The Defense Advanced Research Projects Agency (DARPA) High Productivity Computing System (HPCS) program is developing systems that deliver increased value to users at a rate commensurate with the rate of improvement in the underlying technologies. For example, if the relevant technology was silicon, the goal of such a system would be to double in productivity (or value) every 18 months, following Moore's law. The key questions are how we define and measure productivity, and what the underlying technologies that affect productivity are. The goal of this paper is to synthesize from several different productivity models a single model that captures the main features of all the models. In addition we will start the process of putting the model on an empirical foundation by incorporating selected results from the software engineering and high performance computing (HPC) communities. An asymptotic analysis of the model is conducted to check that it makes sense in certain special cases. The model is extrapolated to a HPC context and several examples are explored, including HPC centers, HPC users, and interactive grid computing. Finally, the model hints at a profoundly different way of viewing HPC systems, where the user must be included in the equation, and innovative hardware is a key aspect to lowering the very high costs of HPC software.