CSIS Discussion Paper No. 124

Endogenous determination of a residential landscape:Asymmetric effects of consumers’ choices and multiple
equilibria

Takaaki Takahashi
Center for Spatial Information Science, The University of Tokyo

November 2013

Abstract
This paper studies the sizes and rents of dwellings determined as a result of the choices of consumers whose utility depends on the level of landscape amenities in their neighborhood. The
level is, in turn, determined by the sizes of dwellings there. It is shown that there emerges a possibility of multiple equilibria if the effect of each consumer’s choice upon landscape amenities isasymmetric.
Keywords: landscape; multiple equilibria; rent gradient; spatial structure of a city; urban amenities; urban mosaic
JEL Classification Numbers: R14; R21
E-mail address: takaaki-t@csis.u-tokyo.ac.jp
This research is partly supported by the Grants in Aid for Research (No. 25285070) by the Ministry of Education, Science
and Culture in Japan (No. 25285070), and those by the Association of Real Estate Agents of Japan.

1 Introduction
One of the most significant features of a city is a spatial diversity within it. Literature in urban economics has traditionally been emphasizing a geographical variation based on the distance from acity center. However, diversity is observed not only for the spaces at different distances but also
for those at quite similar distances. We often encounter, for instance, the situation in which neighboring districts in a city, only divided by a narrow street, exhibit contrasting characteristics withrespect to land use patterns. To give another example, it is not rare that the western part and theeastern part, say, of the same city are thoroughly different from each other in many aspects. For
an illustrative purpose, let us look at the data for the Nagoya metropolitan area in Japan, which is
the third largest metropolitan area in that country. Table 1 shows means, standard deviations andcoefficients of variation for the rents par 1 square meter of floor space for privately-owned rental
apartments and detached houses, classified by the distances from the city center.1 It reveals thata considerable variation exists for dwellings within the same distance classes. Of course, this result is presented only to motivate readers; and if one wanted to demonstrate the spatial diversity
among the spaces at similar distances from a city center more rigorously, it would be necessary to
conduct a more precise analysis such as the one based on a hedonic approach.

Table 1: Rents of privately-owned rental apartments and detached houses in the Nagoya
metropolitan area
distance froma city center(km)

number ofsamples0-10

coefficient ofvariation(yen/m2 )standard
deviation(yen/m2 )317,800

1671.73775.41

0.464

10-20

170,200

1366.35

585.29

0.428

20-30

127,500

1263.51546.26

0.432

30-40

151,500

1252.71543,70

0.434

40-50

31,3001216.85543.41

0.447

mean

Source: Made from Housing and Land Survey by Statistics Bureau (2003).

Most important reasons for such a diversity are arguably differences in income levels andpreferences among the residents of each district (Glaeser et al. (2008) and Bruekner and Rosenthal (2009)). Some neighborhoods, for instance, turn out to attract the rich who can afford to pay
higher rents while some others end up accommodating the poor who pay lower rents, even if they
are located at similar distances from a city center. In other cases, families with kids prefer spacious dwellings while singles and the elderly opt for more compact ones. Notwithstanding, thephenomenon of the “urban mosaic” might have much deeper roots, because casual observations
1 In the original data of Statistics Bureau (2003), rents are divided into 18 classes.

medians of respective classes.

1

We compute statistics using thesuggest that it can emerge even when consumers in the two neighborhoods at similar distancesfrom a city center are more or less homogenous in terms of income and preference. If this is true,
it would be better to understand the phenomenon as a realization of multiple equilibria. This
paper is an attempt to do so.
For that purpose, we pay attention to the role of urban amenities. A number of studies haveargued that they considerably affect consumers’ locational choices in a city (see Rosen (1974),
Rosen (1979), Roback (1982), Bartik and Smith (1987), Papageorgiou and Pines (1999) and Cho(2001) among others). For some of them such as natural or environmental amenities like warm
climate, and historical amenities like old religious buildings, palaces and castles, their levels are
given. For others, in contrast, they are determined as a consequence of consumers’ choices.2 In
particular, there are a type of amenities whose levels directly depend on the sizes of land lots
and dwellings. A typical example is a residential landscape: a sequence of large detached houses
occupying wide land lots with a mass of greenery undoubtedly constitutes an amenity good thatwill raise the utility of nearby residents.
This type of amenity good gives rise to a possibility of multiple equilibria. At one equilibrium,
residents live in large detached houses constructed within large land lots whereas at another equilibrium, they live in small dwellings with small land lots. A higher level of landscape amenities
is generated at the sparsely populated equilibrium than the densely populated one, which favors
consumers at the former equilibrium than the latter, other things being equal. If consumers arehomogeneous, they receive an equal level of utility at an equilibrium. Therefore, the rents ofdwellings and land lots must be higher at the sparsely populated equilibrium, other things being
equal. In other words, the rents at the two equilibria are determined so that the positive effectof a higher level of landscape amenities at the sparsely populated equilibrium exactly offsets thenegative effect of a higher rent at that equilibrium.
To explore this possibility of multiple equilibria, we construct a simple Alonso-Mills-Muth
type model extended to incorporate landscape amenities. Two elements are important. First, thelevel of landscape amenities is determined by the choices of the consumers in a neighborhood.
Second, each consumer makes a decision taking into account the effect of her own choice uponlandscape amenities. Main finding is that whether multiple Nash equilibria exist or not depends
on the nature of landscape amenities. To understand this point, suppose that all the residents
in a neighborhood consume an equal amount of housing and then one consumer changes theamount of housing consumption by one unit. If the landscape amenities are symmetric, that is, if
the rise of amenity level due to the one-unit increase in her consumption is equal to the declineof amenity level due to the one-unit decrease in her consumption, then, there exist no multiple

2 One of the most important examples would be local public goods such as highways, mass transit, museums andgood schools. Many researchers have been studying how their amounts of provision are determined and how it affects
consumers’ location choices in a city. See, for example, Wright (1977), Polinsky and Rubinfeld (1978), Helpman and Pines(1980) and Sullivan (1985).

