Contents

(A) Optimal Risk-Bearing Allocations

(B) Arrow-Debreu Equilibrium

(C) Individual and Social Risks

**(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ﾎ S}p_{s}u(x_{s})

s.t.

・

_{sﾎ S}p_{s}x_{s}｣ ・_{sﾎ S}p_{s}e_{s}

where x_{s} is a state-contingent commodity bundle, p_{s} a set of
state-contingent prices. Recall also that this yields the result that for any commodity i
and any two states s, s｢ ﾎ S:

p

_{s}u｢ (x_{is})/p_{is}= p_{s｢ }u｢ (x_{is｢ })/p_{is｢ }

which was termed the *fundamental theorem of risk-bearing*.

One of the exogenous components of this are the various vectors of state-contingent
prices, p_{s}. 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, x_{1}
and x_{2}. Let there also be two individuals, A and B, with endowment bundle e^{A}
= {e_{1}^{A}, e_{2}^{A}} for agent A and e^{B}_{
}= {e_{1}^{B}, e_{2}^{B}} for agent B. Consequently,
the total size of the economy is {e_{1}, e_{2}} = {e_{1}^{A}
+ e_{1}^{B}, e_{2}^{A} + e_{2}^{B}}. This
will set the dimensions of the Edgeworth box.

In Figure 1, O_{A} represents the origin of agent A's problem and O_{B}
that of agent B. Initial endowment is noted as E in Figure 1. The two 45ｰ 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 e_{1} = e_{2} and the box is square and thus the
45ｰ lines coincide. If, in contrast, there is aggregate
uncertainty, then e_{1} ｹ e_{2} 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ｰ 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 U^{A} and U^{B},
representing preferences over state-contingent commodity bundles x = (x_{1}, x_{2}).
If we *assume* the existence of a state-independent elementary utility functions,
then we can decompose them into their expected utility form, specifically:

U

^{A}(x) = ・/font>_{sﾎ S}p_{s}^{A}u^{A}(x_{s}) = p_{1}^{A}u^{A}(x_{1}) + p_{2}^{A}u^{A}(x_{2})

U

^{B}(x) = ・/font>_{sﾎ S}p_{s}^{B}u^{B}(x_{s}) = p_{1}^{B}u^{B}(x_{1}) + p_{2}^{B}u^{B}(x_{2})

where u^{A} and u^{B} are elementary utility functions and p _{s}^{A} is agent A's subjective probability of
state s and p _{s}^{B} 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:

dx

_{2}/dx_{1}|_{UA}= -p_{1}^{A}u^{A｢ }(x_{1})/p_{2}^{A}u^{A｢ }(x_{2})

dx

_{2}/dx_{1}|_{UB}= -p_{1}^{B}u^{B｢ }(x_{1})/p_{2}^{B}u^{B｢ }(x_{2})

However, along the 45ｰ certainty line stemming from origin
O_{A}, where u^{A}(x_{1}) = u^{A}(x_{2}), then
agent A's indifference curves will have slope -p _{1}^{A}/p _{2}^{A} while along the 45ｰ
line stemming from origin O_{B}, where u^{B}(x_{1}) = u^{B}(x_{2}),
agent B's indifference curves will have slope -p _{1}^{B}/p _{2}^{B}. Once again, there is no requirement that
they be the same.

Figure 1- Optimal Risk-Bearing Allocation with Aggregate Risk

From the endowment point E, where we have indifference curve U^{A} and U^{B},
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 U^{A｢ }and agent B increases utility to U^{B｢ }. Notice that the indifference curves are tangent to each
other at point F. This implies that at F:

p

_{1}^{A}u^{A｢ }(x_{1})/p_{2}^{A}u^{A｢ }(x_{2}) = p_{1}^{B}u^{B｢ }(x_{1})/p_{2}^{B}u^{B｢ }(x_{2})

