In this section we shall link up the Radner equilibrium with the existence of "no arbitrage" asset prices. This, of course, is analogous to what we derived earlier in our heuristic two-period, no uncertainty case. Obviously, with multiple states, assets, etc. we have to redefine our concepts somewhat. Arbitrage opportunities can be defined (adopting the terminology of Ingersoll, 1987: 52-3) as follows:

Arbitrage Opportunity I: there is an "arbitrage opportunity of the first kind" if there is a portfolio a^{h ﾎ }R^{F}such that qa^{h}= ・/font>_{f }q_{f}a_{f}^{h}｣ 0 and Va^{h}> 0.

Arbitrage Opportunity II: there is an "arbitrage opportunity of the second kind" if there is an a^{h}such that qa^{h}= ・/font>_{f }q_{f}a_{f}^{h}< 0 and Va^{h}ｳ 0.

(as our typesetting does not allow us to be more precise about notation,
we should note here that Va^{h} > 0 does *not* mean that all returns are
positive in all states, i.e. it allows for some zero values, while Va^{h} ｳ 0 allows for the possibility that *all* returns are zero).

These definitions are no more complicated than our previous ones. If q_{f}a_{f}^{h}
< 0, then an agent is selling asset f ﾎ F short; if q_{f}a_{f}^{h}
｣ 0, then an agent is either selling asset f short or not
buying it. Summing up over assets, then qa^{h} = ・/font> _{fﾎ F}a_{f}^{h} < 0 implies the agent is
selling short on net whereas if qa^{h} = ・/font> _{fﾎ F}a_{f}^{h} ｣ 0
merely implies that he is either selling short on net or not making any net purchases of
assets. In either case, there is "no commitment", or basically, on net, he uses *none*
of his endowment at time t = 0 to purchase assets. Similarly, Va^{h} ｳ 0 implies that the agent is making no net losses in the future,
while Va^{h}_{ }> 0 in our first definition implies that at least one
return in some state must be positive. Following our previous section, we can define the
condition of "no arbitrage" or "arbitrage free" asset prices q as
follows (again using the definition of arbitrage of the first kind):

Arbitrage-Free: A set of asset prices q ﾎ R^{F}is "arbitrage free" if there is no portfolio a^{h}ﾎ R^{F}such that qa^{h}｣ 0 and Va^{h}> 0.

In other words, there is no portfolio with non-positive commitment (qa^{h}
｣ 0) that yields a non-negative return in every state and a
strictly positive return in some states. The intuitive meaning of these conditions have
already been given earlier in our simple diagrams.

The mathematics of our earlier case generalizes easily to the
multi-good/asset/state case. However, for simplicity, we shall *assume* that all
payoffs are merely in terms of one good (i.e. "gold") which has a spot price of
1 in every state. This, as we shall see later, is a somewhat necessary simplifying
assumption. Thus, in terms of analyzing asset market equilibrium, we have F assets, S +1
states but only one good. With this, we now have S + 1 elements in the net trade vector, t = {t _{0}, t
_{1}, ..., t _{S}} where t
_{s} = p_{s}(x_{s}^{h} - e_{s}^{h}). Thus
our net trade space is S+1 dimensional. Thus, we can define W as the hyperplane or linear
subspace:

W = {t ﾎ R

^{S+1 }| t_{0}= - qa^{h}, t_{s}= V_{s}a^{h}for s = 1, .., S with q, a^{h}ﾎ R^{F}}

Where V_{s} = [r_{s1}, r_{s2}, .., r_{sf}].
Consequently, for "no-arbitrage" we need that W ﾇ R_{+}^{S+1}
= {0}, i.e. the subspace W only shares the origin with the positive orthant R_{+}^{S+1}.
Then, the orthogonal subspace is:

W

^{^}= {y ﾎ R^{S+1}ｽ yW = 0}

where yW = 0 implies that yt = 0 for all t ﾎ W. Thus, as before, we can find a m ﾎ W^ such
that m = [1, m _{1}, m _{2}, ..., m _{S}] where
m _{s} is the "present value" of a unit of
income in state s ﾎ S (notice that the first element remains 1
for normalization). Thus, in general, m W = 0 implies that m ｷ[-q, V]｢ = 0 or simply, that there is
an S-dimensional vector m _{-0} = [m
_{1}, .., m _{S}] such that q = m _{-0}V. Thus, for any particular asset f ﾎ F, then q_{f} = m _{-0}V_{f}
= ・/font> _{sﾎ S} m _{s}r_{fs}, i.e. the price of an asset is some
linear combination of its return in each of the states r_{f} = (r_{f1},
...,r_{fS}) weighted by the state prices (m _{1},
.., m _{S}).

