Psychological Aspects of Market Crashes

where st ,→ st-1 . The left-hand side of (2) represents the purchase of con-
sumption good at price normalized to 1 plus the purchase of new shares of
trees at current prices, and the right-hand side is the endowment plus the
dividends payments from previous holdings plus the proceeds from selling
the current holdings of trees at current prices.

Given the constraints faced by the traders, we also need to rule out the
possibility of rolling over any debt through excessive future borrowing, also
known as Ponzi’s scheme. Consider any vector of prices that is arbitrage-
free. As argued in Hernandez and Santos (1996), when a vector of prices q
is arbitrage-free there exists a sequence of positive numbers {π_st }_st∈St_,t∈N+
with πs₀ = 1 such that

πs_tq_s^j_t =      πsd^j_s ,

s,→st

for every j (j = 1, ..., J) and st (st ∈ S^t and t ∈ N+). We now assume
that every agent cannot borrow more than the present value of her current
endowment at such prices. Formally, we assume that for any vector of prices q
that is arbitrage-free, every portfolio strategy satisfies the wealth constraints

This constraint naturally rules out Ponzi’s scheme, and it is chosen ar-
bitrarily among many others. Hernandez and Santos (1996) gives six other
constraints ruling out Ponzi’s schemes and shows that they are all equiva-
lent when markets are complete. Markets are not assumed to be complete
though.

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