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.