Let us now turn to what has been termed t*he fundamental theorem of
asset pricing*, which has its roots in Stanley Fischer
(1972) and Stephen Ross (1976). This comes in two
parts: the first part establishes the equivalence of arbitrage-free asset prices q and the
existence of non-negative state prices m ; the second part
establishes that the existence of utility-maximizing agents with monotonic preferences
implies that asset prices are arbitrage free. Let us to prove them in turn:

Theorem: (First Fundamental Theorem) an asset price vector q ﾎ R^{F}is "arbitrage free" if and only if there is a semi-positive vector of state prices, m = (1, m_{1}, ..., m_{S}) ｳ 0 where m ｷ[-q, V]｢ = 0 (i.e. q = m_{-0}V).

Proof: (i) ﾞ (ii): Assume all assets have
non-negative prices and returns, thus q_{f} ｳ 0 for
all f ﾎ F and V ｳ 0. Then we can
define W = {t ﾎ R^{S+1} ｽ t _{0} = -qa^{h} and t _{s} = V_{s}a^{h} for all s = 1, .., S, a^{h}
ﾎ R^{F}} which is a convex subspace in R^{S+1}.
Then, define the simplex D = {t ﾎ R_{+}^{S+1} | ・/font> _{sﾎ S+1} t _{s} = 1}. This is
a convex set in the non-negative orthant, thus D ﾌ R_{+}^{S+1}. Note also that 0 ﾏ
D by the definition of the simplex, so D
ﾌ R_{+}^{S+1}\{0}. Now, the
"no-arbitrage" condition implies that W ﾇ R_{+}^{S+1}\{0}
= ﾆ , thus W ﾇ D
= ﾆ . Since both W and D are
disjoint, convex sets then we can apply the separating hyperplane theorem, i.e. there is a
vector r ﾎ R^{S+1} where p ｹ 0 and a scalar b
such that:

r t ｣ b for all t ﾎ W

and:

r t ｳ b for all t ﾎ D

As 0 ﾎ W and 0 ﾏ D , then r t ｣ 0 for all t ﾎ
W and r t > 0 for all t ﾎ D . Also,
because t ﾎ W implies -t ﾎ W, then r t = 0 for all t ﾎ
W, thus r ﾎ W^
. All that remains to show is that r ｳ
0. Suppose not. Suppose there is a state s such that r s < 0. But then consider a t ﾎ D where t s = 1 and t _{s}
= 0 for all s ｹ s . But then, r t < 0. A contradiction. Thus, there
is no s where r s
< 0, i.e. r _{s} ｳ 0 for
all s ﾎ S+1, thus r = (r _{0}, r _{1}, ..., r _{S}) ｳ 0. Finally, define m _{s} = r _{s}/r _{0}. As r ｳ
0, then m = (1, r _{1}/r _{0}, ...., r _{S}/r _{0}) = (1, m _{1}, .., m _{S}) ｳ 0. Furthermore, m ﾎ W^ as W^ is a linear subspace and so m = (1/r _{0})r ﾎ
W^ . Thus, there exists a m = (1, m _{1}, .., m _{s}) ｳ 0 such that m t
= 0 for all t ﾎ W. As [-q, V]｢ ﾎ W, then m
ｷ[-q, V]｢ = 0 or, defining m _{-0}
= (m _{1}, .., m _{S}),
then q = m _{-0}V.

(ii) ﾞ (i): Suppose m
[-q, V]｢ = 0 but nonetheless arbitrage opportunities remain,
i.e. there is an a^{h} such that qa^{h} ｣ 0
and Va^{h} > 0. Now, as q = m _{-0}V, then
post-multiplying by the arbitrage portfolio a^{h}, then qa^{h} = m _{-0}Va^{h}. As m _{-0}
ｳ 0, then as qa^{h} ｣ 0 by
hypothesis, then the only possibility that could make this true is that Va^{h} ｣ 0. But then that contradicts the fact that a^{h} is an
arbitrage opportunity.ｧ

In short, the first part of the "Fundamental Theorem" effectively implies a consistent linear pricing rule, so that the price of asset f is some linear combination of returns. The second part of the "Fundamental Theorem" associates the absence of arbitrage with the existence of utility-maximizing agents with monotonic preferences:

Theorem: (Second Fundamental Theorem) Under the assumption that utility functions are continuous and strongly monotonic (i.e. u^{h}: R^{n(S+1)}_{+}ｮ R is continuous and for any x｢ ｳ x, where x｢ ｹ x, then u^{h}(x｢ ) > u^{h}(x)) then the optimization problem has a solution if and only if there are no arbitrage opportunities on the financial markets, i.e. the following two are equivalent:(i) for every h ﾎ H, consumption plan x

