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Stickiness of Rental Rates and Developers’ Option Exercise Strategies

February 2007
The Journal of Real Estate Finance and Economics 34(1):159-188

February 2007
34(1):159-188

DOI:10.1007/s11146-007-9004-3

Source
RePEc

Authors:

Rose Neng Lai

University of Macau

Ko Wang

City University of New York - Bernard M. Baruch College

Jing Yang

Monash University (Australia)

In this study we incorporate sticky rents into a real options model to rationalize the widely documented overbuilding puzzle in real estate markets. Given the assumption that developers’ objective function is to maximize total revenue by selecting an optimal occupancy level, our model provides a better explanation of the phenomena we observed in the real world than the traditional market-clearance based real options models. We also show that developers’ exercise strategies can be affected by the size and the type of property markets. In other words, developers’ exercise strategies could differ among markets and under different conditions. Copyright Springer Science+Business Media, LLC 2007

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Stickiness of Rental Rates and Developers’Option

Exercise Strategies

Rose Neng Lai &Ko Wang &Jing Yang

Published online: 6 March 2007

#Springer Science + Business Media, LLC 2007

Abstract In this study we incorporate sticky rents into a real options model to

rationalize the widely documented overbuilding puzzle in real estate markets. Given

the assumption that developers’objective function is to maximize total revenue by

selecting an optimal occupancy level, our model provides a better explanation of the

phenomena we observed in the real world than the traditional market-clearance

based real options models. We also show that developers’exercise strategies can be

affected by the size and the type of property markets. In other words, developers’

exercise strategies could differ among markets and under different conditions.

Keywords Sticky rents .Overbuilding .Real options

Introduction

It is well documented that high vacancy rates exist in most types of property

markets. For instance, vacancy rates in the housing rental market and the office

J Real Estate Finan Econ (2007) 34:159–188

DOI 10.1007/s11146-007-9004-3

Submitted to Cambridge—Maastricht 2005 Symposium.

R. N. Lai (*)

Department of Finance and Business Economics, Faculty of Business Administration,

University of Macau, Taipa, Macau, China

e-mail: RoseLai@umac.mo

K Wang

Department of Real Estate, Baruch College (CUNY), New York, NY, USA

e-mail: Ko_Wang@baruch.cuny.edu

K. Wang IJ. Yang

Department of Finance, California State University-Fullerton, Fullerton, CA, USA

K. Wang

e-mail: kwang@fullerton.edu

J. Yang

e-mail: jyang@fullerton.edu

rental market in the United States were exceptionally high in the 1980s and 1990s. In

particular, the former exceeded 10% in 1986 (Belsky & Goodman, 1996) while the

latter reached as high as 18% in 1990 (Grenadier, 1995). The National Association

of Realtors predicts that the vacancy rates will be 14.5–15.5% for offices, 11–11.5%

for industrial spaces, 8–8.5% for retail spaces, and 5.5–6% for multi-family rental

units in year 2005 (Lereah, 2004). A vacancy rate of 40% or higher in the hotel

industry is also very common (see, for example, McAneny, Han, & Gallagher,

2001). All these vacancy levels are too high to be caused merely by costly searching

and matching processes between households and landlords.

While oversupply in property markets has been widely investigated from different

points of view,

only a few studies, such as Grenadier (1996), Wang and Zhou

(2000) and Wang and Zhou (2006), offer theoretical justifications for the results they

present. Grenadier studies the timing of real estate developments using a strategic

option exercise approach based on stochastic demand.

He provides an interesting

rationale for a commonly observed phenomenon: increasing development in the

“cold”real estate markets. He believes that panic in the market can induce

preemptive competition in developments during a market downturn, and terms this

phenomenon “irrational construction.”Wang and Zhou advance the standard

strategic real options approach by incorporating both stochastic demand and

construction cost, and allowing multiple players with unequal production capacities

under various market structures. Wang and Zhou are the first to relate overbuilding

to rent collusion with a game theoretical approach. Their model demonstrates that

under competition developers will start construction whenever there is an

opportunity, and that sticky rents led by implicit collusion can be optimal for

developers.

In other words, under their model framework, excess vacancies could

be an equilibrium solution in real estate markets.

