Demand as the Envelope of Optimization and Duality

Takayama arrived at the substitution properties of demand in Section 2.D directly from properties of the preference orderings (probably without transitivity). He also covered the other approach of using duality, which he derived in Section 1.F with nonlinear programming and separation theory (while MWG could only provide some intuitions of it with normal calculus). Takayama's advocated approach was centered around a 'minimum expenditure funcion', which can be just another way of expressing the supporting hyperplane or support function. Also convexity is included in the conditions of the stated demand properties, which resonate what I have wrote in the other note about the connection between linearity and convexity which supports the approach of solving optimization problems with its dual form.