DYNAMIC MODEL FOR THE SIMULATION OF EQUILIBRIUM

DYNAMIC MODEL FOR THE SIMULATION OF EQUILIBRIUM STATES IN THE
LAND USE MARKET
Francisco Martínez
Universidad de Chile
[email protected]
Ricardo Hurtubia
Universidad de Chile
[email protected]
ABSTRACT
This paper presents a dynamic equilibrium model for the real estate market. Households have
stochastic behavior and compete for quasi-unique locations (real estate goods), which are
assigned to the best bidder through an auction-type mechanism. The producers are modeled as
maximizers of their profits over the long-term through the production of real estate assets,
represented by the present value of future sales. It is assumed that the producers do not
possess complete information about future levels of demand or prices. Rather, it is assumed
that producers are myopic, meaning that they take the actual and historic prices in each period
as the relevant information for their decision-making. A notion of equilibrium is used that
adjusts prices given two situations: supply and demand surplus. In the supply surplus case, the
prices are diminished and supply in the market is reduced until supply equals demand. In the
case of demand surplus, the prices rise and demand diminishes (homeless households) until
demand equals supply. This equilibrium condition yields prices that are jumpy over time,
resembling observations of inventories in the real estate market and the manufacture industry.
KEY WORDS: Dynamic equilibrium, location, real estate supply, residential development.
1. Introduction
A dynamic model of the real estate market is proposed that permits the existence of a transient
surplus in real estate supply and demand. Prices are resolved through an equilibrium
mechanism in each time frame, which adjusts demand to the supply generated some periods
previously. The mechanism adjusts the effective supply in such a way that sales prices comply
with supply restrictions, at the same time that the feasible demand is adjusted considering the
budget restrictions of households. As a result in each period vacant housing units and
homeless households may be generated in each sub-market, while the corresponding prices
have sudden changes (or jumps) when passing from a period of excess o f demand to one with
excess of supply.
The submodel of demand is based on the RB&SM: Random Bidding and Supply Model
(Martínez and Henríquez, 2002), where households (represented by household clusters) that
are seeking a location make bids for the locations (combinations of housing type and zone).
The assignment mechanism is an auction through which the highest bid determines the price
of the housing unit and assigns each unit to the best bidder. The bids of households are
restricted by the clusters’ income level. In this way, the auction is modeled as a discrete choice
process with stochastic bidding using a multinomial logit model, while the income restriction
is modeled with a binomial logit model (see Cantillo and Ortúzar, 2004; Cassetta and Papola,
2001). Locations are valued according to their attributes, including characteristics of the built
environment, such as neighborhood quality, which represent location externalities. These
externalities are evaluated by consumers with one period delay, such that households set their
value using the characteristics of the previous period in each zone. This diminishes, in part, the
complexity of the static equilibrium RB&SM model, which resolves the externalities in the
equilibrium. In each period, new households enter exogenously into the model as a product of
the growing demographics; it is feasible then to also model which households that are already
located may decide to change their locations.
The supply submodel assumes that, in each period, the producers maximize the present value
of the expected profit for the construction and sale of housing. In this case, the decisions are
modeled as a deterministic optimization process. The time taken for the construction of a
housing unit generates a temporal building delay, such that the producers must decide what
type of housing, where and how much to build, without knowing the price levels or the
demand that will affect the sale n periods after construction is initiated. To decide, it is
assumed that available historic information about prices is utilized. This generates a potential
surplus of supply and demand in different periods. The sale is restricted by reserve prices
(minimum price levels) that represent construction costs, such that a housing unit whose
revenue does not reach that minimum value will not be sold and will be taken off the market,
as stock or surplus available in later periods. This restriction is also simulated through a
binomial logit model that determines the proportion of the total housing units that are
effectively in supply.
Here a notion of equilibrium between supply and demand is introduced, which consists of
assuring that in each period the total households located is equal to the total housing units
2
utilized. As discussed, in each period, the number of households and the available supply are
given exogenously, and furthermore, they normally do not coincide. The adjustment of the
equilibrium consists of adjusting prices such that the notion of equilibrium mentioned is
fulfilled. Two cases can be given: an excess of supply in which all households will locate but
only a portion of the supply will be utilized, leaving a surplus; in this case, the adjustment in
the utilized supply will be produced by a decrease in the rents and supply (through the
adjustment factor). In the other case, there is an excess of demand (the supply is insufficient),
in which case prices rise, all housing units are being used while only a fraction of the total
households will be able to find housing, which induces the withdrawal from the market of
those households with a strong income restriction.
This model has a number of principal differences with other dynamic operative models
previously developed. It uses an auction mechanism that has importance in the formation of
prices while the majority of other models use hedonic prices based principally on surplus.
Secondly, new supply is driven by the microeconomic optimal behavior of real estate
producers, while in other models it is a reaction to the surplus of previous periods without
representing suppliers’ behavior explicitly but rather using an econometric function over the
aggregated supply. For examples see Simmonds (1999), Waddell (2002) and Wegener (1985).
In a different approach, Anas and Arnott (1991) use prices that are obtained from the classic
Walrasian equilibrium but with the limitation of not considering supply and demand surplus.
