Duality and consumption decisions under income and price risk

This paper uses duality to provide an analysis of compensated and uncompensated decisions under income and/or price risk. We focus on a two-good model, in which the individual maximizes expected utility subject to a stochastic budget constraint. A Slutsky equation is derived that decomposes the derivative of the optimal decision function with respect to any parameter in the model into an income effect and a substitution effect. The total effect of an increase in risk is controlled by the generalized preference intensity, while the compensated effect of an increase in risk is controlled by the generalized risk premium developed in the literature on aversion to one risk in the presence of another.