Financial System

In a financial system of interlocking balance sheets, the assets of creditors are the liabilities of debtors. A change in the value of underlying assets can ripple through the financial system through valuation changes on balance sheets. Tarski’s fixed point theorem guarantees the existence of consistent valuations. Under mild regularity conditions, there is a unique fixed point. Comparative statics analysis can be used to show how systemic risk propagates.

ABSTRACT INDUCTIVE AND CO-INDUCTIVE DEFINITIONS

In [G. Curi, On Tarski’s fixed point theorem. Proc. Amer. Math. Soc., 143 (2015), pp. 4439–4455], a notion of abstract inductive definition is formulated to extend Aczel’s theory of inductive definitions to the setting of complete lattices. In this article, after discussing a further extension of the theory to structures of much larger size than complete lattices, as the class of all sets or the class of ordinals, a similar generalization is carried out for the theory of co-inductive definitions on a set. As a corollary, a constructive version of the general form of Tarski’s fixed point theorem is derived.