But, in our opinion, the most fundamental difference is that our model adjusts prices through
an equilibrium mechanism, in which there is some similarity only with the Anas and Arnott
model (1991) as far as we know. They are similar in the microeconomic basis, which is
consistent throughout the model, but there are also some interesting differences between the
two models. The Anas and Arnott model introduces the assumption that dynamic equilibrium
is resolved considering that the producers predict the prices of future equilibrium (“perfect
foresight”), assigning residual values to goods over an infinite time period. On the contrary, in
our model the producers are assumed to be myopic, incapable of predicting future prices. In
our model producers assume that actual and historic prices are the only information available
to make decisions on what, where and how much to build. Another important difference is the
equilibrium mechanism, while Anas and Arnott’ model produce smooth price variation along
time, ours generate prices that oscillate between a maximum and a minimum values,
resembling what it is observed in manufacture inventories (Caballero and Engel 1991).
This paper, after setting notation, describes the submodels of demand-auction and supply in
sections 3 and 4. It then tackles the problem of equilibrium in Section 5 and presents test
simulations in Section 6.
2. Model Variables
Given the complexity of the problem, it is necessary to define a large set of variables and
indices for the model.
3
h, v, i, t : category indices of household cluster, real estate type, zone, time period,
respectively. The time period may be considered one year.
t
H h : number of households of type h in period t; this is a variable exogenous to the model
(aggregate cumulative variable). ∆ H ht denotes the variation in t (we assume ∆ H ht ≥ 0 ).
Ĥht : denotes the number of households that are seeking to locate in t; if no households are
relocating, then it represents ∆ H ht plus the homeless from the previous period .
I ht : Available income of households h in period t, in units of income per period; this is a
variable exogenous to the model that arises from the definition of clusters.
K : Number of units of new supply type v in i. The superindex t indicates the period in which
construction is initiated (or the moment in which the decision to construct is made). The
new supply takes n periods of time to be available.
t
E vi : Surplus of supply units (not utilized) of type vi in each t.
t
vi
S vit , svit : Accumulated real estate supply (stock) and available in each t, respectively, by type v
in zone i. If there is no re-location (and therefore no vacancy of housing previously
utilized) svit = K vit −n + Evit −1 .
U vit , u tvi : Number of housing units utilized, accumulated and in the period, for each vi.
t
Bhvi
: Bid or willingness to pay of agent type h for units of supply type vi in each period t.
zvi : Vector of location attributes, depending on the set of characteristics of the housing unit
and on the attributes of the zone. We denote by z •i the vector of attributes of zone i.
rvit : rent, or use value per period of real estate type vi in each t; in monetary units per period.
p vit : price of sale of real estate type vi in each t; in monetary units per property unit.
Pht / vi : probability that household h will be the best bidder in an auction of vi during period t.
Pvit / h : probability that housing unit vi will be chosen by household type h in period t.
t
φ hvi
: budget feasibility factor of household type h.
ρ vit : profit restriction factor for real estate supply.
C vi : construction cost of housing type v in zone i; in monetary units per property unit.
β : discount rate of capital.
3. Submodel of Demand and Auctions.
This submodel is largely taken from the RB&M, fully described by Martínez and Henriquez
(2002). The agents that are located are households differentiated by clusters based on income
and other characteristics of the utility function. Each cluster contains a different number of
households. Real estate assets are assumed to be discrete and differentiable by attribute and
thus since Alonso (1964) it has been assumed that they can be bought and sold in auctions to
the best bidder. To model this process, it is necessary to define bids that agents make for real
estate using the concept of willingness to pay. Agents’ willingness to pay for a real estate
4
asset is derived from the classical consumer problem, which chooses how much to consume
and where to locate to maximize individual profit. Upon optimizing consumption it is possible
to find the function of indirect utility conditional on the location choice, then it is possible to
calculate the price of the real estate asset as a function of the consumer’s income and reference
utility level. The assignment of real estate to agents and its price is obtained through an
auction. Thus the price represents the maximum willingness to pay across bidders (Martínez,
2000).
Then, the location problem consists of finding an h* as the solution of the following problem
that represents the auction of real estate properties:
(
Max B ht 'vi ( z•t i−1 ,V ht )
h∈ H
s.t
B
t
hvi
≤I
)
∀v,i, t
(1)
t
h
where Bhvi is the bid function constrained by the income level of the household Ih in any
auction, which a parameter in this model. It also depends on the vector of location attributes
(z) as they are defined in the previous period equilibrium. The solution to this problem
depends, in each t, on the actual real estate options supplied at auctions (St) and on an
equilibrium condition that fixes the (indirect) utility levels (Vt); both these aspects can be seen
in sections 4 and 5 respectively.
For convenience in calculating equilibrium, we assume that consumers’ utility function is
quasi-linear, or linear in at least one component of the goods vector. This permits us to assume
that the bid in each location is decomposed additive in income and in another three terms, then
t
t
B hvi
( z t•−i 1 , V ht ) = I h − bht (Vh ) + bhvi
( z•t −i 1 ) + b t . The first term, bht , defines the utility level in
t
income units. Term bhvi
is the value of a vector z of location attributes associated (only) with
the location zone i. This vector contains some attributes that are particular to the real estate
asset and others which describe the environment. Term b t is a constant component, in bidders
and location, which only adjusts all bids levels and rents to a reference value.
