When he calls after the contract expires he is an apparently loyal customer and as a result he is likely price insensitive, so they can charge the higher price.
He asks for a lower price but there is no reason for them to do this and so they say no.
But when he says that he is leaving they realise that he either has substitutes lined up or does not need them any more, but whatever the reason not only is his demand price sensitive, but he is also prepared to act on it.
This last bit is very important.
He has to convince them that he is willing to back up words with actions.
They then reduce the price.