The importance of imprisonment in punishing convicted offenders is a major issue in criminal justice, with attention to what statistics best index punishment and historical trends. In a series of articles, Alfred Blumstein and several colleagues have posed and tested a homeostatic model of punishment in society, based on ideas stated by Durkheim. Blumstein's model assumes that forces in society cause a shift in the threshold of criminality to maintain a stable level of crime. Thus, if deviant behaviors become more common, the threshold is moved to punish fewer types of behaviors, but the threshold shifts to punish more behaviors if the opposite occurs. Blumstein supported this theory by analyzing time series of imprisonment rates for the United States between 1926 and 1974, for 47 States during the same period, for Canada between 1880 and 1959, and for Norway from 1880-1964. After a discussion of stationary and nonstationary processes and their implications for time-series modeling procedures, this paper examines Blumstein's models for the United States to show that the data do not necessarily support the contention that the aggregate imprisonment rate remained stable during the period of 1926-70. In an alternative approach, analysis of prison admission rates for the same period also fails to support the stability of punishment hypothesis. As an additional test, California's imprisonment and admission rates are analyzed for the period, 1853-1970, and the results do not validate Blumstein's conclusions. Other objections to the homeostatic model concern its policy implications; use of imprisonment rates; and estimation of flow rates between the prison, criminal, and lawabiding populations. Alternative methods for measuring stabililty in punishment are explored. Blumstein and his colleagues rebut these criticisms by claiming that the empirical implications of their theory have been misinterpreted. Formulas, graphs, and footnotes are included.