Abstract

11
Introduction
This paper investigates a certain property of a differential rent vector in a rental housing market model by Kaneko, Ito and Osawa [9]. In particular, we evaluate the difference between the maximum and minimum values of the differential rent vectors. This model is an application of the generalized assignment market by Kaneko [7],1 where the market participants are divided into households and landlords, the rental housings to be traded are treated as indivisible goods and classified into finite categories, and each household can consume (at most) one housing unit. A differential rent vector (which plays a key role in their model) is a solution of a certain system of equations called "the rent equation". This vector coincides with the maximum competitive rent vector under some reasonable conditions. Thus, a differential rent vector is regarded as a good approximation of the maximum compet- itive rent vector. Kaneko et al. and their following papers (e.g., Ito [4]) presented numerous comparative statics results using a differential rent vector. On the other hand, there is a concern regarding the adequacy of adopting the maximum competitive rent vector to represent the market equilibrium. While there are multiple equi- librium candidates in the market model, we cannot observe which one is actually selected in the real housing market. There may be a gap between the comparative statics results of the model the and real markets. Regarding this problem, Kaneko et al. noted the availability of the minimum instead of the maximum competitive rent vector to represent the market equi- librium. However, a similar concern still remains, even when we adopt the minimum vector. There has been little investigation concerning this problem. One reasonable solution to the above problem is to study the difference between the maximum and minimum differential rent vectors. In this manner, we can identify the gap between the two comparative statics results. Here, we introduce a summary of one of our results, Theorem 3.5 in Section 3.2.  Theorem 3.5 states that the difference between the maximum and minimum differential rent vectors is bounded by the largest income difference between two specific neighboring households.  This implies that the rent difference is not large, and that it tends to smaller as the distribution of household income gets dense. Thus, we could argue that, when we target a considerably large housing market, the difference of the comparative statics results is small whether we use the maximum or minimum vectors. The numerical examples in Section 4.1 confirm our results. We present a more precise discussion on the shrinkage of the difference of the differential rent vectors in Section 4.2. Our results are related to a recent paper by Sai [11], who studied the difference of the maximum and minimum competitive price vectors under weaker conditions. His approach is different from ours in that he characterizes the price difference by the difference in the seller's marginal costs. On the other hand, the cost structure is not easily observable from the real data in comparison to the household income data. Furthermore, the households (buyers) rather than the landlords (sellers) play the main role in our market model. Therefore, we could be say that, for studying the housing market model, our characterization is more helpful in a housing market analysis. We should note that the following two assumptions are important in our market model:  the identical utility function, and the normality of the quality of housing.  The former assumption is necessary for the derivation of the differential rent vectors, and the latter assumption plays a crucial role in the results. These assumptions imply that the difference in housing preferences (i.e., the difference between households) is only characterized by the difference in incomes. These assumptions are salient features of our model because the quasi-linear utility assumption is s still the main-stream view of markets with indivisibilities. Recently, Määttänen and Terviö [5] studied the housing market using a one-sided assignment model that assumed the identical utility function and the normality of the quality of housing, as outlined above. 2 They presented comparative statics results on the relation between income distribution and housing prices. As viewed from the perspective of the urban economics literature, we can find the source of our approach from Alonso’s [1] bid rent theory.  He studied a mechanism of land pricing in urban area with a central business district, where land sizes are treated as continuous variables. Our differential rent approach is based on his bid rent theory. 3 This paper is organized as follows. Section 2 formulates our rental housing market model and gives our definition of competitive equilibrium. Section 3 first introduces two recursive equation systems, the maximum and minimum rent equations, from which we can derive the maximum and minimum differential rent vectors. This section then outlines the main results of our study. Section 4 provides some numerical examples and an application of our theorems. Section 5 presents our conclusions and further remarks.
2
The market model
This section introduces the rental housing market model of Kaneko et al. [9]. In Section 2.1, we give our basic assumptions and the definition of competitive equilibrium. In Section 2.2, we introduce additional assumptions that facilitate our study.
2.1
General formulation
