How inflation and unemployment are related in both the short run and long run is perhaps the key question in macroeconomics. This paper tests various price equations using quarterly U.S. data from 1952 to the present. Issues treated are the following. 1) Estimating price and wage equations in which wages affect prices and vice versa versus estimating "reduced form" price equations with no wage explanatory variables. 2) Estimating price equations in (log) level terms, rst difference (i.e., inflation) terms, and second difference (i.e., change in inflation) terms. 3) The treatment of expectations. 4) The choice and functional form of the demand variable. 5) The choice of the cost-shock variable. The results reject the use of rational expectations and suggest that the best speci cation is a price equation in level terms imbedded in a price-wage model, where the wage equation is also in level terms. The best cost-shock variable is the import price deflator, and the best demand variable is the unemployment rate. There is some evidence of a nonlinear effect of the unemployment rate on the price level at low values of the unemployment rate. Many of the results in this paper are contrary to common views in the literature, but the empirical support for them is strong.