Department of Policy and Planning Sciences
Discussion Paper Series

No.1334

The Impact of Rising Income Inequality on Rent
Distribution in Housing Markets with Indivisibilities

by
Seiken SAI

Oct 2015

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPANThe Impact of Rising Income Inequality on Rent Distributionin Housing Markets with Indivisibilities
Seiken SaiW
University of Tsukuba
October 1, 2015

Abstract
This paper studies the impact of rising income inequality on a rent distribution of rental
housings. The model is based on the assignment market model with a nite number ofhouseholds and apartments. We show that when household income inequality increases,
then (i) a rent rises at every apartment, (ii) rise at apartments of higher quality but declines
at lower quality or (iii) decline at every apartment. These cases are characterized by thelocation of the specic household who divides households into income-increased/declined
groups. This characterization implies (i) is a special case. Numerical examples conrm
our results. We also discuss equitability of competitive allocations in our market model.

1

Introduction

We present the impact of an increase in household income inequality on apartment rents. The
market model we adopt is the rental housing market model by Kaneko, Ito and Osawa (2006).
The model by Kaneko et al. is an application of the assignment market without theassumption of quasi-linear utility functions. In the model, the market participants are divided
into households and landlords. Each household demands at most one apartment unit andeach landlord provides some apartment units. The apartments as indivisible commodities
are classied into nite categories 1, ..., T based on their qualities. The goods other thanapartments are aggregated and consumed as composite good (money). Household utility
function is assumed to be homogeneous, and allows income eect on housing qualities.
It is known that this model guarantees the existence of a competitive equilibrium (Kaneko,
1982; Kaneko and Yamamoto, 1986). In particular, under our assumptions on the utility
functions, a household with a higher income rents an apartment of a better category than ahousehold with lower income at any equilibrium (Proposition 2.1). We can then represents the
maximum competitive rent vector by a solution of a certain system of equations. This system
W

Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai,
Tsukuba, Ibaraki 305-8573 Japan; Tel./Fax: +81-29-853-5580; E-mail: seiken.sai@gmail.com.

1

of equations and its solution are called the rent equation and the dierential rent vector. In
our analysis, we directly consider a dierential rent vector rather than the competitive rent
vector.
Here, we briey introduce our comparative statics result. The eect of an increase inincome inequality is divided by three cases: (i) rise in rent at every category, (ii) decline atevery category or (iii) rise at higher categories and decline at lower categories. Cases (i) and(ii) are counterintuitive because rising income inequality seems to cause decline in rents atworse apartments and rise in rents at better apartments. Indeed, we show that (i) is said tobe an extreme case, while (ii) [and also (iii)] may possible. Three cases (i)-(iii) are associatedwith the location of household who divides the households into the income-increased groupand the decreased group.
Here, we introduce related literatures. Kaneko et al. (2006) studied eects of changes inincomes of boundary households on a competitive rent vector. The boundary household isdened for each category of apartments, and play a crucial role in the model. The authors
showed that when the boundary income dierence is larger (smaller) for a better category ofapartments, the rent dierence forms convex (concave) shape.
Ito (2007) presented the eects of rise in only the boundary household income of category
k on competitive rents, under a more restricted assumption on a utility function. The author
showed that rents are unchanged at k + 1, ..., T , increase at 1, ..., k and a rent dierence ofeach category 1, ..., k  2 is smaller for a better category of apartments.
Määttänen and Terviö (2014) studied the eect of rising income inequality on house prices
in the one-sided assignment model. One-sided means that the agents are potentially seller
and buyer. The authors assume a continuum of agents and housing types (thus, an analyticalmethod is calculus), and the homogeneity and normality on the utility functions. The authors
presented a similar result to our main result with the exclusion of the case (i). Braid (1981)
also studied the eects of parameter changes on rent distributions under the two-sided versionof Määttänen and Terviö’s framework.
This paper is organized as follows. Section 2 formulates our rental housing market modeland gives our denition of competitive equilibrium. Next we introduce the rent equationand dierential rent vector. Section 3 examines the impact of rising income inequality on arent distribution. Numerical examples conrm our theorems. Section 4 studies equitability
property of competitive allocations in our model. Section 5 presents our conclusions and some
remarks.

2

The market model

The rental housing market model (Kaneko et al., 2006) is denoted by (M, N ), where thesymbol M = {1, . . . , m} denotes the set of households, and N = {1, . . . , T } denotes the set oflandlords. The objects of trade are apartments (indivisible) and money (perfectly divisible).