^{h}* and portfolio a^{h}* solves the following:max u

^{h}[x_{0}^{h}, x_{1}^{h}, ...., x_{S}^{h}]s.t. p

_{0}x_{0}^{h}+ qa^{h}｣ p_{0}e_{0}^{h}p

_{s}x_{s}^{h}｣ V_{s}a^{h}+ p_{s}e_{s}^{h}for all s ﾎ S(ii) There is no portfolio a

^{h}ﾎ R^{F}such that qa^{h}｣ 0 and Va^{h}ｳ 0 (but Va^{h}ｹ 0).

Proof: (i) ﾞ (ii): Let the pair (x^{h}*,
a^{h}*) be a solution to the optimization problem, then p_{0}(x_{0}^{h}*
- e_{0}^{h}) = -qa^{h}* and p_{s}(x_{s}^{h}*
- e_{s}^{h}) = V_{s}a^{h}* for all s ﾎ
S. Now, suppose q allows for arbitrage opportunities. Then there is a feasible arbitrage
portfolio a^{h} ﾎ R^{F} such that qa^{h}
｣ 0 and Va^{h} > 0, thus -qa^{h} + Va^{h}
> 0. Thus, there is an x^{h} = [x_{0}^{h}, x_{1}^{h},
.., x_{S}^{h}] such that p_{0}(x_{0}^{h} - e_{0}^{h})
= -qa^{h} - qa^{h}* and p_{s}(x_{s}^{h} - e_{s}^{h})
= V_{s}a^{h} + V_{s}a^{h}*. As a result,

p

_{0}(x_{0}^{h}- e_{0}^{h}) > p_{0}(x_{0}^{h}* - e_{0}^{h})p

_{s}(x_{s}^{h}- e_{s}^{h}) > p_{s}(x_{s}^{h}* - e_{s}^{h}) for all s ﾎ S.

This implies that x^{h} > x^{h}*, or assuming strong
monotonicity, then u^{h}(x^{h}) > u^{h}(x^{h}*), a
contradiction of the individual optimality of x*.

(ii) ﾞ (i): We start by defining a pair of
sets, B_{A} and B_{R}. For the first:

B

_{A}= {y^{h}ﾎ R_{+}^{n(S+1)}| p_{0}(y_{0}^{h}- e_{0}^{h}) + ・/font>_{sﾎ S}m_{s}p_{s}(y_{s}^{h}- e_{s}^{h}) ｣ 0}

which is a closed and bounded (and hence compact) subset of R_{+}^{n(S+1)}
(notice also that this is the budget constraint of an Arrow-Debreu problem); we also
define

B

_{R}= {y^{h}ﾎ R_{+}^{n(S+1)}| p_{0}(y_{0}^{h}- e_{0}^{h}) ｣ -qa^{h}, and p_{s}(y_{s}^{h}- e_{s}^{h}) ｣ V_{s}a^{h}for all s ﾎ S, a^{h}ﾎ R^{F}}

which is a closed set (notice here that this is the budget set of a Radner
problem). Now, if q is arbitrage-free, then there is no arbitrage portfolio a^{h} ﾎ R^{F} such that qa^{h} ｣
0 and Va^{h} > 0. As we saw earlier by the First Fundamental Theorem, this
implies there is a semi-positive vector m = (1, m _{1}, .., m _{S}) ﾎ R^{S+1}_{+} such that q = m
_{-0}V. We can use these to define our set B_{A}. However, if q = m _{-0}V, then we see immediately that if x^{h} ﾎ B_{R}, then it must be true that there exists an a^{h}
such that p_{0}(x_{0}^{h} - e_{0}^{h}) ｣ -qa^{h} and p_{s}(x_{s}^{h} - e_{s}^{h})
｣ V_{s}a^{h} which, if there is no arbitrage
so q = m _{-0}V, consequently implies that p_{0}(y_{0}^{h}
- e_{0}^{h}) + ・/font> _{sﾎ
S} m _{s}p_{s}(y_{s}^{h} -
e_{s}^{h}) ｣ 0, and so x^{h} ﾎ B_{A}. Thus, B_{R} ﾌ B_{A}.
As B_{A} is compact, then B_{R} is a closed subset of a compact set B_{A}
and thus is itself compact. However, notice that B_{R} merely summarizes the set
of budget constraints for a Radner optimization problem. As B_{R} is a compact set
and u^{h} is a continuous function, then by the Weierstrass theorem, it achieves a
maximum over B_{R}. Thus a solution to the Radner optimization problem is defined.ｧ

