Kirchhoff's Law for Renters

Here's a plausible model of renting an apartment:

1. The apartment has several single-tenant rooms that vary in monthly rent depending on area and other factors. Their sum constitutes the total monthly rent of the household. Rent is paid in discrete monthly intervals.
2. Often, landlords will demand a collateral deposit upon moving in, and return it (in the simplest case) upon moving out. Let's say the deposit is proportional to rent (e.g. twice rent).
3. Subletting is a process by which some members move out and others move in without the direct involvement of the landlord. Even if the landlord is not directly involved, we expect that everyone moving out will receive their collateral deposit, and everyone moving in will pay a collateral deposit.

Here are a few scenarios that may arise during the renting process. Think about what ideally should happen, and who owes what amount of money:

1. Someone moves out and a subletter moves in to take their place. This situation is easy to handle, because the subletter's security deposit can be given to the exiting roommate to refund that person's security deposit. (As an aside, note that when subletting, security deposits cease to serve their original purpose as a safeguard because they're immediately consumed during the moving process instead of placed in a bank.)
2. No one moves in or out, but the roommates merely swap rooms with each other. Intuitively, perhaps it shouldn't matter if roommates swap rooms—room swapping isn't "noticeable" to the outside world. It's perhaps noteable that a roommate who moves into a smaller room pays a small security deposit, and if they later swap for a larger room and ultimately move out, they receive a larger security deposit. On the other hand, this isn't a big deal because whatever room they move out of, the incoming subletter will pay the security deposit.
3. The house consists of a large and a small room. Someone moves out of the large room; the person in the small room swaps into the large room; and a new subletter takes the small room. This presents a tough problem actually—do you see why? If the two room rates are \(x > y\), then someone needs to refund the exiting roommate the amount \(x\). On the other hand, the entering roommate is only paying \(y\) —not enough to cover the refund. Where should the remaining money come from? Should the roommate who swapped rooms be forced to pay something? But in the previous scenario, we thought it was alright for roommates to swap rooms with impunity. Why should it suddenly matter that people are swapping rooms?

There is a very simple principle which ensures that everyone is paid what they're owed. I call it:

Kirchhoff's Law for Renters: In addition to the rooms with their own room rates, treat "not living here" as a virtual room whose rent rate is 0. Whenever someone switches rooms, they pay the difference between their new rate and old rate. (Or \(k\) times the difference, if security deposit is proportional to rent by a factor k).¹

As a consequence of this rule:

1. People who move in owe a security deposit. After all, their old rent ("not living here") was 0 and their new rent is x.
2. People who move out are owed a security deposit. After all, their old rent is x and their new rent is 0, and the difference is negative.
3. Swapping rooms, even when no one moves in or out, does involve payment. You pay the difference between your new and old rent to the person whose room you're moving into. In fact, under this scheme, swapping rooms is equivalent to moving out of your old room (and getting refunded) followed by moving into your new room (and paying a new security deposit.) Elegant!
4. If someone moves out and a current tenant moves to take their place, the difference in rent is used to refund the exiting roommate's security deposit. In fact, miraculously, whatever the chain of moves/swaps the exiting roommate is a part of, the payments involved end up perfectly covering the cost of refunding the security deposit.
5. More generally: As long as all rooms are filled each month (through any combination of moving in, moving out, and swapping rooms), the various costs of moving will cover each other perfectly. The net payments will always be zero.
6. Moving out is a limiting case of downsizing. (In fact, in an extreme theoretical case, one could move through a sequence of increasingly tiny, low-rent rooms, recovering almost all of their original security deposit, before imperceptibly moving out and collecting the infinitesimal remainder.)

The analogy with Kirchoff's law for voltage is that security deposits create a kind of “financial potential” analogous to electrical potential, with the ground state naturally being not-living-here. The flow of individual people into and out of the house, or from room to room, constitutes a kind of current. The assumption that all rooms are filled is a condition that there is zero net current between the rooms of the house and ground. "Power" is expended when moving in or upsizing, and regained when moving out or downsizing.