2

The apartments are classied into nite T number of categories by their housing attributes(e.g., a housing size and a commuting time). Each landlord k 5 N supplies units of apartmentsof the k-th category (thus k is the only landlord providing the k-th apartments).1
Each household i 5 M initially has an income Ii > 0 but no dwelling. He wants to live insome apartment by paying rent from his income. Without loss of generality, we can assume
that the households are ordered in their incomes as I1  I2  · · ·  Im . The consumption
set is written by X := {e0 , e1 , . . . , eT } × R+ , where ek is the T -dimensional unit vector withk-th component is 1 (e0 = 0), and R+ is the set of nonnegative real numbers. A consumption
bundle (ek , c) 5 X with k 6= 0 means that household i rents one apartment unit of category k
and enjoys the consumption c = Ii  pk paying rent pk of category k. For k = 0, no apartment
is consumed. An initial endowment of i 5 M is given as (e0 , Ii ) with Ii > 0. Each household
has an identical utility function u : X $ R satisfying the following assumption:
Assumption A. For each x 5 {e0 , e1 , . . . , eT }, u(x, c) is a continuous and strictly monotonefunction of c, and u(e0 , Ii ) > u(ek , 0) for all k = 1, ..., T.
The identical utility function implies that a housing market (M, N ) represents a monocentric city, and every households commute to the same business district. In Assumption
A, continuity and monotonicity on money are standards; the latter inequality means theindispensability of money. We also assume the following: B-D on u(·, ·).
Assumption B. If u(xi , c) = u(x0i , c0 ), and c < c0 , then u(xi , c + ) > u(x0i , c0 + ) for any
 > 0.
Assumption C. If u(xi , c) > u(x0i , c0 ), then u(xi , c) = u(x0i , c0 + ) for some  > 0.
Assumption D. u(e1 , 0) > u(e2 , 0) > · · · > u(eT , 0).
Assumption B is the normality assumption on the quality of apartments in the following
sense. In B, the k-th apartment has a better quality than k0 , since living in k with smaller
consumption c is indierent to living in k 0 with larger c0 . When an income is increased by
the same magnitude  > 0, the household strictly demands better apartment. The normality
implies that even if we assume the identical utility function, households having dierent
incomes demand dierent qualities of apartments. Assumption C means that housing qualityof an apartment is substitutable for money. Assumption D means that the apartment qualities
are strictly ordered by the numerical order.2
We next dene the seller side. Each landlord k 5 N = {1, ..., T } provides apartments ofk-th category. The landlord has a cost function Ck (yk ) : Z+ $ R+ , where Z+ is the set of
nonnegative integers. For each yk 5 Z+ , Ck (yk ) represents the cost (in terms of money) of
1
The original model of Kaneko et al (2006) assume that |N| D T and there are more than one seller
providing apartments of type k (= 1, ..., T ). As far as competitive equilibrium is concerned, we can assume
without of generality that only one seller provides apartments of type k(= 1, ..., T ) (thus the set N becomes
N = {1, ..., T }). See Section 5 of Sai (2014).
2
Assumption D together with Assumptions A, B and C imply that u(e1 , c) > u(e2 , c) > · · · > u(eT , c) for
all c M R+ .

3

supplying yk units of apartments of k-th category. In this study, we employ the following
simple form of Ck (·).
Assumption E. For each k 5 N, Ck (yk ) is expressed as
Ck (yk ) =(ak yk if yk  wk ,
“large” if yk  wk + 1,

In Assumption E, the constant ak > 0 is the marginal cost of providing additional unit.
The “large” is a su!ciently large number. The remaining constant wk is the number of allapartment units owned by landlord k. This cost function means that landlord k supplies units
up to wk with the constant marginal cost ak , while he never supplies more than wk units since
the cost to build a new one is very large relative to the market.3 .
We dene a competitive equilibrium in (M, N ). Let p 5 RT+ be the price vector, x 5{e0 , e1 , . . . , eT }m be the demand vector and y 5 ZT+ be the supply vector. A triple (p, x, y) is
a competitive equilibrium i
P(UM): for all i 5 M, (i) Ii  pxi  0, where pxi = Tk=1 pk xik ;(ii) u(xi , Ii  pxi )  u(x0i , Ii  px0i ) for all x0i 5 {e0 , e1 , . . . , eT } with Ii  pxi  0.
(PM): for all k 5 N, pk yk  Ck (yk )  pk yk0  Ck (yk0 ) for all yk0 5 Z+ .
P
P(BDS): iMM xi = Tk=1 yk ek .

