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Segregation according to household size in a monocentric city
by Theis Theisen Abstract Over the last two centuries, household size has decreased considerably. Within a theoretical model I investigate the relationship between household size and the structure and size of cities. Household utility is assumed to depend on household size, in addition to the consumption of housing and a numeraire good. This basic building block is combined with a Muth-type urban model. The model is used for examining the impact of household size on the sorting of households according to household size, the geographical extension of the city, household utility, forms of housing etc. These issues have previously received some attention in empirical studies, but I am not aware of theoretical examinations. Contact details: Adress: Theis Theisen, Department of Economics and Business Administration, Faculty of Economics and Social Sciences, University of Agder, Servicebox 422, 4604 Kristiansand, Norway. E-mail:

[email protected] Phone: +47-38141526, Fax: +47-38141061.

**1. Introduction **
In industrialized countries, household size has over the last two centuries decreased
considerably. This demographic transition has taken several forms. In an early phase, multi-
generation households split up in separate households, and the number of children in each
family declined. In a later phase, families split up in single-parent and single-person
households, young people lived as singles for a longer period than was usual some decades
ago, and the number of elderly living as singles also increased. In the present paper, the focus
is mainly on the third of these phases. A few numbers may illustrate how dramatic these
changes have been. In 1950, about 15 per cent of the Norwegian households consisted of only
one person. By 2001 this share had increased to 38 per cent. Moreover, in the largest city in
the country, Oslo, the majority of households today contain only one individual.
The effects that arise when multi-person households replace single-person households are
quite complex, both at the household level and at the social level. At the household level, a
single-person household usually will have a lower income than a larger household. If there are
economies of scale in household consumption, individuals in single-person households will
presumably also obtain a lower utility level than they could enjoy in a larger household. These
differences are likely to result in different consumption patterns. In particular, in a city
containing many single-person households a larger number of housing units will be
demanded. Presumably, single-person households will also occupy smaller dwellings than
multi-person households. This will have a strong impact on the land market, population
density, and the geographical extension of the city. One would also expect that households of
different size would settle in separate segments of the city. This raises the question of who
would live in the central parts and who at the outskirts. Another related issue of some interest
is which group will live in apartments in the central parts of the city and who will live in low-
density detached dwellings. To the best of my knowledge, these issues have not previously
been subjected to a thorough theoretical examination, despite the fact that they have a
profound impact on many aspects of people’s lives. In the present paper I therefore examine
the impact on the urban housing sector of the increasing number of single-person households
In empirical research, the impact of household size on the urban housing sector has received
some attention. (Add some references here) Skaburskis (1999) finds that a reduction in
household size leads to a drop in the demand for low-density housing. A better theoretical
understanding of such relationships is however needed, inter alia in order to improve the
specification of empirical models and facilitate the interpretation of the results. In addition, a
better theoretical grasp of the role of household size in the urban housing sector is needed to
guide policy decisions made by local governments. For instance, some local governments
pursue an active housing policy in order to increase the number of multi-person households
with children that live in the central parts of cities. Presumably, the analysis in the present
paper will prove to be useful in judging whether such policies are warranted.
In the next section, we set up the baseline model of household choice between housing and a
numeraire good. The baseline model is the same as the one used by Brueckner (1987), but we
extend the model by introducing households of different size, and derive the condition that
determine whether single-person households will reside in the central parts of the city or in
the outskirts. In Section 3, we set out a very simple model of the supply side of the housing
market. In Section 4 we establish the conditions that must be satisfied in order for the housing
market to be in equilibrium. Section 5 contains a comparative-static analysis of how a change
in the share of households of different size will affect the equilibrium. Section 6 summarize
the main results and provides some ideas for further research.

**2. Spatial segregation of single-person and multi-person households **

We consider a mono-centric city inhabited by individuals with identical preferences. Some
individuals live in single-person households, while others live in two-person households.
Households derive utility from a hicksian composite good (

*c * and from the floor space (

*q *
of their dwelling, where

*n * indicates household size. Let

*v *=

*v*(

*c *,

*q *;

*n * be the household
utility function. Since there may be economies of scale in household consumption, in
particular for housing, household size enters the household utility function. The utility
function is assumed to be strictly quasi-concave in

*c * and

*q *. The members of the household
work a fixed number of hours at a fixed wage rate. This gives the household an exogenous
income,

*y *. The household spends its income on the composite good, on renting floor space,
and on commuting to work in the (spaceless) centre of the city. Total costs of commuting
amounts to

