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1. JOURNAL OF URBAN ECONOMICS 27,188-211 (1990)
A Determination of Bid Rents through
Bidding Procedures
YASUSHI ASAMI’
Department of Urban Engineering, University of Tokyo, Bunkyo-ku, Tokyo I1 3, Japan
ReceivedAugust 6,1986; revised June 1987
Rent profiles in land markets with finitely many participants are derived under
two bidding procedures: free-unit-size bidding and single-unit-size bidding. Free-
unit-size bidding is shown to be basically equivalent to an English auction, whereas
the rent determination problem under single-unit-size bidding reduces to a mathe-
matical programming problem. An application of the latter case is provided, in
which a density model of land use is shown to be a good approximation of
appropriately chosendiscrete models. 0 1990 Academic Press, Inc.
1. INTRODUCTION
In land use theory, a common implicit assumption is that there are
infinitely many potential renters in the land market, and hence that bid rent
always equals the maximum rent anyone can pay at a given location. In
reality, however, the number of renters, even including potential renters, is
often small. To analyze such a land market, it is necessaryto construct a
bid rent theory in economieswith finitely many participants.
In the literature on land use theory (as surveyed in Goldstein and Moses
[ll], Muth [15], and Fujita [lo]), only a few papers analyze land markets
with finitely many participants (see Eckart [7] and Asami [2], among
others). To contribute to the development of such a theory, a simple model
is proposed in this paper which treats a situation in which m renters seekto
rent land units from n landowners, each of whom has one unit of land.
Since land units may not be identical in general, the rent bid for land units
may vary among both land units and renters.*
Two extreme casesof interest are considered.The first case,designated as
single-unit-size bidding, relates to situations in which each renter requires
‘This paper is based on a part of the author’s Ph.D. dissertation written at the Department
of Regional Science,University of Pennsylvania under supervision of Tony E. Smith, Masahisa
Fujita, and Xavier Vives. The author has benefited by their helpful comments and a Dean’s
fellowship at the University of Pennsylvania, which are gratefully acknowledged.
‘Alonso [l] gives a game-theoretic analysis of such a case.In his model, the renters behave
passively, and landowners set rents at the bid rent levels. Though the meaning of “bid” and
the nature of bidding rules are not explicitly stated, it is worth pointing out that the
interpretation of the traditional bid rent concept is based on the caseof a finite economy.
188
0094-1190/90 $3.00
Copyright 0 1990 by Academic Press, Inc.
All rights of reproduction in any form reserved.

2. BID RENTS AND BIDDING PROCEDURES 189
only one unit of land. Typical residential housing markets exemplify this
case.Since each renter needsonly one land unit, his payoff or utility would
not increase by his renting more than one unit of land, provided that the
first unit he rents is the best unit for him. The second case,designated as
free-unit-size bidding, relates to situations in which each renter may rent
more than one unit of land. Land markets involving commercial developers
exemplify this case. For simplicity, it will be assumed that the payoff or
profit obtained from renting one land unit is not affectedby whether or not
the renter rents other land units.
In general, the payoff or utility of renting land units may depend on the
number of other units which the renter rents. The two casesabove are
extremes: single-unit-size bidding being the situation in which the marginal
benefit of renting an additional land unit is zero, and free-unit-size bidding
being the situation in which the marginal benefit of renting an additional
land unit equals the benefit of renting that land irrespective of the number
of land units already rented. Also, it is implicitly assumedthat no extemal-
ity effects are present.
Within this framework, the present analysis shows that free-unit-size
bidding is basically equivalent to an English auction with a single commod-
ity (Holt [12], Milgrom and Weber [14], Vickrey [19], and Wilson [22]), and
the resulting rent is equal to the second-highest reservation offer rent for
each land unit (seeTheorem 1 below). In single-unit-size bidding, the rent
profile depends on the magnitudes of all renters’ reservation offer rents. In
particular, it is shown (in Lemma 3 below) that the resulting rent is
generally lower than the second-highestreservation offer rent. A characteri-
zation of the rent profile is given in Theorem 2. A very similar result to
Theorem 2 is obtained by Demange et al. [6] in the context of multi-item
auctions.3 Several implications of the results are described, and the rent
profiles of somespecial casesare given. In order to develop theseresults, we
begin with the formulation of bidding rules and optimal strategies for
renters in Section 2. In Section 3, a characterization of the rent profile for
both schemes (i.e., free-unit-size bidding and single-unit-size bidding) is
given. One application of the results obtained is demonstrated in Section 4,
where a density model of land use is shown to be a good approximation of
appropriately chosen discrete models when the number of households is
large. Several implications of these results together with the consequences
of the land markets in which landowners actively participate, are discussed
in Section 5.
?n the work of Demange et al. [6], a mathematical model is developed which is closely
related to the single-unit-size bidding casein the present paper (only discovered after writing
this paper). Seealso Footnote 6.

3. 190 YASUSHI ASAMI
2. BIDDING RULES AND OPTIMAL
BIDDING STRATEGIES
Let M and N denote the sets of renters and land units, respectively,
where
M= {l,...,m} (1)
N = {l,..., n}. (2)
The i th renter is assumedto have a reservation ofir rent for each possible
land unit, which can be interpreted as either the profit he can make by
renting that land or the maximum rent he can pay while enjoying a
predetermined utility level. The reservation offer rent for land unit j is
denoted by pj (2 0). It is assumed that P = (@ i E M, j E N) is
exogenous, and in the initial round of the bidding process, p’ =
(pi’: j E N) is known only to i. The reservation acceptance rent of landowner
j (who owns the jth land unit) is denoted by cj ( 2 0), where c =
(cj: j E N) is assumedto be exogenous.
Following the bidding rules to be specifiedbelow, eachrenter i E M bids
a nonnegative rent, q:(t), for each land j E N in round t. Given the bid
rents, Q(t) = (q$ t): i E M, j E N), the maximum bid, rj(t), for each
land unit j E N in round t is
The set, Mi( t), of maximum bidders for land unit j in round t is then
defined by
M,(t) = (i E M: q;(t) = r-) for j E N. (4)
After simultaneous bids are made by all renters, the process of land
assignment follows. Let aj: R”,x R”+ + M U N be the assignment function
for j E N, where R, is the set of nonnegative real numbers. Given the
maximum bid profile, r(t) = (rj(t): j E N) E R”,, and the bid rents,
Q(t) E R”,: aj assignsone renter in M, to land j, i.e., aj(r(t), Q(t)) E
Mj(t) if M,(t) is nonempty and r,(t) > cj, and aj(r(t), Q(t)) = j E N if
Mj(t) is empty. If Mj(t) consists of more than one renter, then the renter
assigned to land unit j in the previous round (t - 1) is assignedif he is in
Mj(t); otherwise the renter i E Mj(t) with smallest index in (1,. .., m} is
chosen to be aj(r(t), Q(t)) [any other well-defined convention is equally
acceptable].

