For Radner equilibrium, we need to assume *complete*
asset markets. This does not require that we have a full set of
"state-contingent" markets as in Arrow-Debreu, but
rather only that the set of assets can "span" the entire
state returns space. However, assuming this does not end all difficulties. Recall that
r_{fs} = p_{s }r_{fs} so that the
return of asset f in state s is equal to the payoff of asset f in state s evaluated at the
spot prices in state s. The fact that returns in different states depend on different
state prices can lead to some rather unpleasant consequences, as pointed out by Oliver D. Hart (1975).

To see this, suppose, for the sake of argument, that we have only two
states S = (1, 2), two goods and two assets, F = (f, g) where the assets pay in *bundles*
of commodities. Suppose they have the following payoff structures:

r

_{f}= [r_{f1}, r_{f2}] = [(1, 0), (0.5, 3)]

r

_{g}= [r_{g1}, r_{g2}] = [(1, 2), (2, 0)]

thus a unit of asset f pays bundle r_{f1}
= (1, 0) in state 1 and bundle r_{f2} = (0.5, 3) in
state 2 whereas a unit of asset g pays bundle r_{g1} =
(1, 2) in state 1 and bundle r_{g2} = (2, 0) in state
2. As a result, at any set of spot prices in state 1, p_{1} = (p_{11}, p_{21})
and spot prices in state 2, p_{2} = (p_{12}, p_{22}) the set of
returns to assets f and g are:

r

_{f}= (p_{1}r_{f1}, p_{2}r_{f2}) = [p_{11}, 0.5p_{12}+ 3p_{22}]

r

_{g}= (p_{1}r_{g1}, p_{2}r_{g2}) = [p_{11}+ 2p_{21}, 2p_{12}].

which, as depicted in Figure 3 are two linearly independent vectors in
state net transfer space. Now, we can see that any net income transfers, such as t * = (t _{1}*, t _{2}*), can be made across both states by constructing
portfolios of assets which are linear combinations of assets f and g. As we saw earlier,
we can purchase a_{f}^{h} units of asset f and a_{g}^{h}
units of asset g so that we obtain the sum vector a_{f}^{h}r_{f} +
a^{h}_{g}r_{g} which yields a particular set of returns in both
states, etc. Thus, by suitable constructions of portfolios of assets f and g, an agent can
undertake any set of transfers of purchasing power between states 1 and 2.

Figure 3- A Hart Problem

However, as Oliver D. Hart (1975)
indicated through a few well-chosen examples, things can go quite awry. The return
vectors, after all, are functions of the spot prices, p_{1} = [p_{11}, p_{21}]
in state 1 and p_{2} = [p_{12, }p_{22}] in state 2. Consequently,
there may be vectors of prices that make the vectors of returns in both states collinear.
Specifically, consider the case where spot prices in state 1 are p_{11}/p_{21}
= 2 and spot prices in state 2 are p_{12}/p_{22} = 6. In this case, the
return vectors become:

r

_{f｢ }= p_{s}r_{f}= [p_{11}, p_{12}]

r

_{g｢ }= p_{s}r_{g}= [2p_{11}, 2p_{12}]

thus, the return vectors are linearly related, as shown in Figure 3.
Consequently, when return vectors become collinear, then they can no longer span the
entire space. Specifically, when state prices are such, we can make construct all sorts of
portfolios with assets f and g, but the possible returns are now restricted to the
hyperplane H that passes through both r_{f｢ }and r_{g｢ }. Any other set of returns in the state space (such as t *) are not obtainable by any portfolio. The dimensionality of the
space of possible returns is severely reduced.

What Hart (1975) went on to demonstrate, such a situation could lead to non-existence of Radner equilibrium. Heuristically, the possibility of such sudden reductions in dimensionality may cause discontinuities in our demand functions - and this resulting "hole" in the demand function may be precisely where the supply function passes through, consequently, it is quite possible that a Radner equilibrium does not exist.

The discontinuity arises because of the structure of our budget
constraint. Recall that we had "consolidated" our budget constraint into p_{0}z_{0}^{h}
+ qV^{-1}p_{-0}z_{-0}^{h} ｣ 0,
which implies that excess demand functions for goods in any state, z_{is}^{h},
are a function of *all* state spot prices. Thus, by Berge's Theorem, for the continuity of excess demand
functions with respect to state spot prices, we require that this "grand" budget
constraint be upper semicontinuous. Let us denote the
grand budget constraint by B(p, e) where p represents the set of spot prices in all states
and e is the particular agents endowment. If B(p, e) is upper semicontinuous, then if we
take a sequence of convergent state-spot prices{p_{n}}, then we can construct a
sequence of convergent excess demand vectors {z_{n}}contained within the sequence
of budget constraints where the limiting demand vector is in the limiting budget
constraint. In other words, p_{n} ｮ p and x_{n}
ｮ x where x_{n} ﾎ B(p_{n},
e) implies that x ﾎ B(p, e).

