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By Philip Ogren

In this first of two blog posts I estimate that there are more than 28,000 empty bedrooms in the city of Boulder using data from the American Community Survey from the U.S. Census Bureau. Most of this blog post is dedicated to how I came up with this number. In part two, I will give commentary about why I think it matters.

Introduction

In November of 2021, voters in Boulder rejected a ballot initiative that would have allowed more than three unrelated people to occupy a house -this is the occupancy restriction currently in effect through most of the city. The provision would have allowed one person per bedroom plus one.  So, for example, a 5-bedroom house could have six unrelated people legally living together. Opponents of the measure argued that this was a giveaway to greedy landlords and developers who would conspire to turn every house in the city into dormitories. A subtext of these arguments was that over-occupancy was already a major problem and was ruining the city. But the notion that Boulder is over-occupied defies my own experience living here and contradicts demographic trends that are well established. My own anecdotal experience suggests that there are vast numbers of large houses that are occupied by just one or two people. A few years ago I attended an economic forum sponsored by the Boulder Chamber. One of the major demographic trends that was highlighted is that Boulder’s population is aging. Various articles highlight the fact that Boulder’s population has declined in recent years and that the school system is projecting significant decreases in enrollment in the next decade. Both of these trends are consistent with an aging population of “empty nesters” who age in place in “family-sized” houses.

The previous provides a partial quick sketch of the milieu from which I have become fixated on a specific question that I would like to know the answer to.  How many empty bedrooms are there in Boulder? It seems like an obvious question to ask for which there should be a ready answer. And so, I keep asking people who I think might know the answer. But the vague hand-wavy answer that I keep getting is some version of “probably a lot but nobody knows.” So, I was surprised when I recently asked an economist about this and got back a relatively specific answer with a number attached to it. At the December 7^th  2022, Housing Advisory Board^[1]of which I am a member. Let me give the standard disclaimer that the opinions expressed here are my own and do not represent HAB. meeting we had a guest speaker that evening named Emily Hamilton who is an economist and senior research fellow at the Mercatus Center which is a libertarian think tank at George Mason University. She gave a great presentation on “light touch density” and talked about what works and doesn’t work in the real world and it sparked a lively discussion which included a chance for me to ask about how would one calculate the number of empty bedrooms in a city.  In a follow-up email she said:

To your question about empty bedrooms:

What’s possible to identify with Census microdata are households that have more bedrooms than people at the level of a Public Use Microdata Area (PUMA). PUMAs are geographic areas that the Census defines for which data can be shared without concerns for being able to identify individuals in the data.

The PUMA that includes the city of Boulder has about 235,000 people, so it’s much larger than the city itself. Within this PUMA, there are about 75,000 “extra bedrooms” defined as a bedroom in a house above the number of people in a house. So a couple living in a three bedroom house would have one “extra bedroom” by this definition, and a single person living in a two bedroom house would also have one “extra bedroom.”

We had a bit of back and forth about how she used the census microdata from ipums.org and I spent a few hours trying to replicate what she reported but I found the data source to be too confusing for me and I abandoned this effort.  My takeaway from this exchange is that there are many empty bedrooms in Boulder – probably 10’s of thousands and that census data is a fruitful place to try to answer my question.  Additionally, it seems reasonable to use her data to get a rough guesstimate of how many empty bedrooms there are in Boulder using a simple proportional approach.  This gives an estimate of between 30K and 33K empty bedrooms depending on whether you use the population in Boulder living in households (93,965) or the total population including residents living in group quarters (104,930).

What follows is my best attempt to answer the question of how many empty bedrooms there are in the city of Boulder using American Community Survey data from data.census.gov. To be clear, I am not a demographer, much less a proficient user of census data. I have reasonably good analytical skills, but I am a first-time user of this resource armed with a confirmation bias to come up with a number based on the preceding estimate to bolster my personal politics which leans heavy towards more inclusive and affordable housing policy that favors walkable neighborhoods, compact housing units, and shared housing. So, take my analysis here with a heavy dose of skepticism and please consider sending me any feedback that might clarify any faulty assumptions I might have made. I also want to confess, hopefully without sounding too conspiratorial, that I believe that data exist to definitively and unambiguously answer this question – I just don’t know where to find it and if I did, I probably wouldn’t have access to it.  For example, I assume an analyst at MLS or Zillow would know how to provide solid numbers on the number of bedrooms in Boulder.  As such, I will try to be humble in my speculative assumptions used in the analysis below and hope that if/when an empirically verifiable answer to my question manifests, that I won’t look too stupid. 

