What kind of mathematical formula would you use for this situation

Factual Questions
1

I haven’t done algebra in a while.

With the loss of about 8 million jobs and a jobless recovery, I’m guessing people are going to find ways to live cheaper.

One possible solution seems to be dormitory style housing. Rather than having 1 person in a 2 bedroom apartment you have 2 in each bedroom and 2 in the living room, so a total of 6. That is how many working class people in expensive cities seem to do it (sharing bedrooms and having people sleep in the living room).

The cost savings come from splitting rent and utilities and (possibly) from bulk purchases of consumables and household goods. However I don’t know if bulk purchases will save money because what is to stop a single individual from doing bulk purchases. Then again perishables will be destroyed too quickly for bulk purchases to be of use. Then there is an issue of storage.

So the cost savings from dormitory style housing seem to follow an L curve where the biggest benefits come from the first few additions of people, but soon afterwards the addition of new people makes a tiny dent.

So if you have 4 people in a $2000/month place to live who also spend $200/month on utilities and another $800/month on household items and food that works out to $750/person per month. However one person living in the same place (who would use maybe $130 in utilities and $350 in household and food) would spend $2480. So a savings of 70%.

However if you get 12 people in the same place and utilities go up to $300 and up to $2100 for household/food then the total cost becomes $4400/month, or about $367/month for a savings of 85%. The difference between a 70% and an 85% decline in price is nowhere near the gain from adding the first new person, which will drop costs by slightly over 50%.

Plus as you add more people you need to consider a larger place with a larger storage area, more bathrooms, a more advanced kitchen. So you have to also add those costs into account.

So I’m wondering if due to having 8 million long term unemployed if dorm housing will become far more common. Tent cities used to spring up, but they were torn down by various governments.

[(rent)+(utilities for one person1.2x)+(household for one person0.8x)]/(x) seems like it might work as a rough estimate with x being the number of people living there. But it doesn’t take into account the breaking point of more people requiring more storage, more bathrooms and more kitchen areas. So costs will go up since you need a bigger place.

You’d assume 5-6 people is the max you can comfortably get into a house designed for 2 people w/o running into problems in those areas.

Is the peak of benefit around 2-4 people in a 2 bedroom apartment before you run into problems? The first addition might drop costs by 55% or so. The second may cut it to 64%, the third to 70% (as guesses).

Is anything beyond the second person overkill? Or does adding a 3rd or 4th matter?

2

I don’t know if my college experience was typical–it probably was for Southern California. At UCSD most students living on campus shared a room with one other person; five rooms would be grouped around a lounge and bathroom shared by all ten people. Yet, even with ten people sharing one bathroom, I don’t remember there ever being much of a problem. The bathroom wasn’t like what you’d find in a private home, but boasted two stalls and a couple of sinks. Like most public restrooms, it could be used by more than one person at a time, and I imagine the plumbing was planned with this in mind. In a house or apartment originally built for one or two people, or a small family, you wouldn’t have that. I suspect that as more people crowded in there would be more frequent breakdowns of the plumbing, to the point where the added expense incurred by that would negate any savings. I also suspect the same is true of the electrical system; five to ten students on any given evening burning lights, watching TV or listening to their stereo systems (remember, this was 1976), is bound to place a greater demand on the system than what a couple of people in an apartment would use.

3

My brother is in UCSD and I noticed his bathroom was build in a way I’d never seen. There were 2 rooms, one had the shower and the other the toilet. And there were 2 sinks that weren’t in the bathroom(s), they were in the hallway outside the bathroom.

4

Here’s your problem.

Percents are a flawed measure, especially in the hands of people who don’t quite know how to work them. The difference between “70% less” and “85% less” is huge. A factor of two. $750 to $367.

Logarithms… we should use them more.

5

Anyway, I think dorms for adults are a great idea. Not only do you save money, but you live a happier, more social life.

And to think how much money and effort we’ve spent taking civilization in the other direction.

6

A true mathematical model is not a straightfoward thing, but linear approximations are often used. A linear model would look like this:

(rent) + (flat utilities) + (variable utilities)*x + (flat consumables) + (variable consumbales)*x

Note in your original formula you tried to combine the flat and variable parts into a coefficient. This is not correct at all. A flat utility is heat. A variable utility is water. Flat consumables might be more rare, but they exist if you think about them. Rent, in theory, could also have a variable component, but in your case that component is zero so is omitted.

7

We learned in environmental engineering class that multi-family housing units use less water (and thus produce less sanitary sewer waste) per capita than single-family dwellings.