To model the intra-cluster idiosyncratic heterogeneity in households behavior, it is assumed
~ t = B t + ε , where the random term ε is
that the bid is a random variable of type B
hvi
hvi
hvi
modeled as an IID Gumbel distribution. The result then of the auction is expressed as the
probability that household type h will be the best bidder in vi, as in the following multinomial
logit expression:
Pht / vi =
t
exp( µBhvi
)
t
∑ exp( µBgvi )
(2)
g ∈Ωgvi
where µ is the scale parameter of the distribution and Ω gvi is the set of consumers who
complies with the feasibility condition Btgvi ≤ I gt imposed in (1).
5
A way of modeling this restriction is to impose that the functional form of the bid comply with
the above-mentioned restriction, a method very demanding in the selection of the function for
representing such behavior at the domain boundary. Another option consists of permitting that
at the domain interior Btgvi is not restricted. In this case, we select a compensatory function
among the attributes of the real estate asset, but then we combine this function with another
function that becomes active only at the domain boundary, such that those agents making
unrealistic bids are eliminated from the set Ω gvi . To avoid discontinuities, and to make
calculations easier, we define the binomial logit probability that the bid is in the domain,
t
denoted by φ hvi
. This is:
t
φ hvi
=
1
t
1 + exp(ω(Bhvi
− I h + θ ))
(3)
which tends to one if bids are far below the income constraint and tends to a constant (small)
probability defined by θ if bids tends to the income constraint.
It should also be noted that equation (2) must be corrected using the McFadden procedure
(1978) to consider the heterogeneity in the size of each cluster presented in the auction. The
number of households of cluster h seeking a housing unit vi and complying with the income
t
restriction is Hˆ ht ⋅φ hvi
. Correcting probability (2) finally we obtain:
Pht / vi =
t
t
Hˆ ht φhvi
exp(µBhvi
)
t
t
t
∑ Hˆ gφgvi exp(µBgvi )
(4)
g
As a result of the same auction, the rent is obtained, determined by the expected value of the
maximum bid among the feasible set Ω •vi , which in this case is given by the following logsum
function:
rvit =

1 
t
t
ln ∑ Hˆ gt φ gvi
exp( µB gvi
) 
µ  g

(5)
Furthermore, using the same consumer theory, it is also possible to calculate the probability
that household h chooses option vi ∈ Ω hvi , Pvit / h , under the assumption that that option
maximizes utility given prices r, as Anas (1982) does, or that the consumer surplus is
t
maximized Bhvi
− rvit , according to Martínez (2000). For this choice process we assume that
t
t
B hvi
− rvit is IID Gumbel distributed given that Bhvi
follows that distribution, and suppose that
consumers observe deterministic values of rvit to calculate their surplus. In this case, denoting
Ŝ vit the number of real estate units type vi available in the market, McFadden’s correction
6
applies those options on the market that also comply with the feasibility requirements given
t
the income restriction, that is Sˆvit ⋅ φhvi
. Then
Pvit / h =
((
((
))
t
t
Sˆ vit φ hvi
exp µ bhvi
− rvit
∑ Sˆ vt'i'φ hvt 'i' exp µ bhvt 'i' − rvt'i '
v 'i '
))
(6)
in that the terms bht and bt of the bids cancel out.
An important observation is that, as the rents that appear in equation (6) are calculated in the
model using equation (5) (i.e. the rents are endogenous variables obtained from the auction). It
can be demonstrated (Martínez, 2000) that the location yielded by a maximum utility (or
surplus) -equation (6)- coincides with that of the auction –equation (4).
Another important observation is that some attributes in vector z represent characteristics of
the built environment, and thus they depend on the dynamic of the location. Such attributes
are thus endogenous to the model, and are denominated as location externalities or economies
of agglomeration (in the case of firms). This type of attribute is assumed to be out delayed by
one period in updating vector z, assuming that this is the time necessary for the changes in
land use to be transformed into available information for agents.
From the previous analysis, it can be concluded that the stochastic auction resolves problems
of location and rents, both of which are conditional on the real estate supply and the utility
levels of households. The subsequent sections will focus on how to model these two sets of
variables.
4. Supply Submodel.
It is assumed that real estate producers are homogeneous and modeled by a representative that
maximizes the present value of future profits. It is also assumed that the producers are
uncertain of the state of the economic or social scenario in which the real estate market will
develop. In this way, they are assumed to be myopic and will make decisions assuming a
steady state of economic variables into the future. This means that they will suppose a constant
growth of price levels.
The producers must decide, in each period, the following: what and how much to sell (how
many housing units of each type in each zone); and how many and where to build. To make a
decision about sales it is assumed that the decision is always to sell when the reserve prices
condition is fulfilled. The decision of how much and where to build K vit , is taken in period t
to be available for sales in period t+n. This generates a temporal delay that produces
disequilibrium between supply and demand in n future periods.
