We study a broad class of dynamic consumer problems and characterize the short and long-run response of the demand for a good to a permanent increase in its market price. Such response can be non-monotonic over time, and the short and long-run price-elasticity of demand may have opposite sign. This is a testable prediction and can arise even in the absence of income effects. Our results are robust to a variety of settings that are commonly used in the economic literature, and have relevant policy implications. We provide illustrative applications to models of human capital and labor supply, addiction, habit and taste formation, health capital, and renewable resources.