G. Thus, the sudden increase in the "risk-aversion" of agent B implies that B is now willing to pay A more in order to induce A to shoulder a greater share of the risk.

(B) Arrow-Debreu Equilibrium

Let us now turn to the general case. Let there be S states of nature. Let there be n physically-differentiated commodities. Thus, the "commodity space" of agent h, Xh, is some subset of RnS. Also for simplicity (although abusive of notation), let S be the set of states of nature with #S = S.

Let H be the set of households (where, again abusively, #H = H) with preferences and endowments which are themselves state-contingent. Namely, h are defined over Xh RnS, i.e. defined over state-contingent commodity vectors. We can make enough assumptions on these preferences so that they are representable by a real-valued, continuous, differentiable, concave function Uh: RnS R. Suppose an agent h H assigns the (objective or subjective) probability p s to a particular state s S occurring but that agents possess state-dependent elementary utility functions over physically-differentiated goods, ush: Rn R. Then, a state-contingent commodity vector xh is preferred to another yh if expected utility is greater, i.e.

xh h yh if and only if ・/font> s S p sush(xsh) ・/font> s S p sush(ysh)

Each household h H has a set of state-dependent endowments, eh = [e1h, ...., eSh] RnS, where esh representing the vector of endowed goods at each state s S. An economy-wide allocation x is defined as a vector [x1, ..., xH] where xh X is the allocation to the hth household and is defined as xh = [x1h, ..., xSh] RnS where each xsh is a vector of commodities the hth household receives in state s S. Thus, xsh is an entitlement receive a particular bundle of goods if a particular state s S occurs. An allocation can also be viewed as a vector of n random variables, x = [x1h, .., xnh] RnS with the ith random variable being a mapping xih: S R, or xih = [xi1h, ..., xiSh] Rs. Thus, the state-contingent vector xh can be viewed as a mapping xh: S Rn, thus a particular state s S leads to n commodities (a bundle) being realized.

We can restrict both allocations and endowments to the positive orthant of RnS if we wish, but this is not necessary. In particular, when considering production, it might not be worthwhile as negative commodities come in as factor supplies. Production itself can be easily linked to the Arrow-Debreu structure. A state-contingent production plan for the fth firm is yf Yf RnS. Thus, not only is the particular production plan undertaken dependent on the state (thus ysf occurs when s S occurs), but the very feasibility of a particular production plan is contingent on the state (as Ysf is the production set when s S).

Let ps = (ps1, ..., psn) Rn+ be the spot prices for goods in a particular state s S. Thus, we can define p = (p1, ..., pS) RnS+ as the vector of state-contingent prices. Markets open at the beginning of the day and all trades must be completed before the resolution of uncertainty - thus we will require that information be symmetric (all agents must know which states can occur and are being contracted for). Let us have a private ownership economy akin to Debreu (1959), except that we shall permit household ownership shares q hf (where ・/font> h H q hf = 1 for all f F) to be constant and state-independent.

The components can be thought as follows: the economy-wide allocation is x = {xh}h H where xh = [x1h, ..., xSh] is the state-contingent commodity vector allocated to agent h H and xsh = [x1sh, x2sh, ...xnsh] is a particular bundle of goods in state s S. Thus, xsh Xsh Rn and xh Xh RnS and, finally, x X RnSH where X = X1 X2 ... XH. Similarly, letting there be F firms, an economy-wide production plan is y = {yf}f F where yf = [y1f, ..., ySf] is a vector of state-contingent production plans and ysf = [y1sf, y2sf, ..., ynsf] is a particular production plan in state s S. Thus, ysf Ysf Rn and yf Yf RnS and, finally, y Y RnSF where Y = Y1 Y2 ... YF is the aggregate production set. Consequently an economy-wide set of allocations is defined as a pair (x, y) X Y RnS(H+F). A particular set of prices, p = [p1, ..., pS] RnS is a vector of state-contingent prices with subvector ps = [ps1, ps2, ..., psn] Rn being a particular set of prices in state s S. We now define an equilibrium:

Arrow-Debreu Equilibrium: an Arrow-Debreu equilibrium is a set of allocations (x*, y*) X Y RnS(H+F) and a set of prices p* RnS such that:

(i) for every f F, yf* satisfies p*yf* p*yf for all yf Yf

(ii) for every h H, xh* is maximal for h in the budget set

Bh = {xh Xh p*xh p*eh + ・/font> hq hfp*yf*}

(iii) ・/font> h H xh* = ・/font> f F yf* + ・/font> h H eh

All the standard results of Arrow-Debreu general equilibrium theory (e.g. existence of equilibrium, Pareto-optimality of equilibrium, etc.) apply without fail in this state-dependent economy. There is a complete isomorphism in the Arrow-Debreu economy with state-contingent markets as in the regular (certainty) case. For more details, consult Debreu (1959: Ch.7).

(C) Individual and Social Risks

There is an interesting proposition regarding the relationship between what can be termed "individual risks" and "social risks" which is worth pursuing. To understand what this means, let us first be clear as to what it does not mean by appealing to our two-agent, two-state, one good Edgeworth box. We claimed there was "aggregate risk" when total endowment in one state was not equal to total endowment in another, i.e. e1 e2 or (e1A + e2B) (e1B + e2B). If e1 = e2, we noted that we had no aggregate risk, but we still had "distributive risk" in the sense that the individuals' consumptions and endowments were uncertain and thus the distribution of the fixed endowment among agents is uncertain and varies from state to state, i.e. (e1A, e1B) (e2A, e2B) and (x1A, x1B) (x2A, x2B).

What we shall term "social risk" corresponds to our old "distributive risk" and not to "aggregate risk" and what we shall term "individual risk" has no direct analogue to our previous case. Individual risks are risks faced by a particular agent (e.g. health or illness); "aggregate risks" are risks faced by all individuals in the economy collectively (e.g. war or peace); "social risks" can be thought of as the collection of individual risks (e.g. "he is healthy, she is healthy" or "he is ill, she is healthy", etc.). Thus, in our definition, an individual state is restricted to a particular agent. What we term "social risks" is what corresponds to distributive risks in our previous context, which can, in the new context, be considered as the distribution of agents among individual risks. Finally, we have no analogue for aggregate risks in the varying total endowment sense. Social risks are merely collections of individual risks: uncertain distributions, which arise from individual risks.

To be more specific, if we have two agents, {A, B} and two individual states {healthy, ill}, then we have four social states: {A healthy, B healthy; A ill, B healthy; A healthy, B ill; A ill, B ill}. As a social state can be thought of as the distribution of agents over different individual states, then if one individual changes the state he is in, then we have a different social state altogether. This set of social states is our Edgeworth box and thus "state-contingent" prices refer to the prices of goods delivered in particular social states.

The reason for this sudden change of notation is to achieve an interesting result regarding the relationship between individual and social risks. It is not incredulous to suggest that most of the uncertainty economic agents face is individual uncertainty After all, insurance companies, financial markets, etc. all arise in response to individuals' particular needs in face of their particular uncertainty (my health, my illness) and we find few cases of insurance against aggregate risks such as "war" or "recessions" (although military spending and welfare programs can be seen as a kind of collective insurance).

Of course, as we noted, how a particular agent reacts to his individual risks will inevitably affect other agents - and we capture this by changing social states. Thus, if we assume an economy without aggregate risk but where agents face individual risks, the question then imposes itself: what is the impact of individual uncertainty on social uncertainty? Or, putting it another way, how much of an impact is a single individual's change of individual state going to have on the economy? We can already start thinking in terms of the law of large numbers here: if the number of agents is very large, then a change in any one agent's individual state will have a negligible impact on the social state - or, more specifically, the resulting state-contingent prices in the new social state ought not to be very different from prices in the old social state.