The rental housing market is denoted by ( M , N ), where M = {1, . . ,m} denotes the set of households, and N = {1', . . , n'} denotes the set of landlords. The apartments are classified into finite categories 1, …., T , and are assumed to be already built. Each household i ∈ M initially has an income I i > 0 but no dwelling. The household wants to rent at most one apartment unit paying rent from his income. Without loss of generality, we can assume that the households are ordered in their incomes as I ≥ I 2 ≥ · ·  ·· ≥ I m . The consumption set is written by X := {e 0 ,e 1 ,...,e T } × R + , where e k is the T-dimensional unit vector with e ξ = 1 (e 0 = 0), and R + is the set of nonnegative real numbers. A consumption vector (e k,c) ∈ X with k̸= 0 means that household ˙ rents one unit of the k-th category of an apartment and enjoys the consumption c = I i — pk , where pk is the rent of the k-th apartment. For k = 0, no apartment is consumed. An initial endowment of i ∈ M is given as (e , I ; ) with I ; > 0. A utility function of household i is given by u i : X → R. We make the following assump- tion. Assumption A . For each i ∈ M and x ∈ {e 0,e 1,. . ,e r }, u i ( x i , c ) is a continuous and strictly The first part of Assumption A allows a utility function to have an "income effect". An inequality in the last part is a boundary condition, meaning that a household prefers keeping his income to rent any apartment by paying all his income. Each landlord j ∈ N provides apartments of exactly one category (say k ), but may provide more than one unit. The landlord has a cost function C j ( Y j ) : Z + → R + , where Z + is the set of nonnegative integers. For each y j ∈ Z +, C j (y j ) represents the cost (in terms of money) of supplying y j units of the k-th category. We make the following assumption for C j ( · ). Assumption B. For each j ∈ N, C j (0) = 0 and C j (y j +2) − C j (y j +1) ≥ C j (y j +1) for all y ∈ Z +. The first part of Assumption B means that no fixed cost is required for no production. The last part is a discrete version of the standard convexity assumption on a cost function, meaning that the marginal cost is increasing. As mentioned above, we assume that the apartments in the market are already built, and thus only the operating costs need to be covered by the landlords. 4 For notational simplicity, we assume that only one landlord k provides apartments in the k -th category. Thus, the set N becomes {1, . . ,T } , and landlord k ∈ N is the only landlord providing the k-th apartments. As far as the competitive equilibrium is concerned, this can be assumed without loss of generality.5 Let (p,x,y) = ((p,...,pr), (x1,...,xm), (y1,...,yyr))) be a triple of p ∈ R , x ∈ Definition 2.1. We say that a triple ( p,x,y ) is a competitive equilibrium iff (1) Utility Maximization under the Budget Constraint (UMC): for all i ∈ M, (i) I i − px i ≥ 0, where px i = ∑ k=1 Pk x ik ; (ii) u i ( x i, I i — r x i ) ≥ u i ( x i, I i — r x i ) for all x i ∈ { e 0 ,e 1 ,. . ,e T } with I i — r x i ≥ 0. (2) Profit Maximization (PM): For all k with 1 ≤ k ≤ T, РкУк − Ск (Ук) ≥ РкУк. − Ск (Ук.) for all Ук. ∈ Z+. (3) Balance of the Total Demand and Supply (BDS): ∑ i ∈ M x i = ∑ k =1 yk e k Under Assumptions A and B, we have a competitive equilibrium in ( M , N ). Theorem 2.1 (Kaneko, 1982; Kaneko and Yamamoto, 1986).  There exists a competitive equilibrium (p,x,y) in a rental housing market (M,N). We say that p = ( p 1, . . , p r ) is a competitive rent vector iff ( p,x,y ) is a competitive equilibrium for some x ∈ {0,e¹,...,e¹ } m and y ∈ Z †. Note that there may be multiple competitive equilibria. In particular, the maximum and minimum competitive rent vectors exist and play an important role in our analysis. 6 Theorem 2.2. There exist the maximum and minimum competitive rent vectors in ( M , N ). Kaneko [8] and Kaneko et al. [9] proved the existence of the maximum competitive rent vector. A complete proof is in Appendix A. We say that ( p,x,y ) is a maximum ( minimum ) competitive equilibrium iff p is the maximum (minimum). By definition, the maximum (min- imum) competitive rent vector is uniquely determined; however, multiple maximum (mini- mum) competitive equilibria may exist. Kaneko et al. and their following subsequent papers (e.g. Ito [4]) adopted the maximum competitive rent vector to represent the market equilibrium rent of their comparative statics. In their analysis, the maximum competitive rent vector is calculated according to a certain system of equations called "the rent equation." On the other hand, it is also possible to use the minimum competitive rent vector in their analysis. As stated in the introduction, we cannot observe what (competitive) equilibrium rent is s elected when the real market achieves equilibrium. Thus, there is room to consider the adequacy of using the maximum competitive rent vector as the market equilibrium.
2.2
Specific assumptions for ( M , N )
In addition to Assumptions A and B, we assume that every household has identical utility function; that is, Assumption C. u i (·, ·) = u j (·, ·) for all i,j ∈ M. By Assumption C, we can simplify the utility function u i as u. In an urban economics context, Assumption C implies that the housing market ( M , N ) represents a mono-centric city, and all the households commute to an identical business district. Thus, under C, each household only is characterized by its initial income level. Some readers may be concerned that C implies an identical apartment preference for each household. However, this concern will be eliminated by the next assumption. Assumption D. If u(x i,c) = u(x i,c ), and c < c , then u(x i,c + δ) > u(x i,c ′ + δ) for any δ > 0. Assumption D is the normality assumption on the quality of apartments. In D, apartment x i is better than x i because a household living in x i with a smaller consumption c is indifferent to living in x e with a larger consumption c . This implies that, for each household, the demand shifts to a better apartment or remains the same if their income increases. The next assumption is regarding the quality of apartments. Assumption E. u(e¹,0) > u(e²,0) > ··· > u(e†,0).7 Thus, the apartments are numbered according to their quality level. The first category is the best one and the T-th category is the worst. These assumptions facilitate our study of using the “differential rent approach”.
3
The rent equation and the evaluation of the differential rent
vector
In Section 3.1, we introduce two systems of equations.  One was formulated by Kaneko et al., which we call the maximum rent equation. Another is newly formulated in a parallel manner, which we call the minimum rent equation. Each solution of the equations is called the maximum/minimum differential rent vector , corresponding to the maximum and minimum competitive rent vectors, respectively.  Using both differential rent vectors, we present two theorems on the evaluation of the difference between the maximum and minimum differential rent vectors in Section 3.2. Section 3.3 gives proofs of two theorems.
