  To study the assumption that the utility maximization hypothesis implicitly
adds to consumer theory, we consider a mathematical representation of
pre-marginal revolution consumer theory based on subjective exchange ratios. We
introduce two axioms on subjective exchange ratio, and show that both axioms
hold if and only if consumer behavior is consistent with the utility
maximization hypothesis. Moreover, we express the process for a consumer to
find the transaction stopping point in terms of differential equations, and
prove that the conditions for its stability are equal to the two axioms
introduced in the above argument. Therefore, the consumer can find his/her
transaction stopping point if and only if his/her behavior is consistent with
the utility maximization hypothesis. In addition to these results, we discuss
equivalence conditions for axioms to evaluate their mathematical strength, and
methods for expressing the theory of subjective exchange ratios in terms of
binary relations.