*t x *, where

*x * is commuting distance, and

*t * is the cost of commuting one
kilometre (roundtrip) for all members of a household of size n. When the composite good is
chosen as the numeraire, the budget constraint takes the form

*y *=

*c *+

*p*
*p*(

*x*) is the rental cost per square metre of floor space located at distance

*x * from the city
centre. Substituting from the budget constraint into the utility function, the utility
maximization problem of a n-person household living at distance

*x * from the city centre takes
This yields the first order condition, where

*c*
*v * denote partial derivates of the utility
Since individual preferences by assumption are identical, all households of a given size will
have identical preferences. We also assume that all households of a given size have the same
income and costs of transportation. Under these assumptions, households of a given size
living at different distances from the city centre must obtain the same utility. Households of
different size will, however, in general differ in income and costs of transportation, and will
therefore enjoy different levels of utility. Hence, we have the restriction:

*v*(

*y *−

*p x q *−

*t x*,

*q *;

*n *=

*v * *n *∈ [ ,
where the bar on the r.h.s. variable indicates that there is a common utility level for all
households of a given size. We return in Section 4 to how these utility levels are determined.
Differentiating Eq. (3) w.r.t.

*x * yields the rent gradient:
Eq. (4) tells us that the rent gradient for floor space in equilibrium drops as one move away
from the city centre. Hence, households with high commuting costs are in equilibrium
compensated through a lower rent on floor space, so that all households of a given size will
enjoy the same utility, irrespective of where in the city they live.
Totally differentiating Eqs. (2) and (3) w.r.t.

*x *,

*y *,

*t *, and

*v *, we can find the impact of
these variables on the price of floor space and the demand for floor space. Since a complete
accord of this can be found in Brueckner (1987), we here just summarize these results in
Table 1 below, where ω < 0 is the slope of the hicksian (constant utility) demand curve, and

*v * is the marginal rate of substitution between the two goods in the utility
Table 1. The impact on floor space and price of floor space of changes in

*x *,

*y *,

*t *, and

*v *
*Modelling spatial segregation according to household size *
As indicated by Eq. (4), the rent gradient will in general be different for households of
different size. Since floor space at a given location will be rented out to the highest bidder, the
household type with the steepest rent gradient will occupy the central parts of the city. 1
Consequently, if single-person households have a steeper rent gradient than two-person
households, single-person households will in equilibrium bid up rents in the central parts of
the city, and occupy the central part, while two-person households will be willing to pay the
highest rents in the outer parts, and will occupy this part. On the contrary, if single-person
household have a less steep rent gradient than two-person households, single-person
households will in equilibrium occupy the outer part of the city, and two-person households
the inner part. In order to examine which of these two cases apply, consider the situation at
~ from the city centre, where we find the borderline between the areas occupied by
the two household types. At this location the rent-functions of two types of households cross,
and they will both have to pay the same rent per unit of floor space. Initially, let us also
assume that the two households have the same income, and that their transportation costs also
are identical (these initial assumptions will be modified below). Preferences are, however,
assumed to differ between household types as stated by:

*v be the marginal rate of substitution between floor space *
*and the numeraire good for a household of size n *∈ { ,
2

*. Preferences satisfy the restriction *
1 For instance, Brueckner (1977) demonstrates that high-income households will settle on large floor space in the outskirts of the city, while low-income households will occupy the central parts of the city. In a similar vein, Solow (1971) demonstrates that business activity will take place in the central parts of the city, while the recidential areas will occupy the less central parts.
In other words, Assumption 1 restricts preferences to be single-crossing. (Ad reference) In the
present context an important rationale behind single-crossing preferences is that there is likely
to be economies of scale in the consumption of housing, while this is likely to be considerably
less prevalent for the numeraire good. In Figure 1 the single-crossing preferences are
illustrated. At point A,

*v * is an indifference curve for a two-person household, while

*v * is an
indifference curve for a single-person household. If the two households have the common
income net of transportation costs represented by the budget constraint FG, the optimum for
the two-person household is at point A, while point B will maximize the utility of the single-
person household. Hence, if the two households have the same incomes and costs of
transportation, the single-person household will with the type of single-crossing preferences
stated in Assumption 1 consume less floor space than the two-person household. At distance
~ from the city centre a single-person household will then have a steeper rent gradient than a
two-person household. Consequently, single-person households will then occupy the inner
The assumption that households containing one and two persons have the same income and
transportation costs is highly unrealistic. Hence, in the sequel the analysis will be based on the