7
Assuming that there is no demolition of housing, the aggregate supply (stock) is defined as:
Svit = Svit −1 + Kvit −n
(7)
and the total surplus at the end of each period is defined as:
E vit = S vit − U vit
(8)
Then the available supply in period t is composed of the surplus of the previous period plus
the newly constructed supply, which is:
svit = K vit −n + Evit −1
(9)
Considering the assumption of no re-location, this implies that the total housing utilized at
period t will be equal to the sum of housing units that began to be used along the period from
t=0 to t. This can be expressed recursively as:
U vit = U vit −1 + u tvi
(10)
Replacing equations (7), (9) and (10) in (8) we obtain the dynamic equation for the surplus:
E vit = S vit − U vit = s vit − u vit
(11)
In other words the available supply minus that which is consumed in the period is equal to the
surplus at the end of the period. This relation generates an interdependence of decisions on
how much to build between the two periods, because the surplus in a period t will necessarily
become a part of the available supply in t+1.
To model the behavior of the representative producer, we define the profit function as the
income from expected sales minus the production cost. That function is:
(
)
(
π K vit , p vit + n , p vit , u vit +n = u vit + n β ( n ) p tvi+n − K vit C K vit , p vit
)
(12)
in which the income depends on the quantity sold in t+n, u tvi+ n , and the selling price of the unit
p tvi+n (discounted by β ); for total cost, this will depend on the number of units that will be
built K vit and their production cost C (K vit , p tvi ) , which will allow for the presence of
economies of scale in production.
The model of the producer’s behavior that maximizes profits over the long term is proposed as
the following problem of deterministic optimization over the present value of profit:
8
[
]
max ∑ β ( t ) π (K vit , pvit , pvit + n , uvit + n )
∞
K vi•
t= 0
s.t .
p ≥c
t
vi
(13)
∀t
t
vi
where the quantity of supply utilized, uvit , is determined in equilibrium. This corresponds to
that fraction of the supply svit that is effectively available when the restriction of no negative
profit from problem (13) is met; otherwise the supply option is taken off the market.
To model the restriction on prices again we use a function of the binomial form, denoted by
ρ vit and defined by:
ρ vit =
1
1 + exp ω ' cvit − p vit + θ '
( (
))
such that if the price of the sale is less than or equal to the minimum price c, the factor tends to
zero, otherwise values near one are assumed. The minimum price of sale c vi is modeled as the
present value of production cost, given by cvit = β − nv Cvit −n .
As will be demonstrated in detail later, in equilibrium the factor ρ vit determines the quantity of
available supply vi that will be used, then we can write:
u vit = svit ρ vit
(14)
Considering the definitions made in (14) and (9) the profit function of the producer can be
conveniently rewritten with the decision variable, K vit , in the argument:
(
) (
)
(
π K vit , p vit , p vit +n , E vit +n −1 = E vit + n −1 + K vit ρ vit +n β ( n ) p vit + n − K vit C K vit , p vit
)
(15)
Thus, the producer’s problem (13) is:
[
]
max ∑ β ( t ) π (K vit , pvit , p tvi+n , Evit +n −1 )
∞
K •vi
t =0
s.t . E
t −1
vi
+K
t− n
vi
− (E
t −1
vi
+K
t −n
vi
)ρ
t
vi
(16)
=E
t
vi
∀t
in which the restriction of surplus is derived by substituting (14) and (9) in (11). The
restriction on sale prices is verified using the factor ρ vit that eliminates from the alternatives
set those housing units that do not meet the restriction.
9
This problem can be reformulated assimilating the real estate producer’s decision to the
dynamic problem of invest and save proposed by Stokey, Lucas and Prescott (1989), which
would write (16) in recursive terms upon defining a function of value V:
( )
[(
)
(
( )]
)
V ξ vit = max Evit + n −1 + K vit ρ vit + n β ( n ) p tvi+n − K vit C K vit , p vit + β V ξ vit +1
K vit
s.t . E
t −1
vi
+K
t− n
vi
(
− E
t −1
vi
+K
t− n
vi
)ρ
t
vi
=E
(17)
t
vi
where ξ vit represents a set of variables ( Kvit , pvit , ptvi+n , Evit + n −1 ).
The variable of function (17) can be reorganized collecting terms of the same period yielding:
( )
[(
)
(
)
)ρ
(
V ψ vit = max Evit −1 + K vit −n ρ vit p vit − K vit C K vit , p vit + β V ψ vit +1
K vit
s .t . E
t −1
vi
+K
t− n
vi
(
− E
t −1
vi
+K
t− n
vi
t
vi
=E
)]
(18)
t
vi
where ψ vit represents the set of variables ( Kvit , Kvit − n , pvit , Evit −1 ).
This dynamic function expresses the value of the real estate industry in terms of the profit in a
period and the present value of profit in future periods. For the producer’s problem, the prices
p vit , the adjustment factors of the supply ρ vit and the surplus of the previous period Evit −1 are
known and are obtained as a result of the equilibrium and the interaction between supply and
demand in the previous period.
To find the optimum supply, the following first-order conditions are derived for problem (18):
t
t
∂V (ψ vit +1 )
∂π
∂V
t
t
t ∂C (K vi , p vi )
+
β
=
−
C
(
K
,
p
)
−
K
+
β
=0
vi
vi
vi
∂K vit
∂K vit
∂K vit
∂K vit
∀ v,i ,t
(19)
Furthermore, it is also necessary to consider the effects that decisions made in period t will
have on the future. For this reason, it is useful to consider the envelope theorem to construct
the following sequence of derivatives of the value function:
(
)
(
∂ V ψ vit + n −1
∂V ψ vit +n
=
β
∂ K vit
∂ K vit
(
)
)
∂ V ψ vit + n
= ρ vit + n p vit + n + λtvi+ n 1 − ρ vit +n
t
∂K vi
(
(20)
)
(21)
where λ is the Lagrange multiplier associated with the restriction in problem (18).