This suggestion is rooted in the famous work of Kenneth J. Arrow and Robert C. Lind (1970) on public investment projects with uncertain outcome but where the burden of the risk was shouldered by many individual taxpayers. What the Arrow-Lind theorem claims is that if the number of individual taxpayers is large enough, then the planners can ignore the uncertain returns of the project. To achieve this, they appealed to a form of the law of large numbers. For example, let there be H identical households, one good ("money") and various states of the world. Let z be a random variable representing social risk (or the project's risk) which takes on positive values in good states and negative values in bad states. We assume E(e ) = 0 and var(z) = s 2 < . Without loss of generality, let us assume that this risk is shared equally among all households, thus a particular individual h faces a random income term e /H (i.e. the project's risk is channeled via equal but uncertain taxes upon individual agents). Having assumed identical households, thus everyone has identical utility function u and identical initial sure income x. Then expected utility of the hth household is E[u(x + e /H)]. Let p h be the premium paid by the hth household to get rid of this risk, thus x - p h is the certainty-equivalent level of income. By definition, for the hth household:

E[u(x + e /H)] = u(x - p h)

Now, taking a Taylor approximation on the left side

E[u(x + e /H)] = E[u(x)] + E[u (x)e /H] + E[u (x)e 2/2H2] + ....

omitting negligible terms and as x is certain, E(e ) = 0 and E(e 2) = s 2, then this becomes:

E[u(x + e /H)] = u(x) + u「「(x)s 2/2H2

Taking now a Taylor approximation of the right side of our earlier term:

u(x - p h) = u(x) - u (x)p h + ...

thus our earlier equation is now (approximately):

u(x) + u (x)s 2/2H2 u(x) - u (x)p h

or simply, solving for p h:

p h [-u (x)/u (x)]s 2/2H2

But recall that p h is the individual h's risk premium. Thus, the "social" risk premium, call it p = ・/font> h H p h is:

p = ・/font> h H p h = ・/font> h H[-u (x)/u (x)]s 2/2H2

or, as we have identical utility functions, x, etc. and because z is not correlated with H, then this becomes:

p = (1/H)[-u (x)/u (x)]s 2/2

so, as H , then p 0. What this means is that as the number of agents among whom risk is shared becomes very large (approaches infinity), the "social" risk premium is reduced to zero. Thus, the law of large numbers suggests that society, as a whole, may ignore that risk and thus pay no risk premium.

The Arrow-Lind theorem has led to some interesting suggestions for extension, particularly in the random preferences scenario of Werner Hildenbrand (1971) and, more directly, in the work of Edmond Malinvaud (1972, 1973). What Malinvaud suggested can be seen as an extension of the Arrow-Lind theorem into our context. Let there be H households, S states and n physically-differentiated goods. Let there be an economy where the aggregate endowment, e is certain (thus no "aggregate risk" by our earlier definition). Now, we can think of our exercise as follows: suppose S represents the individual risks facing a particular individual h and that all other households k H where k h, are only affected indirectly by h's actions. As a consequence, S also constitutes the set of "social risks" except that no one but individual h is actually in a different individual "state" (except indirectly). What we would like to say, then, is that if the number of households is very large, then the indirect impact of the randomness h faces on other agents will be minuscule and, as there is no other source of risk, the social state-contingent prices will be equal to the product of the "sure" prices and the probabilities of the social states.

To see this, let us pretend that endowment is certain in all cases, the only thing that is uncertain is agent h's preferences - thus it is his elementary utility function ush which is state-dependent. Consequently, he has expected utility as E(uh) = ・/font> s S p suh(xsh) where p s, for the sake of simplicity, is the objective probability that state s S emerges and thus that agent h's preferences will be ush. We assume that all other agents k H where k h, face no such randomness and thus have state-independent utility functions. Thus, the expected utility of the kth agent is:

E(uk) = ・/font> s S p suk(xsk)

where, notice, the randomness will emerge because of the impact of agent h's state-dependent trades. Thus, if ei is total aggregate endowment of good i, then the ei - xhsi will be the endowment which will be distributed among the other agents after h has taken his demanded share - thus this is random because of h's actions.

