across market participants about the occurrence of a change in fundamentals,
regardless of whether such anticipations are justified.
In more details, we analyze a standard General Equilibrium economy un-
der uncertainty, where infinitely-lived agents with heterogenous beliefs trade
infinitively-live assets, or Lucas’ trees as in Lucas (1978), in sequential mar-
kets that need not be complete. To rule out the possibility of rolling over
individual debts, we assume that agents cannot borrow more than the net
present value of their future endowments (wealth constraint).
We define an ε-crash (for ε > 0) as an event where the return of every
traded asset paying off positive dividends in this event is below ε. Provided
that agents are not wealth-constrained in equilibrium (this occurs with com-
plete markets for instance), we find that if there is a high enough lower bound
on the probability that every agent assigns to a low enough upper bound on
a next-period drop in aggregate endowment, then a market crash occurs with
positive probability next period. The magnitude of the crash (the “ε”) de-
pends directly on the bounds found above. It is easy to derive from the proof
of this result that if agents are constrained in borrowing in equilibrium, then
low crashes cannot occur because equilibrium assets’ demands are bounded
above, and thus asset prices cannot reach high levels.
We also carry out numerical simulations in the well-known framework of
Mehra and Prescott (1985) to make explicit the direct relationship between
the above bounds sustaining an arbitrary magnitude of crash. In this setting,
we show that for a given level of drop, the higher the level of anticipation the
higher the magnitude of the crash. Highest crash magnitudes are associated