State-Dependent Utility

Abstract

We propose a new approach to utilities in (state) complete markets that is consistent with state-dependent utilities. Full solutions of the optimal consumption and portfolio problem are obtained in a very general setting which includes several functional forms for utilities used in the current literature, and consider general restrictions on allowable wealths. As a secondary result, we obtain a suitable representation for straightforward numerical computations of the optimal consumption and investment strategies. In our model, utilities reflect the level of consumption satisfaction of flows of cash in future times as they are (uniquely) valued by the market when the economic agents are making their consumption and investment decisions. The theoretical framework used for the model is the one proposed in Londoño (2008). We develop the martingale methodology for the solution of the problem of optimal consumption and investment in this setting.

State-Dependent Utilities and Incomplete Markets

“…The problem of optimal consumption and investment for a "small investor" whose actions do not influence market prices in complete markets and where consumers have dependent utility structures has been studied in Londoño [1]. The modern treatment of this problem when the asset prices follow Itô processes started with the seminal works of Merton [2,3].…”

“…Using a "martingale" approach, Cox and Huang [4] and Karatzas et al [5] solved the problem in more general settings in the case of complete markets. Analytical and numerical problems with those earlier solutions and lack of agreement with empirical data motivated alternative treatments; see Londoño [1] for some literature related with problems associated with the standard models as well as some reviews of other approaches.…”

“…The main result of this paper is the solution to the problem of optimal consumption and investment for an agent that does not influence the market. It is assumed that the utility maximization criteria used are based on a statedependent utility functional as proposed in Londoño [1]. For the optimization problem of this paper, it is assumed that consumers are endowed with an initial capital and a hedgeable random endowment where the underlying market is not necessarily assumed to be complete.…”

“…Here, we extend further the approach presented in Londoño [1]. In this model utilities reflect the level of consumption satisfaction of flows of cash in future times as they are valued by the market when the economic agents are making their consumption and investment decisions.…”

“…In this model utilities reflect the level of consumption satisfaction of flows of cash in future times as they are valued by the market when the economic agents are making their consumption and investment decisions. The utilities used in this paper were introduced in Londoño [1] and are equivalent to state-dependent utilities in standard settings, where dependence on the state is through the state price density process (see (7) and Remark 8). The main assumption of the theorem, besides the interpretation of the utility functionals, is that the income of the agent should be hedgeable; in plain English it is required that the employer and employee agree on a labor contract that allows each party to hedge any risk associated with the this contract, and in this way any economic agent would be able to cover the compromises of this contract in exchange for a fair price.…”