There exists a competitive equilibrium (p, x, y) in (M, N) (Kaneko and Yamamoto, 1986),
the maximum and minimum competitive rent vectors (Kaneko et al., 2006; Sai, 2015).4,5 In our
analysis, we focus on the maximum competitive rent vector. This rent vector is calculatedby the solution of a certain system of equations called the rent equation.6 The following
proposition is necessary to dene the rent equation.
Proposition 2.1 (Kaneko et al., 2006). Let (p, x, y) be a competitive equilibrium. Then,0(1) If k < k0 and xi = ek for some i, then pk > pk0 .
0(2) If xi = ek , xj = ek and Ii > Ij for some i, j, then k  k 0 .
This states that in any competitive equilibrium, (1) the price of a better apartment ishigher than a worse one, and (2) a household with a higher income rents a better apartment.
Note that Proposition 2.1.(1) does not exclude the case of yk = 0. The following assumption
eliminates such a case.
Assumption F. Let (p, x, y) be a competitive equilibrium. Then there exists some category
f such that yk > 0 for k = 1, ..., f and yk = 0 for k = f + 1, ..., T .
3

In this sense, our approach is short-run analysis.
A vector p M RT+ is a competitive price vector i (p, x, y) is a competitive equilibrium, and p is the maximum(minimum) competitive price vector i p D p0 (p $ p0 ) for any competitive price vector p0 .
5
Indeed, these existence theorems are guaranteed only under Assumptions A and E.
6
Instead of the maximum one, we may focus on the minimum competitive rent vector. It follows from Sai(2014) and/or Sai (2015) that the dierence between pmax and pmin is rather small when a market is thick withlandlords and/or households.
44

We call this f the marginal category. By Proposition 2.1.(1) and Assumption F, we havep1 > p2 > · · · > pf .
Recall that the households 1, ..., m are ordered by their incomes as I1  I2  · · ·  Im .
We dene the household with the lowest income in each active category. Let (p, x, y) be a
maximum competitive equilibrium. For each category k  f , we dene the household G(k)with the lowest income in the k-th category as:
G(k) :=

k
P

yt .

t=1

For each k, we call G(k) the boundary household of the k-th category.
The rent equation (Kaneko et al., 2006) is dened as the system of equations with unknowns r1 , ..., rf :<u(ef 31 , IG(f 31)  rf 31 ) = u(ef , IG(f 31)  rf ), AA
A
Au(ef 32 , IG(f 32)  rf 32 ) = u(ef 31 , IG(f 32)  rf 31 ), @
..
A
A
.
A
A
>
1
2
u(e , IG(1)  r1 ) = u(e , IG(1)  r2 ).

(3.1)

Note that the rent equation (3.1) has f unknowns, while which is constituted by f  1equations. Eq. (3.1) states that a household G(k) is indierent between renting the k + 1th apartment at rent rk+1 and renting the k-th category at rk . In Eq. (3.1), if the rentof marginal category rf is given, the rst equation of Eqs. (3.1) determines rf 31 . In thesame manner, the remaining rents rf 32 , ..., r1 are recursively determined. We call a solution(r1 , ..., rf ) of Eq. (3.1) a dierential rent vector. Under our assumptions, if rf is given withu(e1 , 0) < u(ef , IG(f 31)  rf ), then a dierential rent vector is uniquely determined andsatises r1 > · · · > rf 31 > rf .
We conclude this section by noting the relation between a dierential rent vector and acompetitive rent vector. Let p = (p1 , ..., pT ) be the maximum competitive rent vector and(r1 , ..., rf ) is a dierential rent vector given by rf  pf . Then, it holds that rk  pk for all
k = 1, ..., f (Theorem 3.1 by Sai, 2015). In particular, if rf = pf and some condition holds,
then rk = pk for all k = 1, ..., f .7 Hereafter, we use a dierential rent vector for a comparative
statics.
7
They are two conditions by Kaneko et al. (2006), Theorem 2.6: (1) IG(k) = IG(k)+1 for each k = 1, ..., f 31;(2) pk < Ck (yk + 1) 3 Ck (yk ) for each k = 1, ..., f 3 1.
Under our assumption E on the cost function, the condition (2) holds because the cost of additional unit
from wk , Ck (wk + 1) =“large”. Even when neither conditions hold, a dierential rent vector can be an
approximation of the maximum competitive rent vector. See Sai (2015), Section 3.1.

5

3The Impact of an increase in Income Inequalities on Competitive Rents

3.1

Comparative statics

In this section, we study the relation between household income distribution and competitive
equilibria. The main purpose is to explain how rising income inequality aects a competitive
rent distribution. Recall that the households M are ordered by their income levels as I1 
· · ·  Im . Here, we consider a new market where only the household incomes change. Precisely,{I1 , · · · , Im } changes to {Ib1 , · · · , Ibm }, but the remaining parameters, the sets M, N , utility andcost functions u(·, ·), ck (·), the marginal category f and the marginal rent rf are unchanged.8
By assumption E, the supply amount of each category 1, ..., f is also unchanged in the market.
P
Therefore, the boundary household G(k) = kt=1 wt (k = 1, ..., f ) remains the same. We
consider the following condition on the household incomes.
Condition InE (Increase in Income Inequality). There exists a household iW 5 M \{m} suchP
that Ii < Ibi for i 5 {1, · · · , iW } and Ii > Ibi for i 5 {iW + 1, · · · , m}, and iMM (Ii  Ibi ) = 0.