2

equilibria, provided that some regularity conditions on consumers’ preference are satisfied. Ifthey are asymmetric and the rise is smaller than the decline, in contrast, there do exist multiple
equilibria. In that case, a certain range of the amounts of housing consumption and that of theprice of housing are supported by an equilibrium.
One implication of the multiple equilibria is that a land rent gradient may become positive,that is, land rent may rise as one moves away from a city center. This possibility has been aninterest of some researchers. Richardson (1977), for instance, attempts to demonstrate it in asetting where a sparse distribution of consumers generates positive externalities. Re-examining
this problem, however, Grieson and Murray (1981) and Tauchen (1981) show that there is no suchpossibility as long as consumers’ preference is well-behaved. Their result corresponds to ourfinding that no multiple equilibria exist when landscape amenities are symmetric.3
In addition to these works, there are several works that discuss the endogenous formationof urban amenities. First, Schall (1976) and Henderson (1985) study the impact of the quality ofneighborhood upon consumers’ choices. In their studies, the quality is defined as an average levelof housing services consumed by neighbors. Second, Brueckner et al. (1999) examine the role ofurban amenities on the location of different income groups within a city. They consider not only
the amenity goods whose amounts are exogenous but also the amenity goods whose amounts
depend on local income levels like restaurants, theaters and sports facilities. Whether the richare located in a central city or a suburb depends on the relative importance of the two types ofamenity goods. Third and finally, Walsh (2007) examines the amounts of open spaces in cities
through an empirical general equilibrium approach, in which open spaces “arise ‘naturally’ as aresult of land market outcomes.”42

Basic framework

We consider a monocentric city in a homogeneous plane, which is made up of a number of smallneighborhoods. Consumers in the city work at a city center to earn a given wage rate, w. The
commuting costs for a consumer living at distance d from the city center are equal to t(d) witht′ (d) > 0 where t′ (·) denotes the derivative of t(·) (we will use similar notations for the derivatives of other variables). Consumers’ income consists of only wage: they do not own dwellings
nor lands and all the rents are received by absentee landlords. Thus, the net income of eachconsumer is y(d) ≡ w − t(d).
Consumers consume housing and a composite good. Here, “housing” contains both a dwelling
and the land lot upon which it is built. Although they are actuallly characterized by many el3 More specifically, they concentrate on the case where the level of landscape amenities depends on an average lot size.

In this case, amenities are symmetric.
4 Yang and Fujita (1983) study the impact of open spaces on the spacial structure of a city; but their distribution isassumed to be exogenous.

3

ements such as size, the number of floors, height, building materials, colors and design for adwelling, and size, shape, location and the width of the road it faces for a land lot, we summarize
them by one variable x. Without loss of generality, it is defined so that larger x is associated with
better characteristics. As a natural interpretation, we regard x as a size of a dwelling.
Consumers’ utility depends not only on the size of a dwelling and the amount of a composite
good but also on the quality of living environment or the amenities in their neighborhood. Thus,
we denote the utility of consumer i as u( xi , zi , a), where xi and zi denote the size of a dwelling
and the amount of a composite good consumed by that consumer, respectively, and a denotes thelevel of amenities in her neighborhood. The utility function is assumed to be increasing in eachargument (a is defined so that a higher value corresponds to a better environment).
We focus on the amenities whose level depends on the sizes of dwellings in a neighborhood.
A typical example is landscape amenities: a sequence of larger dwellings contributes to a creation of better scenery and raises the utility of residents. Then, the level of amenities is written asa( x1 , x2 , ..., xn ), where n is the number of consumers in a neighborhood. Function a(·) is continuous and nondecreasing in each variable (the landscape is no worse when one dwelling is bigger
and the others remain of the same size).
A twist of our model is that the amenity function can be non-differentiable with respect to xi
at xi = x j for j ̸= i, while it is differentiable at any other point. It is often observed, for instance,that a house slightly shabbier than neighbors’ gives a devastating impact on the environment ofa neighborhood although a house slightly more handsome than them does not contribute muchto amenities. Here, consumers value a loss more significantly than the equivalent gain, which isone of the most important observations in the behavioral economics.5 In this case of loss aversion,
the left derivative of a(·) exceeds the right derivative. The opposite case where the left derivativeis smaller than the right derivative is theoretically conceivable; but we do not pay much attention
to that case for the following two reasons. First, it is difficult to find the evidences for the consumers’ behaviors to care about a benefit more than a loss. Second, it turns out that there exists
no equilibrium in that case, as we will see later. Therefore, we concentrate on the case wherelim ai ( x1 , ..., xn ) ≥ lim ai ( x1 , ..., xn ).

