I will be visiting a university in the US to give an invited talk. The organizer told me that I could choose to arrive a couple of days early, and leave a couple of days late. They will be paying for my hotel reservation and airline ticket.
I love to explore new places, and do not have constraints on my time. Hence, I'd love to to be able to stay a couple of extra days in the city and explore. However, this would mean that the department I will be visiting will have to pay more money to the hotel.
What is the etiquette for negotiating this? Will it not be looked upon disfavorably if I tell them that I wish to stay an extra couple of days? Can I tell them that I'd like to pay for my own hotel reservation for the extra days?