Difference in differences require that 0% of observations are treated in the control group and during period 0 (no "always takers") and 100% in the treatment group in period 1 (no "never takers"). Sometimes, the treatment rate increases more in the treatment than in the control group but there are never or always takers. This paper develops results to identify treatment effects in such settings. They only require one common trend assumption on the outcome of interest Y whereas the standard instrumental variable result also requires common trend on treatment D. I derive bounds for treatment effects which are tight when there are no or few always takers. This can be the case in applications considering the effect of an innovation, where by definition no observations are treated in period 0. I derive other bounds that are tight when the treatment rate does not change much between the two periods in the control group, which can be the case in applications considering the extension of a program to a group previously not eligible. I use my results to measure the efficacy of a new drug for smoking cessation.
