What does it cost when you buy a 300-page paperback from Amazon? The price of the book is only the initial cost. When you store it on your bookshelf, it takes space and renting a space costs money. The price of the book might be only $10; the storage cost for 15 years is probably higher than $5.5.
A 1000 square feet apartment costs around $2100/month in the US. Roughly $300 of the cost is fixed cost independent of the size of usable area, including the cost of transaction, management cost, stairs, mechanical rooms, and partly the costs of outdoor areas, utility bills, and property tax. The rest of the rent should grow linearly with area. This leaves $1800/month for 1000 sq ft, or $1.8/month/sq ft, or $0.0125/month/sq inches.
In the short run, buying a book will not increase your rent, but in the long run you are mobile and you will pick a house that’s about the right size by equating your marginal benefit per sq ft per month to around $1.8.
Assume that a book is 8’’ × 5.5’’ × 1’’, and your bookshelf is 29’’ × 12’’ × 72’’ with 6 shelves plus the top. To calculate the minimal needed space, we squeeze as many bookshelves as possible in a library pattern, with aisles that are only 24’’ wide. Half of the area will be bookshelves and half will be aisles since 2 × bookshelf depth = aisle width. A book would cost 1’’ (width) × 12’’ (depth) of space on one of seven shelves (including the top) and an equal share of aisle space. Therefore, one book occupies 1” × 12” ÷ 7 shelves × 2 = 3.42 square inches of space.
Now the cost of storing one book a month is 0.0125 × 3.42 = $0.04286; or $0.5143 a year. Fifteen years? Let’s use a realistic annual discount rate of 4.5% to calculate the present value: $0.5143 / 0.045 × (1 – 1 / 0.045^15) = $ 5.5232. To store a book forever, the cost is: $0.5143 / 0.045 = $11.4286. Next time you bring home a free book from library, beware that you are signing a contract to pay 51 cents each year (adjust for inflation) until you get rid of the book.
We extend the model to calculate the minimum storage cost of any durable good. To maximize the storage capacity of a cabinet, we should build it up to the ceiling, have as many shelves as possible, orient the longest dimension of the item perpendicular to the wall to maximize the depth of the cabinet, and again arrange the cabinets in a supermarket style with aisles being 24’’ wide. Also assume everything is reachable with nothing in front blocking it.
There is inevitably some space between the stored items and the shelf above – about 25% of the height of the items. The thickness of the shelf along with the wasted space below the ceiling when the cabinet is not tall enough is another 15%. So to store a 10’’ tall item, we need 14’’ of space vertically, on average. We probably lose another 4’’ in the depth dimension (including the thickness of the cabinet’s door and back plate) plus half of the aisle – 12’’. Just like storing books, we usually don’t waste any space in the width dimension. Altogether, the share of storage space needed for any item with a given dimension is (depth’’ + 16’’) × (height’’ × 1.4) × (width’’) cubic inches.
From the two-dimensional rent of $0.0125/month/sq inches, and assuming the ceiling is 84’’ high (or we only store things below 84’’), we get the three-dimensional rent of 0.0125 / 84 × 12 = $0.00179/year/cubic inch. Our final generalized model becomes:
Annual storage cost of anything = (depth’’ + 16’’) × (height’’ × 1.4) × (width’’) × $0.00179
A more detailed version: annual storage cost of anything = (depth’’ + 16’’) × (height’’ × 1.4) × (width’’) × ($rent – $300) / (area in sq ft) / 144 * 12 months / (ceiling height”)
Most people only put storage units by the wall instead of in a dense library pattern. This lowers the capacity by 2 to 5 times depending on the size of the room. The owner probably utilizes the spared space in the center of a room for some other activity, so the storage cost of a book can be harder to calculate. On the other hand, if things are stacked densely without aisles in a basement, the storage cost is about 3 times lower.
|Item||Estimated Occupied Space (cubic inch)||Annual Storage Cost||15 Year Storage Cost||Perpetual Storage Cost|
|Book (original formula)||288||$0.51||$5.52||$11.43|
|Book (generalized formula)||241||$0.43||$4.63||$9.58|
|Textbook (600 page hardback)||486||$0.87||$9.34||$19.33|
|Board Game (11.8” × 11.8” × 2.8”)||1,286||$2.30||$24.72||$51.15|
|Jewel CD case||58||$0.10||$1.11||$2.31|
|Kleenex rectangular tissue box||588||$1.05||$11.30||$23.39|
|Cereal 14 oz||499||$0.89||$9.60||$19.86|
Besides storage costs, your decision to purchase may be affected by other factors, many of which are very subtle.
- Transaction cost.
- Delivery and organization cost
- Cost of getting used to buying a lot of things and becoming greedier
- The hassle to remember the location of your goods
- The effort to decide when to get rid of them (can be substantial)
- the possible hazard they bring (is it safe for children?) and the smell they have
- the effort to finally throw them away.
If you accept the fact that you will very likely regret buying things because you are irrational, then most purchases should not happen, but that’s a topic for another day.
The value of space-saving furnitures
A 6-shelf bookcase is priced $130; a similar 5-shelf bookcase is priced $100. Which one should I buy? The former is equivalent to 20% more of the latter (or 16.7% if top is another “shelf”) . The bookcase occupies a space of 29” × 12” × 2 (double it for the aisle space) = 696 sq inches, or $104 per year in rent. Twenty percent of it is $20.88, so the extra shelf is worth $20.88 a year, or $224.24 for 15 years! That’s much higher than the price difference between the two. Even after considering aesthetics and convenience, I see no reason to buy the 5-shelf bookcase. This assumes I’m willing to buy books to fill the 6th shelf and the value of buying and reading the extra books (consumer surplus) is not included.
The same goes for other furnitures such as wall beds, folding chairs and tables, space bags, and DIY cubic storage that can be built to arbitrary height.
Why we don’t have a good intuition for storage costs
Many people tend to buy or collect too many things and only realize they don’t have space to store all of them later on (a.k.a. hoarding). I believe part of it is because humans have not evolved a heuristic to consider the storage cost of things they bring home. Humans had been nomadic during almost all of their evolutionary history and costs were mainly in moving things when migrating. There was plenty of space to store any material and no space consideration was needed and thus such a mindset have not been developed.