What is an empty bedroom?

For the purposes of this blog post I am going to define an empty bedroom as any bedroom that exceeds the number of occupants in a household. We are not going to worry about whether couples (for example) occupy one or two rooms. If there are two people in a household and two bedrooms, then there are no empty bedrooms. If there are two people in a household with four bedrooms, then there are two empty bedrooms. This is consistent with the definition provided by Dr. Hamilton above. 

Some housing units are described as having “no bedrooms”. These correspond to efficiency or studio apartments. Descriptions of the data below are explicit that the counted units described as such are occupied – so we will essentially treat these as having one-bedroom.

Household sizes and numbers of bedrooms per unit

If you poke around on the data.census.gov website, you may marvel at the amount and variety of data published by the American Community Survey.  I scanned over 60 pages of tables looking for relevant tables for this analysis – and I didn’t even finish paging through all that is available.  It should not surprise you to hear that I have not yet found a table that provides statistics about the number of empty bedrooms!  But there does exist data that provides household sizes and the number of bedrooms in housing units.  In short, I am going to do my best to combine these two datasets using what I think are reasonably conservative assumptions to come up with educated estimates.  All data is from the 2021 ACS 5-Year Estimates Subject Tables which is described as more accurate than corresponding 1-Year Estimates.  Table 1 shows the data for household sizes in Boulder. Table 2 shows the data for number of bedrooms. Note that the total number of households of 42,610 in Table 1 is the same as the total number of occupied housing units in Table 2. Also, notable is that we do not know how many 5- or 6-person households (or larger) there are. We just know there are 6,091 4-or-more-person households. Similarly, we do not know how many 2- or 3-bedroom housing units there are. We just know that there are 23,655 units that are “2 or 3 bedrooms”. Also, we do not know how many 5- or 6-bedroom units there are. We just know that there are 10,032 units that are “4 or more bedrooms”.

Household Size	Occupied Housing Units
1-person household	15,244
2-person household	15,313
3-person household	5,962
4-or-more-person household	6,091
Total households	42,610

Number of bedrooms	Occupied Housing Units
No bedroom	1,867
1 bedroom	7,056
2 or 3 bedrooms	23,655
4 or more bedrooms	10,032
Total	42,610

Households and Group Quarters

It is worth pointing out that not everyone in Boulder lives in households. Of the 104,930 people living in Boulder, 93,965 people live in households.  The remaining 10,965 live in “group quarters” which includes “college residential halls, residential treatment centers, nursing facilities, military barracks, prisons, and worker dormitories”.  Because Boulder is a college town and over 30% of the population is undergraduate and graduate students it is probably safe to assume that some large percentage of the 10,965 city residents living in group quarters are college students living in on-campus residence halls. However, I have not been able to find any tables in the American Community Survey that describe the demographics of who those nearly 11,000 residents are. It is possible that there exist empty bedrooms in group quarters but we are not making any attempt to quantify those here.

	Number of People
In Households	93,965
In group quarters	10,965
Total	104,930

Number of bedrooms minus the population living in households

One simple (though blunt) way to estimate the number of empty bedrooms is to calculate the number of bedrooms found in households and subtract the number people living in households. Let’s start by making the most conservative possible estimate of the number of bedrooms found in households using the data in Table 2. Since we don’t know how many of the 23,655 “2 or 3 bedroom units” are 2- or 3-bedrooms, respectively, let’s pretend that there are no 3 bedroom units.  Similarly, this table doesn’t tell us how many 5-, 6-, or 7-bedroom units there are – so let’s pretend that they don’t exist either.  We will count one bedroom for each “No bedroom” unit since they are occupied.  This gives us 96,361 bedrooms as shown in Table 4.  There are 93,965 people in Boulder who live in households.  Subtracting the number of people from the number of bedrooms gives us 2,396 empty bedrooms.  This estimate is shown in the first row of Table 11. 