3.1
The rent equation
To introduce the rent equation, we give more detailed assumptions and some lemmas. The purpose of this paper is to evaluate the difference between the maximum and minimum com- petitive rent vectors. Therefore, hereafter, we consider the case where the maximum com- petitive rent vector (denoted by pmax) and the minimum competitive rent vector (denoted by pmin) satisfy pmin < pmax for all k with 1 ≤ k ≤ T . This assumption together with Theorem 3.1 by Sai (2014) (also Lemma A.2 in the Appendix) imply that for any competitive equilibria (p, x, y) and (p', x', y'), (3.1) yk = Yk for all k with 1 ≤ k ≤ T, that is, for any category k , the equilibrium supply for k is uniquely determined. The following lemma has an important role. Lemma 3.1 (Kaneko et al., 2006). Let (p,x,y) be a competitive equilibrium. Then, (1) If x i = e k , x i = e k  and I i > I , for some i,i ,then k ≤ k . This lemma states that in any competitive equilibrium, (1) the price of a better apartment is higher, and (2) a household with a higher income rents a better apartment. Note that in (1), it may be possible that no one rents an apartment in the k ′ -th category, while the k -th apartment is rented by someone. To eliminate this case, we assume that there is a category f dividing the apartments into active categories and inactive categories. That is, Assumption F. There exists some category f such that for any competitive equilibrium (p,x,y), yk > 0 for k with 1 ≤ k ≤ f and yk = 0 for k with f ≤ k ≤ T. We call this f the marginal category. Note that Assumption F includes the same f for each competitive equilibria. The condition of equation (3.1) guarantees such a treatment. In the literature of urban economics, the marginal category corresponds to the marginal land in Ricardo’s differential rent theory. Recall that all the households M = {1, ..., m} are ordered by their incomes as I ≥ I 2 ≥ ··· ≥ I m . We next define the household with the lowest income in each active category. This household plays a crucial role in the rent equation. Let ( p,x,y) be a competitive equilibrium. For each category k ( ≤ f ), we define the household G ( k ) with the lowest income in the k -th category as:
For each k , we call G(k) the boundary household of the k-th category.  Note that by equation (3.1), G(k) is uniquely determined for each k . We now introduce the rent equation. The maximum rent equation is defined as the system of equations with the unknowns (ﾃ1 , ...,ﾃf ). The formulation is from Kaneko et al. In a parallel manner, the minimum rent equation is defined with the unknowns ( Σ 1 , ……, Σ f ). Definition 3.1.  (1) (Kaneko et al, 2006):  We call the following system of equations the maximum rent equation:
(2) We call the following system of equations the minimum rent equation :
The difference between equations (3.2) and (3.3) is the replacement of the boundary in- come I G (k) by I G (k)+1 for k = 1, …., f − 1. In equation (3.2), ifﾃf is given with u (e¹,0) < u(ef , IG(f-1)-Ff), then the unknown Ff-1 is uniquely determined by the first equation. In the same manner, the remaining unknowns r 1 , ..., r f –2 are recursively and uniquely determined, and it holds thatﾃﾞｭ> · · >ﾃﾞƒ−ı >ﾃﾞƒ .8 Similarly, a solution ( r 1 , …., x ƒ−ı) is uniquely deter- mined if r f is given with u(e 1,0) < u(e f , IG(f-1) - x f ) and it holds that x 1 > · · > x f-1 > x f-1 > x f We say that ( F 1 , ..., F f ) is a maximum differential rent vector iff it is a solution of the rent equation (3.2) and ( x 1 , ..., x f ) is a minimum differential rent vector iff it it is a solution of the rent equation (3.3). We then have the following relationships for the vectors ( r 1 , ..., r f ), ( r 1 , ..., r f ) and the maximum and minimum competitive rent vectors. Theorem 3.1. (1) Let (p,x,y) be a maximum competitive equilibrium and ( r 1,..., r f ) with Tf = pf be a solution of equation (3.2). Then Tk ≥ pk for all k with 1 ≤ k ≤ f − 1. (2) Let (p,x,y) be a minimum competitive equilibrium and (r 1 ,….,r f) with r f = p f be a solution of equation (3.3). Then r k ≤ p k for all k with 1 ≤ k ≤ f − 1. Proof is in Appendix B. In our study, although the competitive rent p f of the marginal category is endogenously determined, the rent p f of the marginal category f is exogenously given (e.g., Kaneko et al. adopted an estimated rent pf f from the real data). The meaning of Theorem 3.1.(2) is that if such a pf is close to the maximum competitive rent, the vector ( ¯ 1, …., ¯ f ) derived from equation (3.2) corresponds to an upper bound of the set of competitive rent vectors. Similarly, (2) means that if pf is close to the minimum competitive rent, the vector ( r 1 , ..., r f ) derived from equation (3.3) corresponds to an lower bound of the set of competitive rent vectors. An illustration of Theorem 3.1 is given by Fig. 1, which depict a positional relationship of a differential and a competitive rent vectors. Kaneko et al. provided two sufficient conditions for the maximum differential rent vector to coincide with the maximum competitive rent vector (Theorem 3.2).  We can also expect a similar condition for the minimum competitive rent vector to coincide with the minimum differential rent vector (Theorem 3.3). Theorem 3.2. (Kaneko et al., 2006) Let (p,x,y) be a maximum competitive equilibrium. If at least one of the following holds: (1) I G ( k ) = I G ( k )+1 for each k with 1 ≤ k ≤ f − 1; (2) pk < Ck (yk + 1) − Ck (yk) for each k with 1 ≤ k ≤ f − 1, then the maximum differential rent vector (ﾃ1 , ...,ﾃf ) determined byﾃf = p f coincides with (P1, …., Pf). Theorem 3.3.  Let (p,x,y) be a minimum competitive equilibrium.  If at least one of the following holds: (1) IG(k) = IG(k)+1 for each k with 1 ≤ k ≤ f − 1; (2) pk > Ck (Yk) − Ck (Yk − 1) for each k with 1 ≤ k ≤ f − 1, then the minimum differential rent vector ( r 1 , ..., x f ) determined by r f = p f coincides with ( P 1 , … , P f ) . The proof is in Appendix C. Each theorem has the same Condition (1), which states that the income of the last household in the k-th category coincides with the income of the first household in the k + 1-th category. This implies that, when the number of households is large and the income distribution is more or less dense (i.e., Condition (1) holds approximately), then the maximum and minimum differential rent vectors can be regarded as approximations of the maximum and minimum competitive rent vectors, respectively. Condition (2) of each theorem states that a profit maximization condition strictly holds for the k-th category, in other words, a competitive price p k does not coincide with a marginal cost for the k-th category. Note that under condition (1) of Theorems 3.2 and 3.3, if the competitive rent p f of the marginal category is determined uniquely, then by the rent equations, the maximum differential rent vector (determined by ¯ f = p f) coincides with the minimum differential rent vector (determined by r f = p f ). When conditions (1) and (2) fail, however, the difference between the maximum and minimum differential rent vectors is an open question.
