Dynamics differ when you negotiate more favorable rates with firms you already use significantly than when you negotiate with new firms that want your business. With incumbents, the benefits can come more quickly because they are already handling matters and rate reductions kick in at once. For the same reason – familiarity and entrenchment can strengthen resistance – you might end up with lower discounts. Whereas, a hungry newcomer might concede deeper or better discount arrangements but the payoff lies farther off since they have to be assigned matters and work them before those rate breaks help you.
On a different measure, incumbents know more targets to lobby within your company and can muster more appeals to “look at all we’ve done for you.” Even so, they can probably put on the table insightful cost-saving measures that do not rest on rates. Untested contender firms play the card that they can bring to bear all sorts of talent and you don’t really know whether the attorneys will perform as advertised. Lack of baggage may be an advantage. More generally, the challengers will probably out-market the incumbent.