Differential Topology and General Equilibrium with Complete and Incomplete Markets

Abstract

The goal of this publication is to provide basic tools of differential topology to study systems of nonlinear equations and to apply them to the analysis of general equilibrium models with complete and incomplete markets. The main content of general equilibrium analysis is to study existence, (local) uniqueness and efficiency of equilibria. To study existence Differential Topology and General Equilibrium with Complete and Incomplete Markets combines two features. First order conditions (of agents’ maximization problems) and market clearing conditions, instead of aggregate excess demand function. Then, the application to that “extended system” of a homotopy argument, which is stated and proved in a relatively elementary manner. Local uniqueness and smooth dependence of the endogenous variables from the exogenous ones are studied using a version of a so-called parametric transversality theorem. In a standard general equilibrium model, all equilibria are efficient, but that is not the case if some imperfection, like incomplete markets, asymmetric information, strategic interaction, is added. Then, for almost all economies, equilibria are inefficient, and an outside institution can Pareto Improve upon the market outcome. Those results are proved showing that a well-chosen system of equations has no solutions. \ud \ud The target audience of Differential Topology and General Equilibrium with Complete and Incomplete Markets consists of researchers interested in Economic Theory. The needed background is multivariate analysis, basic linear algebra and basic general topology

Idiosyncratic risk and financial policy

“…19 For simplicity of notation, price vectors and gradients of utility functions are taken as row vectors, whereas quantities are taken as column vectors. 20 The unique optimizer for individuals of type i, (x i , y i ) is characterized by the following first-order conditions: for a vector …”

“…With these properties, for any economy in a generic set, there is associated a lower-dimensional 22 See, also, [20].…”

On the orientability of the asset equilibrium manifold

“…Secondly, recall that if f : X → Y is a continuous and compactly rooted mapping 11 between orientable manifolds of the same dimension, then one can define deg(f ), which denotes the standard topological degree of f (see [13] p.196). Notice that deg(f ) depends on the orientation of X and Y .…”

No-arbitrage, state prices and trade in thin financial markets

“…11 Intuitively, consistent price impact matrices reflect the price changes needed to clear the market for any small deviationΘ i by trader i, given that other traders respond rationally to market prices in any subequilibrium. This means that each trader correctly estimates her price impact, to a first order, and incorporates this information into her individual decision problem.…”

“…Suppose that R is nonsingular. 13 Then, (P ,Θ,M ) is an equilibrium in thin financial markets for economy {u, e, A} if, and only if, 11 Strictly speaking, the intuition given before the definition is slightly weaker than the definition itself:…”