For every i, we define the budget set Bi (q) faced by agent i at prices q
as follows. If q is arbitrage-free, the budget set Bi (q) is the set of sequences
(c, θ) that satisfy conditions (2)-(4) above. If now the vector of prices has an
arbitrage opportunity, define Bi (q) as the set of sequences (c, θ) that satisfy
conditions (2)-(3) only.
Definition 1 An equilibrium is a sequence (ci,θi)i and a system of prices q
such that
1. taking prices qc as given, for every i the vector (cci, θci) is solution to the
program consisting of maximizing (1) subject to (c, θ) ∈ Bi(qc), and
2. for every history st we have that i ccist = wst and i θcist = 1.
The above definition requires that, taking prices as given, every agent
sequentially chooses consumption plans and portfolio holdings so as to max-
imize her expected utility, and markets for consumption good and trees all
clear in every history. It is also straightforward to see that the equilibrium
prices are arbitrage-free. Indeed, if otherwise then every agent will have an
infinite demand for at least one tree in at least one history, and Condition 2
in the above definition will always be violated. By a similar reasoning, it is
easy to check that equilibrium prices must be strictly positive.
2.2 Market crashes
We next describe the notion of market crashes occurring in financial markets.
This notion focuses on arbitrarily low returns on traded trees. For every
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