Then replacing (21) in (20) and thus successively until reaching t+1 the following is obtained:
10
(
)
∂ V ψ vit +1
= β ( n −1) ρ vit +n p vit + n + λtvi+ n 1 − ρ vit + n
t
∂ K vi
(
(
))
(22)
which is replaced in equation (22) obtaining the following Euler equations for the producer’s
problems:
(
)
− C K vit , p tvi − K vit
(
)
∂ C K vit , p vit
+ β ( n ) ρ vit + n p vit + n + λtvi+n 1 − ρ vit +n = 0 ∀ v, i, t
t
∂ K vi
(
(
))
(23)
In this equation, λ is interpreted as the benefit of postponing the sale of a housing unit (or the
benefit associated with having a unit of supply in stock). For example, if the sale is postponed
from period t to t+1, λ tvi will take the present value of the sale in period t+1, that is:
λ tvi = β p vit +1 .
It is clear that producers would prefer to sell in the period that shows the highest present value
of sale price. Then the problem is to select between receiving p vit in t or receiving λ tvi in
another later period. The difficulty for the producer in making this decision is found in
knowing the value of the sale price in t+n, for which certain assumptions must be introduced.
The assumption that is made on this issue substantially defines the dynamic of the supply
model. An elegant assumption , used by Anas and Arnott (1991) is to assume that producers
have “perfect foresight”, implying that they can analyze the future development of the real
estate market to calculate prices. We consider this assumption extremely demanding of
information and full of forecasting assumptions, which makes it unlikely to represent the real
behavior of producers.
We prefer another alternative. This is to assume that the producers do not have the ability to
predict future prices, thus without better information, they will assume that the sales price of
an available supply unit in t and sold in t+m (with m ≥ 1 ) will be identical in real terms to the
price in t. That is:
p
t +m
vi
p vit
= ( m)
β
(24)
This assumption can be complemented, without changing the structure of the model, making
the more general assumption that producers estimate future prices as a function of past prices,
that is pvit +m = β ( −m ) (αpvtt + (1 − α ) pvit −1 ), α ∈ [0,1] . Given that all these prices are known at the
moment of deciding on production, this does not introduce further complexity into the model’s
algorithm.
Under the previous assumption (24), the present value of postponing production, λtvi , will be
equal to the actual price of the unit p vit :
11
λtvi = β ( m )
p tvi
= pvit
β ( m)
(25)
and the Euler equation, which describes producer behavior can be rewritten as:
∂C(K vit , pvit )
− C (K , p ) − K
+ β ( n ) p tvi+n = 0 ∀ v,i ,t
t
∂Kvi
t
vi
t
vi
t
vi
p vit +n =
in which
p vit
β ( n)
(26)
(27)
such that equation (26) can be reformulated as:
(
)
C K vit , p vit + K vit
(
)
∂ C K vit , p vit
= p tvi
t
∂ K vi
∀ v, i, t
(28)
This equation indicates that, considering that the present value of future sales in any period
m ≥ t + n is equal to the price at t, producers will produce a quantity of supply K vit such that
the marginal cost of production is equal to the sale price at t. This behavior reflects the
competitive character of the market, because it reproduces the classic economic result for
markets in perfect competition.
5. Equilibrium
The notion of equilibrium used here accepts that the general state is that in which the supply
and demand are different in each period, i.e. there is a surplus of supply or demand. This
notion can be explained as “the total number of households located must be equal to the
number of housing units utilized”, which allows for an unsatisfied demand or an unused
supply. In this definition, surpluses occur by components of the supply and demand vectors,
such that in a given period there could be a supply surplus in a sub-market vi, for example
because in that market the condition of positive profit was not satisfied, while there is
simultaneously a demand surplus for a segment of households that could not access those
surplus units because of budgetary constraints.
The supply levels are calculated for each period by the supply submodel, before reaching
equilibrium, thus they are parameters for the location equilibrium process. The aggregate level
of demand is exogenous because population by cluster is defined by demographic growth thus
making supply and demand, in general, different.
The equilibrium state defines a set of sale prices -and rents- such that the following condition
is met: all households actively in the market (with a budget restriction which is feasible for the
12
equilibrium prices) locate in some housing unit and all housing units available (with sales
prices that obtain non-negative profit) are utilized. This condition is written as the following:
∑ Hˆ
φ Pvit / h = ∑ s tvi ρ vit Pht / vi
t t
h hvi
vi
(29)
vi
The term on the right represents the effective demand aggregated by location; the term on the
left represents the total housing units utilized during the period.
For this equilibrium equation to have a solution, there must be at least one set of values of
utility level i.e. of vector bht , ∀h . To study whether this set exists, we solve (29) for bht
obtaining:
( (
t
t
 ∑ svit ρ vit Hˆ ht φ hvi
exp µ bhvi
− rvit

1
bht = I h + ln vi
µ 
∑vi Hˆ ht φhvit Pvit / h

)) 
∀h , t



(30)
where the income is a cluster constant. We can observe that the rents are the expected value of
the maximum bid, then equation (30) is a fixed-point equation with the known logsum form.