Number of Bedrooms in Unit	Units	Bedrooms
No bedroom	1,867	1,867
1 bedroom	7,056	7,056
2 bedrooms	23,655	47,310
4 bedrooms	10,032	40,128
Total	42,610	96,361

Incorporating household sizes

One of many problems with our initial estimate is that we completely ignored what we know about household sizes from Table 1.  For example, we can deduce that there are some number of 1-person households living in 2-bedroom (or larger) units simply because there are fewer 1-bedroom units than there are 1-person households.  For this estimate, we will assign households as efficiently as possible so that, for example, for each 1-bedroom unit we will assign a 1-person household if possible.  For this estimate we will ignore the fact that 5- and 6- person (and larger) households exist since this will not increase our estimate of empty bedrooms.

Number of Bedrooms	Number of people	Households	Empty Bedrooms
No bedrooms	1-person	1,867	0
1-bedroom	1-person	7,056	0
2-bedroom	1-person	6,321	6,321
2-bedroom	2-person	15,313	0
2-bedroom	3-person	2,021	0
4-bedroom	3-person	3,941	3,941
4-bedroom	4-person	6,091	0
Total		42,610	10,262

Note that in Table 5 if you sum the first three rows which correspond to 1-person households that you will get the corresponding number of 1-person households found in Table 1. Or if you sum the third, fourth, and fifth rows corresponding to 2-bedroom units, you will get the respective number found in Table 2. The only rows that correspond to empty bedrooms in this table are 2-bedroom units occupied by 1-person households and 4-bedroom units occupied by 3-person households. Again, we have precluded the possibility of a 2-person household residing in a 4-bedroom unit simply because we are making the most efficient assignments possible. The 10,262 empty bedrooms estimate (shown in the second row of Table 11) given by this approach is indeed very conservative.

Let’s assume 3-bedroom, 5-bedroom, and 6-bedroom units exist

The previous two estimates assume, for the sake of being conservative, that there do not exist any 3-bedroom units or for that matter anything larger than a 4-bedroom unit. I know from personal experience that both exist and are quite common, but I don’t know how to estimate the number of the 23,655 “2-or-3-bedroom” units are 2 and 3 bedrooms, respectively.  I think that assuming that there are more 3-bedroom units than 2-bedroom units is too aggressive and will unreasonably inflate our estimate of empty bedrooms. Conversely, I think to assume there are more than twice as many 2-bedroom units than 3-bedroom units would swing the pendulum too far in the other direction. I think a conservative estimate is that 60% of the units in question have two bedrooms and the remaining have three bedrooms. 

Similarly, let’s guesstimate the number of 5- and 6- bedroom units. In this case, I think it is reasonable to conservatively assume that there are many more 4-bedroom units than the number of 5- and 6-bedroom units combined. What I’m going for here is something that resembles rapid exponential decay. Let’s say 70% of the units are 4-bedroom, 20% are 5-bedroom, and 10% are 6-bedroom. We will pretend that houses larger than 6 bedrooms do not exist. Table 6 shows an estimate of the number of bedrooms in Boulder given what I hope are reasonably conservative assumptions.

Number of Bedrooms in Unit	Units	Bedrooms
No bedrooms	1,867	1,867
1-bedroom	7,056	7,056
2-bedroom	14,193	28,386
3-bedroom	9,462	28,386
4-bedroom	7,022	28,088
5-bedroom	2,006	10,030
6-bedroom	1,004	6,024
Total	42,610	109,837

Again, by subtracting the population of people living in households (93,965) from the estimated number of bedrooms (109,837) we get an estimated 15,872 empty bedrooms which is shown in row 3 of Table 11. 