4. BID RENTS AND BIDDING PROCEDURES 191
In the initial round of the bidding procedure (called round 0), the initial
maximum bid, r(O), is set to be the landowners’ reservation acceptancerent
c, i.e., r(0) = c; and the bid rents, Q(O),aresetto be zeros.In eachround 1,
after the assignment of renters and land units, the maximum bid profile,
r(t), and the assignment pattern, a(r(t), Q(t)) = (aJr(t), Q(t)): j E N),
are announced to all the renters. If none want to change their bids, i.e., if
r(t) = r(t - l), then r(t) and a(r(r), Q(f)) are realized and the bidding
procedure ends. Otherwise, all the renters bid again in the next round
(t + 1) and the maximum bid profile, r(t + l), is calculated on the basis of
the new bids, Q( t + 1). The assignmentprocessthen proceedsand r( t + 1)
and a(r(t + l), Q(t + 1)) are announced. If the maximum bids do not
change, i.e., r(t + 1) = r(t), then r(t + 1) and a(r( t + l), Q(t + 1)) are
realized. Otherwise this processis repeated until the maximum bids do not
change. In each round, f, of the single-unit-size bidding process,only one
q;(t) for each renter, i E M, is allowed to be positive, and all others are
zero.4 This case is an instance of a multi-object auction environment
(Engelbrecht-Wiggans and Weber [9], Maskin and Riley [13], Palfrey [16],
and Weber [21]). In each round, t, of the free-unit-size bidding process,
each q;.(t) is assumedonly to be nonnegative. It should be emphasizedthat
the landowners are passivein the sensethat they do not participate in the
bidding, and hence are not involved in the rent determination process.(A
relaxation of this assumption is discussedin Section 5.) It is also assumedin
both bidding processesthat once i E M is assignedto one land unit j, he is
committed to bid (at least) that rent level on land unit j until he is outbid
by another renter.5
Let r* = (ri*: j E N) and u* = (u~(T*, Q*): j E N) denote the final
rent profile and the assignment pattern; respectively. Let J, be the set of
land units to which i is assigned,i.e.,
J, = {j E N: uj(r*, Q*) = i}. (5)
The puyof, ui, for i E M in the single-unit-size bidding process is then
defined to be
such that j E J, ifJ,# 0
ifJ,= 0.
40ne may allow 4; to be nonnegative in single-unit-size bidding. In equilibrium, however,
the resulting rent profile will be the same as that of our case by letting e + 0.
‘This assumption guarantees that the maximum bid, r, will not decrease as bidding
progresses. Relaxations of this assumption may require a complicated tatonnement process to
reach the equilibrium.

5. 192 YASUSHI ASAMI
The payoff, ui, for i E M in the free-unit-size bidding process is defined
to be
i
C (pj - rj*) if J, # 0
u, = .i’Jl (7)
0 if Ji = 0 .
It is assumed that the bidding environment described above is common
knowledge.
Given this bidding environment, we now establish an optimal bidding
strategy for each renter. To do so, the following notation is introduced. Let
E (’ 0, = 0) be a minimal monetu~ unit (e.g., It). Define the sets K;(t)
and N,(t) for i E M by
K;(t) = (j E N: p; - r;(t) 2 E)
Ni(t) = (j E N: p; - rj(t) = n&p; - &)) 2 0). (9)
Recall the assumption that once i E M is assignedto one land unit j E N,
he must continue to bid at least that rent until he is outbid by another
renter. It is convenient to introduce terminology to treat such a situation.
Let Q(t) and r(t) be the current bid rent and maximum bid profile,
respectively, and a(r( t), Q(t)) be the current assignment pattern. If
aj(r(t), Q(t)) = i with rj(t) > 0, then i E M is said to be bound to j. If
i E M is assigned to j (i.e., uj(r(t), Q(t)) = i) with a positive rent (i.e.,
rj( t) > 0), then by assumption i must bid at least r,(t) on land unit j in
the next round (t + 1). Since in the single-unit-size bidding process, i can
bid a positive rent for only one unit of land, it follows that i is bound to
land unit j in the sensethat he can bid a positive rent only on j. It is
evident that if every renter is bound to someunit of land, then the present
set of bids must be a final configuration, unless some renters want to
increase their own rents, which is unlikely to happen. If i E M is currently
bound to someunit of land, then i is said to be bound, and this land unit is
denoted by b(i). (This notation is used for single-unit-size bidding process.)
Thus if b(i) is not empty, then i is bound to b(i). With this terminology,
an optimal bidding strategy under the single-unit-size bidding is derived as
follows.
In any round t, if i is bound, then i is better off by not raising his bid in
the next round (t + 1). If i is not bound, then there are severalpossibilities.
Let 1. 1denote the cardinality of a set. First, if INi( = 0, then (since no
land unit is profitable for i) qj(t + 1) = 0 is the optimal strategy. Second,
if 1
N,(t) I = 1, then i should stick to renting the land unit in Ni( t). As long

