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Additional resources for Microeconomics: A Modern Treatment
E. x 0 is not preferred to x 1 or that x 1 is not preferred to x 0 or the fact that both x0 is not preferred to x 1 and x 1 is not preferred to x 0 ; in other words, you cannot The Theory of Choice 25 have both x 0 preferred to x 1 and x 1 preferred to x 0 , but in saying this we do not rule out indifference. This meansP cannot give a strong ordering on X· However, we have established that with the aid of the equivalence relation, /, we can establish an induced strong ordering. So far we have the situation that the choice set x can be divided into the attainable set A and the unattainable set A.
In so far as utility is a function defined over a commodity space we must be clear on which theorems in mathematics we can call upon to discuss our economic problem. For example, can the commodity space be unconnected (contain holes)? Chapter 3, therefore, contains a certain amount of mathematics which allows us to be precise on the properties possessed by the commodity space; and secondly, by what types offunctions will be suitable representations of utility which satisfy the axioms previously discussed.
But there is no reason to suppose that there will be no two or more elements ranked the same. If this is the case then we have a weak ordering. But this is not sufficient; we require the ordering to be transitive so that if X; is ranked higher than xi and xi is ranked higher than xk we can take it as reasonable (consistent) to suppose X; is ranked higher than xk. Given the attainable set and a transitive relation defined on it then it will be possible to choose one element of A which is ranked at least as high, or at least as preferred, as any other element in A.