This condition states that in the new market, an income increases at upper households
than iW + 1 and declines at lower households than iW , preserving the gross income.
r1 , · · · , rbf 31 , rf ) be dierential rent vectors in the original andLet (r1 , · · · , rf 31 , rf ) and (bnew markets determined by rf with u(e1 , 0) < u(ef , IbG(f 31)  rf ). In the next theorem, we
examine how the new rent vector (br1 , · · · , rbf 31 ) changes under Condition InE (the proof will
be given in Section 3.2).
Theorem 3.1 (The Possible Cases of Rent Change). Under Condition InE, either (1), (2)or (3) holds:(1) rk < rbk for k = 1, ..., f  1.

;
W
A
? rk < rbk for k = 1, ..., k  1,
(2) There exist a category kW ( f  2) such thatrkW  rbkW ,A=
rk > rbk for k = kW + 1, ..., f  1.

(3) rk > rbk for k = 1, ..., f  1.

This theorem shows three possibilities of rent change when income inequality increases.
Theorems 3.1.(1) and (3) are straightforward: (1) [(3), respectively] states that a rent rises(declines) at every category 1, ..., f  1 in the new market. Thus, an average rent rises (falls).
The remaining (2) states that a rent increases at upper categories 1, ..., kW , declines at lower
categories kW + 1, ..., f  1. The illustration of (2) is depicted in Fig. 1.
One may think Theorem 3.1.(1) and (3) are counterintuitive: it is natural that rising income inequality causes decline in rents at lower categories and rise in rents at upper categories[case (2)]. Indeed, in the next theorem we show (1) is an extreme case; on the other hand, we
8

We assume {Ie1 , · · · , Iem } also satises Ie1 D · · · D Iem

6

Figure 1: An illustration of Theorem 3.1.(2).
also show (3) is a common case.
Theorem 3.2 (Location of Household iW and Rent Change). Under Condition InE, the following holds:(1) G(f  1)  iW implies Theorem 3.1.(1).

(2) G(1)  iW < G(f  1) implies Theorem 3.1.(2) or (3).

(3) iW < G(1) implies Theorem 3.1.(3).

This theorem characterizes three cases of Theorem 3.1 by the location of household iW of
Condition InE. In Theorem 3.2.(1), the inequality G(f  1)  iW implies that a boundary
income of every category rises, i.e., IG(k) < IbG(k) for every k = 1, ..., f  1. In this case, adierential rent rbk rises for every k = f  1, ..., 1. (3) is also understood in a similar manner.
The remaining (2) is the case that household iW is located below G(1) and above G(f  1). In
this case, there are two possibilities: (i) decline in rents at every category or (ii) rise in rents
above some category k W and decline below kW .
The condition G(f  1)  iW of Theorem 3.2.(1) is an extreme case in that every income
declined household is assigned to the marginal category f and the income declined segment{Ibl+1 , ..., Ibm } is irrelevant to the determination of rents rb1 , ..., rbf 31 . The condition iW < G(1) of
Theorem 3.2.(3) is another extreme case in that every income increased household is assigned
to the rst category 1. Theorem 3.2 states that even if we eliminate the case iW < G(1),Theorem 3.1.(2) still possible.
Now, we compare our results and other related studies. Kaneko et al. (2006) studied eectsof changes in boundary incomes on a dierential rent vector. In particular, they considered
the case: IbG(f 31) IG(f 31)  IbG(f 32) IG(f 32)  · · ·  IbG(1) IG(1) , i.e., the boundary income
increment is larger for a better category of apartments.9 We can apply their condition to our
9They also considered the opposite case: IeG(f 31) 3 IG(f 31) D IeG(f 32) 3 IG(f 32) D · · · D IeG(1) 3 IG(1) .

7

Condition InE as follows:
IbG(f 31)  IG(f 31)  · · ·  IbG(k)  IG(k) < 0 < IbG(k31)  IG(k31)  · · ·  IbG(1)  IG(1)for some k 5 {2, ..., f  1}.

This could be understood as the income inequality signicantly increases. Then, by Theorem
5.2.(1) and Corollary 6.2.(1) (Kaneko et al., p.160 and p.162), the rent dierences form convex
shape
0 < rbf 31  rf 31 < · · · < rbk1  rk1 = · · · = rbk2  rk2 < · · · < rb1  r1 ,where k  k2  k1  f  1,that is, the decrement of rbk to rk gradually gets large from category f  1 to k1 (and takes
maximal from k1 to k2 , and gradually decrease as
Määttänen and Terviö (2014) also studied the eect of rising income inequality on house
prices by using the one-sided assignment model. Their model assume a continuum of agentsand housing types, and the homogeneity and normality on the utility functions. Their main
result (Proposition 4, p.391) is essentially the same as our Theorem 3.1 with the exclusion of
the case (1).10 Nevertheless, their analytical method is dierent from ours in that they uses
a calculus for analyses, while our model is based on niteness.