6. BID RENTS AND BIDDING PROCEDURES 193
as the land unit in Ni(t) is profitable (i.e., pi’ - rj(t) 2 a), i should bid
qj(t + 1) = rj + E. Finally, if INi( > 1, then i should stick to any one
land unit j’ chosen from 4. As long as it is profitable, i should bid
rj,(t) + E for land j’ in the next round (t + 1) expecting that i will be
assigned to j’ in round (t + 1).
This observation is summarized in Lemma 1.
LEMMA 1. Given (r(t), a(r(t), Q(t))) in round t, an optimal bidding
strategy for i E M in round (t + 1) under single-unit-size bidding is as
follows:
[A] Zf i is not bound, then
(A-l) if ]II$(r)] = 0, then for all j E N, set qj(t + 1) = 0;
(A-2) if I&(t)/ = 1, then for allj E N, set
q;(t + 1) =
i
r,(t) + E ifj E N;(t) andpj - r,(t) 2 E
0 otherwise;
(A-3) if I&(t)1 2 2, then choose one land unit j’ E N,(t) randomly
and for all j E N, set
q;(t + 1) = z(t) + &
(
ifj = j’ andpj - rj(t) 2 E
otherwise.
[B] Zf i is bound then for j E N
i
rj(t>
qxt+l)= 0
ifj E b(i)
otherwise
To clarify the procedure and the strategy described, an example is now
constructed under the assumption that every renter follows the strategy
given by Lemma 1.
EXAMPLE 1. m=n=3, p’=(2,0,4), p*=(2,6,1), p3=(0,1,3),
E= 0.5:
[Round 0] r = c = (l,l, l), a = (A, B, C) (i.e., landowners)
[Round l] q’ = (0, 0, 1.5) q* = (0, 1.5, 0), q3 = (0, 0, 1.5)
r = (1,1.5,1.5), a = (A,2,1)
[Round 21 q1 = (O,O,1.5) q* = (0,1.5,0), q3 = (0,0,2), r = (1,1.5,2),
a = (A,2,3)
[Round 31 q1 = (0, 0, 2.9, q* = (0, 1.5, 0), q3 = (0, 0, 2)
r = (1,1.5,2.5), a = (A,2,1)
[Round 41 q1 = (0,0,2.5), q* = (0,1.5,0), q3 = (0,0,3), r = (1,1.5,3),
a = (A,2,3)
[Round 51 q1 = (1.5, 0, 0), q* = (0, 1.5, 0), q3 = (0, 0, 3) r =
(1.5,1.5,3), a = (1,2,3)

7. 194 YASUSHI ASAMI
Hence r* = (1.5,1.5,3), a* = (1,2,3). If E= 0, then r* = (1,1,3) and
u* = (1,2,3).
Turning to free-size-unit bidding, an optimal strategy can be derived as
follows. In each round t, if pj - r,(t) 2 E,then i is better off renting land
unit j. Hence i should bid qj( t + 1) = rj( t) if i is currently assignedto j,
and qj(t + 1) = rj( t) + Eif not, expecting that i will be assignedto j next
round. If pj - rj( t) I 0, then qj.(t + 1) = 0 is an optimal strategy when i
is not bound. If i is bound, then qi(t + 1) = r,(t) is the best strategy.
Summarizing:
LEMMA 2. Given (r(t), a(r(t), Q(t))) in round t, an optimal bidding
strategy for i E M in round (t + 1) under free-unit-size bidding is us follows:
For all j E N, set
i
r,(t) + E if i z uj undj E Ki
qj<t + 1) = rj(t) ifi = a,
0 otherwise.
The following example illustrates this strategy.
EXAMPLE 2. Parametersare the sameas Example 1:
[Round 0] r = c = (1, 1,l), a = (A, B, C) (i.e., landowners)
[Round l] q1 = (1.5, 0, 1.5) q2 = (1.5, 1.5, 0), q3 = (0, 0, 1.5)
r = (1.5,1.5,1.5), a = (1,2,1)
[Round 21 ql = (1.5, 0, 1.5) q* = (2, 1.5, O), q3 = (0, 0, 2)
r = (2,1.5,2), a = (2,2,3)
[Round 31 q1 = (0, 0, 2.5), q2 = (2, 1.5, 0), q3 = (0, 0, 2),
r = (2,1.5,2.5), a = (2,2,1)
[Round 41 q1 = (0,0,2.5), q2 = (2,1.5,0), q3 = (0,0,3), r = (2,1.5,3),
a = (2,2,3)
[Round 51 q1 = (0, 0, 3.5) q2 = (2, 1.5, 0), q3 = (0, 0, 3)
r = (2,1.5,3.5), a = (2,2,1)
[Round 61 q1 = (0, 0, 3.5), q2 = (2, 1.5, 0), q3 = (0, 0, 0),
r = (2,1.5,3.5), a = (2,2,1)
Hence r* = (2,1.5,3.5) and u* = (2,2,1). If E5: 0, then r* = (2,1,3)
and u* = (1,2,1) or (2,2,1).
3. CHARACTERIZATION OF THE RENT PROFILE
Given the bidding environment defined above, several characterizations
of rent profiles are developed in this section. Since free-unit-size bidding is
easier to analyze, we will begin with this case.

8. BID RENTS AND BIDDING PROCEDURES 195
To describe our first result in a convenient manner, the following nota-
tion is introduced. Let Y = { yi: i E I} be a set of real numbers with the
index set I. The secondmaximum of Y is then denoted by
The Kth maximum of Y is inductively defined by
k = 2,..., K. (11)
With this notation, the rent profile resulting from free-unit-size bidding can
be characterized as follows.
THEOREM 1. Under free-unit-size bidding, if E + 0, then the limiting rent
pro$le r* is given for all j E N by
r-* =
i
max( y-FM2 ( 4 1’ 5}
if[Ml >l
J 02)
cJ
iflMl = 1.
Prooj SeeAppendix 1.
Theorem 1 implies that under free-unit-size bidding, the rent of eachunit
is given by the larger of the second-highest reservation offer rent and the
landowner’s reservation acceptancerent, as reflected by Example 2. This
result is exactly the sameasthe high-bid auction (Riley and Samuelson[17]
and Wilson [22]), and it suggeststhat under the free-unit-size bidding the
rent of each unit can be independently treated. In other words, the land
markets for each land unit are completely separable. This is due to the
assumption of independent reservation offer rents. One direct consequence
of this theorem is the following.
COROLLARY 1. Under free-unit-size bidding, if E + 0, then in the limit
the following three properties hold for all j E N:
(i) land unit j is rented IY maxi EMpj 2 cj
(ii) ‘j* = cj if maxf, Mpj I cj
(iii) r* > cj ifl maxfE,pj > cJ.
Several implications of the results above are discussedin Section 5.
Next turning to the single-unit-size bidding case, we first derive a
preliminary result which provides us with upper and lower bounds on the
rent profile.