Consequently, we can trace out a "contract curve" from O_{A} to O_{B}
representing the series of allocations in the Edgeworth box where the indifference curves
of O_{A} and O_{B} are tangent - thus each point on the contract curve is
Pareto-optimal or an "optimal risk-bearing allocation" (or ORBA). As the 45ｰ lines do *not* coincide (i.e. there is "aggregate
uncertainty"), then notice that the contract curve will lie *between* the 45ｰ 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 (x_{1}^{A}, x_{2}^{A}, x_{1}^{B},
x_{2}^{B}) and a vector state-contingent prices (p_{1}, p_{2})
where markets clear and all agents are at their individual optimum, in this case where:

(1) (x

_{1}^{A}, x_{2}^{A}) ﾎ argmax p_{1}^{A}u^{A}(x_{1}) + p_{2}^{A}_{ }u^{A}(x_{2}) s.t. p_{1}x_{1}^{A}+ p_{2}x_{2}^{A}｣ p_{1}e_{1}^{A}+ p_{2}e_{2}^{A}(A's optimum)

(x

_{1}^{B}, x_{2}^{B}) ﾎ argmax p_{1}^{B}u^{B}(x_{1}) + p_{2}^{B}u^{B}(x_{2}) s.t. p_{1}x_{1}^{B}+ p_{2}x_{2}^{B}｣ p_{1}e_{1}^{B}+ p_{2}e_{2}^{B}(B's optimum)

(2) x

_{1}^{A}+ x_{1}^{B}= e_{2}^{A}+ e_{1}^{B}(market for good 1 clears)

x

_{2}^{A}+ x_{2}^{B}= e_{2}^{A}+ e_{2}^{B}(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/dx

_{1}= p_{1}^{A}u^{A｢ }(x_{1}) - l^{A}p_{1}= 0

dL/dx

_{2}= p_{2}^{A}u^{A｢ }(x_{2}) - l^{A}p_{2}= 0

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

dL/dx

_{1}= p_{1}^{B}u^{B｢ }(x_{1}) - l^{B}p_{1}= 0

dL/dx

_{2}= p_{2}^{B}u^{B｢ }(x_{2}) - l^{B}p_{2}= 0

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

p

_{1}^{A}u^{A｢ }(x_{1})/ p_{2}^{A}u^{A｢ }(x_{2}) = p_{1}/p_{2}= p_{1}^{B}u^{B｢ }(x_{1})/ p_{2}^{B}u^{B｢ }(x_{2})

thus the indifference curves are tangent to each other and to the equilibrium price
line (shown in Figure F with slope -p_{1}/p_{2}). 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 _{1}^{A}/p _{2}^{A} = p _{1}^{B}/p _{2}^{B}, then the tangency condition on the
contract curve reduces simply to u^{A｢ }(x_{1})/u^{A｢ }(x_{2}) = u^{B｢ }(x_{1})/u^{B｢ }(x_{2}), or cross-multiplying:

u

^{A｢ }(x_{1})/u^{B｢ }(x_{1}) = u^{A｢ }(x_{2})/u^{B｢ }(x_{2})

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 e^{A} = (1, 0) and e^{B} = (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 {e_{1}, e_{2}} = {e_{1}^{A}
+ e_{1}^{B}, e_{2}^{A} + e_{2}^{B}} = (1,
1) so aggregate e_{1} = e_{2}. Thus, the 45ｰ
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ｰ line connecting O_{A} and O_{B}. This implies that
at any optimal risk-bearing allocation, both agents will "insure" completely,
i.e. both lie on the 45ｰ 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 U^{A}(F) and B achieves utility U^{B}(F).