3.2
The difference between the maximum and the minimum differential
rent vectors
As presented in the previous section, the maximum/minimum differential rent vectors coin- cides with the maximum/minimum competitive rent vectors under a certain condition. We may use either of the maximum or minimum differential rent vectors for comparative statics; however, our results may differ, depending on which one we use. This section provides an answer to this question. In particular, we evaluate the difference between the maximum and minimum differential rent vectors. The proofs of theorems are in Section 3.3. The following theorem concerns the relationship between the income difference and the rent difference. Theorem 3.4.  Let ( F 1 , …. F f ) and ( Σ 1 , …. L f ) be the maximum and minimum differential rent vectors determined by ¯ ƒ = ¯ ƒ and r ƒ = ¯ ƒ with p ƒ ≤ ¯ ƒ, and k with 1 ≤ k ≤ ƒ. Then,ﾃﾞk −  Γ k ≤ I G(k−1) − I G(k−1)+1 if and only ifﾃﾞk −  Γ k ≤ﾃﾞk−1 −  Γ k−1
Note that ≤ of equation (3.4) can be replaced by ≥, >, < or =. The form of Theorem 3.4 is similar to the "Basic comparative statics theorem" of Kaneko [8] and Kaneko et al. [9]. Nevertheless, the meaning is quite different. Theorem 3.4 states that the rent difference of the k-th category is smaller than the income difference of two neighboring households numbered G(k − 1) and G(k − 1) + 1 if and only if the rent difference of k − 1 is greater than the rent difference of k . This implies that we can reduce the comparison of the differences Fk − r k and F k −1 − r k −1 to the comparison of the differences F k − r k and I G ( k − 1) − I G ( k − 1)+ 1 · Fig. 2 depicts three examples of a shape of rent differences ¯ k − r k (1 ≤ k ≤ f). Fig.2.(1) explains the case of (3.4) holds for each k . In this case, the difference ¯ k −  r k gradually increases as k reaches 1. Fig.2.(2) explains the case where the opposite inequality of (3.4) holds for each k . In this case, the difference rk — rk gradually decreases as k reaches 1. The remaining Fig.2.(3) explains the case where there is a category l (1 ≤ l ≤ f) such that an inequality of (3.4) switches at l : the differenceﾃk − τ k gradually increases for k with l ≤ k ≤ f and decreases for k with 1 ≤ k < l . Numerical examples are given in Section 4.1. The next theorem evaluates the rent difference by the income difference. Theorem 3.5.  Let ( F 1 , ..., F f ) and ( r 1 , ..., L f ) be the maximum and minimum differential rent vectors determined by ¯ f = ¯ f and r f = ¯ f with ¯ f ≤ ¯ f f . Suppose that ¯ f − r f ≤ IG(f-1) - IG(f-1)+1・ Then,
Theorem 3.5 states that if the rent difference Ff — Tf of the marginal category f is less than the income difference I G(f −1) − I G(f −1)+1 of two neighboring households, then the rent difference of the k-th ( k < f ) category is bounded by at most the largest income difference I G(l) − I G(l)+1 (k ≤ l < f). In our differential rent approach, the rent of the marginal category f is regarded as a uniquely determined value. 9  This implies that the maximum and minimum differential rent vectors are determined by the same p f (=  f f =  f ); thus, the supposition of Theorem 3.5 holds. Under this situation, the theorem implies that the rent differences ¯ k −  k for each k are rather small. In particular, when we target a considerably large housing market with a dense household income distribution (i.e., the equality I G (k) = I G (k)+1 approximately holds), the difference can be approximated by zero.  Consequently, the comparative statics results are not very different, whether or not we use the maximum or minimum differential rent vectors. Theorem 3.5, together with Theorem 3.1 in Section 3.1, imply the following:
that is, the difference of the maximum and minimum competitive rents of k -th category is also bounded by the largest income difference I G (l) − I G (l)+1 with k ≤ l < f . This implies the shrinkage result on the competitive rent vector set, which will be presented in Section 4.2.
3.3
Proofs of Theorems 3.4 and 3.5
Here, we prove Theorems 3.4 and 3.5. Proof of Theorem 3.4. ( Only if ) By (3.3), we have u ( ek−1 , IG(k−1)+1−1 k−1) = u ( k , IG(k−1)+1− r k ). Let δ = I G(k-1) − r k −( I G(k-1)+1 − r k ) ≥ 0. By Assumption D, u(e k-1 , I G(k-1)+1 − r k-1 + δ) ≥ u(e k, IG(k-1)+1 − rk + δ), that is,
This inequation together with Assumption A imply I G (k − 1) − Lk − 1 − Tk + Lk ≥ I G (k − 1) − Tk − 1, ● that is,ﾃﾞk — ঙk ニ r k —  ঙk — 1· (If ) We prove the contraposition of the claim. Suppose that ¯ k − ˙ L k > IG(k−1) − IG(k−1)+1 By (3.2), we have u(e k −1 ,I G(k −1) − rk−1) = u(e k ,I G(k −1) − rk). Let δ = I G(k −1)+1 − rk −(I G (k −1) − ¯ rk) ≥ 0. By Assumption D, u (e k −1, I G (k −1) − rk−1 + δ) > u (e k , I G (k −1) − rk + δ), that is,
This inequation together with Assumption A imply IG(k-1)+1 - Tk-1 - Tk ++ Tk > IG(k-1)+1 - Tk-1, that is, Tk — Ck > Tk-1 — Ck-1 Proof of Theorem 3.5.  We proof this by mathematical induction over k = f − 1, f − 2 ,…., 1. Let k = f − 1. By the hypothesis and monotonicity (Assumption A), we have u(e f , I G(f − 1)+1 − L f ) ≤ u(e f , I G(f − 1) − ¯ f ). The left hand side is equal to u(e f − 1, I G(f − 1)+1 − Σf−1) by (3.3), and the right hand side is equal to u(e f−1, IG(f−1) − Ff−1) by (3.2). Hence, we have u(ef-1, IG(f-1)+1 - Lf-1) ≤ u(ef-1, IG(f-1) - rf-1). This and Assumption A imply