While looking for a cite, I ran across this paper which explains that ‘common area laundry rooms’, such as a laundry room for an apartment complex, can save water.

8

Wes, is the question you’re asking about finding the optimal number of renters, or is it something a bit more complicated than that?

If it is indeed about the optimal number of renters in a given space, which it seems to be, then you’d need to, as Alex suggested, describe a number of things as a function of x, where x is the number of renters.

Some things would be benefits from each additional renter - paying a smaller share of the rent, for example, is a benefit. Some things, as your intuition indicated, would be costs - for example, you’d have less space, and everyone would receive a smaller share of a bedroom. Basically, you have a marginal product of the additional renter, and it’s either positive or negative - as you intuited, you’ll get a high positive marginal product for the first few renters, because you’ll face high benefits but a low cost. Slowly, that marginal product will lower, and eventually it’ll become more costly (not financially but in terms of space and general unhappiness). Unfortunately, the best way to analyze this once you’ve estimated your function is by using calculus.

You could estimate this using a logistic-type differential equation, which is often used to model population growth with a carrying capacity (read: adding renters to a space of fixed size). Again, though, that’s that filthy calculus stuff.

I’m saying this not to discourage you but to let you know that what you want to do is certainly analyzable at a more exact level than Alex’s perfectly reasonable suggestion of linear approximation.

9

Which college? UCSD has grown a lot since my day, but I think Revelle and Muir are pretty much the same as they ever were, allowing for updated technology. I was in the Revelle “Mud Huts”, which I’m told were once military quarters. The bathroom wasn’t two rooms, but more like an ordinary public washroom with a shower. The shower wasn’t completely separated, but was in a sort of alcove of its own.

Biffy The Elephant Shrew was there in the 1970s too. I must have known him by sight but he hasn’t as yet revealed his identity to me.

10

Nonetheless, my point was that the benefits seem to follow an L-curve and the biggest benefits come from the first 1 or 2 people.

So if cost of living is $2500/month per person for one person, it gets cut down to about $1400 per person for two people. With three people it goes down to roughly $1000/month.

So you save as much money in real dollars and a percentage from adding the first person (going from 1 to 2) as you do from adding 10 more after that (going from 3 up to 12). Because with 12 you are looking at about $400/person for rent, utilities, household and consumables. With 1 it is $2500 and with 2 it is roughly $1400/person.

One thing that made me curious about this was looking at rent in a high cost of living area. I noticed it didn’t really matter if 10 people were living in a house/apartment or 4, the amount people were charging for rent and utilities was pretty much the same.

As far as variables based on population, I think most utilities would vary, but not by much. The only flat utility is probably water, many charge a flat rate. Others like electric, phone (you may need 2-3 lines), internet, heating and cooling would vary. If you have 1-2 people then the heating and cooling would be turned off for most of the day. But with 12 people there are going to be at least some people at home 24/7, so the heating and cooling bills (as well as electric in general) go up. With the internet, with 1-3 people you can get by on 3Mbps DSL connections. With 6 people you may need a 7.5Mbps DSL. With 12 people you might need a 10-20Mbps FiOS line.

Water is the only flat utility I can think of, and even that isn’t flat in some cities as they charge by the quantity.

11

He lives in Mesa grad housing up in La Jolla.

12

I think you’re still not getting it. 1->2 is 1.8x less. 2->3 is 1.4x less. You see it as a big drop-off in benefit. But 1->2 was a doubling in population, while 2->3 is a 1.5x increase. Apples-to-apples would be to examine 2->4. In fact, justing using your numbers, you get more benefit going from 2->3 (1.5x improvement / 1.5x increase in ppl) than 1->2 (1.8x / 2.0x).

Of course, the benefits will still drop off, but more gradually that you’re interpreting.

13

You don’t need 20Mbps for 12 people to browse the web. If everyone is torrenting porn, then yeah. But you can share the porn. Preferably all at the same time. Don’t you get communism, man?

14

I think one way to examine this is to plot 2 different curves. A $ Cost curve - which will go down over time - more steeply at the beginning and then ultimately approaching a flat line. This curve, as others have mentioned would need to include both fixed and variable costs. And then you need a second curve, called something like Emotional Costs or Dissatisfaction, or whatever. This curve would probably be more of an S curve, relatively flat and low with the addition of 0, 1 or 2 - then pretty steep when you get to 7+, then at some point, flat again when everyone is as pissed off as they can get.

Then - to optimize your solution, you look for the intersection of these two curves. At least, that’s how economists would look at this problem.