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

However, if subjective probability assignments are different, say p _{1}^{A}/p _{2}^{A}
> p _{1}^{B}/p _{2}^{B},
then the tangency conditions for ORBA remain [p _{1}^{A}/p _{2}^{A}][u^{A｢ }(x_{1})/u^{A｢ }(x_{2})] = [p _{1}^{B}/p _{2}^{B}][u^{B｢ }(x_{1})/u^{B｢ }(x_{2})] - which implies that u^{A｢ }(x_{1})/u^{A｢ }(x_{2})
< [u^{B｢ }(x_{1})/u^{B｢ }(x_{2})], or:

u

^{A｢ }(x_{1})/u^{B｢ }(x_{1}) < u^{A｢ }(x_{2})/u^{B｢ }(x_{2})

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:

u

^{A}(x_{1})/u^{B}(x_{1}) > u^{A}(x_{2})/u^{B}(x_{2})

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ｰ
certainty line and allow for different allocations in different states, e.g. x^{A}
= (3/4, 1/4) and x^{B} = (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, U^{A}(G), is much steeper at the 45ｰ line than the indifference curve of B, U^{B}(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ｰ
line in this case but will be below it, shown in Figure 2 as the curve connecting O_{A}
and O_{B} lying below the 45ｰ 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ｰ 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

_{1}^{A}/p_{2}^{A}][u^{A｢ }(x_{1})/u^{A｢ }(x_{2})] = p_{1}/p_{2}= [p_{1}^{B}/p_{2}^{B}][u^{B｢ }(x_{1})/u^{B｢ }(x_{2})]

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

[p

_{1}^{A}/p_{2}^{A}][u^{A｢ }(x_{1})/u^{A｢ }(x_{2})] < p_{1}^{A}/p_{2}^{A}

and:

[p

_{1}^{B}/p_{2}^{B}][u^{B｢ }(x_{1})/u^{B｢ }(x_{2})] < p_{1}^{B}/p_{2}^{B}

which implies that p _{1}^{A}/p _{2}^{A}, p _{1}^{B}/p _{2}^{B} > p_{1}/p_{2}.

Suppose for the moment that both agents share the same probability assignments. Then,
in equilibrium, p _{1}/p _{2}
> p_{1}/p_{2}. Now suppose further that p _{1}
= p _{2} = 1/2 (i.e. equal probability), then this
condition implies that p_{2} > p_{1}. 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 e_{1} > e_{2} (so aggregate endowment of good in state 1 is
greater than aggregate endowment of good in state 2), then p_{2} > p_{1}
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. e_{1} = 20, e_{2} = 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 p_{2} > p_{1},
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 U^{A}(H) and
agent B has (convex) indifference curve U^{B}(H). As H is on B's certainty line,
then B is being insured completely while A is taking a massive amount of risk.

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 U^{A}
and agent B has utility U^{B}. 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 U^{B} to V^{B}. Note that now with U^{A} and V^{B},
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 V^{B} as he had at F but now agent A's utility
has increased to U^{A｢ }. 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.

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, X^{h},
is some subset of R^{nS}. 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, ｳ _{h}
are defined over X^{h} ﾍ R^{nS}, 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 U^{h}: R^{nS} ｮ R. Suppose an
agent h ﾎ H assigns the (objective or subjective) probability p _{s} to a particular state s ﾎ
S occurring but that agents possess state-dependent elementary utility functions over
physically-differentiated goods, u_{s}^{h}: R^{n} ｮ R. Then, a state-contingent commodity vector x^{h} is
preferred to another y^{h} if expected utility is greater, i.e.

x

^{h}ｳ_{h}y^{h}if and only if ・/font>_{sﾎ S}p_{s}u_{s}^{h}(x_{s}^{h}) ｳ ・/font>_{sﾎ S}p_{s}u_{s}^{h}(y_{s}^{h})

Each household h ﾎ H has a set of state-dependent
endowments, e^{h} = [e_{1}^{h}, ...., e_{S}^{h}] ﾎ R^{nS}, where e_{s}^{h} representing the
vector of endowed goods at each state s ﾎ S. An economy-wide
allocation x is defined as a vector [x^{1}, ..., x^{H}] where x^{h}
ﾎ X is the allocation to the hth household and is defined as x^{h}
= [x_{1}^{h}, ..., x_{S}^{h}] ﾎ
R^{nS} where each x_{s}^{h} is a vector of commodities the hth
household receives in state s ﾎ S. Thus, x_{s}^{h}
is an entitlement receive a particular bundle of goods if a particular state s ﾎ S occurs. An allocation can also be viewed as a vector of n random
variables, x = [x_{1}^{h}, .., x_{n}^{h}] ﾎ R^{nS} with the ith random variable being a mapping x_{i}^{h}:
S ｮ R, or x_{i}^{h} = [x_{i1}^{h},
..., x_{iS}^{h}] ﾎ R^{s}. Thus, the
state-contingent vector x^{h} can be viewed as a mapping x^{h}: S ｮ R^{n}, thus a *particular* state s ﾎ S leads to n commodities (a bundle) being realized.