*Assumption 2: Each adult person earns the same income, y. *
Assumption 2 implies that income per capita in the city will be independent of city population
and of how individuals group themselves together in households. At least as a first
approximation, this seems reasonable. Moreover, since Wheaton (1974), Brueckner (1987)
and many other have found that average per capita income has a profound impact on the
equilibrium of the urban housing sector, we want to neutralize this source of difference
between cities. Assumption 2 serves this purpose.
Figure 1 Single-crossing preferences for households of different size.
In order to be able to make precise inference of how the income difference between
households of different size affects the consumption of housing, we also incur the following
assumption, which is frequently made in different variants of the mono-centric urban model:

*Assumption 3: Housing is a normal good*
Next. it is reasonable to assume that costs of transportation differ systematically between
households of different size. If each (adult) individual in the household holds a full job, and if
all costs of transportation to the city-centre were related to job-commuting, it would be
reasonable to assume

*t *= 2

*t .* Households incur, however, transportation costs also when
they travel to the city centre for shopping, or for visiting cultural amenities, restaurants, etc.
When measured on a per person basis, such transportation costs are likely to be much higher
for individuals living as singles than for individuals in two-person households. One reason for
this is that one individual in a two-person households often may do shopping for the whole
household. Another reason is that a single-person household is likely to incur more
commuting costs than a two-person household in order to obtain social contact with other
individuals at restaurants, cultural amenities, etc. For a two-person household much of the
social contact is obtained within the household, without any costs of transportation.2 Based on

*Assumption 4: The costs of transportation for households of size 1 and 2 are related as *
*follows* *t *= α

*t , where the parameter *α ∈ (
Assumptions 2 and 4 combined implies that income net of transportation costs for a single-
person household living at distance

*x*
~ from the city centre, will be less than half of what it is
for a two-person household at the same location. The budget constraint of a single-person
household is in Figure 1 shown as the line IJ.
2 At the cost of a substantially more complex modell it would be possible to include social contact in the utility function and to let the consumer choose endogenously the volume of this good. The simpler approach taken in the present paper correspons well, however, to the assumption that work participation as well as working hours and number of commuting trips are tken as exogenous.
Under Assumptions 1, 2, 3, and 4 the optimum of a single-person household living at distance
~ from the city centre will be at point C in Figure 1, while it for a two-person household will
be at A. From Figure 1 we can then conclude that the consumption of floor space for a single-
person household compared to a two-person household, both living at distance

*x*
city centre, is affected negatively by the difference in preferences as well as by the drop in
The analysis so far tells us that a single-person household living at distance

*x*
centre under reasonable assumptions will have a lower consumption of floor space than a two-
person household. However, since the single-person household also has lower transportation
costs, additional restrictions are needed in order to determine which household type will have
~ . For this purpose, let us rewrite the rent-gradient for a single-
is the relative reduction in demand due to different preferences for a
household of size 1 compared to a household of size 2, i.e. the move from A to B in Figure 1.
is the relative drop in demand due to Assumption 2, i.e. that a single-person
household has a lower income than a two-person household. If we in Eq. (5) have γ > 1 , a
single-person household will have a steeper rent gradient than a two-person household. This

*Proposition 1. If *α > (1 − β 1 − β

*, single-person households have at x*
*gradient than two-person households. When this condition is fulfilled, single person *
*households will occupy the dwellings closer to the city centre than x*
*households will live further from the city centre than x*
In order to assess the implications of the condition in Proposition 1, let us plug in the
borderline values α = 0.5 and β = 0 . We then obtain the condition β >
with our assumption of how household income changes when a two-person household is split

*E *, in the demand for floor space must
not be less than 1. Empirical estimates of this elasticity often lie around 1. Notice, however,
that this result is calculated on the very extreme assumptions that α = 0.5 and β = 0 . If we
plug in the more realistic assumptions α = 0.6 and β =

*E *≥ 0.50 . Since vast majority of
estimated income elasticities for housing are higher than this, we conclude that the condition
stated in Proposition 1 is likely to be satisfied, and that single-person households therefore in
equilibrium is likely to outbid two-person households for the dwellings closer to the city
~ . This situation is illustrated in Figure 2. The analysis in the sequel is based on
Figure 2 Spatial segregation of single-person and two-person households

