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שורה 32:
שורה 32: − − ===Completeness and Strong axiom=== − The strong axiom of revealed preferences (SARP) is equivalent to the weak axiom of revealed preferences, except that the consumer is not allowed to be indifferent between the two bundles that are compared. That is, if WARP concludes <math>\mathbf{a} \succeq \mathbf{b}</math>, SARP goes a step further and concludes <math>\mathbf{a} \succ \mathbf{b}~</math> . − − If A is directly revealed preferred to B, and B is directly revealed preferred to C, then we say A is ''indirectly'' revealed preferred to C. It is possible for A and C to be (directly or indirectly) revealed preferable to each other at the same time, creating a "loop". In mathematical terminology, this says that [[Transitive relation|transitivity]] is violated. − − Consider the following choices: <math>C(A,B)=A</math> , <math>C(B,C)=B</math> , <math>C(C,A)=C</math>, where <math>C</math> is the choice function taking a set of options (budget set) to a choice. Then by our definition A is (indirectly) revealed preferred to C (by the first two choices) and C is (directly) revealed preferred to A (by the last choice). − − It is often desirable in economic models to prevent such loops from happening, for example if we wish to model choices with [[utility function]]s (which have real-valued outputs and are thus transitive). One way to do so is to impose completeness on the revealed preference relation with regards to the situations, i.e. every possible situation must be taken into consideration by a consumer. This is useful because if we can consider {A,B,C} as a situation, we can ''directly'' tell which option is preferred to the other (or if they are the same). Using the weak axiom then prevents two choices from being preferred over each other at the same time; thus it would be impossible for "loops" to form. − − Another way to solve this is to impose the ''strong axiom of revealed preference'' (SARP) which ensures transitivity. This is characterized by taking the [[transitive closure]] of direct revealed preferences and require that it is [[Antisymmetric relation|antisymmetric]], i.e. if A is revealed preferred to B (directly or indirectly), then B is not revealed preferred to A (directly or indirectly). − − These are two different approaches to solving the issue; completeness is concerned with the input (domain) of the choice functions; while the strong axiom imposes conditions on the output. −
←Completeness and Strong axiom
: <math>p \cdot x(p',m') \leq m ~ \wedge ~ x(p',m') \neq x(p,m) ~\Rightarrow ~ p' \cdot x(p,m) > m'~</math>.
: <math>p \cdot x(p',m') \leq m ~ \wedge ~ x(p',m') \neq x(p,m) ~\Rightarrow ~ p' \cdot x(p,m) > m'~</math>.
==ביקורת==
==ביקורת==