As mentioned above, under the assumption of quasi-linear utility, the values for bh yielded by
this equilibrium condition defines the clusters’ utility levels (in relative monetary terms),
which measures the equivalent income variation of consumers at each period.
The second level of adjustment of equilibrium prices is verified on the aggregate of equation
(29) over the whole population, such that:
∑ Hˆ
hvi
t
h
t
Pvit / h φhvi
= ∑ s tvi Pht / vi ρ vit
(31)
hvi
which means that the total number of households located (in all the vi pairs) is equal to the
total number of housing units utilized (by all types of households). This condition allows us to
define values for the constant term of bids, b t , in equilibrium.
Solving equation (31) for b t we obtain:

Hˆ ht Pvit / h
∑
t
1  hvi exp − ωb t + exp ω b ht + bhvi
− I ht + θ
bt =
ln
2ω 
svit
t
t
t
t
 ∑
 hvi exp ω b + exp ω cvi + b − rvi + θ
(
)
( (
( )
( (
))
))







(32)
which represents the second fixed point of the equilibrium. The values obtained from this
condition define the absolute values of bids, rents and prices.
13
Finally, the sale prices (p) of the supply must be related to rents (r) of the auction-demand
model. For this, the following simple relationship is assumed (DiPasquale and Wheathon,
1996), which defines the price as the present value (with an infinite time frame) of future
rents, supposing that these will maintain their value constant over time according to (24):
rvit
i
p vit =
(33)
where i represents the interest rate or some discount factor; in this model, i = β is assumed.
It is useful to understand the price adjustment mechanism, which is verified using the values
of b t generated by equation (32), and its repercussion on the effective supply, using ρ vit , and
t
a feasible demand, using φ hvi
. Considering that the parameters of equilibrium are the available
t
supply svi and the levels of population Ĥ ht , in each period t, these parameters can define three
cases that are interesting to analyze:
i) Excess of aggregate supply,
∑ Ĥ
t
h
h
t
< ∑ svit , where equation (32) will generate low values
vi
t
for b , denoted as b , associated with lower bound prices derived from the positive profit
constraints. This induces bid levels to diminish homogeneously for all types of households
and housing units, with the consequent reduction in rents and prices, which reduces the
effective supply until equilibrium is reached. This is modeled by a reduction of the supply
adjustment factor ρ vit , which diminishes with prices, because it increases the probability of
violating the profit restriction in some markets vi, which exit the market. This process
unfolds until the available “feasible” supply equals the effective demand.
ii) Excess of aggregate demand
∑s
t
vi
vi
t
< ∑ Hˆ ht , where equation (32) will generate high values
h
t
for b , denoted as b , associated with upper bound prices derived from budget constraints.
This induces homogeneous increase of bid levels which will make the demand adjustment
t
t
factor φ hvi
take lower values (φ hvi
<< 1 ), while the supply adjustment factor approaches to 1
( ρ vit → 1 ). In other words, all the real estate supply will be used while some households are
unable to locate since rents increase violates the income restriction in some household
clusters.
iii) equality between aggregate supply and demand
∑s
t
vi
vi
= ∑ Hˆ ht , which generates a
h
discontinuity in the equilibrium, that must be addressed by a reformulation of the
equilibrium equations. This occurs because of the inability of equation (32) to find a value of
t
and ρ vit simultaneously and for all
bt that can reach values infinitesimally close to 1 for φ hvi
14
hvi, since for this to occur, we must have bt → ∞ and b t → −∞ at the same time, which is
clearly a contradiction.
This contradiction reflects an interesting situation for the model. The equality between supply
and demand represents the limit case of the previous cases: s t > Hˆ t with b t = b t , and s t < Hˆ t
with b t = b t . Then the case s t = Hˆ t can be resolved for any value of bt ∈ b t , bt , leaving its
value undefined. However, this indetermination of the model never really occurs because the
equality condition are only theoretical, but what is relevant for this analysis is that when the
equilibrium changes from the excess supply to the excess demand, prices and rents jump from
lower bounds to upper bounds, which generates a non smooth output from the model.
[ ]
6. Application of the model
An application requires that we define two functions, the cost of construction C vit ( K vit , p vit ) and
t
the component of bids associated with attributes bhvi
(z vit−1 ) . The simulations that are presented
utilize the the Cobb-Douglas function of unitary cost:
(
)( )
C vit ( K vit , p vit ) = C v + qp tvi K vit
γ
(34)
where q represents the fraction of the price of the real estate that corresponds to the value of
the land as production input cost, C v is the fixed cost of constructing a housing unit type v,
and γ defines economies of scale, with γ > 0 implying decreasing economies of scale.