However, it is fair to say that these three models, while interesting, have certain

weaknesses. The models in Grenadier (1996), and Wang and Zhou (2006) are

basically market clearance models. This type of model is not suitable for explaining

the excess vacancy puzzle because it assumes that any excess space will be absorbed

by adjusting rental rates. Consequently, under these two model frameworks, the

occupancy rate of properties should always be at the 100% level. While Wang and

Zhou (2000) do not impose the market clearance condition, their two-stage model

focuses only on the vacancy rates of markets with the assumption that market rental

rates could be sticky. A major weakness of the paper is that the model does not

For instance, Fickes (2001), Johnson (1998), and McAneny et al. (2001) argue that, as long as financing

is readily available and zoning or tax systems are in favor of construction, there is always room for hotel

building. Kummerow (1999) suggests that when a developer considers what her/his rivals will decide, her/

his “rational”development decision may lead to oversupply. Mueller (1999) concludes that rental growth

rates of different real estate markets can be predicted from occupancy level cycles. Hendershott (2000)

points out that the Sydney office market developers failed to incorporate mean reversion in property price

movements, and hence created market cycles.

In this model framework, construction decisions can be considered as exercising real options (see for

example Titman, 1985; Williams, 1991 and 1993, for pioneering researches on real options).

Pagliari and Webb (1996) also suggest rental properties can be operated at a higher rental rate with a

higher vacancy rate. Additionally, Blackley and Follain (1996) link the increasing rental rate to user costs,

which are in turn related to long-run supply elasticity.

160 R.N. Lai et al.

describe developers’behavior in a dynamic setting. Given this, it is difficult to

understand what drives developers’investment decisions at various stages of the

game. In other words, while their result can be used to predict the vacancy level of a

market, the model does not address the detailed conditions affecting the timing of

developments in different markets.

Our paper attempts to build a model that incorporates the sticky rent argument

developed by Wang and Zhou (2000) into the dynamic options exercise model

setting of Grenadier (1996) and Wang and Zhou (2006). To the best of our

knowledge, this model is the first in the literature that has incorporated sticky rents

into a real options framework. We will then demonstrate that, in a sticky rental rate

setting, our model performs better than traditional market clearance based real

options models in terms of interpreting the essence of overbuilding and vacancies.

Our model starts with an assumption that developers will seek a revenue-

maximizing occupancy level and will allow an optimal excess vacancy rate in the

market. This assumption differs from that used by Grenadier (1996) and Wang and

Zhou (2006), for which developers will adjust rent levels to eliminate all excess

supplies. Our model also differs from Grenadier in that we use the expected mean

time to estimate development intensity, as opposed to the expected median time used

by Grenadier. The use of expected mean time allows us to obtain closed form

solutions (as opposed to simulations in Grenadier, 1996) for the timing of

developments and the time lag between developers’construction decisions. With

closed form solution, it is easier to analyze how the various factors might affect a

developer’s supply decision in terms of magnitude and direction.

Our model provides two important implications that differ from the existing

literature. First, we show that an increase in demand variance will decrease the speed

of developments. It should be noted that, with identical stochastic factors in both

models, our result contrasts to the prediction in Grenadier (1996) that an increase in

demand variance will increase development activities.

Second, we show that the

impacts of demand elasticity on developers’exercise decisions can be different when

developers adopt different negotiating strategies. When developers collude with each

other to set an optimal rental rate, an increase in demand elasticity will delay overall

development activities. This result contrasts to the model prediction provided by

Wang and Zhou (2006). Under the market clearance framework, they predict that an

increase in demand elasticity will increase overall development activities.

This paper is organized as follows. The next section presents the model

framework. In “Developers’Game with a Market Clearance Condition,”we will

derive a duopolistic market equilibrium where rents are adjusted to clear the market

so that all the excess capacity is eliminated. In “Strategic Exercises with

Overbuilding,”we will deduce the optimal collusive rent that developers should

adopt and the conditions when overbuilding is preferred. We also generalize the

explanation of overbuilding when rigid rates exist. “Market Clearance versus

Overbuilding”compares the overbuilding model with the market clearance model.

Finally, “Conclusion”concludes.

However, our result using one stochastic factor conforms to the result derived by Wang and Zhou (2006)

using both demand and construction cost as the stochastic factors in the model.

Stickiness of Rental Rates and Developers' Option Exercise Strategies 161

Model Framework

Our model relies on several well known assumptions in the field. First, we use the

standard assumptions for financial options that investors prefer more to less, the

demand shock process follows a random walk with a known growth rate and variance,

and the risk-free rate is constant and known. Second, we impose a risk neutrality

condition, which basically states that the discount rate is equal to the risk-free rate (see

Cox & Ross, 1976; Harrison & Kreps, 1979; and Merton, 1975, for the discussion of

this assumption). Finally, we assume an incremental demand function that is

inversely related to the rental rate, or

DX tðÞ;RtðÞ½¼aX tðÞbR tðÞ ð1Þ

where aand bare constants, X(t) is the multiplicative demand shock at time t, which

may represent changes in the number of households or in the standard of living due

to changes in market conditions, consumers’taste, and other factors affecting

demand. R(t) is the rental rate at time t.