xi → x −j

xi → x +j(1)Here, ai denotes the partial derivative of a(·) with respect to the i-th argument (similar notations
will be used for the partial derivatives of other functions).
We limit our analysis to the symmetric equilibria at which consumers choose dwellings of thesame size. Thus, consider, without loss of generality, the decision problem of consumer 1 when
all her neighbors consume the dwellings of sizes x0 . Then, the amenity function can be written as()
a x, x0 , · · · , x0 ≡ α( x, x0 ). Since a(·) is nondecreasing in each variable, the same remark applies| {z }n −1

to α(·). Moreover, the assumption on the differentiability of a(·) mentioned above implies that
5 This point is mainly discussed by the prospect theory originating from Kahneman and Tversky (1979) and Tversky

and Kahneman (1992). For the overview of the topic, see Starmer (2000).

4

α( x, x0 ) is differentiable with respect to x at any point with x ̸= x0 , but may not be so at x = x0 .
If the amenity function satisfies (1), the following equation holds.
lim α1 ( x, x0 ) ≥ lim α1 ( x, x0 ).

x → x 0−(2)x → x 0+We impose several additional assumptions on the amenity function. First, when all the consumers choose dwellings of the same size, the level of amenities is given by α( x, x ) ≡ A( x ). Our
assumption is that A(·) is differentiable, that is,lim α1 ( x, x0 ) + lim α2 ( x0 , x ) = lim α1 ( x, x0 ) + lim α2 ( x0 , x ) = A′ ( x0 )x → x 0−x → x 0+(3)x → x 0−x → x 0+for any x0 . Because α(·) is nondecreasing in each argument, A(·) is a nondecreasing function.
Second, we assume that lim α1 ( x, x0 ) and lim α1 ( x, x0 ) are differentiable with respect to x0 .
x → x 0−x → x 0+Third and last, α( x, x0 ) is concave in x, that is, α11 ( x, x0 ) ≤ 0, where α11 ( x, x0 ) ≤ 0 is a partial
derivative of α1 ( x, x0 ) with respect to the first argument (similar notations will be used for theother second-order partial derivatives).
Before completing the description of a basic setting, to take a look at two examples of theamenity function would help us understand the framework.
Example 1(

As a benchmark, consider the amenity function given by a( x1 , ..., xn ) = f

n

/

)

∑ xi n , wherei =1

f (·) is differentiable, increasing and concave. In this example, the level of amenities is determined by the average size of dwellings in a neighborhood. We refer to this type of amenity
function as an average-value amenity function. This(function )is differentiable with respect to each/n1xi : lim ai ( x1 , ..., xn ) = lim ai ( x1 , ..., xn ) = f ′ ∑ xi n for i = 1, ..., n and j = 1, ..., n with+nxi → x −
x
→
x
i
i =1
j
j
(
)
[
]/
j ̸= i. Furthermore, α( x, x0 ) = f x AVE where x AVE ≡ x + (n − 1) x0 n. Thus, lim α1 ( x, x0 ) =x → x 0−/lim α1 ( x, x0 ) = f ′ ( x0 ) n. In addition, A( x ) = f ( x ), which is differentiable, increasing and conx → x 0+/
cave for any x by assumption. Finally, α11 ( x, x0 ) = f ′′ ( x AVE ) n2 ≤ 0.
Example 2
Next, we turn to another example which illustrates well the unique feature of our model. The
amenity function is given by()
a( x1 , ..., xn ) = g min[ x1 , ..., xn ] ,(4)where g(·) is differentiable, increasing and concave. In this example, the level of amenities isdetermined by the size of the smallest dwelling in a neighborhood. It depicts an extreme casewhere one blot spoils everything. We refer to this type of amenity function as a minimum-value
amenity function. Three observations follow. First, let m−i ≡ min [ x1 , ..., xi−1 , xi+1 , ..., xn ]. Then,ai ( x1 , ..., xn ) is equal to g′ ( xi ) > 0 if xi < m−i , and to 0 if xi > m−i . Consequently,
lim ai ( x1 , ..., xn ) = g′ ( xi ) > 0 = lim ai ( x1 , ..., xn )xi → m −−ixi → m +
−i

5

(
)
and therefore, (1) is satisfied. Second, α( x, x0 ) = g min[ x, x0 ] , which implies that α1 ( x, x0 ) isequal to g′ ( x ) if x < x0 , and to 0 if x > x0 . Consequently,
lim α1 ( x, x0 ) = g′ ( x0 ) > 0 = lim α1 ( x, x0 ).

x → x 0−x → x 0+Third, A( x ) = g( x ), which is differentiable, increasing and concave for any x. Last, α11 ( x, x0 ) ≤ 0
since it is equal to g′′ ( x ) if x < x0 , and to 0 if x > x0 .
Returning to the basic framework, let r be the rent of a dwelling and take the composite goodas a numéraire. The budget constraint of the i-th consumer in a neighborhood at distance d fromthe city center is given by
rxi + zi = y(d).
Finally, consumers can freely migrate among neighborhoods, and the city is small and open.
Therefore, the level of utility they enjoy is a constant and given. We denote it by ū.

