This paper derives conditions for two key Keynesian propositions in a simple IS-LM model: (a) the paradox of thrift, and (b) the crowding-in of private investment ex- penditures by government expenditures. A linear specification of the model is then presented as a special case that can be used for empirical analysis. Using data for the US economy for the period 1959--2009, time series estimation of the linear model using instrumental variables regression shows that the paradox of thrift and crowding-in are real possibilities, especially in the sub-period, 1974 -- 2009, that excludes the Golden Age of capitalism.