3.2

Proofs of Theorem 3.1 and Theorem 3.2

It su!ces to prove Theorem 3.2.
Proof of Theorem 3.2.(1). Suppose G(f 1)  iW , i.e., IG(k) < IbG(k) for every k = 1, ..., f 
1. We prove this by mathematical induction over f  1, ..., 1. Let  = IbG(f 31)  IG(f 31) > 0.
the rent equation (3.1) and the normality assumption (Assumption B) implyu(ef 31 , IG(f 31)  rf 31 + ) > u(ef , IG(f 31)  rf + ),that is,u(ef 31 , IbG(f 31)  rf 31 ) > u(ef , IbG(f 31)  rf )= u(ef 31 , IbG(f 31)  rbf 31 ) by Eqs. (3.1).

This inequality and the monotonicity (Assumption A) imply IbG(f 31)  rf 31 > IbG(f 31)  rbf 31 ,that is, rf 31 < rbf 31 .
Suppose rk < rbk for k with 1 < k  f  1. Then we show this relation also holds for k  1.
Let  = IbG(k31)  rbk  (IG(k31)  rk ) and suppose  > 0. Then, Eqs. (3.1) and Assumption
10

Their condition on the income distribution excludes the occurrence of antecedents of Theorem 3.2.(1) and

(3).

8

B implyu(ek31 , IG(k31)  rk31 + ) > u(ek , IG(k31)  rk + ),that is,u(ek31 , IbG(k31)  rk31  rbk + rk ) > u(ek , IbG(k31)  rbk )= u(ek31 , IbG(k31)  rbk31 ) by Eqs. (3.1).

This inequality and Assumption A imply IbG(k31)  rk31  rbk + rk > IbG(k31)  rbk31 , that is,
rbk31  rk31 > rbk  rk > 0.
Suppose the other case   0. Then, Assumption A imply u(ek , IG(k31) rk )  u(ek , IbG(k31) rbk ). Since the left hand side equals u(ek31 , IG(k31)  rk31 ) and the right hand side equals
rk31 ) by Eqs. (3.1), we have u(ek31 , IG(k31) rk31 )  u(ek31 , IbG(k31) b
rk31 ).
u(ek31 , IbG(k31) b
Again, by Assumption A, we have IG(k31)  rk31  IbG(k31)  rbk31 , that is, rbk31  rk31 
IbG(k31)  IG(k31) > 0. Hence we obtain rk31 < rbk31 .

Proof of (2). Suppose G(1)  iW < G(f  1) and let k  = min[k : iW < G(k)]. We rst
prove the inequality rk > rbk holds for k = k , ..., f  1 by mathematical induction. Let
 = IG(f 31)  IbG(f 31) > 0. The rent equation (3.1) and Assumption B imply
that is,u(ef 31 , IbG(f 31)  rbf 31 + ) > u(ef , IbG(f 31)  rf + ),u(ef 31 , IG(f 31)  rbf 31 ) > u(ef , IG(f 31)  rf )= u(ef 31 , IG(f 31)  rf 31 ) by Eqs. (3.1).

rf 31 > IG(f 31) rf 31 , that is, rf 31 > rbf 31 .
This inequality and Assumption A imply IG(f 31) b
Suppose the inequality rk > rbk holds for k with k  < k  f  1. We show this also holds
for k  1. Let  = IG(k31)  rk  (IbG(k31)  rbk ) and suppose  > 0. Eqs. (3.1) and Assumption
B implyu(ek31 , IbG(k31)  rbk31 + ) > u(ek , IbG(k31)  rbk + ),that is,u(ek31 , IG(k31)  rbk31  rk + rbk ) > u(ek , IG(k31)  rk )= u(ek31 , IG(k31)  rk31 ) by Eqs. (3.1).

This inequality and Assumption A imply IG(k31)  rbk31  rk + rbk > IG(k31)  rk31 , that is,
rk31  rbk31 > rk  rbk > 0. Hence we obtain rk31 > rbk31 .
rk )  u(ek , IG(k31) Suppose the other case   0. Then, Assumption A imply u(ek , IbG(k31) brk ). Since the left hand side equals u(ek31 , IbG(k31)  rbk31 ) and the right hand side equals9u(ek31 , IG(k31) rk31 ) by Eqs. (3.1), we have u(ek31 , IbG(k31) b
rk31 )  u(ek31 , IG(k31) rk31 ).
Again, by Assumption A, we have IbG(k31)  rbk31  IG(k31)  rk31 , that is, rk31  rbk31 IG(k31)  IbG(k31) > 0. Hence we obtain rk31 > rbk31 .
From the above discussion, we have rk > rbk holds for k = k , ..., f  1. We next showeither rk > rbk or rk  rbk holds for k = 1, ..., k   1. Furthermore, we show that once rk  rbk
appears for some kW  k  1, then it holds that rk < rbk for k = 1, ..., kW  1.
Let  = IbG(k 31)  rbk  (IG(k 31)  rk ). By condition InE, we have
IbG(k 31)  rbk  (IG(k 31)  rk ) > 0.