We can restrict both allocations and endowments to the positive orthant of R^{nS }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 y^{f} ﾎ Y^{f} ﾌ R^{nS}. Thus, not only is the particular production plan
undertaken dependent on the state (thus y_{s}^{f} occurs when s ﾎ S occurs), but the very *feasibility* of a particular
production plan is contingent on the state (as Y_{s}^{f} is the production
set when s ﾎ S).

Let p_{s} = (p_{s1}, ..., p_{sn}) ﾎ
R^{n}_{+} be the spot prices for goods in a particular state s ﾎ S. Thus, we can define p = (p_{1}, ..., p_{S}) ﾎ R^{nS}_{+} 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ﾎ H q }^{hf} = 1 for all f ﾎ F)
to be constant and state-independent.

The components can be thought as follows: the economy-wide allocation is x = {x^{h}}_{hﾎ H} where x^{h} = [x_{1}^{h}, ..., x_{S}^{h}]
is the state-contingent commodity vector allocated to agent h ﾎ
H and x_{s}^{h} = [x_{1s}^{h}, x_{2s}^{h},
...x_{ns}^{h}] is a particular bundle of goods in state s ﾎ S. Thus, x_{s}^{h} ﾎ X_{s}^{h}
ﾌ R^{n} and x^{h} ﾎ
X^{h} ﾌ R^{nS} and, finally, x ﾎ X ﾌ R^{nSH} where X = X^{1}
ｴ X^{2} ｴ ... ｴ X^{H}. Similarly, letting there be F firms, an
economy-wide production plan is y = {y^{f}}_{fﾎ F}
where y^{f} = [y_{1}^{f}, ..., y_{S}^{f}] is a
vector of state-contingent production plans and y_{s}^{f} = [y_{1s}^{f},
y_{2s}^{f}, ..., y_{ns}^{f}] is a particular production
plan in state s ﾎ S. Thus, y_{s}^{f} ﾎ Y_{s}^{f} ﾌ R^{n}
and y^{f} ﾎ Y^{f} ﾌ
R^{nS} and, finally, y ﾎ Y ﾌ
R^{nSF} where Y = Y^{1} ｴ Y^{2} ｴ ... ｴ Y^{F} is the aggregate
production set. Consequently an economy-wide set of allocations is defined as a pair (x,
y) ﾎ X ｴ Y ﾌ
R^{nS(H+F)}. A particular set of prices, p = [p_{1}, ..., p_{S}] ﾎ R^{nS} is a vector of state-contingent prices with
subvector p_{s} = [p_{s1}, p_{s2}, ..., p_{sn}] ﾎ R^{n} being a particular set of prices in state s ﾎ S. We now define an equilibrium:

Arrow-Debreu Equilibrium: an Arrow-Debreu equilibrium is a set of allocations (x*, y*) ﾎ X ｴ Y ﾌ R^{nS(H+F)}and a set of prices p* ﾎ R^{nS}such that:(i) for every f ﾎ F, y