Replacing this equation and its derivatives in equation (28) the optimum production is
obtained, given by:


p vit

K = 
t

 C v + qp vi (1 + γ ) 
(
t
vi
1
γ
)
(35)
We observe that this production is undefined for γ = 0 because in this value the optimum
quantity level is indifferent. Furthermore, we observe that the production depends on the price
level according to:
∂ K vit
1
Cv
> 0 if γ > 0
=
=
t
t
t
∂ p vi γ p vi C v + qpvi
< 0 if γ < 0
(
)
(36)
The willingness to pay function in this application is assumed to be linearly dependent on two
zone attributes, the average income of population in the zone (in the previous period) and the
average density of land use. That is:
15
I g H gvt −1' i
ϕ hv' S vt ' i
b (z ) = θh ∑
+∑
t −1
t
g ,v ' ∑ H g 'v 'i
v ' ∑ Sv ' i
t
hvi
t −1
•i
g ' ,v'
(37)
v'
where θ h and ϕ hv' are parameters that indicate relative value of the attributes: average zonal
t
income and density by housing type, respectively. The terms H hvi
and S vit are variables
endogenously calculated by the model.
Furthermore,
assumption that producers are myopic is represented by
p = α p + (1 − α ) p ; α ∈ [0 ,1] , which smoothes out abrupt price variations in each period,
but also has the effect of making prices dependent on the complete history.
t
vi
t
vi
the
t −1
vi
To resolve the dynamic equilibrium problem set in the model the following algorithm is
applied (see Figure 1). The variables( H , S , I , B ) t =0 are initialized and a modeling timeframe,
T in years, is defined along with growth rates of income and population for each period. The
levels of construction are also defined ( K vit , ∀t = − n ,.., −1 ), considering the delay between the
initiation of construction and availability to the market. The model advances sequentially over
time from t=1 until t=T, calculating equilibrium and new production in each period. The
equilibrium defines the surpluses (in supply and demand), the (probabilities of) location, the
zonal attributes, and the rent levels. The latter define sales prices which are input to the
supply submodel to determine the amount of real estate units to be built in the period and
which will be available n periods in the future. Figure 1 shows the diagram of the algorithm.
To illustrate some features of the model mechanism, we present a set of simulation results of
selected scenarios of the model parameters. These scenarios were arbitrarily defined but they
represent the model behavior on large space of parameters studied; an application to a real
data set is pending for future studies. We consider 3 household clusters and 4 location
alternatives vi. The base scenario is defined by clusters’ income I=(250, 200, 150), costs by
building type and zone Cvi=(60, 40, 30, 15). Income and population increase by 1% per period
for all clusters. Parameters are n=1, q=0.7, µ =0.6, ω =0.7, γ =0.1 and η =0.01. The
simulation considers scenarios in which variations of the cost of land inputs (q) and the rate of
income increase. Figures 2 to 6 depict curves of average prices (pt) and aggregate values of
available supply (st), population variation (∆H t ) and excess supply or building stock (Et).
The base scenario (Figure 2) shows an initial adjustment followed by a stationary state at
which, except for local oscillation in prices, available supply and excess supply, equilibrium is
attained when total demand equals total supply. This stationary case is the result of the
selection of parameters. The oscillations occur whenever excess supply changes into excess
demand; prices jump because bids change from the lower bound (defined by building cost) to
the upper bound (defined by household budget), followed by a period of decreasing stock and
prices.
16
The effect of land costs is analyzed by changing q, first increased to 0.8 (Figure 3) and then
reduced to 0.6 (Figure 4). The result is that higher land costs induce higher prices and a
reduction in supply, followed by permanent excess of demand (structural homeless
population) because some poor households can not afford prices. Conversely, a reduction in
the q parameter induces producers to over-react by deciding to produce large amounts of
building thus generating a large building stock. This is followed by several periods where
production and stock decrease and prices are low, up to a point when stock vanishes and prices
jump to reach the upper bounds; this cycle repeats although the increase in income and
population induce changes in the cycle’s period and amplitude (see Figure 4). Similar effects
were obtained changing the scale economies parameter γ .
The variation in income is shown to be dependent on the relative change of income and
population. Increasing the rate of income change by period from 1% to 1.5% (Figure 5)
induces an increase in the amplitude of price jumps, because upper bounds are increased. As a
result, peaks in the building stock curve (Et) tend to increase over time, a tendency not
observed in the base scenario; this is also caused by the increase in the size of price peaks.
Reducing the rate of income change by period to 0.5% (Figure 6) induces price reductions
followed by a reduction of supply and an increase in homeless population.
7. Conclusions
The model generates a stable dynamic pattern, which does not diverge over time. The
examples presented show a range of model behavior, with initial transients followed by
alternative market developments that include, in some cases, periodic cycles on the set of
endogenous variables: prices, real estate supply and stock. The cycles are determined by a
sequence of periods alternating between excess of supply, where endogenous variables drop
until a minimum value, and a short period of high production to recover stock at higher prices.
The effects of economies of scale and land costs are similar, with prices increasing as a
reaction to, either the increase in land costs or the reductions in scale economies; in both cases
the model may produce permanent excess of demand. Conversely, in scenarios where land
costs are reduced or scale economies are higher, periodic cycles on the endogenous variables
are obtained. Cyclic behavior is also produced at high rate of income increase, while a
reduction in the income increase rate leads to excess of demand and an increase of the
homeless population. Of course, cyclic behavior emerges conditional on the rest of parameters
not discussed here.
A notable feature of the model is that it yields cycles in some scenarios, initiated by a lumpy
production at high prices, which then evolves consuming the real state stock and reducing
prices. When stock vanishes, the cycle restarts with a jump of prices up to a maximum value.
