  In the canonical persuasion model, comparative statics has been an open
question. We answer it, delineating which shifts of the sender's interim payoff
lead her optimally to choose a more informative signal. Our first theorem
identifies a coarse notion of 'increased convexity' that we show characterises
those shifts of the sender's interim payoff that lead her optimally to choose
no less informative signals. To strengthen this conclusion to 'more
informative' requires further assumptions: our second theorem identifies the
necessary and sufficient condition on the sender's interim payoff, which
strictly generalises the 'S' shape commonly imposed in the literature. We
identify conditions under which increased alignment of interests between sender
and receiver lead to comparative statics, and study a number of applications.