Let δ = IG(f-1) − IG(f-1)+1 ≥ 0.  Since u(ef-1,IG(f-1) − rf-1) = u(ef,IG(f-1) − rf) by (3.2), we have, by Assumption D, u(e f−1 , IG(f−1) − r f−1 − δ) ≤ u(e f, IG(f−1) − r f − δ). This inequation is restated as
11 This and Assumption A imply I (f −1)+1 − r f −1 ≤ I G(f −1)+1 − r f −1, that is, r f −1 ≥ r f −1. By this and (3.6), we have the relation 0 ≤ r f−1 − x f−1 ≤ IG(f−1) − IG(f−1)+1. Suppose that for k = j with 1 < j ≤ f − 1,
Then, for k = j − 1, (i) Supposeﾃj − Γ j ≤ IG(j−1) − IG(j−1)+1. Then, u(e j, IG(j−1)+1 − τ j ) ≤ u(e j, IG(j−1) − τ j ). The left hand side is equal to u(ej-1,IG(j-1)+1 - tj-1) by (3.3), and the right hand side is equal
to u(ej-1,IG(j-1) - tj-1) by (3.2). Hence, u(ej-1,IG(j-1)+1 - tj-1) < u(ej-1,I This and Assumption A imply I G ( j −1)+1 − t j −1 ≤ I G ( j −1) − t j −1, that is,
Let δ = I G(j-1) - I G(j-1)+1 ≥ 0. Since u(e j-1, IG(j-1) - ī j-1) = u(e j, IG(j-1) - ī j) by (3.2), we have, by Assumption D, u(e j −1, IG(j −1) − π j −1 − δ) ≤ u(e j , IG(j −1) − π j − δ). This inequation is restated as
This and Assumption A imply I G(j-1)+1 − T j-1 ≤ I G(j-1)+1 − L j-1, that is, T j-1 ≥ t j-1. By this and (3.8), we get (3.9) 0 ≤ r j −1 − ≥ j −1 ≤ IG(j −1) − IG(j −1)+1. (ii) Suppose ñ = L j > IG(j-1) − IG(j-1)+1. Then, u(e j, IG(j-1)+1 − L j ) > u(e j, IG(j-1) − r j ). This together with (3.2) and (3.3) we have u(e j −1 , IG(j −1)+1 −1 j −1 ) > u(e j −1 , IG(j −1) − r j −1 ). This and Assumption A imply I G ( j −1)+1 − 1 j −1 > I G ( j −1) − r j −1, that is, (3.10) T j −1 − t j −1 > I G (j −1) − I G (j −1)+1. Let δ = ¯ j − 1 − ˙ r j − 1 − ( IG(j − 1) − IG(j − 1)+1) ≥ 0. Since u(e j − 1, IG(j − 1) − ¯ j − 1) = u(e j , IG(j − 1) − ¯ j ) by (3.2), we have, by Assumption D, u(e , I G(j − 1) − ¯ j + δ) ≤ u(e − 1, I G(j − 1) − ¯ j − 1 + δ). This is restated as
This and Assumption A imply I G ( j −1 )+1 − F j + F j −1 − L j −1 < I G ( j −1 )+1 − L j , that is, F j −1 − L j −1 <ﾃﾞj − x j . By this and (3.10), we get I G(j−1) − I G(j−1)+1 < π j−1 − x j−1 < π j − x j . This together with (3.7) implies
This inequation together with (3.9), we have Hence, for all k with 1 ≤ k ≤ f − 1, we have 0 ≤ ¯ k − ¯ k ≤ l ∈{ k,….; − 1} IG(l) − IG(l)+1}.
4
Numerical examples and the application
In this section, we give three examples of the rental housing market model. The examples con- firm our results in Section 3.2.  As an application of Theorem 3.5, we also present the shrinkage result on the difference between the maximum and minimum differential rent vectors.
4.1
Numerical examples: Calculation of maximum and minimum differen-
tial rent vector
First, we give common settings for our examples. Suppose that there are six categories of apartments (T = 6). Let w k (k = 1, …. 6) be the number of apartment units for rent in the k-th category.  We assume that the number of households and apartments for rent are the same, and that all the apartments are ultimately rented. That is, the marginal category is f = 6 and the number of households is m = ∑ k =1 wk . Assume that each household has the following utility function:
where h 1 = 9, h 2 = 7, h 3 = 5, h 4 = 4, h 5 = 3, h 6 = 2 and h 0 = 0. Assume that the income of each household is uniformly distributed over the interval [100,500] . (Then this utility function satisfies Assumption A, C,D and E). The settings for the following Examples 4.1 and 4.2 are the same, except for the number of apartment units and households. These examples show the difference between the maximum and minimum differential rents, as well as the smaller difference in a market with a large number of households compared to a market with a small number of households. Example 4.1. Let w k = 5 for each k = 1, ..., 6. This implies IG(k) − IG(k)+1 ≥ 13.8 for each k. Let F 6 = r 6 = 20. Then, we can calculate the maximum and minimum differential rent vectors by the rent equations (3.2) and (3.3). Table 4.1 shows the calculation results of r k, Lk and rk — Lk
By Table 4.1, an inequality r k − r k ≤  j ∈{ k,..,,f − 1 } C( j ) − I C( j )+1 } of Theorem 3.5 holds for each k . Furthermore, we observe π k − π k < π k − 1 − τ k − 1 for each k . This is consistent with Theorem 3.4 because rk − rk < IG(k−1) − IG(k−1)+1 for each k (which correspond to Fig. 2.(1) in Section 3.2). To sum up, the difference rk − rk is significantly smaller than I G (k) − I G (k)+1 for each k ; however, the difference tends to increase as k reaches 1. Example 4.2.  Let w k = 20 for each k = 1, …, 6.  This implies I G (k) − I G (k)+1 ≈ 3.4 for each k. Let r 6 = r 6 = 20. Table 4.2 shows the calculation results of r k , r k and r k = r k
As with Example 4.1, the difference rk − rk is smaller than I G (k) − I G (k)+1 for each k ; however,the difference gradually becomes larger as k reaches 1. We can also observe that for each k , the difference r k − r k is significantly smaller than Example 4.1. Next, we give another example where the hypothesis of Theorem 3.5 fails. This example shows that whereas the rent difference rk — rk exceeds the income difference IG(k) — IG(k)+1, the rent difference tends to decrease as k goes to 1. Example 4.3.  Let wk = 20 for each k .  This implies IG(k) − 1G(k)+1 ~ 3.4.  Let ñ 6 = 20 and r 6 = 15. Then, we have r 6 = 5 > 3.4 ~ I G (5) − I G (5)+1; that is, the hypothesis of Theorem 3.5 fails. Table 4.3 shows the calculation results of ī k, ī k and, ī k − ī k .