9. 196 YASUSHIASAMI
LEMMA3. Under single-unit-size bidding, if m > n, then the limiting rent
projile r*, when E + 0, satisfies the following condition for all j E N:
If m I n, then the limiting rent profile r* satisfies the following for all j E N:
r* = cj
J
if 1441= 1. (15)
Proof SeeAppendix 2.
Lemma 3 states that under single-unit-size bidding, the rent is bound by
the (n + 1)th maximum of reservation offer rents. Hence, the resulting rent
is, generally, lower than that under free-unit-size bidding. This is because
under single-unit-size bidding the renters rent only one unit of land and the
second potentially highest bidder doesnot necessarily have an opportunity
to bid on the land in question. It is also of interest to note that if there are
many renters with more reservation rent than landowners’, then the final
rents tend to be higher than the reservation acceptancerents of landowners
anywhere. Combining these observations together with Theorem 1 and
Corollary 1, we have the following corollaries.
COROLLARY2. Under single-unit-size bidding, if E + 0, then for all
j E N:
(i) a necessary condition for land unit j to be rented is that
max(pj) 2 cj;
;cM
(ii) a st@cient condition for land unit j to be rented is that
m2n and ~G~x(p~) 2~~;
(iii) a necessary condition for rj* = cj is that
(iv) a suficient condition for r* = cj is that
(16)
(17)
08)
m>1andmax2

10. BID RENTS AND BIDDING PROCEDURES 197
(v) a necessarycondition for rj* > cj is that
m>l and yE%2(p;) >Cj; (20)
(vi) a suficient condition for rj* > c, is that
m > n and IIXIX”~~(pj) > Cj.
iEM (21)
COROLLARY
3. Let r* be the rent projile under single-unit-size bidding
and r** be the rent projile underfree-unit-size bidding. If E-+ 0, thenfor all
j E N, cj I rj* I rj**.
To state the first characterization lemma of single-unit-size bidding, let xi
be an assignment indicator where xj = 1 if i is assigned to j, and xi = 0
otherwise. Also define the variables
x; = 1 - c xj jEN (22)
iEM
x6=1- xx; i E M. (23)
jEN
Note that if x/” = 1, then for any i E M, land unit j is not assignedto any
renter, or in other words, that land unit is assigned to its landowner. By
letting 44, = M U (0) and N, = N U {0}, one may characterize the limit-
ing rent profile as follows.
LEMMA4. Under single-unit-size bidding, if E-+ 0, then the limiting rent
profile r* is a solution to thefollowing problem:
[Pl]: Find (xi: i E M,, j E N,; r,: j E N) such that
xfl E (0, l} iEM,,, jENO (24)
x;(F’(r) -pj + rj) = 0 iEM, jEN (25)
x&F’(r) = 0 iEM (26)
xp<q - c,) = 0 jEN (27)
5-cj20 jEN (28)
xx:=1 jEN (29)
ICM,
C xf=l iEM, (30)
i=No

11. 198 YASUSHI ASAMI
where
F’(r) = max( rpsa$ p; - rk),O) i E Al. (31)
Proof: See Appendix 3.
The meaning of Lemma 4 is explained in the proof in Appendix 3. It can
be shown that
LEMMA 5. There exists a solution to [Pl].
ProofI SeeAppendix 4.
With these observations, our main result is to show that the limiting rent
profile, r*, under single-unit-size bidding is given by the “smallest” solution
to [Pl].
THEOREM 2. Under single-unit-size bidding, if E + 0, then the limiting
rent projile r* is given for j E N by
r.* =
J
min rj,
(x. r)EX
(32)
where
X = {(x, r): (x, r) isasolution to [Pl].} (33)
Proof .6 SeeAppendix 5.
In other words, Theorem 2 assertsthat in the limit as E+ 0, the rent
profile under single-unit-size bidding is given by the minimum rent possible,
namely by the solution to problem [Pl]. Severalremarks are worth making.
First, each component, rj*, of rent protile r* is determined to be the
minimum for land unit j. Since r* itself is also a solution to [Pl], we can
restate r*, for example, as
(34)
which provides us one way to find the rent profile by a programming
approach as is shown below.
6This theorem is mathematically similar to Theorems 3.1 and 3.2 in Demange et 01. [6]. The
present proof can be regarded as an alternative proof to theirs.

12. BID RENTS AND BIDDING PROCEDURES 199
COROLLARY
4. Under single-unit-size bidding, by letting E -+ 0, the limit-
ing rent projle r* is obtained by solving the following program:
[MPl] min: ZjcNrj such that
xj E {OJ}
x;(F’(r) -pj + rj) = 0
xhF’(r) = 0
x,“(r, - c,) = 0
rj - c, 2 0
c x;=1
iEM
c xf=l
.jEN,
i E M,,, j E N, (35)
iEM, jEN (36)
iEM (37)
jEN (38)
jGN (39)
jEN (40)
i E M. (41)
Next, it is evident from Theorem 2 that r* is unique, for each r,* is
determined by the minimum feasible value of 5. Hence it follows that the
programming problem [MPl] yields the unique solution for r.
4. SIMPLE APPLICATION
In land use theory, the equilibrium rent profile is one of the main issues
in many models. Equilibrium rent is usually determined by the maximum of
“bid rents”. The concept of bid rent was first introduced by von Thiinen
[20] in the context of agricultural land use,and it waslater applied to urban
land use by Alonso [l]. The bid rent represents the maximum rent that a
particular economic agent can pay while enjoying a specified utility or
profit level at each location. This concept has been applied to many
situations in land use literature (Fujita, [lo]).
Alonso [l] provides somegame-theoretic analysis to justify this concept.
In his model, renters behave passively, and landowners set the rents.
Though the meaning of “bid” and the bidding rules are not stated, it is
worth pointing out that the interpretation of the traditional bid rent
concept is based on the case of an economy with a finite number of
participants.
In many subsequent articles, bid rent hasbeen reinterpreted asa rent bid
by renters within a perfectly competitive land market. Assuming complete