Incorporating household sizes (again)

Now before we do the exercise of efficiently assigning households to units based on household size and number of bedrooms, let’s acknowledge the fact that 5- and 6-person households exist. In fact, a bit of simple math tells us that the average household size for the 6,091 households with 4 or more people is just a hair under five. Since we have conservatively capped the number of bedrooms to six we are going to similarly cap household size to six to avoid inflating the number of empty bedrooms by allocating more people to large households which would have the effect of allocating fewer people to smaller households and therefore push up the number of empty bedrooms. We have less wiggle room here for estimating the number of households of different sizes because we know the total population. My intuition suggests that there’s more 4-person households than 5-person households and that the number of 5-person households is more than 6-person households. However, there is no way to have these numbers exhibit exponential decay and have the total population come out right without introducing larger household sizes which we are trying to avoid so as not to inflate our numbers. Table 7 shows estimates of the number of households with 4, 5, and 6 people with numbers that are close to linearly decreasing. 

Household Size	Occupied Housing Units	Population in Housing Units
1-person household	15,244	15,244
2-person household	15,313	30,626
3-person household	5,962	17,886
4-person household	2,150	8,600
5-person household	2,037	10,185
6-person household	1,904	11,424
Total	42,610	93,965

Now we will assign the n-person households shown in Table 7 to the n-bedroom units shown in Table 6 as efficiently as possible to get an estimate of empty bedrooms using the same technique we used for the results in Table 5.  This estimate of 17,703 empty bedrooms is shown in Table 8 and Table 11 as row 4.

Number of Bedrooms	Number of people	Households	Empty Bedrooms
No bedrooms	1-person	1,867	0
1-bedroom	1-person	7,056	0
2-bedroom	1-person	6,321	6,321
2-bedroom	2-person	7,872	0
3-bedroom	2-person	7,441	7,441
3-bedroom	3-person	2,021	0
4-bedroom	3-person	3,941	3,941
4-bedroom	4-person	2,150	0
4-bedroom	5-person	931	0
5-bedroom	5-person	1,106	0
5-bedroom	6-person	900	0
6-bedroom	6-person	1,004	0
Total		42,610	17,703

Not-perfectly-efficient assignment

I did some guesswork as it relates to the number of 2-, 3-, 4-, 5-, and 6-bedroom households and similarly for 4-, 5-, and 6-person households and this does, in fact, increase our estimates.  However, I think we are still badly underestimating the number of empty bedrooms.  The “efficient assignment” technique we have used results in zero 2-person households residing in 1-bedroom units.  Again, my personal experience suggests that 2 people living in a 1-bedroom unit is common.  Similarly, my personal experience contradicts the notion that a 1-person household never resides in a 5-bedroom house.  So, let’s try to make some reasonably conservative estimates here and see what happens to the numbers.  In particular, we will assume that two-thirds of 1-bedroom apartments are occupied by a 1-person household and the other third is occupied by a 2-person household.  We will then have the remaining 1-person households exponentially decay as a function of assignment to increasing bedrooms.  That is, after we make assignments for all of the 1-bedroom units we will assign 80% of the remaining 1-person households to 2-bedroom units.  After that we will assign 80% of the remaining 1-person households to 3-bedroom units and so on.  We will repeat this process for 2-bedroom units by assuming that 2/3 of the remaining units have 2-person households and the remaining 1/3 have 3-person households.  After that, the remaining 2-person households will exponentially decay across 3-, 4-, 5-, and 6- bedroom units.  We will continue this pattern as faithfully as we can until the constraints imposed by the totals require us to fill in cells with remainders.^[2]I seriously hope that Hal Gabow, who made me read Václav Chvátal in grad school, doesn’t read this blog post as he would be ashamed of me for not framing this as a linear program.  This approach results in an estimate of 24,115 empty bedrooms as show in Table 9 and row 5 of Table 11.