^{f}* satisfies p*y^{f}* ｳ p*y^{f}for all y^{f}ﾎ Y^{f}(ii) for every h ﾎ H, x

^{h}* is maximal for ｳ_{h}in the budget setB

^{h}= {x^{h}ﾎ X^{h}ｽ p*x^{h}｣ p*e^{h}+ ・/font>_{hq }^{hf}p*y^{f}*}(iii) ・/font>

_{hﾎ H}x^{h}* = ・/font>_{fﾎ F }y^{f}* + ・/font>_{hﾎ H}e^{h}

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. e_{1} ｹ e_{2} or (e_{1}^{A}
+ e_{2}^{B}) ｹ (e_{1}^{B} + e_{2}^{B}).
If e_{1} = e_{2}, 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. (e_{1}^{A}, e_{1}^{B})
ｹ (e_{2}^{A}, e_{2}^{B}) and
(x_{1}^{A}, x_{1}^{B}) ｹ (x_{2}^{A},
x_{2}^{B}).

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}/2H^{2}] + ....

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}/2H^{2}

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}/2H^{2}ｻ u(x) - u｢ (x)p^{h}

or simply, solving for p ^{h}:

p

^{h}_{ }ｻ [-u｢ ｢ (x)/u｢ (x)]s^{2}/2H^{2}

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

p = ・/font>

_{hﾎ H}p^{h}= ・/font>_{hﾎ H}[-u｢ ｢ (x)/u｢ (x)]s^{2}/2H^{2}

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 ｮ ･ , then p ｮ 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 ﾎ H where k ｹ
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 u_{s}^{h}
which is state-dependent. Consequently, he has expected utility as E(u^{h}) = ・/font> _{sﾎ S} p
_{s}u^{h}(x_{s}^{h}) where p _{s},
for the sake of simplicity, is the objective probability that state s ﾎ
S emerges and thus that agent h's preferences will be u_{s}^{h}. We assume
that all other agents k ﾎ H where kｹ
h, face no such randomness and thus have state-independent utility functions. Thus, the
expected utility of the kth agent is:

E(u

^{k}) = ・/font>_{sﾎ S}p_{s}u^{k}(x_{s}^{k})

where, notice, the randomness will emerge because of the impact of agent h's
state-dependent trades. Thus, if e_{i} is total aggregate endowment of good i,
then the e_{i} - x^{h}_{si} 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 ﾎ H, where k ｹ 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｢ ﾎ S:

p

_{s}u^{k｢ }(x_{is})/p_{is}= p_{s｢ }u^{k｢ }(x_{is｢ })/p_{is｢ }

or, rearranging:

u

^{k｢ }(x_{is})/u^{k｢ }(x_{is｢ }) = (p_{s｢ }/p_{s})ｷ(p_{is}/ p_{is｢ })

where the term on the left is agent k's marginal rate of substitution between the same
good in two states, p_{is}/p_{is｢ }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. e_{i}
- x^{h}_{is} 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 ｹ 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.
u^{k｢ }(x_{is})/u^{k｢
}(x_{is｢ }) ｻ 1. As a
result, by the fundamental theorem of risk-bearing:

p

_{is}/p_{s}ｻ p_{is｢ }/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
p_{i} = ・/font> _{sﾎ S}p_{is}.
Pre-multiplying this sure price by the probability of a particular state s, p _{s}, yields:

p

_{s}p_{i}= p_{s・/font> sﾎ S}p_{is}= p_{s}p_{i1}+ p_{s}p_{i2}+ p_{s}p_{i3}+ ..... + p_{s}p_{iS}

But as our new fundamental theorem implies that for any s｢
, that p _{s}p_{is｢ }ｻ p _{s｢ }p_{is},
then this becomes by substitution:

p

_{s}p_{i}ｻ p_{1}p_{is}+ p_{2}p_{is}+ p_{3}p_{is}+ ..... + p_{S}p_{is}= ・/font>_{s｢ ﾎ S}p_{s｢ }p_{is}

or, as ・/font> _{s｢ ﾎ S} p _{s｢
}= 1, then:

p

_{s}p_{i}ｻ p_{is}

so the state-contingent price for good i in state s, p_{is}, 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, p_{i}, 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, p_{i}, 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 p_{is} = p _{s}p_{i}, where p_{i} is the
"sure" price and p _{s} the probability of
social state s ﾎ S.

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.