Psychological Aspects of Market Crashes

where for every history st the real number μ > 0 is the Lagrange multiplier
associated with the Constraint (2). Taking the first-order conditions with
respect to every variable yields the following relationships for every history
st-1 and asset j

^dPit-1 ■ ^βt 1 ∙ ui(cit-1 )  =  μst-1 ^and                      (7)

∑ μit ∙ [^djit + q^jt]  =  μit-1 ∙^qjt- 1 >                     ⁽⁸⁾

st,→st-1

Rearranging terms gives

^X  dPsⁱ_t ∙β^ti∙u⁰i(cst)∙ [d^js_t + qs^j_t] = dPsⁱ_t-1 ∙ βi^t-1 ∙ u⁰i(cst-1) ∙ qs^j_t-1,     (9)

st,→st-1

and by (5) and some simplifications we obtain the desired relationship

^X Psⁱ_t ∙ βi ∙ u⁰i(cst) ∙ Rs^j_t = u⁰i(cst-1).                    (10)

st,→st-1

With the above relationship, we can prove our result. Fix ε > 0 and a
history s_t. It is easy to see that, for every δ > 0 such that Pi > δ for every
i, there exists an agent, denoted by δ(i), such that for the history st_-1 such
that st → st_-1 we have that c^δ(-)₁ ≥ w∣ ¹ in equilibrium.

Since ui satisfies the Inada conditions for every i, this last remark implies
that the expression u⁰_δ(_i)(c^δ(-_-⁾1 ) is bounded away from + ∞ for every δ > 0.

Also, since ci(ⁱ) ≤ w_st and by the Inada conditions, a low enough value of
aggregate endowment w_st in history st will increase the left-hand side of (10)
above to an arbitrary high level. Thus, as δ converges to 1 and w_sst converges
to 0, for (10) to hold for agent δ(i) it must be true that R_s^j_st converges to
0 for every j such that d^j_sst > 0. Thus, it is straightforward to find the two

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