From Table 4.3, we have r k − r k >        max − 1 { IG(j) − IG(j)+1} for each k (Theorem 3.5 fails). On the other hand, the difference rk − r k tends to decrease as k reaches 1. This is consistent with Theorem 3.4 because r k − r k > IG(k−1) − IG(k−1)+1 for each k (corresponding to Fig . 2.(2) in Section 3.2). Note that this example does not explain the necessity of the condition rf − rf ≤ 1G(f−1) − 1G(f−1)+1 for Theorem 3.5. It may be possible that Theorem 3.5 holds but rf − rf > IG(f-1) − IG(f-1)+1.
Application: Shrinkage of differential rent vectors with a large number
4.2
of households
In their pioneering work of assignment games, Shapley and Shubik [6] conjectured that a set of competitive price vectors tends to shrink as the number of traders becomes large. However, they did not suggest a formulation for a suitable limit procedure. For an answer to their argument, Kamecke [12] and Gretsky, Ostroy and, Zame [3] presented a limit model of the linear assignment model with continuum of traders. Recently, Sai [11] showed that without the quasi-linear utility assumption, the set of competitive price vectors shrinks to a unique point as the number of sellers become large (though they did not deal with a limit model). The point is that Sai's result required no condition for the buyers' set.11  According to our Theorem 3.5, we also obtain the same shrinkage result when only the buyers' set is large. Here, we consider a sequence of rental housing markets {( M ν , N ν )} ∞ 1 . We consider the situation where for a large ν , the market has many households and their income distribution gets dense. This is formalized by the following condition. Condition 4.1. There is some constant α > 0 such that for any ν, {( M ν, N ν)} ∞ 1 satisfies (1) |M ′ | → ∞ as ν → ∞ ; (2) Let I i ' < α for all i ∈ M ' ; (3) maxı≤i≤m−ı[I i − I i +ı] ≤ α/ | M i |. Condition 4.1.(2) together with (1) implies that, although the number of households be- comes large, the income of each household is bounded; while (3) implies that an interval of two adjacent incomes tends to be small as the number of households becomes large. For ν ≥ 1, let f ν be a marginal category in the market ( M ν, N ν ) and ( π 1, ..., π f ν ), ( Σ 1 , …, Σ f ν ) be the maximum and minimum differential rent vectors determined by ¯ f ν = ¯ f ν and r fv = Pfv with 0 ≤ Pfv − Pfv ≤ IG(fv−1) − IG(fv−1)+1. Theorem 4.1.  Suppose that {( M ν , N ν )} ∞ = 1 satisfies Condition 4.1.  Then, ∑ k = 1 τ k − τ k → 0 as ν → ∞. Proof.  By Theorem 3.2, we have 0 ≤ ∑ k=1ﾃﾞk − τ k ≤ ∑ k=1  ιε{ k,., This inequality and Condition 4.1.(3) imply the consequence. Theorem 4.1 together with equation (3.5) in Section 3.2 imply that a competitive rent vector ( p 1 , …, p f ) of a relevant part also shrinks to a unique point ( ∑ k =1 pmax − pmin → 0 as ν → ∞).12 Thus, we obtain the same shrinkage result for the competitive rent vectors only under the condition for the household income distribution.
5
Conclusion
We have evaluated the difference between the maximum and minimum differential rent vectors in a rental housing market model by Kaneko et al., where the identical utility function for household and the normality for apartment quality are assumed. The differential rent vector is a solution of a certain system of equations called "the rent equation," and it is regarded as a representative of the competitive rent vector in the model. Our main result (Theorem 3.5) is that the rent difference of the k-th category is smaller than the largest income difference between specific neighboring households numbered G ( l ) and G(l) + 1 (k ≤ l ≤ f − 1). This implies that the rent difference can be regarded as rather small and consequently, the difference between the maximum and minimum competitive rent vectors is also small. Furthermore, the difference shrinks to zero as the market becomes larger and the household income distribution becomes denser. A precise discussion of the shrinkage result is presented in Theorem 4.2 of Section 4.2. Another result (Theorem 3.4) indicates that we can reduce the comparison of two rent differences of the k -th and k — 1-th categories into a comparison of the rent differences of the k -th category and the income differences of neighboring households numbered G ( k − 1) and G ( k − 1) + 1. Our results argue that, when we study considerably large housing markets, the difference in the comparative statics results is small, whether or not we use the maximum or minimum differential rent vectors. We conclude with a remark on a future empirical study.  It may be possible to study whether our shrinkage result on the rent difference is actually viewed from a real housing market by comparing real apartment rent data of a densely populated area and a sparsely populated area.
Acknowledgements
I am very grateful to Professor Mamoru Kaneko for his helpful guidance and suggestions.
