Complete and incomplete financial markets in multi-good economies

Abstract

We investigate conditions for endogenous incompleteness and completeness in continuous-time financial markets driven by diffusion processes with multiple consumption goods and heterogeneous agents. We show that for a class of utility functions the financial market is endogenously incomplete. A sufficient condition for market completeness is that the dividend diffusion matrix in units of the numeraire good is invertible. Further, financial market completeness can depend on the choice of the numeraire good since changing the numeraire good implies a change of the risk-free asset and the asset structure.

The Completeness and Incompleteness of Financial Markets in Economies Driven by Diffusion Processes

“…Specifically, we present two types of utility functions that give rise to the incompleteness of financial markets. The first one is similar to that of Ehling and Larsen [7]. If the utility function satisfies a partial differential equation, we find that one of the stock prices is replicable.…”

“…We extend parts of the results in Hugonnier et al [4] and Ehling and Larsen [7]. Specifically, we present two types of utility functions that give rise to the incompleteness of financial markets.…”

“…Note that the processes of stock prices are different from those in Ehling and Larsen [7], in which the process of prices is subject to a geometric Brownian motion.…”

“…Ehling and Larsen [7] pay attention to conditions for the incompleteness of markets and get a class of utility function for incomplete financial markets in which stock prices follow the process of a geometric Brownian motion. Ehling and Larsen [7] demonstrate that this class of utility function leads to the incompleteness of markets.…”

International Capital Markets with Time-Varying Preferences

“…However, under the assumption of constant log-linear preferences financial markets are typically incomplete (see Cass and Pavlova (2004), Berrada et al (2007), Ehling and Heyerdahl-Larsen (2015), Serrat (2001) and Kollmann (2006).) Curatola (2017) shows that when preferences depend on the popularity of traded good i) financial markets are complete even when agents are equipped with log utility and ii) the equilibrium obtained by solving the decentralized economy and the equilibrium obtained by solving the social planner problem are equivalent.…”