13. 200 YASUSHI ASAMI
information and a perfectly competitive land market, it does not matter
whether we adopt Alonso’s interpretation or the latter interpretation. How-
ever, if one or both assumptions are violated, then this argument is no
longer valid, and the relevant equilibrium rents may be determined very
differently. In particular, if the number of renters is finite, somegame-theo-
retic consideration is necessary in order to derive the equilibrium rent.
While this case is important for understanding actual land markets, this
problem has been neglected because of its analytical difficulty. Hence
researchershave for the most part simply assumedthat perfect competition
is a good approximation. A natural question arisesasto whether the perfect
competition model closely approximates the imperfect competition model
as the number of participants becomeslarge.
It is important to note that in the classical perfect competition model the
notion of land is very different from that in the finite model. In particular,
the bid rent function is defined at each point in space,In other words, land
is treated as a point which has no size. On the other hand, if we analyze a
finite number of renters, then the size of land is crucial. Each plot must
have somepositive measure.Thus the traditional bid rent approach neglects
the effects of contiguity of land, and is here designated asthe density model
of land use. In the finite case, where each plot of land has a positive
measure,the resulting model is here designatedasthe discrete model of land
use. It has been pointed out that one must be careful in incorporating the
effect of contiguity of space within an extended version of the density
model (cf. Berliant [5] and ten Raa and Berliant [18]).
We have developed a model in which a finite number of renters bid rents.
The bidding environment was similar to the English ascending auction
(Milgrom and Weber [14]), in which renters bid for land units, and in which
each land unit goesto the highest bidder. Following the results of Section 3,
we can obtain corresponding bid rents for the caseof a finite number of
participants. By comparing theseresults with those of the density model, it
is possible to investigate the approximation question. In particular, it is
possible to consider conditions under which the discrete model convergesto
the density model. Along this line, the present section analyzesthe limiting
property of the discrete model, i.e., land markets with a finite number of
bidders.
Suppose that n identical households are trying to locate in a monocen-
tric, closed city which is represented by a half line L = [0, cc) with I = 0
being the CBD. Each household commutes to the CBD to work and shop.
Let T(Z) be the transport cost associated with all such trips for each
household locating at 1. If a household occupies a site [1 - s/2, I + s/2),
then we can approximate the transport cost of this household by T(I),
where s is the sizeof the site. The utility function of a consumer is U(z, s),
where z is the amount of composite good consumed. Each household is

14. BID RENTS AND BIDDING PROCEDURES 201
assumed to maximize its utility subject to a budget constraint, that is, to
solve the problem
s.t.
where p is the price of the composite good, R(y) is the rent at y E L, and
Y is the income of the household. For simplicity, we assumethat s = 1
(fixed), p = 1 (i.e., a numeraire), and that both land and the composite
good are normal goods. Then the problem above can be written as
m/axz= Y- T(I) - j’+1’2R(y) dy s.t.I2 i/2. (44)
l-1/2
If we assume further that land is partitioned into land units [j - 1, j)
(i E Z,,) of unit length, then without loss of generality we can treat this
problem as
max 2 = Y - T(j - l/2) - rj,
.iEZ++
where
'j = J::(Y) dY.
(45)
Since the transport cost function T( .) is given, this problem can be
viewed as a single-unit-size bidding problem with a reservation offer rent,
Y - T(j - l/2) - zO,where z0 is a minimal level of the composite good
required for living. In view of Theorem 2 above, the equilibrium rent
profile, r* = (rj*: j E Z,,), when the minimal monetary unit E--) 0 is
obtained by solving the following mathematical program:
[MP2]rj* = min
(x9T)EX
rj, (47)

15. YASUSFII ASAMI
202
where
[Cl] :
x;.( F’(r) -
X = {(x, r): (x, r) satisfies [Cl])
H= {l,...,n}
Xf E {OJ} iEH,,jE&
(Y - T(j - l/2) - ZJ + q) = 0
xf)F’(r) = 0
xyrj = 0
rj 2 0
c x;. = 1
iGH,
c x;=1
je%
H,, = H u (0)
VI = z++u PI
(48)
(49)
(50)
i E H, j E Z,,
(51)
iEH (54
j E Z++ (53)
j E Z++ (54)
j E 6, (55)
iEH (56)
(57)
(58)
I’ = max( kzF [(Y - T(j - l/2) - za) - rk],O) i E H. (59)
++
Since we are dealing with a closed city model, it is necessaryto assume
that it is feasible for all n households to live in the city, i.e., that
Y - qn - l/2) - z0 2 0. (60)
Under this assumption it is not difilcult to see that the solution, r*, to
problem [MP2] is given by
T(” - l/2) - 7lj - l/2) jln-1
j 2 n.
(61)
Next we derive the bid rents for an associateddensity model (i.e., with
perfect competition among a continuum of households). In this case, the
lot-size, s, is still assumedto be 1. However, this should be understood asa
point density, i.e., the associated lot does not have positive length in L.
Then the bid rent, r**, is given by
r**(z) = Y - T(Z) - z**, (62)

16. BID RENTS AND BIDDING PROCEDURES
where z** is determined by a boundary condition
r**(n) = 0.
It follows that
203
(63)
r**(l) = T(n) - T(l). (64)
Having characterized the equilibrium rent profiles under both the discrete
model and density model, we now show that the density model approxi-
mates the discrete model under certain assumptions.To do so,let q,* (1) be
defined by
4,*t1)
= C ‘j*X[j-1, j)(l),
jsH
(65)
where x F(,) is a characteristic function defined by
XFW =
1 ifIEF
0 otherwise. (66)
Let q,*(1) be the rent profile under the discrete model. Since r**(I) given
by (64) depends on n, it is denoted by qz *. The ratio of q,*(l) and q,* *(I),
i.e.,
is well defined for all 1E [0, n). Note also that q,*(n) = q,**(n) = 0. It
follows from (61) and (64) that
dn(1)
= C [iTtn- 1/2) - T(j - 1/2)1/[T(n) - T(~)llX[j-l, j)(l)
jGH
= 1 + c [[(T(n - l/2) - T(n))/n + (T(I) - T(j - 1/2))/n]
jEH
/[(T(n) - T(r))/~IIX,j-l,j,(‘). (68)
Assume that (i) T( *) is differentiable; (ii) there exists t > 0 such that
T’(l) 2 t for I 2 0; and (iii) there exists 52 _tsuch that T’(I) < j for I 2 0.