** **

3. Segregation of households in different building types

The supply side of the housing market is represented by entrepreneurs who rent dwellings to
households. The supply-side model has previously been set out by Brueckner (1987). Housing
entrepreneurs produce housing, measured in square meters of floor space, from land (

*l *)

* *and
physical capital (

*K *) . The housing production function,

*H *(

*K *,

*l *)

*,* is assumed to be concave,
and to exhibit constant returns to scale. An entrepreneur’s revenue from renting out the
dwellings contained in the building sitting on a piece of land under his command is

*pH *(

*K*,

*l*)

*.*
In order to make the analysis tractable, we assume that housing entrepreneurs rent land and
physical capital from landlords and capital owners living outside the city under consideration.
Let

*r * denote the endogenously determined, spatially variable, rent per square meter of land,
and let

*i * denote the exogenously given, spatially invariant, rent per unit of physical capital.
The entrepreneur earns a profit (

*pH *(

*K *,

*l*) −

*iK *−

*rl*) from renting out the structure, including
the land affiliated with it, to households. With constant returns to scale the profit can also be
written

*l*(

*pH *(

*K l *)
, −

*i K l *−

*r*) , and introducing the capital-land ratio (

*S *=

*K l*)

*, *which
hereafter is denoted the structural density, the profit expression can be simplified to:

*l*(

*ph*(

*S *) −

*iS *−

*r *) (6)

* *
In this equation,

*h*(

*S *) ≡

*H *(

*S *)
,

* *measures floor space per square-meter of land. We assume
and second order derivates of the production function

*H *(

*S *)
,

* *w.r.t its first argument. For a lot
at a specific location, the entrepreneur maximizes profit through choosing the optimal
structural density (building height), taking the land-rent at the specific location as
exogenously given. This yields the first-order condition stating that in optimum the marginal
revenue from renting out the building structure must be equal to the marginal cost of physical
With constant returns to scale, producer profit must in equilibrium also be equal to zero. This

*ph*(

*S *) −

*iS *=

*r *, (8)
which tells that the profit net of the rent that the entrepreneur must pay for the physical capital
embedded in the building structure in optimum must be equal to the land rent. The land-rent at
each location must in equilibrium adjust to the levels that fulfil condition (8).
In Section 2 we found that the price per square meter floor space is a function of
ϕ =

*x*,

*y *,

*t *,

*v *. Taken together with Equations (7) and (8) this means that structural density
(

*S *) and land rent (

*r*)

* *will be functions of the same variables, plus the rent (

*i*) for a unit of
physical capital (which we assume exogenous and fixed throughout the analysis). Totally
differentiating Eqs. (7) and (8) w.r.t. ϕ =

*x*,

*y *,

*t *,

*v *, and solving for the impact of these
variables on structural density and land rent yields:
(ϕ =

*x*,

*y *,

*t *,

*v * (9)
(ϕ =

*x*,

*y *,

*t *,

*v * (10)
Since

*h*′ > 0 , and

* h *′ < 0 , we obtain by using the results for ∂

*p*
Hence, the land rent and buildings are lower the further one come from the city centre.
Comment on relationship with household size in this Section or in the next?

**4. Equilibrium conditions for a city with two household types **

Let

*M * be the number of single-person households, while

*M * is the number of two-person
households. We consider a city with an exogenously given population

*N *, i.e. the closed city
case. Hence, the total population of the city is

*N *=

*M *+ 2

*M .* We want to examine how a
decrease in the number of two-person households which is matched by an increase in the
number of single-person households, conditional on the population of the city being fixed,
affects the equilibrium of the city. That is, we will be interested in the effect on the
geographical extension of the city, the distance from the city centre to the borderline between
the areas occupied by the two households types, the rent level, the floor size of dwellings,
structural density, population density, and the utility of each of the two household types.
In order to examine these issues we have to add three equilibrium conditions to the model set
out in Sections 2 and 3. First, the two-person households living at the border of the city,

*x *,
must, through the price they pay for renting floor space at that location, be willing to pay a
land-rent that matches the rent that landlords can earn from renting out the land to agriculture
(assuming that agriculture is the alternative land-use that would be willing to pay the highest
rent). Denoting the exogenously given land rent in agriculture by

*r *, this gives the condition:

*r*(

*x*,

*y *,

*t *,

*v*
Since we in Section 3 found that land rent declines as one move away from the city centre,
two-person households will, when the condition in Proposition 1 is satisfied, outbid
agricultural land use between

*x * and

*x*
The second equilibrium condition states that at the borderline of the area occupied by single-
person households and two-person households, the two types of households must be willing to
pay exactly the same rent for a unit of land. This condition is formalized as:
=