3Equilibrium in a general setting

Because the analysis is limited to symmetric equilibria, it is useful to denote the utility function()
by v( x, x0 , r ) ≡ u x, y(d) − rx, α( x, x0 ) . The Nash equilibrium is defined as a pair ( x ∗ , r ∗ ) thatsatisfies the following two conditions:v( x ∗ , x ∗ , r ∗ ) ≥ v( x, x ∗ , r ∗ )(5)v( x ∗ , x ∗ , r ∗ ) = ū.

(6)

for any x ≥ 0 and

The first condition requires that each consumer maximize her utility given the neighbors’ choices.
The second prescribes that the resulting utility level equal the one prevailing in the economy.
Note that each consumer takes into account not only the effects of neighbors’ choices but also theeffects of her own choice upon the amenity level. This is a significant departure from the existing
works mentioned in the introduction, which implicitly assume that consumers disregard theirown effects.
First of all, since α( x, x0 ) is differentiable with respect to x at x ̸= x0 ,v1 ( x, x0 , r ) = u x − ruz + u a α1 ( x, x0 )for x ̸= x0 .

Here, u x , uz and u a denote the partial derivatives of u( x, z, a) with respect to x, z and a, respec()
tively, evaluated at ( x, z, a) = x, y(d) − rx, α( x, x0 ) . To avoid unnecessary complications, we
assume that function v( x, x0 , r ) is strictly concave in x as long as x ̸= x0 . In other words, thefollowing assumption is imposed for v11 ( x, x0 , r ).
Assumption 1
v11 ( x, x0 , r ) < 0 for any x ̸= x0 .
6This assumption says that the marginal utility of housing consumption, inclusive of the effects of theinduced change in the amenity level, decreases as the dwelling becomes bigger.
For x = x0 , on the other hand, it is necessary to distinguish the left and right derivatives: lim v1 ( x, x0 , r ) = u x − ruz + u a lim α1 ( x, x0 ) ≡ λ( x0 )

x → x 0−x → x 0−lim v1 ( x, x , r ) = u x − ruz + u a lim α1 ( x, x0 ) ≡ µ( x0 ),0x → x 0+(7)x → x 0+()
where u x , uz and u a are evaluated at ( x, z, a) = x0 , y(d) − rx0 , α( x0 , x0 ) . (Although λ(·) and µ(·)
depend on r, we suppress r to make the presentation clearer.)
We are now ready to discuss the utility maximization problem of (5). Because v( x, x0 , r ) is differentiable with respect to x and strictly concave in x for any x ̸= x0 , the necessary and sufficientcondition for the utility maximization is that v1 ( x, x ∗ , r ) > 0 for any x < x ∗ and v1 ( x, x ∗ , r ) < 0
for any x > x ∗ . By construction, these conditions hold if and only if
λ( x ∗ ) ≥ 0 andµ( x ∗ ) ≤ 0.

(8)

We have mentioned earlier that no equilibrium exists unless (1) is satisfied. This can be readily
verified: λ( x0 ) < µ( x0 ) for any x0 and no x ∗ satisfies (8), if (2) does not hold.
To characterize the solutions, we derive the derivative of λ(·):λ′ ( x0 )= lim v11 ( x, x0 ) + lim v12 ( x, x0 )x → x 0−x → x 0−= lim v11 ( x, x ) + lim α2 ( x, x0 ) · lim ω a ( x, x0 ) + u a lim α12 ( x, x0 ),0x → x 0−x → x 0−x → x 0−x → x 0−where ω k ( x, x0 ) is the change in the marginal utility of housing consumption caused by a rise ink ∈ { x, z, a}: ω k ( x, x0 ) ≡ u xk − ruzk + u ak α1 ( x, x0 ). To understand this equation, let us focus on aparticular consumer thinking of buying a dwelling whose size is almost equal to the neighbors’.
First, when neighbors’ dwellings are bigger, the dwelling she has a mind to buy is also bigger,
which affects her marginal utility. This own effect, which is negative due to Assumption 1, isrepresented by the first term in the second line. Note that it includes the effect of the amenity
improvement stemming from the increase in her own housing consumption, namely, an own
amenity improvement effect.6 Second, the increase in the sizes of neighbors’ dwellings directly
raises the amenity level in the neighborhood. This effect is captured by the second term in thesecond line. Third and last, the sizes of neighbors’ dwellings affect the magnitude of the own
amenity improvement effect mentioned above. This is represented by the last term in the secondline. Because the signs of the last two terms describing cross effects are ambiguous, we cannot
determine the direction of the total effect. In this paper, however, we confine our attention to thecase where the own effect is strong enough to dominate the cross effects when λ( x0 ) = 0. In otherwords, we assume that λ( x ) is decreasing at the value of x that solves λ( x ) = 0. We assume asimilar property for µ(·).
6 Here,lim v11 ( x, x0 , r ) =x → x 0−[lim

x → x 0−]
ω x ( x, x0 ) − rω z ( x, x0 ) + ω a ( x, x0 )α1 ( x, x0 ) + u a α11 ( x, x0 ) . The last term in thesquare brackets represents the own amenity improvement effect.

7

Assumption 2λ′ ( x ) < 0 if x solves λ( x ) = 0;and µ′ ( x ) < 0 if x solves µ( x ) = 0.

