Valuation bubbles and sequential bubbles

Price bubbles in an Arrow-Debreu valuation equilibrium in infinite-time economy are a manifestation of lack of countable additivity of valuation of assets. In contrast, known examples of price bubbles in sequential equilibrium in infinite time cannot be attributed to the lack of countable additivity of valuation. In this paper we develop a theory of valuation of assets in sequential markets (with no uncertainty) and study the nature of price bubbles in light of this theory. We consider an operator, called payoff pricing functional, that maps a sequence of payoffs to the minimum cost of an asset holding strategy that generates it. We show that the payoff pricing functional is linear and countably additive on the set of positive payoffs if and only if there is no Ponzi scheme, and provided that there is no restriction on long positions in the assets. In the known examples of equilibrium price bubbles in sequential markets valuation is linear and countably additive. The presence of a price bubble indicates that the asset's dividends can be purchased in sequential markers at a cost lower than the asset's price. We also present examples of equilibrium price bubbles in which valuation is nonlinear but not countably additive.