[susan53]

Just trying to give a bit of context to the example - language doesn't exist in a vacuum, the context is always important. The point is that "reimburse" means "pay back" -ie give someone the money for something they are not responsible for but have bought/paid for when the payment is the responsibility of another person. In your situation it's not Peter who wants the a/c but John, so the payment is John's responsibility, even if Peter benefits from it too. Reimburse could only be used in this situation if Peter agreed to pay half the cost - and therefore accepted the responsibility. Hence the context of You pay for it when you're in town and I'll reimburse you/pay you back later. Peter, in this situation, is authorising John to pay his (Peter's) half of the cost on his behalf.