17. 204 YAWSHI ASAMI
Then we have
jcH
Given 6 > 0 ( = 0), if we let N = [(i/r)/S2], then for n > N and x E
LO,
41 - @I
Ido - II s Ci/!) C X[j-l,j)(z)/(n6)
jcH
< ’ C X[j-1.j) (I) I s. (70)
jEH
Thus we have shown that:
PROPOSITION 1. If T( -) satisjies:
(i) T( *) is difirentiable;
(ii) 3i > 0, such that T’(I) 2 t for 1 2 0;
(iii) 3i 2 _t,such that T’(I) I i for I 2 0,
then Vi3 > 0, 3N E Z, such that Vn > N, VI E [0, n(1 - S)],
(71)
This proposition means that if T’(I) is bounded above and below by
positive constants, then the ratio in (67) convergesuniformly to 1 except for
those I with I/n close to 1. It is also easyto seethat the ratios of total rents
under both models converge to 1 under the sameconditions. To state this
result, let D,, be the ratio of total rents, i.e.,
PROPOSITION 2. If T(a) satisjies
(i) T( .) is diferentiable;
(ii) 3_t> 0, such that T’(1) 2 t for I 2 0;
(iii) 3t 2 _t,such that T’(1) I i for I 2 0,

18. BID RENTS AND BIDDING PROCEDURES
then
lim D, = 1.
“-CO
205
(73)
ProoJ: SeeAppendix 6.
If n + cc, then the transport cost becomesinfinite. To consider this limit,
we must thus assumethat income increasesas n increases.
To clarify the significance of conditions (ii) and (iii) in the propositions
above, two examples are provided in which either (ii) or (iii) is violated.
EXAMPLE 3. T(x) = 1 - exp(-ax) (a > 0):
lim D,, = u/( eai2 - epa12) > 1.
n--to0
EXAMPLE 4. T(x) = exp(ax) - 1 (a > 0):
lim D n =e-a/2<1.
n+w
5. IMPLICATIONS AND EXTENSIONS
In the preceding sections,the limiting rent profiles (as E+ 0) under both
free-unit-size bidding and single-unit-size bidding were characterized in
Theorem 1 and Theorem 2 (with its corollary). The present section is
devoted to a discussion of the implications and possible extensions of these
results. Throughout the section, it is assumedthat E+ 0.
First, under free-unit-size bidding, the rent profile was shown to corre-
spond to the second-highestreservation offer rent. This implies that if there
are two or more renters with identical reservation offer rents, they both
must pay their reservation offer rents, provided that they rent land units.
This suggestsone possible justification for the common practice of using
bid rent as the relevant rent profile, when households are assumedto have
identical utility functions and identical incomes. If we attempt to analyze
land markets with finitely many heterogeneousrenters (such as in spatial
competition among land developers), we have to take this result into
account. An analysis of such a caseis reported in Asami [3].
Second, as a simple application of single-unit-size bidding, a bid rent
profile in a monocentric land market with finitely many households
was derived, and its limiting property was examined in Section 4. If each
lot-location is fixed, and if the transport cost function satisfies the

19. 206 YASUSHI ASAMI
bounded-slope condition, then the classical density model of bid rent is a
good approximation to a discrete model when the number of households
becomes large. This property can be extended to a more general model, in
which the location of eachlot is not predetermined (though its sizeis fixed)
(Asami [4]).
So far it has been assumedthat the landowners are passive in the sense
that they cannot influence the rent profile except through their predeter-
mined reservation acceptancerents, c. If they have somebargaining power
in the rent determination process, then the rent profile derived in the
present paper can be thought of as the greatestlower bound of all feasible
rents. An upper bound is given by the maximum reservation offer rents (i.e.,
bid rents) under free-unit-size bidding, and the maximum solution to [Pl],
with the restriction that the number of land units rented remains the same
as the present caseunder single-unit-size bidding. One interesting extension
would be to examine cases in which landowners are allowed to set c
strategically.7
Finally it is worth noting that in either bidding schemethe rent profile
will converge to the bid rent if the number of identical renters increases.
Hence the traditional approach using the bid rent concept can be regarded
as the limiting caseas the economy becomeslarge.
APPENDIX 1. PROOF OF THEOREM 1
From Lemma 2, it follows that for any land unit j, if the set {i E M:
j E Kj} is nonempty and not a singleton, then there exists i E M such that
i # aj and j E K,, implying that rj must increase in the next round.
Noting that E-+ 0, .rj stops increasing when this set becomesa singleton,*
i.e., rj = max’{ pj} if ]M] > 1 and rj = cj if ]M] = 1. If this set is empty
initially, then no renter will rent land unit j and rj* = cj. By construction,
rj 2 cj, and hence the result follows. Q.E.D.
APPENDIX 2. PROOF OF LEMMA 3
By construction, rj* 2 cj in both cases.Suppose that M > n. If rj* <
maxi:,,,{ pi}, then there are (n + 1) renters whose reservation offer rents
are higher than rj*. Since the land units can be assigned to at most n
renters, there is at least one renter, i, who is not assignedto any land, and
has a reservation offer rent pj > rj*. Moreover since the rent profile r*
satisfies the condition that i does not have an incentive to bid higher rents
‘Engelbrecht-Wiggans [8] provides two examples in the context of single-object auctions
which are inspired by a similar motivation.
8Strictly speaking this may become empty, in which case by letting PJ be this set one round
before, one renter in P, can have the same reservation offer rent for land unit j. Then the rent
of land unit j is determined by the reservation offer rent, which is equal to the second
maximum of reservation offer rents.