# of Bedrooms	# of people	Households	Explanation	Empty Bedrooms
0	1	1,867	One person per efficiency/studio	0
1	1	4,704	2/3 of 1-bedrooms have 1-person	0
1	2	2,352	1/3 of 1-bedrooms have 2-person
2	1	6,938	80% of remaining 1-person households	6,938
2	2	4,836	2/3 of remaining 2-bedroom have 2-person	0
2	3	2,419	1/3 of remaining 2-bedroom have 3-person	0
3	1	1,388	80% of remaining 1-person households	2,776
3	2	6,500	80% of remaining 2-person households	6,500
3	3	1,049	2/3 of remaining 3-bedroom have 3-person	0
3	4	525	1/3 of remaining 3-bedroom have 4-person	0
4	1	277	80% of remaining 1-person households	831
4	2	1300	80% of remaining 2-person households	2600
4	3	1,995	80% of remaining 3-person households	1,995
4	4	1,625	Remaining 4-person households	0
4	5	1,825	Value depends on previous four	0
5	1	56	80% of remaining 1-person households	224
5	2	260	80% of remaining 2-person households	780
5	3	399	80% of remaining 3-person households	798
5	4	0	No remaining 4-person households	0
5	5	169	80% of remaining 5-person households	0
5	6	1,122	Value depends on previous 5 values	0
6	1	14	Remaining 1-person households	70
6	2	65	Remaining 2-person households	260
6	3	100	Remaining 3-person households	300
6	4	0	No remaining 4-person households	0
6	5	43	Remaining 5-person households	43
6	6	782	Remaining 6-person households	0
Total		42,610		24,115

Untethered from reality

I think Table 9 represents an interesting thought exercise that shows how easily one can estimate more than 20,000 empty bedrooms in Boulder using available data and what I think are very conservative assumptions.  Let’s make explicit some of the assumptions that Table 9 makes:

• no more than one person ever occupies a bedroom.  
• 60% of “2 or 3 bedrooms” units are 2-bedroom units
• 70% of “4 or more bedrooms” units are 4-bedroom units
• there are no 7-bedroom units (or larger)
• there are no 7-person households (or larger)
• 4 or more people never live in a 2-bedroom unit
• 5 or more people never live in a 3-bedroom unit
• 6 or more people never live in a 4-bedroom unit
• 78% of 6-bedroom units have 6-person households in them
• 64% of 5-bedroom units have 5-person households in them
• 67% of all empty bedrooms can be found in 2- or 3-bedroom units
• less than 4% of 4-bedroom units have only 1 person in them

I would not bet large amounts of money that any of these assumptions are true but I would definitely bet good money on most of these being false – by a mile.  And each of these assumptions, if false, could increase the number of empty bedrooms. Since you have indulged me in coming up with an estimate that is untethered from reality that makes many unrealistic assumptions – let me make one more audacious estimate. I don’t want to claim that it is any more tethered in reality than the previous estimate – but I’m putting my stake in the ground, and I’m betting all my chips that the following estimate is still less than the true number if we were to actually count the empty bedrooms. I challenge you to prove me wrong with hard data!

Intuitions about empty bedrooms

If you live in Boulder, I’d like you to think about the housing units near you or those of people you know that you have visited and do a little mental inventory of their homes.  For example, when I walk through my neighborhood, I can point out many 3-, 4-, and 5-bedroom houses that have empty rooms in them.  For example, a nearby neighbor lives alone in a 5-bedroom house and another nearby couple lives in a 3-bedroom house. A few years ago, I was part of a men’s group where the average age was significantly older than me (I am in my late 40’s) and I shared a meal in more than a dozen different houses over the course of a couple of years. Most were large houses occupied by one or two people who had raised a family in that house and their children had long since moved out. These anecdotes are consistent with a demographic trend of aging empty nesters who have never “right sized” – so to speak. This is why BVSD schools in Boulder have been having downward trending enrollment for a decade (See pages 18 and 19 of this DAC presentation.)  Anecdotally, I’ve compared notes with many others on this subject and there seems to be an informal consensus that there’s at the very least a strong perception that there are many large houses in Boulder that have one or two occupants. It’s curious to me that I can’t actually think of any large houses that I personally know of that don’t have empty bedrooms. 

One statistic that jumps out at me when I study the census data is the large number of large households in Boulder. Again, while I had to guesstimate the number of 5- and 6-person households, I did that subject to the constraint that the total population is known. One possibility for the large number of large households is that there are many multi-generational families living together in our Latinx communities and that many of these families may not be living in large houses.  In fact, for example, many of these large households could perhaps be found in manufactured home neighborhoods.  If the census data obfuscates the fact that many of our large households in Boulder live in 2- or 3-bedroom housing units, then that would suggest that there are more empty bedrooms than what I’ve estimated above.