Appendix
A. Proof of Theorem 2.2. To prove Theorem 2.2, we need the following lemma. Lemma A.1. Let (p,x,y) and (p ′ ,x ′ ,y ′ ) be any competitive equilibria and suppose that there is i ∈ M  such that x i = ek  and x i = el, k̸= l.  Then, (1): Pk < Pk if and only if pl < Pl; (2):  pk = pk if and only if pl = pl; (3): Pk > Pk if and only if pl > Pl. The proof of Lemma A.1 needs the following lemma from Sai [11]. Lemma A.2 (Sai, 2014).  Let (p,x,y) and (p',x',y') be any competitive equilibria and let k be an integer with 1 ≤ k ≤ T. Then, pk < pk implies yk = Yk Proof of Lemma A.1. ( If part of (1) ): Assume p t < p t . It follows from the assumption and household i ’s UMC, u i (e k , I i − p k ) ≥ u i (e l , I i − p l ) ≥ u i (e k , I i − p k ). Thus we have u i (e k , I i — p k ) > u i (e k , I i — p k ), which implies p k < p k . ( Only if of (1)): Suppose pk < pk . Suppose, on the contrary, pl ≥ pl . By lemma A.1, it holds that yk = yk .  On the other hand, in equilibrium ( p ′ , x ′ , y ′ ), one household i switches his housing choice from k to l . This implies that at least one household j ( ≠  i ) switches his housing choice from m(≠ k) to k . By the assumption and household j 's UMC, u j (e m , I j − P m ) ≥ u j (e k , I j − p k ) > u j (e k , I j − p k ) ≥ u j (e m , I j − p m ). This inequality derives p m < p m , which implies m̸= l . In the same manner with the above discussion, p m < p m implies ym = Ym r and in equilibrium ( p ′ , x ′ , y ′ ), at least one household switches his choice from n( ≠  m ) to m. This also derives p n < p n , n̸= l and y n = Y n , so this process continues. Since M is finite, this process does not finish even with all the possible household switched. This implies the hypothesis p l ≥ p l is false. Thus, we obtain p l < p l . ( If of (2)): Suppose p l = p l . By only if part of (1), it is enough to show that p k ≤ p k . It follows from the supposition and buyer i’s UMC, u i (e k , I i − pk) ≥ u i (e t , I i − pk) = u i (e t , I i − pk) ≥ ( Only if of (2)): Suppose pk = pk . By if part of (1), it is enough to show that pk ≤ pk . Suppose, on the contrary, pι > p ι . By lemma A.1, it holds that γι = ψ . On the other hand, in equilibrium ( p ′ , x ′ , y ′ ), one household i switches his housing choice from k to l . This implies that at least one household j ( ≠ i ) switches his housing choice from l to m ( ≠ l ). By the supposition and household j 's UMC,u j (e m , I j − l m ) ≥ u j (e ', I j − p ( ) > u j (e ', I j − p l ) ≥ u j (e m, I j − p m ). This inequality derives p m > p m , which implies m̸= k . In the same manner with the above discussion, pm > p m implies ym = ψ m, and in equilibrium ( p ′ , x ′ , y ′ ), at least one household switches his choice from m to n( ≠  m ). This also derives p n > P n , n̸= k and y n = y n , so this process continues. Since M is finite, the process does not finish even with all the possible household switched. This implies the hypothesis p t > p t is false. Thus, we obtain pi ≤ pi ( (3) ): It is immediately proved by using Lemma A.1.(1) and (2). ● Proof of Theorem 2.2.  Let ( p, x, y ) and ( p ′ , x ′ , y ′ ) be any competitive equilibria and suppose that p k < p k and p k > p k for some k, l . Then we construct a tuple ( p, x, y ) such that (m-1): p k = min{p k ,p k } for k with 1 ≤ k ≤ T ;
(m-3):  for k = 1, . . . , T, Y, = Yk. Note that the above x is well defined: Indeed, by Lemma A.1, each i ∈ M chooses at most one category k in x . In the following, we show that a tuple ( p,x,y ) satisfies competitive equilibrium conditions UMC, PM and BDS. UMC: Let i ∈ M . We have three cases. (Case 1): π i = x i = e k . By (m-1), l k = p k . It is obvious that u i (e k , I i − p k ) ≥ u i (e m , I i − P m , for all m with p m = pm . Let l be the category which household i chooses in ( p ′ , x ′ , y ′ ). By Lemma A.1, we have p t ≤ p t . By this inequality and the UMC, u i (e k , I i − p k ) ≥ u i (e l , I i − p l ) ≥ u i (e ′ , I i − p ′ i ) ≥ u i (e m , I i − p m ) for all m with l m = p m . Thus i satisfies UMC in ( p, x, y ). (Case 2): π i = π i = e k . By Then p k = π k . It is obvious that u i (e k , I i = p k ) ≥ u i (e m , I i = l m ) for all m with p m = p m . Let l be the category which household i chooses in ( p,x,y ). By Lemma A.1, we have p t > p t . By this inequality and the UMC, u i (e k , I i − p t ) ≥ u i (e l , I i − p ( ) > u i (e l , I i − p i ) ≥ u i (e m , I i − p m ) for all m with p − = p m . Thus i satisfies UMC in ( p, x, y ). (Case 3): x i = 0. by (m-2), we have x i = x i = 0. Hence i satisfies UMC in ( p, x, y ). PM and BDS: Let k with 1 ≤ k ≤ T. If Pk = pk , landlord k maximizes his profit with production y k = y k . By (m-2), x i = x i = e k for all i ∈ M k . This implies L i ∈ M , x i = LieM k x i = ykek = ykek, that is, BDS holds for category k.  Otherwise ( pk = pk), the landlord k maximizes his profit with production y k = y k = y k (by Lemma A.2)."The balance of total demand and supply is inherited from the equilibrium ( p ′ , x ′ , y ′ ). The vector p satisfies p ≤ p and p ≤ p ′ . Since the set of competitive rent vectors is a compact set, there is the minimum competitive rent vector in the market ( M, N ). In the dual manner, we can also prove the existence of the maximum competitive rent vector. ● B. Proof of Theorem 3.1. Proof.  of (1).  