The present paper is devoted to the question of the dynamic stability of a perfectly competitive market with the price adjustment rate proportionate to excess demand. Although the problem dates back to the nineteenth century, a general answer is not yet available. In none of the cases studied has the system been found to be unstable. The same is true of the further results presented in "On the Stability of Competitive Equilibrium, II," by K. J. Arrow, H. D. Block, and L. Hurwicz scheduled to appear in the next issue of Econometrica. The latter paper extends the results of the present article in several directions.