Despite some schools of thought, it's generally to your advantage to name a price first in negotiations. This is thanks to the anchoring effect, where presenting a value skews later judgments towards it. There is plenty of evidence that setting salary for a new role is influenced by relevant anchors, such as the applicant stating their previous pay or expectations for this job. But decision-making research suggests that estimates and attributions can be influenced by even arbitrary and extreme anchors. Todd Thorsteinson at the University of Idaho set about seeing how crazy numbers might also shape take-home pay.
206 psychology students were asked to make a salary suggestion for a desirable job applicant question. Participants were presented with the applicant's description including two anchors: a realistic one of the applicant's previous salary ($29,000), and an unusual one of either $100k or $1, embedded within a joking statement they made about their salary expectations. The joking context was considered necessary to allow the unusual anchor to be presented without triggering other effects, like being considered overly arrogant or having poor judgment. Participants given the high unusual anchor awarded a higher salary than both those given the low unusual anchor and a control condition with just the realistic anchor.
A second experiment asked its 150 participants to additionally record their perceptions when reading about the applicant, and introduced an even more extreme anchor: one million dollars. Participants were not put off by the extreme anchor, perceiving it as just as plausible and influential as the $100k reference, and in both cases ended up offering the applicant a higher salary than when these high anchors were absent. So, just as in the literature on estimation, even radically inappropriate anchors can sway decisions. It's worth noting too that the unusual anchors had their effect despite being presented alongside realistic ones, as some studies have suggested that in such situations we may simply defer to the more plausible. That wasn't the case here.
There are risks to naming a salary first, such as underselling yourself or pricking the sensibilities of the hirer. So using a joke to introduce an anchoring value may be a safer bet. Organisations may of course respond: using clearly defined pay ranges and clear criteria to shape a fair financial offer for a desired candidate. Both parties should take seriously the power of framing the financial borders of a negotiation.