We proof this by mathematical induction over k = f − 1, f − 2 , ..., 1. Let k = f − 1. By utility maximization condition and (3.2), we have u ( ef− 1, I G ( f− 1) − pf-1) ≥ u(e f , IG(f-1) − pf) and u(e f-1, IG(f-1) − ñf-1) = u(e f , IG(f-1) − ñf). These together withﾃƒ = pƒ imply u(eƒ−¹, IG(ƒ−¹) − Pƒ−¹) ≥ u(eƒ−¹, IG(ƒ−¹) −ﾃƒ−¹). This inequality and Assumption A imply IG(f-1) - Pf-1 > IG(f-1) - rf-1, that is, rf-1 > rf-1. Suppose that for k = l, 1 < l ≤ f − 1, π l ≥ p l and let k = l − 1. By utility maximization con- dition and (3.2), we have u(e l−1 , I G (l−1)−Pl−1) ≥ u(e l , I G (l−1)−Pl) and u(e l−1 , I G (l−1)−Tl−1) = u(e ′ , I G (1–1) –ﾃi ). On the other hand,ﾃl ≥ p l and Assumption A imply u(e ′ , I G (1–1) – p l ) ≥ u(e ′ , I G (1–1) − Fi). This inequality together with previous inequalities imply u(e ′− 1 , I G (1–1) − Pl−1) ≥ u(e l−1, IG(l−1) − π l−1). This and Assumption A imply IG(l−1) − Pl−1 ≥ IG(l−1) − π l−1, that is,ﾃl − 1 ≥ p l − 1 . Therefore we haveﾃk ≥ p k for all k with 1 ≤ k ≤ f − 1 . ● Proof of (2). It is proved by the dual manner with (1). Let k = f − 1. By utility max- imization condition and the first equation of (3.3), we have u (ef-1,Ic(f-1)+1 - Pf-1) < u(ef , IG(f-1)+1 − pf) and u(ef-1, IG(f-1)+1 − xf-1) = u(ef , IG(f-1)+1 − xf). These together with rf = pf imply u(ef-1, IG(f-1)+1 − Pf-1) ≤ u(ef-1, IG(f-1)+1 − xf-1). Thus, we have ½ƒ−ı ≤ pƒ−ı· Suppose that for k = l,1 < l ≤ f − 1, r 1 ≥ p t and let k = l − 1. By utility maxi- mization condition and (3.3), we have u(e l−1,IG(l−1)+1 − Pl−1) ≤ u(e l ,IG(l−1)+1 − pl ) and u(e l−1, I G (l−1)+1 − r 1−1) = u(e l, I G (l−1)+1 − r 1). On the other hand, r 1 ≤ p l and Assumption A imply u(e l , IG(l–1)+1 − pl) ≤ u(e l , IG(l–1)+1 − r l ). This inequality together with previous in- equalities imply u(e l−1, I G (l−1)+1−Pl−1) ≤ u(e l−1, I G (l−1)+1−T l−1). Thus, we have r l−1 ≤ pl−1. ● Therefore we have r k ≤ p k for all k with 1 ≤ k ≤ f − 1. C. Proof of Theorem 3.3. By the definition of G( k ) and the utility maximization condition for households G( k ) and G(k + 1), it holds that
Suppose that condition (1) of Theorem 3.3 holds. Then, By the above inequalities, we have
that is, the rent equation holds. Suppose that condition (2) of Theorem 3.3 holds. We prove by contradiction. Suppose that there is a category t with 1 ≤ t ≤ f − 1 such that u(e k , I G (k)+1 − Pk) = u(e k +1, I G (k)+1 − Pk+1) for k with 1 ≤ k ≤ t − 1; u(e t , I G (t)+1 − Pt) < u(e t+1 , I G (t)+1 − Pt+1). Then, we can decrease p t and p t − 1, …, p 1 slightly into p t and p t − 1, …, p 1 such that
We now let the new rent vector p* as In the following, we show a tuple ( p*, x, y ) is also a competitive rent vector: this is a contra- dictory claim since p is the minimum competitive rent vector. Since ( x, y ) is a competitive allocation, the balance of total supply and demand condition is satisfied. Furthermore, by the bottom of (C.1) and (C.2), each landlord's profit maximization condition holds with ( p *, y ). The utility maximization condition of households is checked by as follows. Let i ∈ M with x i = e k . We easily find u (e k , I i − p k ) ≥ u (e k , I i − p k ) for price unchanged categories k' = t + 1, ..., T . The remaining part is shown by the following case analysis: (i) The case of k ≥ t + 1. By the definition of G ( k ), we have I i ≤ IG ( t )+1. This together with the top of (C.1) and Assumption E imply u(e k , I i — p k ) > u(e t , I i — p k ). Furthermore, this in- equality together with the top of (C.2) and Assumption E imply u (e k , i − p k ) > u (e t , i − p t ) > u(e t − 1 , I i − P t − 1 ) ≥ ··· ≥ u(e 1 , I i − P t ). (ii) The case of k < t + 1. Let k ′ with k < k ′ < t + 1. By the definition of G( k ), we have I i ≥ IG(k) ≥ IG(k)+1.  This together with (C.2) and Assumption E imply u(e k , I i − p k ) ≥ u(e k , I i − p ). Furthermore let k ′′ with 1 < k ′′ < k . By the definition of G(k), we have I i ≤ I G (k −1)+1. This together with (C.2) and Assumption E imply u (e k , I i − p k ) ≥ u (e k ′′ , I i − p k ,,). ● Combining them, we have u(e k , I i − p k ) ≥ u(e l , I i − p k ) for all l = 1, …, t. [1] Alonso, W., Location and Land Use, Harvard Univ. Press, Cambridge, 1964. [2] Fujita, M., Urban Economic Theory: Land Use and City Size, Cambridge Univ. Press, Cambridge, 1989. [3] Gretsky, N., Ostroy, J.M., Zame, W.R., Perfect competition in the continuous assignment model, J. Econ. Theory 88 (1999), 60-118. [4] Ito, T., Effects of quality changes in rental housing markets with indivisibilities, Re. Sci. Urban Econ. 37 (2007), 602-617. [5] Määttänen, N., Terviö, M., Income distribution and housing prices: An assignment model approach, J. Econ. Theory 151 (2014), 381-410. [6] Shapley, L.S., Shubik, M., Assignment game I: The core, Int. J. Game Theory 1 (1972), 111–130. [7] Kaneko, M., The central assignment game and the assignment markets, J. Math. Econ. 10 (1982), 205–232. [8] Kaneko, M., Housing market with indivisibilities., J. Urban Econ. 13 (1983), 22–50. [9] Kaneko, M., Ito, T., Osawa, Y., Duality in comparative statics in rental housing markets with indivisibilities, J. Urban Econ. 59 (2006), 142–170. [10] Kaneko, M., Yamamoto, Y., The existence and computation of competitive equilibria in markets with an indivisible commodity, J. Econ. Theory 38 (1986), 118–136. [11] Sai, S., The structure of competitive equilibria in an assignment market, J. Math. Econ. 51 (2014), 42-49. [12] Kamecke, U., On the uniqueness of the solution to a large linear assignment problem, J. Math. Econ. 21 (1992), 509-521.