The amenity function discussed in Example 2 satisfies this assumption as long as Assumption 1
holds. We will discuss this point later. Figure 1 describes λ( x ) and µ( x ) as a left derivative curveand a right derivative curve, respectively. Because λ( x ) ≥ µ( x ) for any x, the former curve lies
above or on the latter.
Figure 1: Left and right derivative curves
Assumption 2 implies that λ( x ) = 0 (µ( x ) = 0, resp.) has at most one solution, which wedenote as a function of r by x̄ (r ) (x (r ), resp.). Two comments are worth adding. First, λ( x ) ≥ µ( x )( )( )implies that x r ≤ x̄ r . Second,x̄ ′ (r ) =uz<0λ′ ( x )Furthermore, (8) is equivalent toandx ′ (r ) =uz< 0.
µ′ ( x )[]x ∗ ∈ x (r ∗ ), x̄ (r ∗ ) .

(9)

The interval in (9) is the set of the sizes of dwellings that are compatible with consumers’ optimizing behaviors given their rents, and thus referred to as a compatibility set. Figure 2 shows the lower
and upper limits of this set in an x-r space as lower and upper compatibility curves, respectively.
Both curves have negative slopes.

Figure 2: Compatibility curves and sustainability curve

Example 2-2Here, it would be instructive to revisit the example with the minimum-value amenity function, (4). Because lim α2 ( x, x0 ) =x → x 0−lim α12 ( x, x0 ) = 0 for this function, we have λ′ ( x0 ) =x → x 0−lim v11 ( x, x0 , r ). Therefore, as long as Assumption 1 is satisfied, Assumption 2 holds. Further-x → x 0−more, (8) becomes −u a g′ ( x ∗ ) ≤ u x − r ∗ uz ≤ 0, where all the partial derivatives are evaluated atx = x ∗ . x̄ (r ∗ ) is a solution to u x − r ∗ uz = −u a g′ ( x ) while x (r ∗ ) is a solution to u x − r ∗ uz = 0.
Next, let us turn to the second condition for an equilibrium, (6). It implicitly yields x ∗ as afunction of r ∗ :x ∗ = s (r ∗ ).

8

(10)

Here, function s(·) gives the size of a dwelling that brings a consumer the utility level of ū for agiven rent. If ( x, r ) does not satisfy x = s(r ), it invokes consumers’ migration from or into thecity. In this sense, (10) assures the sustainability of a city’s population.
To see that the sustainability function s(·) is increasing, let us introduce an additional notation,V ( x, r ) ≡ v( x, x, r ). It represents the utility level of the consumer who chooses a dwelling of thesame size as her neighbors’. Then, we haves ′ (r ∗ ) =x ∗ uzx ∗ uz=> 0.
V1 ( x ∗ , r )u x − r ∗ uz + u a A′ ( x ∗ )(11)

The denominator, which measures the effects of an increase in x ∗ upon the utility level, is decomposed into two parts. First, u a A′ ( x ∗ ) represents an indirect effect through the improvementof amenities. This amenity effect is positive. Second, u x − r ∗ uz represents a direct effect. The sec()
ond inequality in (8) implies that u x − r ∗ uz ≤ 0 at ( x, z, a) = x ∗ , y(d) − r ∗ x ∗ , α( x ∗ , x ∗ ) because
lim α1 ( x, x0 ) ≥ 0 (see (7)): the direct effect is nonpositive. The sign of the denominator is deter-x → x 0+mined by the relative sizes of the two effects. However, (3) implies that A′ ( x ∗ ) ≥ lim α1 ( x, x ∗ ).
x → x ∗−

Therefore, it follows from the first inequality of (8) that the denominator is nonnegative, that is,V1 ( x ∗ , r ∗ ) ≥ 0: the amenity effect offsets or more than offsets the direct effect. Hence, the sustainability function is increasing and, in Figure 2, described by an upward sloping curve, whichis referred to as a sustainability curve.
The set of the equilibrium pairs, namely, the ( x ∗ , r ∗ )’s that satisfy both (9) and (10), is represented by a part or parts of the sustainability curve between the upper and lower compatibility
curves. Because both the compatibility curves are downward sloping and the sustainability curve
is upward sloping, the latter cuts each of the former at most once. We denote these intersectionsby ( x l , r l ) and ( x u , r u ), respectively (see Figure 2), that is, ( x l , r l ) solves x l = x (r l ) and x l = s(r l ),whereas ( x u , r u ) solves x u = x̄ (r u ) and x u = s(r u ). Then, the set of equilibrium pairs is given by{}E ≡ ( x, r ) x = s(r ), x ∈ [ x l , x u ] .
Thus, x l and r l give the lower bounds of the equilibrium values of x and r, respectively, whereas x u
and r u give their upper bounds.
Set E does not degenerate to a point as long as the lower compatibility curve always lies
below the upper compatibility curve. Instead, if the two curves coincide with each other, the set
is reduced to a point. Thus, we have established the following result.
Proposition 1 (Multiple equilibria)i) Suppose that α( x, x0 ) is not differentiable with respect to x, that is, (2) holds with the inequality,
at x = x0 for any x0 . If there is an equilibrium, there always exists another equilibrium.
ii) Instead, suppose that α( x, x0 ) is differentiable with respect to x, that is, (2) holds with theequality, at x = x0 for any x0 . Then, the equilibrium, if it exists, is unique.
The second part of this proposition says that without the non-differentiability of the amenity
9

function, merely introducing externalities in amenity formation does not produce multiple equilibria, provided that the utility function and the amenity function satisfy the regularity conditions(Assumptions 1 and 2). In short, the multiple equilibria necessitate the non-differentiability.
Suppose that there are multiple equilibria. Then, two equilibrium pairs of ( x, r ) must be on
the same sustainability curve, which is upward sloping. Therefore, r is higher at the equilibriumwith larger x than at the equilibrium with smaller x.
Proposition 2 (Relationship between the sizes and rents of a dwelling)