(3.2)

Eqs. (3.1) and Assumption B implyu(ek 31 , IG(k 31)  rk 31 + ) > u(ek , IG(k 31)  rk + ),that is,u(ek 31 , IbG(k 31)  rk 31  rbk + rk ) > u(ek , IbG(k 31)  rbk )= u(ek 31 , IbG(k 31)  rbk 31 ) by Eqs. (3.1).

This inequality and Assumption A imply IbG(k 31)  rk 31  rbk + rk > IbG(k 31)  rbk 31 , that
is,
(3.3)
rk  rbk > rk 31  rbk 31 .

On the other hand, Eq. (3.2) and Assumption A imply u(ek , IbG(k 31)  rbk ) > u(ek ,IG(k 31) rk ). Since the left hand side equals u(ek 31 , IbG(k 31)  rbk 31 ) and the right handside equals u(ek 31 , IG(k 31)  rk 31 ) by Eqs. (3.1), we have u(ek 31 , IbG(k 31)  rbk 31 ) >u(ek 31 , IG(k 31)  rk 31 ). Again, by assumption A, we have IbG(k 31)  rbk 31 > IG(k 31) rk 31 , that is, rk 31  rbk 31 > IG(k 31)  IbG(k 31) . By this and Eq. (3.3), we haverk  rbk > rk 31  rbk 31 > IG(k 31)  IbG(k 31) .

Since rk > rbk and IG(k 31) < IbG(k 31) , there are two cases: rk 31 > rbk 31 or rk 31  rbk 31 .
If the latter case, the category k W of Theorem 3.1.(2) is kW = k   1.
Let k with 1 < k  k   1.
(Case rk > rbk ): By Condition InE, IbG(k31) > IG(k31) . Thus, we have IbG(k31)  rbk  (IG(k31) rk ) > 0. In the same manner with the above discussion, we haverk  rbk > rk31  rbk31 > IG(k31)  IbG(k31) ,and there may be two cases rk31 > rbk31 or rk31  rbk31 . If the latter case, the category k W of
Theorem 3.1.(2) is kW = k  1.
rk (IG(k31) rk ) > 0. Eqs. (3.1) and Assumption(Case rk  rbk ): Suppose that  = IbG(k31) b
10

B implyu(ek31 , IG(k31)  rk31 + ) > u(ek , IG(k31)  rk + ),that is,u(ek31 , IbG(k31)  rk31  rbk + rk ) > u(ek , IbG(k31)  rbk )= u(ek31 , IbG(k31)  rbk31 ) by Eqs. (3.1).

This inequality and Assumption A imply IbG(k31)  rk31  rbk + rk > IbG(k31)  rbk31 , that is,
rk31 < rbk31 .
rk (IG(k31) rk )  0. This inequality and Assumption A
Suppose the other case IbG(k31) bkkbrk )  u(e , IG(k31) rk ). Since the left hand side equals u(ek31 , IbG(k31) 
imply u(e , IG(k31) b
rbk31 ) and the right hand side equals u(ek31 , IG(k31)  rk31 ) we have u(ek31 , IbG(k31)  rbk31 ) u(ek31 , IG(k31)  rk31 ). Again, by Assumption A, IbG(k31)  rbk31  IG(k31)  rk31 . Since
IbG(k31) > IG(k31) , we obtain rk31 < rbk31 .
Proof of (3). The proof is the same as the early part of the proof of (2).

3.3

Numerical Examples

In this section, we conrm our comparative statics results by numerical examples. In examples,
we nd that rising income inequality possibly causes (i) decline in rents at every category or(ii) rise in rents above some category k W and decline below kW [the statement of Theorem
3.2.(2)].
Suppose that there are 6 categories of apartments (T = 6) and T = f . For each categoryk = 1, ..., 6, wk amount of apartments are already built and owned by landlord k for sale.
Here, suppose that w1 = w2 = 200, w3 = w4 = 300 and w5 = w6 = 500. We assume thesame number of households are coming to the market to seek the dwelling, and that all thePPapartment units are traded in the end. Therefore, m = kt=1 wt and G(k) = kt=1 wt for
k = 1, ..., 6.
Each household has the following utility function:u(ek , c) = hk +sc (k = 0, ..., 6),

where h1 = 5.1, h2 = 4.4, h3 = 3.7, h4 = 3, h5 = 2, h6 = 1 and h0 = 0. We assume thata household (monthly) income is lognormally distributed.11 In this example, we adopt the11

