Economists often analyze cross-sectional data to estimate the value people implicit place on attributes of goods using hedonic methods. Usually strong enough assumptions are made on the functional form of utility to point identify individuals' willingness-to-pay (WTP) for changes in attribute levels. Instead, this paper develops a new way to partially identify WTP under a weak set of conditions on the shape of individual indifference curves. In particular, indifference curves are assumed to be increasing and convex in an attribute-cost space that is finitely bounded above. These shape restrictions provide informative partial identification without relying on functional form restrictions for utility. Identification given general, potentially discrete, as well as smooth price functions is analyzed. To illustrate this method, we contribute to the literature on the value of a statistical life (VSL) by analyzing labor market data to study people's willingness to pay (WTP) for reductions in levels of fatal risk. The paper contrasts VSL estimates from conventional analysis with the bounds obtained under this new approach using a common data set. The data are shown to be consistent with a wide range of WTP values even given equilibrium and credible shape restrictions. This suggests that conventional estimates may be driven by functional form restrictions imposed on utility rather than by the data or properties of equilibrium.