Suppose that ( x ′ , r ′ ) and ( x ′′ , r ′′ ) are equilibrium pairs for given d. Then, r ′{ > }x ′ = x ′′ .
<{ > }= r ′′ if<

In a standard setting with no amenities, in order to keep the utility at the same level, it is necessary
to reduce housing consumption when its price rises. However, this is not always the case in ourmodel. Despite the rise in the rent, consumers may increase housing consumption because doing
so improves amenities: the effect of the decrease in the consumption of the other goods may be
more than compensated by the effect of the amenity improvement.
It is worth adding a comment on the level of social welfare. As far as only the consumers
living in the city are concerned, the multiple equilibria give the same level of welfare, because
they enjoy the same level of utility. However, if we take into account absentee landlords, the levelof social welfare will depend on the rents of dwellings: the equilibrium with a higher rent (and alarger dwelling) yields a higher welfare level.
How does a space affect the range of the equilibria? To answer this question, note that in ourmodel, the distance from the city center affects the equilibrium range only through the level ofdisposable income, y(d). Therefore, to discuss the effects of a change in d, it suffices to examine
the effects of a change in y(d).
To begin with, note that since y(·) is a continuous function, all x l , r l , x u and r u are also continuous with respect to d. Consquently, if multiple equilibria exist at a location, then, they also exist
at sufficiently close locations. Therefore, a rent gradient can be positive in a sufficiently narrow range
of geographical space. This would be one of alternative explanations for the positive rent gradient discussed by many researchers (Schall (1976), Richardson (1977), Grieson and Murray (1981),
Tauchen (1981) and Henderson (1985)), although the problem of indeterminacy arises.
Furthermore, totally differentiating x l = x̄ (r l ) and x l = s(r l ) with respect to x l , r l and y(d),//and solving the two resulting equations, we derive dx l dy(d) and dr l dy(d). Similarly, totally//differentiating x u = x̄ (r u ) and x u = s(r u ) yields dx u dy(d) and dr u dy(d):
dx k
dx k
t ′ ( d ) u z s ′ (r k )= −t′ (d)
=−
dd
dy(d)
x k Γk10and)t′ (d) (dr k= − k k Γk + uz
dd
x Γfor k ∈ {l, u}, where Γl ≡ µ′ ( x l )s′ (r l ) − x l lim ω z ( x, x l , r ) − uz ,x→xl +u′u′u Γ ≡ λ ( x )s (r ) − x u lim ω z ( x, x u , r ) − uz .
x → x u−andFirst, let us examine the changes in x l and x u . Becuase µ′ ( x l ) < 0 by Assumption 2 ands′ (r l ) > 0, Γl is negative if lim ω z ( x, x l , r ) is nonnegative. Even if lim ω z ( x, x l , r ) is negative,x→ xl +
x→ xl +
lhowever, Γ is still negative as long as it is not too large in the absolute value. In such cases, we/have dx l dd > 0: the lower bound of the equilibrium sizes of dwellings rises as we move farther
away from the city center. The similar remark applies to the upper limit.
Next, we examine the changes in r l and r u . When lim ω z ( x, x l , r ) is nonnegative or its absox→xl +/lute value is sufficiently small, Γl + uz as well as Γl becomes negative. In that case, dr l dd < 0:

the lower bound of the equilibrium rents declines as we go farther away from the city center.
Once again, the similar remark applies to the upper limit. Here, note that ω z ( x, x k , r ) represents
the change in the marginal utility of housing consumption when the income level y changes. Ourfindings are summarized as follows:
Proposition 3 (Spatial variation)

Suppose that the marginal utility of housing consumption increases or only slightly decreases as
the income level rises. Then, as we move farther away from the city center, both the lower andupper bounds of the equilibrium sizes of dwellings rise and both the lower and upper bounds ofthe equilibrium rents decline.
We can explain these results using Figure 2 above. First of all, look at the sustainability curve.
By totally differentiating (6) with respect to x ∗ and y while fixing r ∗ , we obtaindx ∗t ′ ( d ) s ′ (r ∗ )
=
> 0,
dd
x∗
where (11) is used. Therefore, the sustainability curve shifts to the right as d increases. Next, we( )examine the compatibility curves. Remember that x is defined as a solution to µ x = 0 (in therest of this section, the arguments r’s of x and of x̄ are suppressed for the sake of clarity). Totallydifferentiating this equation with respect to d yieldsµ′ ( x )dx− t′ (d) lim ω z ( x, x, r ) = 0.
dd
x→ x+

