Psychological Aspects of Market Crashes

constants δ and γ as described in Proposition 3 satisfying RS_t < ε for every
j as above.

We have thus shown that, for the constants δ and γ found above, an
ε-crash occurs in history s_t. The proof is now complete.

B An inequality for numerical simulations

We now derive an upper-bound on equilibrium returns that depends on the
parameters described in Proposition 3, under the assumptions of Section 4.
This uniform upper-bound readily allows for the numerical simulations given
in this last section.

Fix any history s_t- 1, and let s_t → s_t- 1 and s_t → s_t- 1 be defined as in
Section 4. Consider any security j such that Equation (10) holds for those
histories. Given the shape of our utility function, and since the consumption
of the representative agent must be the aggregate endowment in every history,
Equation (10) rewrites as

Pst μɪ) ^α RSt+(1 - Pst) μɪ) ^α Rst=ι μɪ) ^α, <ιι>
^t wsSt        ^{t t}    wst       ^st    β ws_t-1

for every security j as described above. Rearranging terms gives

Rj∙=⅛ ( w-∙∙)^α ∙ β μ ■ )^α -⁽¹ - Pst) μ ɪ )^α RSt ] ■      <^ι2)

sst          β wst-1                     wst

Moreover, since in equilibrium it must be true that R_s^j _t > 0, we obtain the
following inequality

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