A theoretical model of a consumer who faces a price that varies with the number of units bought, and who faces random future changes in his demand for the good. An example is cumulative deductibles in health insurance policies. The problem is treated as a dynamic program involving medical demand under an insurance policy with a deductible. In this model, the perceived price of care falls (following a nonlinear path) as the consumer approaches the deductible. The model suggests: (1) Because demand and administrative costs are likely to be insensitive to the size of the deductible above a certain range, deductibles above that range will not be optimal; they add risk with no return. (2) Demand estimates will be biased if insurance policies in the sample contain deductibles, and if the dependent variable is annual medical demand. (3) Demand analysis by episode of illness is the appropriate framework in such circumstances.