*r x*,

*y *,

*t *,

*u *. (13)
Again, when the condition in Proposition 1 is satisfied, single-person households will outbid
two-person households closer to the city centre than

*x*
Next, let the housing density (number of households per unit of land) at distance x from the
city centre be

*D*(

*x*,

*y *,

*t *,

*v *. The two last equilibrium conditions require that each of the
population segments (households of size 1 and 2) must fit within the areas of the city that they
occupy. When θ denotes the number of radians that are available for housing, at each

*x *≤

*x *,
∫ θ (

*x*,

*y *,

*t *,

*v dx *=

*M *, (14)
∫ θ (

*x*,

*y *,

*t *,

*v dx *=

*M *. (15)
Eqs. (14) and (15) can be modified by writing housing density as follows:

*h q *= − (∂

*r *∂

*q t*

*h q *= − (∂

*r *∂

*q*
Substituting this into Eqs. (14) and (15) we obtain:
Integrating by parts on the l.h.s. of Eqs. (17) and (18) we obtain:
where

*r*~ is the land rent at distance

*x*
The four Equations (12), (13), (19), and (20) determinine the following four endogenous
variables: The extension of the city,

* *(

*x *) , the distance between the city centre and the
borderline (

*x*~)

* *between the areas where the two population segments live, and the utility
levels of the two population segments (

*v *,

*v *.

**5. Comparative static analysis of a city with two household types **

Totally differentiating Eqs. (12), (13), (19), and (20) w.r.t. χ =

*M *,

*M *,α , β , taking costs of
transportation,

*i *, and

*r * as fixed, and using the results from Sections 2 and 3, we can find the

*x*,

*x *,

*v , *and

*v *. First, differentiating Eqs. (12) and (13) we obtain:
In these equations, the subscripts on the land-rent functions indicate household type. Next,
differentiating Eqs. (19) and (20) w.r.t. χ =

*M *,

*M *,α , β and substituting from Eq. (22) in
From this we can find the impact on the utility level of the two household groups of an
increase in the share of single-person households. Once the impact on the utility levels are
found, Eqs. (21) and (22) give the effect on the border of the city and the border between the
areas occupied by the two households types. Next we can find the impact on rents, building
heights, etc. The derivation of these results, which is very much inspired of Brueckner (1983)
and Wheaton (1976), is quite technical, and is therefore relegated to the appendix which will
be included in the final version of the paper. The results can be summarized as follows:

*Proposition 2: When the number of two-person households drops, and the members of these *
*households form new single-person households, the following changes will occur: (I) the *
*utilities of both household types will decline, (2) the city border will be pushed outwards, (3) *
*the border between the inner part of the city occupied by single-person households and the *
*outer part occupied by two-person households will be pushed outwards, (4) land rent and the *
*rent of floor space will increase at each location, (5) bilding height will increase at each *
*location, and (6) housing density and population density will increase at each location.*
**6. Concluding remarks **

Within a framework where all

*individuals* have identical preferences, but where economies of
scale gives rise to different single-crossing preference maps for single-person and two-person
households, we demonstrated that single-person households will occupy the central parts of
cities, while two-person households will live in the outer parts. It is noteworthy that these
results basically follow from economies of scale in households and competition for land.
Single-person households will enjoy a lower utility level than two-person households. Hence,
single person households have a strong incentive to form two-person households, but this is
counteracted by dissolution of two-person households. The household formation and
dissolution processes were taken as exogenous in our analysis. We studied the consequences
of a net increase in the share of households containing only one person, in a situation where
the city’s total population was fixed, and where also per capita income was fixed. We found
that this would lead to a decline in the utilities of both household types, and therefore also a
decline in the welfare of city’s inhabitants. Due to the increase in the number of households,
the city border will be pushed outwards. Likewise, the border between the inner part of the
city occupied by single-person households and the outer part occupied by two-person
households will be pushed outwards. Through increased competition for land the land rent
will increase at each location, and this in turn induces an increase in building height at each

**References **
Brueckner, J.K. (1987): The structure of urban equilibria: a unified treatment of the Muth-
Mills model, pp 821-845 in Mills, E. S. (ed.): Handbook of regional and urban economics.
Wheaton, W.C. (1976): On the optimal distribution of income among cities, Journal of Urban

Source: http://www.propertyfinance.it/sitoeres/contents/papers/id81.pdf

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