Duke 2008 | ASHE | Equity and Efficiency in Health and Healthcare

Analyses of expected utility problems are commonplace in economics, as they are useful in exploring a variety of insurance, investment, and consumption decisions. Surprisingly, there are two ways of modeling budget constraints for uncertain lifetimes in the literature, and these two approaches have different implications. This does not appear to be widely appreciated. Indeed, the norm appears to be to use one approach or the other with little or no discussion, even if the choice of budget constraints drives the key results of the analysis. The discussion has been hampered by the absence of a vocabulary to describe these approaches, so the paper offers one. A conditional budget constraint describes the resources available to the consumer given survival. An expected budget constraint describes the consumer's resources weighted by the probability of being in a particular state. This paper will show that a conditional budget constraint is appropriate for a standard expected utility problem.

The first section of the paper examines the decision to consume when survival is uncertain, a problem considered Barro and Friedman (1977) using an expected budget constraint. Their key result, that consumption plans do not depend on survival probabilities, can be generated using a very simple two-period, one-good model. This set up makes it easy to show rigorously that 1) an expected budget constraint does not describe the consumer's choice set accurately; 2) a conditional budget constraint does; and 3) their key result does not hold if a conditional budget constraint is used. None of these points have been noted in the several responses to this paper.

The second section of the paper examines a more complex problem that was analyzed separately by Garber and Phelps (1997) and Meltzer (1997). Both analyzed an expected utility maximizing consumer's choices of medical interventions. Garber and Phelps, who used a conditional budget constraint, concluded that the consumer should not consider unrelated future costs, but Meltzer, who used an expected budget constraint, concluded that the consumer should. A very simple set up shows that 1) Meltzer's expected budget constraint does not describe the consumer's choice set properly and 2) the conclusion that consumers should attend to unrelated future costs does not emerge when one uses a conditional budget constraint.

The third section of the paper briefly looks at the colloquy between Johansson (2001 and 2002) and Blomqvist (2002). Again an important question is the structure of the budget constraint, with Johansson using a conditional budget constraint and Blomqvist using an expected budget constraint. Blomqvist offers a rationale for the expected budget constraint, which the paper analyzes in some detail.

The final section concludes.