Thus, the second half of the Fundamental Theorem of Asset Pricing connects
the concept of an "arbitrage-free" q with the solution to the optimization
problem. The meaning of this theorem can be gauged by examining the multi-state,
multi-asset analogue to our earlier two-dimensional Lagrangian solution. Recall that if q
is arbitrage-free, then there is a semi-positive vector m _{-0}
= (m _{1}, ..., m _{S})
ｳ 0 such that q = m _{-0}V
implying that for any f ﾎ F:

q

_{f}= ・/font>_{sﾎ S}m_{s}r_{fs}

Thus, as we saw earlier, can assign multiplier values m
_{s} to returns in different states so that the price of a unit of asset f ﾎ F is equal to the weighted sum of the value of returns across
states. Thus, we can think of m _{s} as shadow price of
the state-contingent commodity that pays a unit of account if state s occurs and nothing
otherwise. To illustrate this more clearly, let us assume that our agent has a
differentiable von Neumann-Morgenstern utility function ・/font> _{sﾎ S+1p s}^{h}u_{s}^{h}(x_{s}^{h}),
where p _{s}^{h} is the probability
(subjective/objective) of a particular state and u_{s}^{h} is the
elementary utility function that emerges in a particular state. Thus, our optimization
problem for agent h ﾎ H becomes:

max ・/font>

_{sﾎ S+1 p s}^{h }u_{s}^{h}(x_{s}^{h})s.t.

p

_{0}x_{0}^{h}+ qa^{h}｣ p_{0}e_{0}^{h}p

_{s}x_{s}^{h}｣ V_{s}a^{h}+ p_{s}e_{s}^{h}for all s ﾎ S

and consequently the Lagrangian would looks like:

L = ・/font>

_{sﾎ S+1 p s}^{h}u_{s}^{h}(x_{s}^{h}) - l_{0}^{h}[qa^{h}_{ }+ p_{0}(x_{0}^{h}- e_{0}^{h})] + l_{1}^{h}[V_{1}a^{h}- p_{1}(x_{1}^{h}- e_{1}^{h})] + ..... + l_{S}^{h}[V_{S}a^{h}- p_{S}(x_{S}^{h}- e_{S}^{h})]

where we have S+1 Lagrangian multipliers, l _{0}^{h},
..., l _{S}^{h}. Let us substitute our
utility-function u_{s}^{h}(x_{s}^{h}) with an indirect
utility function, y _{s}^{h}(p_{s}, m_{s}^{h})
where income is defined as m_{s}^{h} = p_{s}e_{s}^{h}
+ V_{s}a^{h}* = p_{s}e_{s}^{h} + ・/font> _{fﾎ F} r_{fs}a_{f}^{h}*,
the income from the sale of endowments in state s (p_{s}e_{s}^{h})
and the returns from the chosen portfolio in state s (V_{s}a^{h*}).
Consequently, the Lagrangian can be rewritten as:

L = ・/font>

_{sﾎ S+1 p s}^{h}y_{s }^{h}(p_{s}, m_{s}^{h}) - l_{0}^{h}[qa^{h}]

where all that remains is the choice of the optimal portfolio, a^{h}*.
The first order conditions for a maximum implies that for any asset f:

dL/da

_{f}^{h}= ・/font>_{sﾎ S+1}p_{s}^{h}(ｶ y_{s}^{h}/ｶ m_{s}^{h})(ｶ m_{s}^{h}/ｶ a_{f}^{h}) - l_{0}^{h}_{ }q_{f}= 0

which holds true for all assets f = 1, ..., F. Recognizing that ｶ m_{s}^{h}/ｶ a_{f}^{h}
= r_{fs}, we can rewrite this as:

(1/l

_{0}^{h})・/font>_{sﾎ S+1}p_{s}^{h}(ｶ y_{s}^{h}/ｶ m_{s}^{h})r_{fs}= q_{f}for f = 1, ..., F.

so the expected marginal utilities of F assets are proportional to the
vector of asset prices, q. Letting m _{s}^{h} =
(p _{s}^{h }/l _{0}^{h})(ｶ y _{s}^{h}/ｶ m_{s}^{h}), then this reduces to:

・/font>

_{sﾎ S+1}m_{s}^{h}_{ }r_{fs}= q_{f}for f = 1, ..., F.