20. BID RENTS AND BIDDING PROCEDURES 207
at other land units, it follows that i has an incentive to raise his bid rent
higher than rj*,
rnaxl:L{ pi}.
which contradicts the definition of r*. Hence r,* 2
As in the proof of Theorem 1, if the set { i E M: j E K, }
becomesa singleton, rj stops increasing,’ which implies from the argument
in the proof of Theorem 1 that rj* I max?, ,,,{ pj} if maxfE,,,{ pj} 2 c,
and rj* = cj otherwise in both cases.Since if lM( = 1, then there is no
competition in the bidding process,it must be true that rj* = cj. Q.E.D.
APPENDIX 3. PROOF OF LEMMA 4
The proof is given by interpreting the conditions. The first condition (24)
is definitional. In the secondcondition (29, if xj = 1, i.e., if i is assignedto
j, then F’(r) = p; - rj. Since F’(r) is the maximum payoff that i can get
under r, this implies that i has no incentive to deviate. If xj = 0, the
condition is met. In the third condition (26), if xi = 1, i.e., if i is not
assigned to any land unit, then F’(r) = 0, i.e., i can only obtain a
nonpositive payoff under r. This may be interpreted asa no-entry-incentive
condition. If xd = 0, then the condition is met. In the fourth condition (27)
if xj” = 1, i.e., if the land unit j has no renter, then rj = cj. If x9 = 0, then
this condition is met. The fifth, sixth, and seventh conditions (28)-(30) are
again definitions. Since r* must satisfy precisely the conditions described
above, the lemma is established. Q.E.D.
APPENDIX 4. PROOF OF LEMMA 5
It suffices to show that a rent profile, r*, under single-unit-size bidding
exists. If E> 0, then since r(t) is monotone increasing with the number of
rounds, t, it follows from Lemmas 1 and 3 that the bidding processends in
a finite number of repetitions. Letting E+ 0, the resulting rent profile r*
still remains a finite nonnegative n-tuple (cf. Lemma 3). Q.E.D.
APPENDIX 5. PROOF OF THEOREM 2
(1) Let r* be the resulting rent profile of the optimal bidding strategy
in Lemma 1 when E+ 0. Supposethere exists a rent profile, r, such that
(x, r) E X and there exists j E N with rj < 7*. Without loss of generality,
we may assumethat j = 1. If n = 1, then slop to (6).
(2) Since r: > rl 2 ci, land unit 1 is assignedto someone i under r*.
Without loss of generality, we can assumethat i = 1. Since renter 1 can
obtain a positive payoff by locating at land unit 1 under r, he must be
assigned to some land unit j’ under r.
(2A) If j’ = 1, then it must be true that renter 1 was obliged to raise
his rent bid on land unit 1 to rl*, which implies that someonei’ must have
‘The remark in Footnote 8 holds here.

21. 208 YASUSHI ASAh
bid rr*( - E) on land unit 1 in the process of bidding. If renter i’ is not
assigned under r *, then F”(r*) = p[ - r;” = 0. Since ri* > rr or pr -
rr > 0, i’ must be assignedto some land unit j” under r. It follows then
that pj;, - 5% 5 pc - rl* < pr - rl I p$ - Q,, or ri* - rr I Q?,- r/,,. If
renter i’ is assignedunder r*, say to j”, then i’ must be indifferent between
locating at 1 or j” under r*. It then follows that pi:, - rjTT2 pc - rl > pi
- r;” 2 p$, - rj?: or rl* - rl I q!: - ~j,,.In either case,there exists another
land unit (other than 1) which has a higher rent under r*, and moreover the
difference between the two maximal bids at this land unit is larger than that
at land unit 1. Without lossof generality, we can assumesuch a land unit to
be 2.
(2B) If j’ # 1, then without loss of generality we can assumej’ = 2.
In this case,it must be true that pt - rz* _<pi - r;” < pi - r, I pi - r2 or
*
r1 - rl I r2* - r2. If we let x* be the associatedassignmentunder r*, then
in summary we have
(a) XT’ = xi = 1,0 < r: - r, < r2* - r,; or
(b) XT’ = x2* = 1, 0 < rl* - r, 5 rz* - r2.
If n = 2, then skip to (5).
(3) Since r2* > r, 2 c2, some renter must be assigned to land unit 2
under r*. Without loss of generality, we can assume that renter 2 is
assigned to land unit 2 under r *. For if not then m = 1, which yields a
contradiction. But if m 2 2, then since pz - r2 > pi - r2* 2 0, renter 2 is
assigned to someland unit under r.
(3A) Suppose(a) described above is the case.If renter 2 is assigned
to land unit 2 under r, then a parallel argument to (2A) leads us to
conclude that there exists a land unit j other than 1 and 2 which has a
higher rent under r*, and q* - 5 2 r2* - r,. Without loss of generality, we
may assumej = 3.
(3B) Suppose(b) described above is the case.If renter 2 is assigned
to land unit 1 under r, then it follows that pi - r2* I pi - rl* < pi - rl I
pi - r2 and pf - r: I pi - rt < pz - r, < pf - r,, implying that ri* -
rl = r2* - r,, p:-r,*= pi - r2*, pi - r, = pi - r2, p: - r;” = pz - r2*
and pf - r, = pf - r2. Hence we can reassign XT’ = xT2 = xi = xi = 1
without changing the equilibrium. By an argument similar to that in (3A)
above, we conclude that there exists somej other than land units 1 and 2
such that q* - 5 2 r2* - r,. Without loss of generality we can assumethat
j = 3. If renter 2 is assignedto another land unit j (other than 1 and 2),
then it follows from an argument parallel to that in (2B) that rj* - r, 2
*
r2 - r,. We may assumej = 3 without loss of generality.

22. BID RENTS AND BIDDING PROCEDURES 209
In summary, we have 0 < r;” - rl I r2* - r, I rj* - r3 and
(1) x:’ = -& = x; = x2’ = 1;
(2) XT’ = xz2 = x: = xi = 1; or
(3) xt”’ = x2*2= x; = x: = 1.
If n = 3, skip to (5).
(4) We can continue this argument to the sth step, so that we have
0 c rl* - rl I *. . I r,* - r, and
(1) x:’ = . . . = x,*-“;’ = x: = . . . = x,s-; = 1;
(2) xy = . . . = x,*-“y = x; = . . . = x,sI; = x,“-l = 1;
(3) x:’ = . . . = x,*-“;’ = x; = . . . = x,s-; = x,“+f = x,“-’ = 1;
. ; or
(s)xp . . . =x;-“,‘=x:‘i . . . =x,“-‘=l.
Since r,* > r, r c,, some renter, (who will be designated by s) must be
assigned to land unit s under r *. If m < s, then this yields a contradiction,
and we can conclude that such a rent profile, r, does not exist. Now
suppose m 2 s. Parallel arguments lead us then to conclude that without
loss of generality 0 < r;” - rl I: *+* I r,*+i - r,+ i and
(1) xy = . . . = x:” = x; = . . . = x: = 1;
(2) x:’ = . . . = X:” = x: = . . . = x;-; = x,“-1 = 1;
. or
... )
(s+l)x+ . . . =x$s=x;= . . . =x;+l=l.
By mathematical induction, the above structure holds for any s.
(5) Now recalling that the number of land units is n (which is finite),
we must modify the argument at the nth step. Since r,* > r,, 2 c,, some
renter (designated as n) must be assigned to land unit n under r*. If
m < n, then we obtain a contradiction, and can hence conclude that such a
rent profile does not exist. Suppose m 2 n. Since p,” - r” > p,” - r,,* 2 0,
renter n must be also assignedto someland unit under r. But now there is
only one land unit left to be assigned.Let this land unit be denoted by j, so
that we now have xc’ = . . . = x,*~, xi = - . . = X/I: = 1, ~j+~ = . . .
=.X ‘-’ = 1, 0 -Crl* - r, I . . . _< r* - r,,, and x7 = 1. Hence, by an
argu”ment parallel to that in (3B), it ‘must be true that rj* - ‘I= ... =
r * - r,,, and we can reassignrenters as XT’ = . . . = x;” =
n x;= . . . =
x” = 1, without influencing the equilibrium condition. Therefore we can
akume x* = x and 0 < rl* - r, 5 . . . I r * - r, without loss of general-
n
ity.