This cycles resembles the evidence in manufacture inventories, where producers follow a
policy that allow stock to fall freely until a lower critic level (s) and then they adjust stock by a
lumpy production to hit an upper critic level (S); this production cycle is called (S,s) rule
(Caballero and Engel, 1991).
17
The cyclic behavior only appears under the condition of excess of supply. This case
represents an expanding economy where producers anticipate the expanding demand by a
lumpy production that exhaust the advantages of scale economies or land prices effects.
Conversely, in a contracting economy incentive to produce are reduced, yielding shortage of
supply with come consumers leaving the market. This market situation represents a time
period affected by an economic shock that reduces demand for real estate properties. Thus, the
model responds to macroeconomic conditions combining a steady production under
constrained demand and a cyclic production under an expanding economy.
These results motivate further research. First, to apply the model to observed data under
different macroeconomic conditions and market sizes, in order to validate the model
performance on a wider context. Second, to complement the model including other important
features, for example, an alternative behavior for suppliers like rational expectations. The
model can accommodate, without affecting the algorithm performance, some features not
explicitly described, as for example, voluntary movers and a construction lead time
differentiated by building type.
Acknowledgements
This research was partially funded by Fondecyt 1010422 and by ICM Sistemas Complejos de
Ingeniería. We thank E. Rossi-Hansberg for his fruitful comments.
References
Alonso, W. (1964). Location and Land Use: Towards a General Theory of Land Rent.
Harvard University Press, Cambridge, Mass.
Anas, A. (1982). Residential Location Markets and Urban Transport ation, Academic
Press, London.
Anas, A. and Arnott, R. (1991) Dynamic housing market equilibrium with taste heterogeneity,
idiosyncratic perfect foresight, and stock conversions. Journal of Housing Economics 1, 232.
Caballero, R. And Engel, E. (1991) Dynamic (S,s) economies. Econométrica 59, 6, 16591686.
Cantillo, V. and Ortúzar, J. D. (2004) A semi-compensatory discrete choice model with
explicit attribute thresholds of perception. Transportation Research 39B, 641-657.
Cascetta, E. and Papola, A. (2001) Random utility models with implicit availability/perception
of choice alternatives for the simulation of travel demand. Transportation Research 9C, 249263.
18
DiPasquale, D and Wheathon, W. (1996) Urban Economics and Real Estate Markets.
Prentice-Hall, New Jersey.
Martínez, F. (2000) Towards a land use and transport interaction framework. In D. Hensher
and K. Button (eds), Handbooks in Transport – Handbook I: Transport Modelling,
Elsevier Science Ltd., 145-164.
Martínez, F.J. and Henríquez R. (2002) Modelo de equilibrio en localización urbana
asumiendo comportamiento idiosincrático de agentes. XII Congreso Panamericano de
Ingeniería de Tránsito y Transporte, Quito, Ecuador.
McFadden, D.L. (1978) Modelling the choice of residential location. In A. Karlquist, L.
Lundquist, F. Snickars and J. W. Weibull (eds.), Spatial Interaction Theory and Planning
Models, North-Holland, Amsterdam, pp 75-96.
Simmonds, D.C. (1999) The design of the DELTA land use modeling package. Environment
and Planning 26B, 665-684.
Stockey, N., Lucas, R. and Prescott, E. (1989) Recursive Methods in Economic Dynamics.
Harvard University Press, Cambridge, Mass.
Waddell, P. (2002) UrbanSim: Modeling urban development for land use, transportation and
environmental planning. Journal of the American Planning Association 68, 297-314.
Wegener, M. (1985) The Dortmund housing market model: a Monte Carlo simulation of a
regional housing market. In K. Stahl (ed), Microeconomic Models of Housing Markets,
Springer Verlag, Berlin.
19
Figure 1: Diagram of the algorithm
Read initial values:
0
S vi0 , K vi−n ...K vi−1 , E vi0 , b hvi
, t=0
H ht , I ht
∀h ,t ,
Update location
externalities
{
Solve location equilibrium b , ∀h ; b
and calculate price
t
vi
t
h
( )
t
hvi
r B
t
}
∀ v, i
(
t
bhvi
P•t/ vi
)
and excess
supply
Evit −1
Calculate real estate
production
avaliable at t+n:
( )
K t rt
Stop
yes
t=T
no
t = t+1
20
Figure 2. Base Scenario
∆H s E
t
t
pt
t
25
375
p
t
300
20
15
225
st
150
10
∆H
t
75
5
E
t
0
0
10
20
30
40
50
60
70
80
90
Figure 3. Increase in land costs
∆H s E
t
t
t
pt
30
450
pt
20
300
∆H
t
150
10
Et
st
0
0
10
20
30
40
50
60
70
80
90
Figure 4. Reduction in land costs
∆H s E
t
t
pt
t
160
320
pt
240
120
s
t
80
160
40
80
∆H t
E
t
0
0
10
20
30
40
50
60
70
80
90
21
Figure 5: Increasing the rate of income change by period
∆H t s t E t
pt
60
600
pt
40
400
20
200
st
∆H t
Et
0
0
10
20
30
40
50
60
70
80
90
Figure 6: Decreasing the rate of income change by period
pt
∆H t s t E t
15
225
pt
10
150
∆H t
st
5
75
Et
0
0
10
20
30
40
50
60
70
80
90
22