Duality in the optimization context basically means the existence of an equivalent representation of the optimization problem in linear form, coming from the connection between linearity and convexity. Basically if a person's decision behavior is more rational (convexity of consumption set), it will be more regular and predictable (convertable to linear optimization of the dual problem). A deeper root of such a property lies in the property of binary ordering.
Showing posts with label 《Microeconomic Theory》. Show all posts
Showing posts with label 《Microeconomic Theory》. Show all posts
第12页
Relationship between preference relations and choice rules is investigated in this section, but in fact, it is the rational preference relations and the consistent choice rules that is being compared. What about the question of whether every choice rule, consistent or not, can be explained by a preference relation, and whether every preference relation, rational or not, can be realized in a choice rule? The second question obviously has an answer yes, and with multiple possibilities of realization. 举一反三. The answer to the first question is more like the mathematical / philosophical question of whether there is a solution / truth to a particular problem or not. For example, can the second case in Example 1.C.1 be explained by a preference relation while the choice rule itself is not consistent? Can it be explained by a rational preference relation? If it can then there are cases not consistent but rational, which is not impossible intuitively, suggesting that the choice rule is not exactly a generalization of preference relations and we cannot say that one assumption is weaker or stronger than the other. This should be a more fundamental question related to the structures behind these two different ways of storing behavior, of decision making or of choice.
The authors in the book actually touched on this point while introducing on Page 14 a different way of realizing choice from a rational preference relation, i.e. a different definition 1.D.1., which allowed for choosing less than one's optimal choices and leaving behind some of those one is indifferent to, suggesting that, in an extreme case, one can be rationally indifferent among all alternatives which trivially serves as an explanation for any choice behavior, including those inconsistent ones, and in this sense, consistency is actually an additional restriction rather than a relaxation. In a sense those difficulties in preference theory actually also arise from this more restrictive view of how behavior is connected to rationality, and the Condorcet paradox on Page 8 can be readily solved using the indifferent preference relation.
The authors in the book actually touched on this point while introducing on Page 14 a different way of realizing choice from a rational preference relation, i.e. a different definition 1.D.1., which allowed for choosing less than one's optimal choices and leaving behind some of those one is indifferent to, suggesting that, in an extreme case, one can be rationally indifferent among all alternatives which trivially serves as an explanation for any choice behavior, including those inconsistent ones, and in this sense, consistency is actually an additional restriction rather than a relaxation. In a sense those difficulties in preference theory actually also arise from this more restrictive view of how behavior is connected to rationality, and the Condorcet paradox on Page 8 can be readily solved using the indifferent preference relation.
Preference and Choice
Takayama oriented his Mathematical Economics stressing the set-theoretic approach to behavioral modeling, in contrast to the traditional calculus approach. In this Oxford book however, it is the rationality assumption (completeness and reflexibility of preference relations), present in earlier times including Takayama's for ease of modeling, that the authors now try to substitute with the new concept of choice rule, which basically weakened the assumption of complete and transitive rationality to instantaneous and pairwise rationality (WARP). How would such a simplification change the mathematical formulations? Maybe it changed nothing, but just extended the application of these formulations to less rational conditions, like the extension of KTCQ of optimization which is calculus based thus requiring differentiability, to non-differentiable cases using non-linear programming.
Like differentiability, completeness and transitivity are also properties of functions / correspondences from X, to >() if not a function (not continuous), and to R if represented by utility. The structure (X, >()) is a correspondence commonly restricted by the axiom of rationality, while the structure (B, C()) is a correspondence commonly restricted by the axiom of WERP (consistency in behavior rather than in the intellect, i.e. people are, assumed, to be more motivated or even forced to behave consistently in a social context, which reflects some rationality of mind in the decision making process, but such a process can also be skipped to arrive at the same behavior.).
However, less assumptions usually means more information to be considered and more complex analysis. Preference relations relaxed differentiability but generally went back to utility functions (assuming continuity and very often even back to differentiability) in analyzing practical problems. The choice approach will also do the same.
Theories are simplifications of the world, and relaxing assumptions generally brings theories down by introducing a more complex world, which however is also a way of testing how powerful and brilliant the theories are. Everyone can have their own theories, just some theories are based on relatively less information and more assumptions in the view point of those who have more information, and these theories are usually just called prejudice, presumption, or cliche if information come with time. In this sense those economic theories that do not pass the test of 'choice rule' might then be considered as prejudice, presumption, or cliche, but can still be true and influential within their circles or within their time.
What then, do the assumptions of continuity and differentiability mean exactly, if preference relations means rationality and choice rule means consistent behavior? It seems the two calculus related concepts have something to do with the comparative statics and thus the thermodynamical equilibrium concept.
Like differentiability, completeness and transitivity are also properties of functions / correspondences from X, to >() if not a function (not continuous), and to R if represented by utility. The structure (X, >()) is a correspondence commonly restricted by the axiom of rationality, while the structure (B, C()) is a correspondence commonly restricted by the axiom of WERP (consistency in behavior rather than in the intellect, i.e. people are, assumed, to be more motivated or even forced to behave consistently in a social context, which reflects some rationality of mind in the decision making process, but such a process can also be skipped to arrive at the same behavior.).
However, less assumptions usually means more information to be considered and more complex analysis. Preference relations relaxed differentiability but generally went back to utility functions (assuming continuity and very often even back to differentiability) in analyzing practical problems. The choice approach will also do the same.
Theories are simplifications of the world, and relaxing assumptions generally brings theories down by introducing a more complex world, which however is also a way of testing how powerful and brilliant the theories are. Everyone can have their own theories, just some theories are based on relatively less information and more assumptions in the view point of those who have more information, and these theories are usually just called prejudice, presumption, or cliche if information come with time. In this sense those economic theories that do not pass the test of 'choice rule' might then be considered as prejudice, presumption, or cliche, but can still be true and influential within their circles or within their time.
What then, do the assumptions of continuity and differentiability mean exactly, if preference relations means rationality and choice rule means consistent behavior? It seems the two calculus related concepts have something to do with the comparative statics and thus the thermodynamical equilibrium concept.
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