We say that a (positive) random variable X is lognormally distributed with parameters > and j2 i
Y = ln X is normally distributed with mean > and variance j 2 . The lognormal distribution is denoted by\(>, j 2 ). The probability density function of X ; \(>, j2 ) is given by(ln x 3 >)2
1
exp 3
f (x) = I
(x > 0).
2j2
2Zjx
The mean E, variance V , median M and mode D of \(>, j2 ) are given by E = exp(> + 12 j2 ), V =11Figure 2: Probability density distributions of lognormal distributions.

k
1
2
3
4
5
Gini

G(k)
200
400
700
1000
1500

Changes in IG(k)IG(k)
IbG(k)
371.6
356.7
341.8
329.3
310.1
0.05

514.7
433.8
356.1
305.2
231.4
0.22

bb
I G(k)

660.5
470.1
346.0
264.3
167.5
0.36

Dierences in IG(k)bb
I G(k)  IbG(k)IbG(k)  IG(k)

143.1
77.1
14.3
24.1
78.6

145.8
36.3
10.1
40.9
63.9

Table 1: Boundary incomes in the example
following three lognormal distributions: the mean of lognormal distribution is xed as E =330, and variances are V1 = 1000, V2 = 20000 and V3 = 80000. Fig. 2 depicts probability
density distributions.
In Fig. 2, the most highest graph corresponds to the mean E = 330 and the varianceV1 = 1000, the second highest one corresponds to E = 330 and V2 = 20000, and the remaining
one is E = 330 and V2 = 80000. We generate three sets of 2000 random numbers following
each distribution. We suppose the initial income distribution is V1 = 1000, and it changes into
V1 = 20000 (denote the new incomes by hats); and it also changes into V1 = 80000 (by doublehats). Table 1 gives boundary incomes and Gini coe!cients of each generated incomes.
Table 1 shows that income inequality increases as the variance increases. The table also
shows the magnitude of income dierence is monotonically increasing. Locations of householdb
iW of Condition InE is as follows: (i) G(3)  ibW < G(4) and (ii) G(2)  ibW < G(3). Both satisfy
Condition of Theorem 3.2.(2).
Let the marginal rent r6 = 50. We then calculates dierential rent vectors (r1 , ..., r6 ),
exp(2> + j2 ) exp(j 2 ) 3 1 , M = exp(>) and D = exp(> 3 j2 ). By them, we have D < M < E, and thus,\(>, j 2 ) has a long-tail form. These dinitions and properties are due to Crow and Shimizu (1988). The
lognormal distribution is aften used as an approximation of an income distribution.

12

Changes in rk
rk
rbk
173.2
177.2
153.0
150.9
132.5
126.9
111.8
105.2
81.3
75.9
50
50

k
1
2
3
4
5
6

Dierences in rkbrbk  rbk
rbk  rk
4.0
1.0
2.0
6.1
5.6
7.8
6.5
7.7
5.3
5.3
0
0

b
rbk
176.1
114.8
119.1
97.5
70.7
50

Table 2: Dierential rents in the example
R ent
177.2
176.1

rk
V =1000
153.0

173.2

r̂k
V =20000

150.9

V =80000
r̂ˆk

132.5
126.9

144.8

111.8
105.2

119.1

81.3

97.5

75.9
70.7
50.0

1

2

3

C ategory

4

5

6

Figure 3: Rent changes in the example.
rb1 , ..., b
rb5 , r6 ) by Eq. (3.1). The calculation result and its illustration are(b
r1 , ..., rb5 , r6 ) and (bgiven in Table 2 and Fig. 3.
As seen from Table 2, the rst income change causes decline in rents except the rst
category and rise in rent at the rst category, that is, Theorem 3.1.(2). On the other hand,the second change causes decline in rents at every category [Theorem 3.1.(3)]. These results
are consistent with Theorem 3.2. In sum, it can be said that increase of an income inequality
often lowers a rent at every category (as well as average rent).

4

Income inequality and equitability of competitive allocations

Here, we briey mention the equitability property of competitive allocations in our market
model. Foley (1967) and Varian (1974) developed the theory of equitability (or fairness) inmarkets with perfectly divisible goods. Svensson (1983), Alkan, Demange and Gale (1991)
and Sakai (2007) also studied equitability/fairness in the indivisibility framework. However,their models are dierent to ours in that their models consist of (i) only buyers (ii) the13

