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 framework into a partial single-commodity framework. 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. This pins down an exact relationship between general equilibrium analysis and partial equilibrium analysis.