23. 210 YASUSHI ASAMI
(6) Finally since the process described in Lemma 1 starts from the
initial rent profile c, and r < r*, it follows that somepoint of time in the
process there must exist somej’ E N such that
cj < rj’ = rj for j = j’
cj I rjt 5 rj forj#j’,
where r’ is the current rent profile. We claim that no other renter than j’
has an incentive to raise the rent of land unit j’. For if i is any renter who
is not to be assignedunder r, then pJ:,- rj, = pj, - rj! I 0. Thus i will not
raise the rent of land unit j’. Moreover, if i is any renter who is assigned
under r, and if i # j’, then pi - r;’ 2 pj - ri 2 pi, - ri, 2 pi, - rj;. Hence i
will not raise the rent of land unit j’, and the claim ISestablished. If j’ is
currently assigned to land unit j’ under r’, then j’ is bound to j’ and has
no incentive to raise the rent. If renter j’ is not assigned to land unit j’
under r’, however, there are two casesto consider.
(i) If j’ is not assignedat all under r, then j’ may raise the rent of
land unit j’ to rj + E.But after this, no increasewill occur with respect to
the rent of land unit j’.
(ii) If j’ is assignedto another land unit j” (Z j’), then j’ doesnot
have an incentive to raise the rent of land unit j’ until others outbid j’ on
j”. In such a casej’ may raise the rent of land unit j’ to rj + E; but after
this, no one has an incentive to raise the rent. Since E+ 0, we conclude
that the rent will not rise beyond r to reach r*, which is a contradiction.
Hence the r defined initially does not exist, which establishesthe theorem.
Q.E.D.
APPENDIX 6. PROOF OF PROPOSITION 2
From (61), (64) and (72)
-n(T(n) - T(n - l/2))
Hence we have
lim 10, - 11 = lim 25-/(tn) = 0. Q.E.D.
n+m n-rcc

24. BID RENTS AND BIDDING PROCEDURES 211
REFERENCES
1. W. Alonso, “Location and Land Use,” Harvard Univ. Press,Cambridge, MA (1964).
2. Y. Asami, “Admissible Allocations with an Application to a Land-Development Game,”
Working Paper No. 107,Department of Regional Science,University of Pennsylvania
(1986).
3. Y. Asami, “A Price-Location Duopoly of Land Developers Leads to an Optimal Equilib-
rium,” Discussion Paper No. 28, Department of Urban Engineering, University of
Tokyo (1987).
4. Y. Asami, “Game-theoretic approaches to bid rent,” Ph.D. dissertation, Department of
Regional Science,University of Pennsylvania (1987).
5. M. C. Berliant, A characterization of the demand for land, J. Econ. Theo9 33, 289-300
(1984).
6. G. Demange, D. Gale, and M. Sotomayor, Multi-item auctions, J. Polit. Econ. 94,
863-872 (1986).
7. W. Eckart, On the land assemblyproblem, J. Urban Econ. 18, 364-378 (1985).
8. R. Engelbrecht-Wiggans, “On Optimal Reservation Prices in Auctions,” BEBR Faculty
Working Paper, No. 1241,College of Commerceand BusinessAdministration, Univer-
sity of Illinois, Urbana-Champaign (1986).
9. R. Engelbrecht-Wiggans and R. J. Weber, An example of a multi-object auction game,
Manage. Sci. 25, 1272-1277 (1979).
10. M. Fujita, “Urban Economic Theory,” Cambridge Univ. Press,Cambridge (1989).
11. G. S. Goldstein and L. N. Moses,A survey of urban economics, J. Econ.Lit. 11,471-515
(1973).
12. C. A. Holt, Jr., Competitive bidding for contracts under alternative auction procedures,
J. Polit. Econ. 88, 433-445 (1980).
13. E. Maskin and J. Riley, “Optimal Multi-unit Auctions,” mimeograph, Department of
Economics, MassachusettsInstitute of Technology (1984).
14. P. R. Milgrom and R. J. Weber, A theory of auctions and competitive bidding, Economet-
rica SO,1089-1122 (1982).
15. R. F. Muth, Models of land use, housing, and rent: An evaluation, J. Reg. Sci. 25,
593-606 (1985).
16. T. R. Palfrey, Multi-object, discriminatory auctions with bidding constraints: A game-the-
oretic analysis, Manage. Sci. 26, 935-946 (1980).
17. J. G. Riley and W. F. Samuelson,Optimal auctions, Amer. Econ.Rev. 71,381-392 (1981).
18. T. ten Raa and M. Be&ant, General competitive equilibrium of the spatial economy: Two
teasers, Reg. Sci. Urban Econ. 15, 585-590 (1985).
19. W. Vickrey, Counterspeculation, auctions, and competitive sealedtenders, J. Finance 16,
8-37 (1961).
20. J. H. von Thtinen, “Der isolierte Staat in Beziehung auf Landwirtschaft und
Nationaliikonomie,” Hamburg and Restock (in 3 volumes, 1826-1863).
21. R. J. Weber, Multi-object auctions, in “Auctions, Bidding, and Contracting: Uses and
Theory” (R. Engelbrecht-Wiggans et al., Eds.), pp. 165-191, New York Univ. Press,
New York (1983).
22. R. Wilson, A bidding model of perfect competition, Rev. &on. Srud 44, 511-518 (1977).