occur so that anticipations can have a significant effect on prices formation.
This view is somewhat consistent with that in Lee (1998) where crashes ar
driven by successive releases of public information on the actual state of the
economy.
Finally, the fact that with incomplete markets agents must not be wealth
constrained for arbitrary level of crashes to occur suggests a natural policy
recommendation. Indeed, limiting agents’ borrowing abilities when bad times
are largely anticipated appears as a natural way to reduce the magnitude of
market crashes.
A Proof of Proposition 3
We next prove our result. The strategy of our proof goes as follows. We first
find an equilibrium relationship between beliefs, aggregate endowments and
equilibrium returns. We then derive our result by simply using the Inada
conditions to generate an arbitrarily high marginal utility as endowments
drop, forcing equilibrium returns to drop as well.
Consider the program of any agent i, consisting of maximizing (1) subject
to (c,θ) ∈ Bi (g) and taking as given any arbitrage-free and strictly positive
asset prices. Since we assume that Constraint (4) does not bind, and since
we know by the Inada conditions that Constraint (3) does not bind as well,
the Lagrangian to this program rewrites as
L = Σ,dP'it β iui ( cst )+∑ μs.
st st
wsi + djs θjs - cst + gsj (θjs - θjs )
st st-1 st t st st st-1
jj
19
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