This is about the properties of the 'envelope' of optimization problems (solved readily with nonlinear programming), a correspondence between the solutions of optimization processes and its conditions, i.e. properties of a projection from others behavior to an individual's reactions which is determined by properties of another correspondence confined within one individual, under the optimizing principle. The behavior of this 'envelope' correspondence under some principles like equilibrium is what people want to know, and the behavior is determined by properties such as continuity, elasticities, and as related to its differentials.
These properties are traditionally derived with the preference ordering representation of behavior, elaborated in Takayama's book. Yet as emphasized in the more recent Microeconomic Theory, they can also be derived from the choice representation. This connection is essentially due to the fact that the axiom of transitivity of the preference orderings is generally irrelevant to properties of the demand function, thus can be relaxed in investigating demand, reducing the behavioral representation to something more or less equivalent to the choice approach.
About the detailed results from demand theory. All properties are derived directly from the preference representations, and the internal optimization process inside the envelop is taken as already finished (leading to equivalence with choice approach, as pointed out in Takayama's footnote 4 on Page248).
The compensated demand function is a function from the desired demand to the demand that a rational consumer adjust to so that one can still enjoy the same utility but achieved within one's budget. In a very strong sense, it is an envelope closing at an indifference curve. In this perspective, an envelope is just a tangent line that is not tangent at any specific point. Such an concept enabled the description of a tangent or sloping relationship without being restricted to one point, i.e. envelope is the tangent line of a set, and all the efforts with separation theories or minimum expenditure function are just to mathematically express an envelope. Then under certain conditions, the envelopes form a dual relationship with the demand function, and substitution properties of the demand function which is an implicit function involving optimization can be conveniently transferred to the substitution properties of its envelope.
These properties are traditionally derived with the preference ordering representation of behavior, elaborated in Takayama's book. Yet as emphasized in the more recent Microeconomic Theory, they can also be derived from the choice representation. This connection is essentially due to the fact that the axiom of transitivity of the preference orderings is generally irrelevant to properties of the demand function, thus can be relaxed in investigating demand, reducing the behavioral representation to something more or less equivalent to the choice approach.
About the detailed results from demand theory. All properties are derived directly from the preference representations, and the internal optimization process inside the envelop is taken as already finished (leading to equivalence with choice approach, as pointed out in Takayama's footnote 4 on Page248).
The compensated demand function is a function from the desired demand to the demand that a rational consumer adjust to so that one can still enjoy the same utility but achieved within one's budget. In a very strong sense, it is an envelope closing at an indifference curve. In this perspective, an envelope is just a tangent line that is not tangent at any specific point. Such an concept enabled the description of a tangent or sloping relationship without being restricted to one point, i.e. envelope is the tangent line of a set, and all the efforts with separation theories or minimum expenditure function are just to mathematically express an envelope. Then under certain conditions, the envelopes form a dual relationship with the demand function, and substitution properties of the demand function which is an implicit function involving optimization can be conveniently transferred to the substitution properties of its envelope.
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