same number of buyers and indivisible units (iii) no initial endowments and (iiii) without
homogeneous preference assumption. Here, we focus on competitive equilibria, and consider
equitability of competitive allocations in our model.
We rst give some notations (denitions are due to Foley, 1967). Recall that the consumption set of households are given by X = {e0 , e1 , . . . , eT } × R+ . Let an m-tuple a =(a1 , ..., am ) 5 X m be a consumption allocation. We say that i envies j at a 5 X m iu(aj ) > u(ai ). We say that a 5 X m is the equitable (envy-free) allocation i u(ai )  u(aj ) for
every i, j 5 M . Note that in our framework, this condition can be translated by u(ai ) = u(aj )for every i, j 5 M .
The following proposition holds in our market model.
Proposition 4.1. Let (p, x, y) be a competitive equilibrium and let i, j 5 M. Then, Ii  Ij if
and only if u(xi , Ii  pxi )  u(xj , Ij  pxj ) (note that  is replaced by , >, <, or =).
Proof. (Only If ) By the antecedent Ii > Ij and utility maximization condition, we haveu(xi , Ii  pxi )  u(xj , Ii  pxj ) > u(xj , Ij  pxj ). (If ) Suppose, on the contrary, Ii  Ij .
Then, we obtain the contradictory inequality by utility maximization condition: u(xj , Ij pxj ) u(xi , Ij  pxi )  u(xi , Ii  pxi ).
This proposition means that if there exist two households having dierent incomes, thenthe lower-income household envies the higher-income household in any competitive allocations;conversely, if some household envies the other in a competitive allocation, then the income ofthe envied household is higher. Furthermore, if incomes of some two households are the same,then their utility levels also the same in any competitive allocations; conversely, if utility levelsof some two households are the same in a competitive allocation, then their incomes also thesame.
The following corollary follows from the proposition.
Corollary 4.2. Let (p, x, y) be a competitive equilibrium. Then every household has the same
income if and only if an m-tuple ((x1 , I1  px1 ), ..., (xm , Im  pxm )) is an equitable allocation.
Thus, when the household income distribution has even a little inequality, any competitive
allocation does not satises equitability (conversely, if a competitive allocation does not satises equitability, then the income distribution has an inequality). Theorems 3.1,2 and Corollary
4.2 imply that rising income inequality tends to cause both dampening the equitability on
household allocations and a decline in landlord revenues. Note that since any competitive
equilibrium is Pareto e!cient in our market model, an equitable competitive allocation isa fair allocation.12 Note also that the only-if part of the corollary holds without identicalutility function assumption, whereas the if part does not holds without this assumption. The
12

Svensson (1983) and Sakai (2007) gave a result related to Corollary 4.2. According to them, a consumption
allocation ((x1 , c1 ), ..., (xm , cm )) M X m is a Walrasian allocation from equal income i there exist p M RT+ andI M R+ such that ci = I 3 pxi for all i M M and every household maximizes his utility, where I is the implicit
imcome. They showed that the set of equitable allocations coincides with the set of Walrasian allocations fromequal income.

14

next example shows a case that income inequality exists but a competitive allocation satises
equitability.
Example 4.3 (Equitable competitive equilibrium with income inequality exists). Suppose that
there are two households 1 and 2 with incomes I1 = 150 and I2 = 100, two dierent apartments 1 and 2 (with reservation prices 50 and 36). Suppose that their utility functions are
given as;;ss
A 0 + c for k = 0,
A 0 + c for k = 0,?
?
ssu1 (ek , c) =4 + c for k = 1, u2 (ek , c) =1 + c for k = 1,
A
A
s
s
=
=
1 + c for k = 2,
4 + c for k = 2.

This setting explains, for example, the following situation: the apartment 1 is a relatively
large one located in a suburban area and the apartment 2 is a small one located in a central
city. Household 1 with higher income prefers the apartment 1 to 2, while the household 2
prefers the apartment 2 to 1.
Let p = (p1 , p2 ) = (50, 36). Then, u1 (e1 , I1  p1 ) = 14 > u1 (e0 , I1 ) > u1 (e2 , I1  p2 ) andu2 (e2 , I2  p2 ) = 12 > u2 (e0 , I1 ) > u2 (e1 , I1  p1 ). Hence, a triple (p, (e1 , e2 ), (1, 1)) is acompetitive equilibrium. On the other hand, u1 (e1 , I1  p1 ) = 14 > u1 (e2 , I2  p2 ) = 9 andu2 (e2 , I2  p2 ) = 12 > u2 (e1 , I1  p1 ) = 11. Hence, this equilibrium satises equitability but
income inequality exists.

5

Conclusions

We have studied the comparative statics analysis based on the assignment market model. In
particular, we present how rising income inequality aects a competitive rent distribution. The
key assumptions of the model are homogeneous and normality assumptions on the household
utility functions. A competitive rent vector can be then calculated by a system of equations.
Our main comparative statics result is Theorem 3.1, stating that an increase in income
inequality eects three cases on the competitive rent vector: (i) rise at every category, (ii) riseat higher categories and decline at lower categories or (iii) decline at every category. Another
Theorem 3.2 implies that (i) is a special case, while (iii) [as well as (ii)] is possible in a generalsituation. Numerical examples facilitated our comparative statics results. We also mentioned
the equitability property of competitive allocations in our market model.

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15

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16