But this upper semicontinuity, under the Hart problem, may not be true.
Because collinear returns reduce the dimensionality of income transfers across states,
then the budget constraint defined by the prices which yield those collinear returns will
be drastically different from the budget constraint defined at prices which yield
linearly-independent returns. Now, as we can envision from Figure 3, we can take a
sequence of prices {p_{n}} which yield linearly independent returns r_{f}
and r_{g} and approach a limiting set of prices p which yield the collinear
returns, r_{f｢ }and r_{g｢
}. We denote the budget constraint formed by the former B(p_{n}, e) and the
budget constraint formed by the latter B(p, e). Taking a sequence of excess demand vectors
{z_{n}} in B(p_{n}, e), such that z_{n} ｮ
z, we can imagine how the drastic reduction in the possibility of state returns at the
limiting prices p can easily imply that z ﾏ B(p, e).

For instance, suppose that at *every* set of prices in {p_{n}},
we demand the same constant amount so z_{n} = z for all n and thus the desired
transfer of income across states is t * is the same for every
set of prices, p_{n}. Now, in Figure 3, t * is *off *the
hyperplane defined by the collinear returns r_{f｢ }, r_{g｢ }as t * can be constructed by the
linearly-independent returns implied at every p_{n} in the sequence. But if this
is true, then this transfer t * is *no* longer available
in the limit when prices are p and returns are collinear and thus the budget set is B(p,
e). Thus, as z_{n} implies t * and z implies t *, then x ﾏ B(p, e), i.e. p_{n}
ｮ p, z_{n} ｮ z where z_{n}
ﾎ B(p_{n}, e) does *not* imply z ﾎ (B, p). The budget constraint is not upper semicontinuous and thus
the demand functions are not necessarily continuous - and we have a serious problem that
can undermine the existence of a Radner equilibrium.

There are (roughly) three answers to the Hart problem. The first is that,
in general, Hart counterexamples are rare. As was shown for the complete markets case by
Repullo (1986) and Magill and Shafer (1990) - and in the incomplete markets case by
Duffie and Shafer
(1985, 1986) and Hirsch, Magill and Mas-Colell (1987) - Radner equilibria exist
generically, with Hart-type counterexamples being rather exceptional. A second answer,
proposed initially by Kenneth J. Arrow (1953) but
more forcefully forwarded by John Geanakoplos and
Herakles Polemarchakis (1986), is to express all
payoffs in terms of a *numeraire* good ("gold"). A third answer, pursued by
David Cass (1984), Jan Werner (1985) and Darrell Duffie (1987) was to propose that we construct a
nominal unit of account ("cash") which is independent of prices and have asset
payoffs expressed in these.

The first genericity argument is quite involved and we refer elsewhere (e.g. Magill and Shafer , 1991; Magill and Quinzii, 1996) for more extensive reviews. That leaves the numeraire and unit of account options. Both of these, effectively, eliminate real assets - i.e. assets that pay bundles of commodities - in favor of assets that only pay one type of "thing".

Of the two types of reductions, the Arrow-Geanakoplos-Polemarchakis
normalization via a *numeraire* good is the easiest to handle. Let us take good 1
as the numeraire good (call it "gold") and assume that *all* asset payoffs
are in units of "gold". This simplifies our problem tremendously as the return
to asset f in state s is merely r_{fs} = p_{1s}r_{fs}
where r_{fs} is the *number* of units of gold
asset f pays out in state s while p_{1s} is the price of gold in state s. With no
loss of generality, we normalize the price of gold to 1 in each states, i.e. p_{1s}
= 1 for all s ﾎ S. Thus, spot price vector in any state is p_{s}/p_{1s
}= [1, p_{2s}/p_{1s}, p_{3s}/p_{1s}, .., p_{ns}/p_{1s}].
Notice that as the spot price of gold may be *different* in different states, then
the normalization depends on the state of nature that emerges. The main result of the gold
normalization is that returns can be written r_{fs} = r_{fs}
which is merely a number. As a result, the Hart problem will not arise: all the
dimensionality problems are sidestepped: prices between different real goods in a given
state can change all they wish - they will not affect the relative returns of assets in
that state.

