Psychological Aspects of Market Crashes

for every history st .

A portfolio θⁱ for every agent i is a vector (θs_t)_st∈St_,t∈N+ of shares held
of the J trees in every history s_t, where θ_st = (θ^j_st )j is the vector of holdings
in history st and θ^j_st is the holding of j in this same history st . Every agent
i has an initial portfolio θⁱ₀ at date 0.

Every agent i does not have any information about P , the true probability
measure from nature draws the states; however agent i has a subjective
belief about nature represented by a probability measure Pⁱ on (S^∞, Γ). We
assume that dP_tⁱ (s) > 0 for every infinite history s and every period t, to
avoid problems of existence as pointed in Araujo and Sandroni (1999).

Every agent derives some utility in any history from consuming the only
consumption good present in the economy. We assume that agent i ranks
all the possible future consumption sequences c = (c_st )_st∈St_,t∈N+ according to
the utility function

Uⁱ(c) = E^{Pi    X} (βi)^tui(ct)   ,                        (1)

t∈N+

where β_i ∈ (0, 1) is the intertemporal discount factor, ui is a strictly increas-
ing, strictly concave and continuously differentiable function. We assume
that uⁱ satisfies the Inada condition, namely (ui)⁰(c) 7→ ∞ as c 7→ 0.

The budget constraint faced in every history st by agent i is

cst+^Xqs^jθ^js ≤ wsⁱ +^Xd^js θ^js +^Xqs^jθ^js            (2)

t               st st              st               st st_-1               st st_-1

j   jj

cst ≥ 0,                                                   (3)

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