Since µ′ ( x ) < 0 by Assumption 2, dx/dd ≤ 0 as long as lim ω z ( x, x, r ) ≥ 0: the lower compatix→ x+

bility curve shifts to the left as d rises. Similarly, becauseλ′ ( x̄ )dx̄
− t′ (d) lim ω z ( x, x̄, r ) = 0,ddx → x̄ −the upper compatibility curve also shifts to the left as long as lim ω z ( x, x̄, r ) ≥ 0. The curves
x → x̄ −associated with higher d are shown by the two downward-sloping dotted lines. Consequently, if
11lim ω z ( x, x, r ) ≥ 0 and lim ω z ( x, x̄, r ) ≥ 0 hold, the intersections of each compatibility curvex → x̄ −x→x+

and the sustainability curve move downward. Thus, the upper and lower bounds of the equilibrium rents decrease. However, the intersections may move to the left or to the right dependingon the relative magnitudes of the shifts of these curves. Proposition 3 nevertheless demonstrates
that the sustainability curve shifts relatively more compared to the compatibility curves, and as aresult, the intersections move to the right. Therefore, the upper and lower bounds of the equilibrium sizes of dwellings increase.

4

Equilibrium for specified preferences

In this subsection, we explore the equilibria assuming a special functional form for the utility
function as well as the amenity function. The purpose is not only to illustrate the above findings in a familiar setting but also to derive further implications. We consider a utility functionu( x, z, a) = x θ z1−θ aν for θ ∈ (0, 1) and ν ∈ (0, 1), and a minimum-value amenity functiona( x1 , ..., xn ) = κ (min[ x1 , ..., xn ])η for κ > 0 and η > 0. The following function summarizes
necessary information:()ϕ
v( x, x0 , r ) = κ ν x θ z1−θ min[ x, x0 ] ,where ϕ ≡ ην > 0.
The associated second-order derivative is given by v( x, x0 , r ) −q( x )
if x < x02 z20xv11 ( x, x , r ) =v( x, x0 , r )θ (1 − θ )y2 −< 0 if x > x0 ,x 2 z2where q( x ) ≡ θ (1 − θ )y2 + 2ϕz(rx − θy) + ϕ(1 − ϕ)z2 . Here, z is evaluated at y − rx and theargument of y, i.e., d, is dropped from the expression for the sake of exposition (these two qualifications are applicable throughout the rest of the paper). To judge the sign of q( x ), note, first,
that q( x ) takes a minimum value in the interval x ∈ [0, y/r ] either at x = y/r or at x = 0, because q( x ) is strictly concave in x. Second, q(y/r ) = θ (1 − θ )y2 > 0. Third and last, q(0) =(1 − θ − ϕ)(θ + ϕ)y2 , which is positive if
θ + ϕ < 1.

(12)

These observations altogether imply that v11 ( x, x0 , r ) < 0 for any x < x0 and Assumption 1 issatisfied, if (12) holds. The qualification requires that when a consumer’s dwelling is smaller
than the neighbors’, her marginal utility of housing consumption inclusive of the own amenity
improvement effect be sufficiently small. In the rest of the paper, we assume this.
The compatibility set is given by[]x (r ∗ ), x̄ (r ∗ ) =

[

12

]
θy (θ + ϕ)y,.
r ∗ (1 + ϕ )r ∗

Three observations follow. First, because x ∗ ≥ x (r ∗ ), the share of spending on housing is nosmaller than θ. In the standard model of a consumer behavior without amenity goods, the share
would become equal to θ. In our model, however, the share can exceed it because the rise in theshare enhances the utility through the amenity improvement. Second, since ϕ > 0 and θ ∈ (0, 1),/
θ < (θ + ϕ) (1 + ϕ); and therefore, the interval has a positive measure. Third and finally, bothx (·) and x̄ (·) are decreasing functions.
Moreover, the sustainability curve is derived by solving the following equation:( x ∗ )θ +ϕ (y − r ∗ x ∗ )1−θ = ū′ ,(13)where ū′ ≡ ū/κ ν . Applying (11) yieldss ′ (r ∗ ) =(1 − θ ) ( x ∗ )2.
( θ + ϕ ) y − (1 + ϕ )r ∗ x ∗The denominator is positive if x ∗ ≤ x̄ (r ∗ ). Therefore, the part of the sustainable curve in thecompatibility set is indeed upward sloping.
Then, substituting x (r ∗ ) (x̄ (r ∗ ), resp.) into x ∗ in (13) and solving the resulting equation for r ∗ ,we can derive x l and r l (x u and r u , resp.):()( x l , r l ) = Kxl y−ΘΨ , Krl yΦΨ