In contrast, the Cass-Werner "*unit of account*" or
"*cash*" structure is more complicated. In a cash economy, the resulting
equilibrium is not strictly a "Radner equilibrium"
and the Arrow-Debreu equivalence does not really hold
precisely and, perhaps most significantly, there is a severe indeterminacy problem. Why
this is so can be conceived in the following way: let g _{fs}
be the *cash* return of asset f in state s. Then, the *real* return of asset f
in state s is r_{fs} = g_{s}/P_{s}
where P_{s} represents the "price level" in state s (some index of the
spot prices, p_{s}). To fix ideas fully, we could make it comparable to our
previous case by letting P_{s} = p_{1s} so only good 1 ("gold")
is included in the price-index. Thus, the real return of a cash-paying asset f in state s
is r_{fs} = g _{s}/p_{1s}. Thus note
now that returns are *dependent* on the price-level p_{1s}. This, of course,
should come as no surprise. In a non-cash situation - as in a "gold" or real
asset economy - a rise in prices in any state implies that returns rise; but if we
restrict payoffs to units of account, then the value of the return will fall as prices
rise, i.e. there is an "inflation" element in cash economies.

The problem with cash economies, as first noted by Cass (1985) and proved by Geanakoplos and Mas-Colell
(1989) and Balasko and Cass (1989), is that Radner equilibria will be severely
*indeterminate*. We can follow the "Walras-Cassel"
test of counting equations and unknowns to see intuitively why this arises. Let us have H
agents, n physical goods, S+1 states and F assets. Consequently, let us count the
equations of this model. Then the number of demand functions for goods is H[n(S+1)], the
number of demand functions for assets is HF. We also have n(S+1) goods markets clearing
conditions and F asset market clearing conditions. Thus the total number of equations in
the economy is (H+1)[n(S+1) + F]. However, recall that there are H(S+1) budget constraints
met thus, in aggregate, S+1 Walras' Law constraints exist which permit us to drop S+1
market-clearing conditions (the "nth" in each state s ﾎ
S+1). Thus, we now have (H+1)[n(S+1) + F] - (S+1) equations.

How about unknowns? We need to determine how much of each good in each
state will be allocated to each household, x_{i}^{h}, thus we have
H(n(S+1)) of these; we need to determine the assets allocated to each household, a_{f}^{h},
thus we have HF of these. We have n(S+1) goods prices p_{is} to discover and F
asset prices q_{f} to determine. Thus, the total number of unknowns is
(H+1)[n(S+1)+F]. Now, conventionally, we would just suspect that normalizing prices would
allow us to strike out another S+1 expressions: as we only need relative prices in each
state, then normalizing one of the prices p_{is} in each state s ﾎ S and one of the q_{f} in state s = 0. In this case, it is
easy to notice that the number of unknowns would be (H+1)[n(S+1) + F] - (S+1), which would
be equal to the number of equations.

However, while this last normalization step is indeed true for the
real-asset or numeraire asset case, it is *not* true when we have a cash economy.
Price normalizations *do* affect real demands. The intuition is effectively the
following: suppose spot prices doubled in a particular state s but not in any other state.
Obviously, then, *relative* spot prices *within* state s have not changed; *but*
the purchasing power of the payoff to an asset which delivers cash in state s has
declined. Consequently, one would expect the agent to make changes in his net transfers
across states to compensate for the rise in spot prices in state s. Thus, a rise in p_{s}
is *not* "neutral" if payoffs are in cash: it forces agents to shift their
transfers between states. The "grand" budget constraint, then, is affected by
changes in the price level in state s.

Of course, if *all* prices in *all* states double (and the
nominal price of asset halves in compensation), then there will be no shifting about - in
this case, changes in the price level is neutral. Similarly, if price levels double in the
*initial* states, s = 0, there will also be no change. Thus, in these two exceptional
cases, the budget constraint will not be affected - thus we are *at least* permitted
to remove two of the unknowns in our system by appropriate normalization - e.g. set ・/font> _{sﾎ S} p_{1s} = 1 and
set p_{01} = 1 to deal with the first and second cases respectively. However, that
leaves us with (H+1)[n(S+1) + F] - 2 unknowns whereas we still have (H+1)[n(S+1) + F] -
(S+1) equations - thus we have S-1 *more* unknowns than we have equations. Thus, as
we can anticipate, there will be *indeterminacy* of Radner equilibrium in a cash
economy.

Some economists (e.g. Magill and Quinzii, 1992, 1996) have jumped on this indeterminacy as an avenue by which one may "reintegrate" monetary theory into general equilibrium theory - a task long left dangling since the early 1970s. While an interesting area of research, this takes us a bit far off our trajectory here and shall refer to the standard references for further details.

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