Consumption Selection on Incomplete Markets

The question how to allocate capital best is as old as financial markets themselves. Maximizing expected gains only might be a good approach but cannot be the best answer because usually high expected gains are driven by highly speculative and risky investments. In this thesis we study economic agents who subordinate expected gains to other plans. The essentials idea is justified and generally accepted in the common literature, since each portfolio insurance strategy subordinates expected gains to capital guarantees. Following an approach of Dybvig (1995), we consider agents whose plans are afflicted with future expenditures and mirror decisions based on expenses. Agents driven by those plans will be more careful withdrawing money today because high expenses narrow the capital stock and impede further enduring expenditures. The entire expected utility maximization problem is embedded in a semimartingale model for incomplete markets in the line of Kramkov and Schachermayer (1999). More precisely, on the basis of the Filtered Bipolar Theorem (Zitkovic, 2002), we join the models introduced in Bouchard and Pham (2004) and Karatzas and Zitkovic (2003) to get a more general approach. While utility is gained form a rate of consumption process as in Karatzas and Zitkovic, evaluation of the consumption process bases on a distribution function F (Bouchard and Pham, 2004), which weights the intertemporal utility function over time. This enables us to set up a general model on intertemporal consumption choice without causing exhausting calculations since the intertemporal utility function itself does not change over time. As the main results in this section we prove existence and uniqueness of primal and dual optimizers and list the main properties of primal and dual value function. Necessary and sufficient for solvability of both the primal and the dual problem will be introduced and discussed as well as main properties of primal and dual value function. Like in many other optimization problems on semimartingale models these assertions hold even if the intertemporal utility function u does not satisfy the usual condition on asymptotic elasticity (Kramkov and Schachermayer, 1999). Some special cases and examples are discussed as well. We derive a nice result on constrained consumption selection in the line of Cvitanic and Karatzas (1992, Theorem 10.1). In that paper they solved a utility maximization on constrained portfolio choice (cf. incomplete markets) and unconstrained consumption choice via auxiliary complete markets. Moreover they verified that the value of that (portfolio) constrained optimization problem corresponds to the value of that auxiliary market with minimal (dual) value. We found a version of this theorem on incomplete markets when consumption selection is constrained. Finally we study the case of consumption ratcheting (Dybvig, 1995) in more detail. As usual consumption selection takes place on a set of progressively measurable processes. This time it turns out that it suffices to consider the smaller set of optional processes. Thus, we are able to apply the theory developed in Kauppila (2010). We will show how optimal consumption on complete markets looks like and state further properties of the (dual) value function. Finally we relate the optimization problem for consumption ratcheting agents to similar problems (resp. similar individual likings).