One final estimate

For this final estimate, I’m going to make assumptions that are more consistent with my own intuitions about the occupancy of Boulder’s housing stock. Here I add the assumption that 1/3 of large households (4-, 5-, and 6-person households) are in 3-bedroom units and the other 2/3 of large households are in larger units. I also Table 10 shows the order in which I filled out the table and gives an explanation for each entry. After assigning large households I then assign 1-, 2-, and 3-person households to each n-bedroom sized house type such that they are distributed according to the remaining distribution of 1-, 2-, and 3-person households. as closely as possible. For example, the 6-bedroom houses are given 187, 279, and 130 households of 1-, 2-, and 3-persons, respectively. This distribution is very close to the respective distribution for 2-bedroom units. This approach results in an estimate of 28,242 empty bedrooms and is shown in row 6 of Table 11.

# of Bedrooms	# of people	Households	Explanation	Empty Bedrooms
0	1	1,867	1: One person per efficiency/studio	0
1	1	4,704	2: 2/3 of 1-bedrooms have 1-person	0
1	2	2,352	3: 1/3 of 1-bedrooms have 2-person	0
2	1	4,462	14: remaining 1-, 2-, and 3-person households are assigned proportionally to the remaining housing units	4,462
2	2	6,669	14: see above	0
2	3	3,062	14	0
3	1	2,336	14	4,672
3	2	3,491	14	3,491
3	3	1,608	14	0
3	4	715	4: 1/3 of 4-person households are in 3-bedroom units	0
3	5	678	5: 1/3 of 5-person households are in 3-bedroom units	0
3	6	634	6: 1/3 of 6-person households are in 3-bedroom units	0
4	1	1,313	14	3,939
4	2	1,962	14	3,924
4	3	903	14	903
4	4	1004	7: 70% of remaining 2/3 of 4-person households are in 4-bedroom units	0
4	5	951	8: 70% of remaining 2/3 of 5-person households are in 4-bedroom units	0
4	6	889	9: 70% of remaining 2/3 of 6-person households are in 4-bedroom units	0
5	1	375	14	1,500
5	2	560	14	1,680
5	3	259	14	518
5	4	287	10: 20% of remaining 2/3 of 4-person households are in 5-bedroom units	287
5	5	271	11: 20% of remaining 2/3 of 5-person households are in 5-bedroom units	0
5	6	254	12: 20% of remaining 2/3 of 6-person households are in 6-bedroom units	0
6	1	187	14	935
6	2	279	14	1,116
6	3	130	14	390
6	4	144	11: remaining 4-person households are in 6-bedroom units	288
6	5	137	12: remaining 5-person households are in 6-bedroom units	137
6	6	127	13: remaining 6-person households are in 6-bedroom units	0
Total		42,610		28,242

Estimate Approach	Number of Empty Bedrooms
Conservative Number of bedrooms – population living in households (see Table 4)	2,391
Conservative Number of bedrooms incorporating household sizes (see Table 5)	10,262
Estimated Number of bedrooms – population living in households (see Table 6)	15,872
Estimated Number of bedrooms incorporating estimated household sizes (see Table 8)	17,703
Empty bedrooms assuming 2/3 of 1-bedroom households have 1 person in them, and the remaining 1-person households are spread out over the other sized households using exponential decay.	24,115
Empty bedrooms assuming 1/3 of 4-, 5-, and 6-person households are in 3-bedroom units and 1-, 2-, and 3-person households are distributed in each housing unit size according to remaining distribution of n-person sized households	28,242

Conclusion

In this post I have generated several estimates of the number of empty bedrooms by starting with very strict assumptions licensed by the data and, step-by-step, updating the set of assumptions using my best intuitions and guesswork.  My goal here is to show that it is quite likely to that 10’s of thousands of empty bedrooms exist in this city. I am putting a stake in the ground and making the best possible guess I can.  My best guess is that there are over 28,000 empty bedrooms. Despite my liberal use of guesswork I would bet good money that 28,000 is an underestimate.  Remember that all of the estimates above assume that each resident occupies one room by themselves.  It isn’t hard to imagine a couple occupying a single room in a 2-bedroom unit or a family of four occupying 2 bedrooms of a 4-bedroom house. I hope this thought exercise generates lively community debate and that this will lead to a more accurate accounting of how many bedrooms there are and how many of them are occupied by people.