Smallness of a Commodity and Partial Equilibrium Analysis

Partial equilibrium analysis has a conceptual dilemma that its object should be negligibly small in order to be free from income effect but then the consumer does not care for it and the notion of willingness to pay for it does not make sense. In the setting of a continuum of commodities, we propose a limiting procedure which transforms the general many-commodity environment into a partial single-commodity environment. In the limit, willingness to pay for a commodity is established as a 'density' notion and it is shown to be free from income effect. As an application, we provide a general equilibrium foundation of partial equilibrium analysis