Actually modems & other networking gear go into a "Green" mode when they no longer see packets coming from connected equipment. The amount of power that the modem uses, is just pennies a month.
Unlike Dial-up modems that you could turn off after using. High Speed modems are required to stay on 24/7, 365 days, so that the ISP knows that a customer has an active connection. If the ISP's equipment is not seen online for anywhere from 1 to 7 days, you get placed into a "Walled Garden" state. Because the ISP's equipment considers that the gear is no longer working.
IP leases have a certain time period. Even on a LAN, if the Router does not see an IP connected to the system, it will purge that Dynamic IP out of the system, so that it is available for another device. The only time a router does not purge an IP, is if the IP is considered Reserved or Static on the router.