and

(
)
( x u , r u ) = Kxu y−ΘΨ , Kru yΦΨ ,where Θ ≡ 1 − θ, Φ ≡ 1 + ϕ, Ψ ≡ (θ + ϕ)−1 and all the K’s are positive constants.7 Several
observations follow. First, x u (r u , resp.) is proportional to x l (r l , resp.); we denote the proportional///coefficient by Cx ≡ x u /x l (Cr ≡ r u /r l , resp.). Second, dx l dd > 0, dx u dd > 0, dr l dd < 0 and/
dr u dd < 0. Third, let us take a look at an equilibrium density at each point in a city, which//
is an inverse of the size of a dwelling. Note that d2 (1/x l ) dd2 and d2 (1/x u ) dd2 are positive
if the following two conditions are satisfied. For one thing, θ and/or ϕ is sufficiently small that2θ + ϕ < 1, or equivalently, ΘΨ > 1. Furthermore, t′′ (d) is negative or sufficiently small when
it is positive.8 In other words, the lower and upper bounds of the equilibrium density become
convex functions of the distance from a city center as long as consumers’ inclination for housing
and amenities is not too strong and the commuting costs do not increase with distance too rapidly.
This result is consistent with the well-known empirical fact of a declining density gradient since
Clark (1951).9
How do various factors affect the range of the multiple equilibria? To see this, we look at thechanges in the ratios of upper and lower bounds, Cx and Cr .
First, let us examine a change in θ. For one thing, it is readily verified that d ln Cx /dθ =//[]−(1 + ϕ) ln(1 + ϕ) (θ + ϕ)2 < 0. Furthermore, note that d ln Cr /dθ = k1 θ (θ + ϕ)2 , where
7 Note that K l ≡x
8 We obtain

[

ū/ΘΘ

]Ψ

[
]Ψ
[
]Ψ
[
]Ψ
, Krl ≡ θ ΘΘ /ū , Kxu ≡ Φ−1 ūΦΦ /ΘΘ and Kru ≡ Ψ−1 ΘΘ /ūΦΦ .

d2 (1/x l )
ΘΨ
= l
dd2
xy

[

(ΘΨ − 1){t′ (d)}2− t′′ (d)y]andd2 (1/x u )d2 (1/x l )
= Cx
.
2
dd
dd2

9 For surveys, see Mills and Tan (1980), McDonald (1989) and Anas et al. (1998) among others.

13

[
]
k1 ≡ −ϕ(θ + ϕ) + θ (1 + ϕ) ln(1 + ϕ). Because k1 < θ (1 + ϕ) ln(1 + ϕ) − ϕ < 0, we conclude
that d ln Cr /dθ < 0.10 Therefore, when consumers put a relatively higher value on housing compared to the composite good, the range of multiple equilibria, measured by either the price or theamount of housing, is narrower.

//[]
Second, we examine a change in ϕ. Note that d ln Cx dϕ = k2 (1 − θ ) (1 + ϕ)(θ + ϕ)2 , wherek2 ≡ θ + ϕ − (1 + ϕ) ln(1 + ϕ). The sign of k2 is ambiguous. It is positive when θ is sufficiently
large and ϕ is sufficiently small (it increases with θ and decreases with ϕ). Indeed, we can showthat it is unambiguously positive if θ > ϕ, or, in other words, if amenities are not too important/for consumers compared to housing.11 In such a case, d ln Cx dϕ > 0: as amenities become
relatively more important, the range of the amounts of housing consumption associated withmultiple equilibria expands. On the other hand, the effect upon Cr is unambiguously positive://d ln Cr dϕ = (1 − θ ) ln(1 + ϕ) (1 + ϕ) > 0. Thus, the rise in the relative importance of amenities
results in a wider range of the housing prices associated with multiple equilibria.
One of the other variables worth exploring is the share of the expenditure on housing, h ≡rx/y. The lower and upper bounds of the share become equal to hl ≡ r l x l /y = θ and hu ≡/r u x u /y = (θ + ϕ) (1 + ϕ), respectively. Both bounds increase with θ and the upper bound in/[]creases with ϕ. The ratio of the two bounds of the share, Ch ≡ hu /hl = (θ + ϕ) θ (1 + ϕ) ,decreases with θ and increases with ϕ.

5Concluding remarks

This paper studies the sizes and rents of dwellings determined as a result of the choices of consumers whose utility depends on the level of landscape amenities in their neighborhood. The
level is, in turn, determined by the sizes of dwellings there. It is shown that there emerges a possibility of multiple equilibria if the effect of each consumer’s choice upon landscape amenities isasymmetric and the negative effect of consuming a slightly smaller dwelling dominates the positive effect of consuming a slightly larger dwelling. If the landscape amenities are symmetric, incontrast, there is no multiple equilibria, provided that some regularity conditions are satisfied.
One implication of the multiple equilibria is that the land rent gradient may become positive.

10 One can prove that β ( ϕ ) ≡ ln(1 + ϕ ) − ϕ < 0 for any ϕ > 0 from two observations: first, β (0) = 0 and second,β′ (ϕ) = −ϕ/(1 + ϕ) < 0 for any ϕ > 0.
11 This is explained as follows. Note that k = θ − ϕ + γ ( ϕ ) where γ ( ϕ ) ≡ 2ϕ − (1 + ϕ ) ln(1 + ϕ ). It is readily verified
2
that γ(ϕ) > 0 for any ϕ < 1 since γ(0) = 0 and γ′ (ϕ) = 1 − ln(1 + ϕ) > 0 for ϕ < 1. Therefore, θ > ϕ implies that k2 > 0.

14

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16

v1(x, x0,r)

x0

x(r)

Ο

x(r)

v1(x, x0,r)xleft derivative curve: λ(x)right derivative curve: µ(x)Figure 1: Left and right derivative curvesrlower compatibility curve: x = x(r)upper compatibility curve: x = x(r)sustainability curve: x = s(r)

ru
rl

Ο

xl

xu

x

Figure 2: Compatibility curves and sustainability curve