For all agents k H, where k h, maximizing expected utility subject to a conventional budget constraint yields the fundamental theorem of risk-bearing where, for any good i and any two states s, s S:

p suk (xis)/pis = p s uk (xis )/pis

or, rearranging:

uk (xis)/uk (xis ) = (p s /p s)キ(pis/ pis )

where the term on the left is agent k's marginal rate of substitution between the same good in two states, pis/pis are the ratio of state-contingent prices and p s /p s the ratio of probabilities of states s , s. Malinvaud's basic suggestion is that if there is a large number of agents, then agent h's demand will be very small and thus the amount of endowment left undemanded by agent h in social state s (i.e. ei - xhis for each good i) will be pretty much the same as in any other social state s . Consequently, the amount of consumption by agent k h will be pretty much independent of the state of the world, thus the resulting marginal rate of substitution will be approximately 1, i.e. uk (xis)/uk (xis ) 1. As a result, by the fundamental theorem of risk-bearing:

pis/p s pis /p s

so the state-contingent price of good i weighed by the probabilities of their respective states emerging is approximately the same.

Now, let us turn to define "sure" prices. If we wish to have a unit of good i in all states, then the cost of this will be the sum of all state-contingent prices for that good i. In other words, the price of a sure unit of good i can be denoted pi = ・/font> s Spis. Pre-multiplying this sure price by the probability of a particular state s, p s, yields:

p spi = p s・/font> s Spis = p spi1 + p spi2 + p spi3 + ..... + p spiS

But as our new fundamental theorem implies that for any s , that p spis p s pis, then this becomes by substitution:

p spi p 1pis + p 2pis + p 3pis + ..... + p Spis = ・/font> s S p s pis

or, as ・/font> s S p s = 1, then:

p spi pis

so the state-contingent price for good i in state s, pis, is equal to the product of the sure price of that good and the probability of that state. Thus, state-contingent prices are directly proportional to the probability of the social state in which they occur. This means that if we know the sure price, pi, and the probabilities, then we can easily determine what the state-contingent price is.

The implications of this can be understood as follows: in principle, in the Arrow-Debreu economy, when we have n goods and S states, we need nS markets in order to determined nS prices. This is particularly troublesome when considering individual risks as it seems to imply that there are separate markets and prices for "delivery of eggs when I am ill" and "delivery of bread when he is ill", etc. which are far too particular. However, what Malinvaud's theorem implies is that when there is a large number of agents, then the price of "eggs when I am ill" becomes merely the sure price of eggs (determined in the sure market, pi, which delivers in all states) multiplied by the probability of "I am ill". Consequently, instead of nS markets, we merely need n sure markets and S insurance markets (or Arrow securities, if we wish - see our section on Radner equilibria) which deliver individual purchasing power in the case of particular states. Thus the number of necessary markets gets reduced considerably when the number of agents is large.

Of course, this particular example assumes that only one agent faces randomness directly. What if all agents face individual risks of some sort? In this case, we need to specify "social states" as distributions of specific agents over different individual states, (e.g. "I am ill, you are ill, he is healthy, she is healthy" would be one social state, while "I am ill, you are healthy, he is healthy, she is healthy" would be another social state). However. with S now redefined this way, we can resurrect the fundamental theorem of risk-bearing and, by the same law of large numbers argument, achieve the same result, i.e. that pis = p spi, where pi is the "sure" price and p s the probability of social state s S.

Selected References

M. Allais (1953) "L'extension des théories de l'équilibre 馗onomique général et du rendement social au cas du risque", Econometrica, Vol. 21, p.269-90.

K.J. Arrow (1953) "The Role of Securities in the Optimal Allocation of Risk-Bearing", Econometrie; as translated and reprinted in 1964, Review of Economic Studies, Vol. 31, p.91-6.

K.J. Arrow and R.C. Lind (1970) "Uncertainty and the Evaluation of Public Investment Decisions", American Economic Review, Vol. 60, p.364-78.

G. Debreu (1959) Theory of Value: An axiomatic analysis of economic equilibrium. New Haven: Yale University Press.

W. Hildenbrand (1971) "Random Preferences and General Economic Equilibrium", Journal of Economic Theory, Vol. 3, p.414-29.

E. Malinvaud (1972) "The Allocation of Individual Risks in Large Markets", Journal of Economic Theory, Vol. 5, p.312-28.

E. Malinvaud (1973) "Markets for an Exchange Economy with Individual Risks ", Econometrica, Vol. 41 (3), p.383-410.