Customers will offer what they feel will stop the initial call and get you to go away. If the customer knows you can take a minimum amount and that will stop the call and conversation for today, they will offer that to you. Helping the customer understand the importance and advantage of paying more today is how you help them.
The initial offer often does not fully resolve the account, only a portion. When you take a partial and don’t negotiate for more, you are only creating more call backs to the customer to resolve the remaining. This cost both the customer and the company.