We can think of m _{s}^{h} as
the marginal utility of state s for agent h ﾎ H weighted by
the personal proportionality factor (p _{s}^{h}/l _{0}^{h}) which obviously depends on the subjective
probability of state s (p _{s}^{h}) and the
personal marginal utility of wealth at t = 0 (l _{0}^{h}).
Thus, m _{s}^{h} is the ratio of expected
utility of one extra unit of wealth at state t = 1 and state s (i.e. p
_{s}^{h}(ｶ y _{s}^{h}/dm_{s}^{h}))
to the actual utility of an extra unit of wealth at t = 0 (l _{0}^{h}).

It might be useful to notice the analogy between this last rule and the characterizations familiar to us from the Arrow-Debreu model. Notice first that our result implies that for any two assets f, g ﾎ F that:

・/font>

_{sﾎ S+1}m_{s}^{h}_{ }r_{fs}/q_{f}= ・/font>_{sﾎ S+1}m_{s}^{h}_{ }r_{gs}/q_{g}for all f, g ﾎ F

which can be seen as a financial analogue of the individual optimum of the
state-preference approach. It effectively states that at the individual optimum, with
given asset prices, q_{f} and q_{g}, the individual will adjust his asset
holdings until the expected marginal utility per dollar in each asset is equal. There is
also an analogue to the *fundamental theorem of risk-bearing* in Arrow-Debreu, namely
for any two individuals h, k ﾎ H and for any pair of assets f,
g ﾎ F:

[・/font>

_{sﾎ S+1}m_{s}^{h}_{ }r_{fs}]/[・/font>_{sﾎ S+1}m_{s}^{h}_{ }r_{gs}] = q_{f}/q_{g}= [・/font>_{sﾎ S+1}m_{s}^{k}_{ }r_{fs}]/[・/font>_{sﾎ S+1}m_{s}^{k}r_{gs}]

Notice that the term on the left, agent h's ratio of expected marginal utilities of assets f and g, can be conceived of as the marginal rate of substitution between assets f and g. Thus, this states that the agent h's marginal rate of substitution is equal to agent k's marginal rate of substitution - the familiar Pareto-optimality condition, now in a financial context.

Notice that q_{f} and r_{fs} are *given* in this
result and thus are exogenous to the consumer. As a result, it must be true that this
holds true for *an*y household h ﾎ H. This implies that m _{s}^{h} = m _{s}
for *all* households h ﾎ H, so that ・/font>
_{sﾎ S+1} m _{s }r_{sf}
= q_{f} for f = 1, ..., F. Thus, marginal utilities of states will be equal across
agents at each state at the rate m _{s}. However, it
must be stressed here that this relies on the existence of complete asset markets, i.e.
that there are at least S linearly-independent assets.

To see why, notice that the linear pricing rule ・/font>
_{sﾎ S+1} m _{s}^{h}_{
}r_{sf} = q_{f} can be rewritten as m _{-0}^{h}V_{f}
= q_{f} where m _{-0}^{h} = [m _{1}^{h}, m _{2}^{h},
.., m _{S}^{h}] and V_{f} is the fth
column of the V matrix. Consequently, completing the matrix m _{-0}^{h}V
= q which is a simple linear equation. As a result, as V and q are given, we are trying to
solve for m _{-0}^{h}, then there are three
possible results: either there is no solution, there is a unique solution or there is an
infinite number of solutions. The first case of no solution can be ruled out as rank(V, q)
= rank(V) because, by the linear pricing rule we obtained from no-arbitrage, q is a linear
combination of the vectors in V.

Now, for a unique solution, we require that rank(V) = S, so that we need
it that the number of linearly-independent assets equal the number of states. Notice that
this implies that V is a square matrix (or can be made one by dropping the linearly
dependent assets) and is non-singular (|V| ｹ 0), thus
inverting, m _{-0}^{h} = V^{-1}q,
exactly as we predicted earlier. Because the solution is necessarily unique, then any
other solution m _{-0}^{k} = V^{-1}q
will necessarily imply that m _{-0}^{k} = m _{-0}^{h}, thus the marginal values of the states
are equal across all agents k, h ﾎ H.

Can we have an infinite number of solutions? We will if the rank(V) <
S, or the number of linearly independent assets fall below the number of states. We refer
to such a situation as one of *incomplete* asset markets.
Notice that this implies that it is no longer necessarily true that we have m _{-0}^{k} = m _{-0}^{h}
in equilibrium Some commentators have noted that m _{-0}^{h}
ｹ m _{-0}^{k}_{ }can
be regarded as the "basic property of incomplete markets" (Magill and Quinzii,
1996: p.97). We shall return to incomplete markets later, but, for now, we make note of
the fact that this "basic property" immediately implies that we do *not*
have Pareto-optimality as the marginal rates of substitution are no longer equated to each
other across agents, i.e. we will not have an "optimal risk-bearing allocation."

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