The multiplicative demand shock X(t)inEq.1follows a geometric Brownian motion

dX ¼mXdt þsXdw;ð2Þ

where wis a random variable following a Wiener process, that is, with a normal

distribution with E(dw)=0, and Var(dw)=dt.μis the constant instantaneous growth rate

of the demand shock per unit time, and σis the constant instantaneous standard

deviation per unit time with respect to w. Equation 2states that the instantaneous change

in demand shocks governing the rental rates is its rate of return plus the standard deviation

times the instantaneous change in the Wiener process governing its randomness.

Next, consider a simple duopoly market in which each of the two developers own

a parcel of land for construction.

Both are said to own construction options that can

be exercised whenever they see a fit. Unlike financial options that generate perpetual

dividend yields immediately after exercise, there may be a few years of construction

before rental income can be realized. Hence, we assume it takes δyears to complete

the construction. A developer who decides today to build the rental unit at time t

will receive rental income only after time t+δ. We assume that the two developers

will exercise their developments in sequence. We term the first developer as the

Leader and the second developer as the Follower. Both developers will be indifferent

to being the Leader or the Follower if the equilibrium payoff (to be solved by the

model) of the Leader is the same as that of the Follower.

Other types of shocks can certainly be accommodated. For instance, Wang and Zhou (2006) work on

two stochastic processes, one describing demand and another describing the construction costs. Riddiough

(1997) incorporates the probability of regulatory changes into the land option value. Martzoukos and

Trigeorgis (2002), on the other hand, work on mixed jump-diffusion process to describe the impact of rare

events on the value of the underlying assets. However, most real option models use a geometric Brownian

motion to describe the shock probably because it is easier to derive a closed-form solution using this

particular process.

A duopoly, the simplest form of oligopoly, is assumed for the purpose of simplicity. Wang and Zhou

(2006) relax this assumption to an oligopoly and a competitive market, finding that the results are very

similar to a duopoly case.

162 R.N. Lai et al.

The value of the building option for each unit that a developer supplies at the time

of decision-making is equal to the discounted perpetual rent series received after the

rental unit is built (analogous to a constant-growth dividend discount model), that is

Vt;XðÞ¼sup

Et;XðÞerterμðÞδRX tþδðÞ½

rμðÞ

I



;ð3Þ

where ris the risk-free rate, and tis the date when construction commences. The

first term within the inner parentheses in Eq. 3represents the perpetual rent flow

discounted to time t; while the second term represents any costs incurred during

construction and assumed paid in one lump sum at time t. (Note that r−μis the

capitalization rate for a constant growth income stream.)

Applying Ito’s Lemma, the option value before exercising possesses an

instantaneous rate of change of

dV ¼1

2s2X2@2V

@X2þmX@V

@Xþ@V



dt þsX@V

@Xdw:ð4Þ

If the time of valuation is assumed as today (t=0), the payoff function will be

invariant with time, and becomes

VXðÞ¼sup

EXðÞerμðÞδRX tðÞ½

rμðÞ

I



:ð5Þ

As there is no interim payoff from the bare land before construction, the expected

rate of return from holding the development option should be equal to the risk-free

rate according to the risk neutrality argument. Hence, equating the expectation on

Eq. 4to the risk-free rate (we assume a flat yield curve for simplicity),

we obtain an

equation governing the rental flow as

2s2X2d2V

dX 2þmXdV

dX rV ¼0:ð6Þ

In the following section, we will deduce the optimal construction conditions in a

sub-game Nash equilibrium, where the rental rate considered is the ordinary market

equilibrium rate that clears the market. Sticky rates will be imposed later in

“Strategic Exercises with Overbuilding.”

Developers’Game with a Market Clearance Condition

Development Strategies

When a developer begins construction, she/he will pay the construction cost and

receive rent flow when the building is completed. At equilibrium with market

clearance, the rent will always be adjusted until all units are occupied. This means

that all the existing space, denoted as M, is already fully occupied. Therefore, the

Brennan and Schwartz (1980) and Kim, Ramaswamy, and Sundaresan (1993) have shown that constant

interest rate models yield negligible calculation errors. Given this, it is common to assume a flat yield

curve when using a real option model.

Stickiness of Rental Rates and Developers' Option Exercise Strategies 163

incremental demand level in Eq. 1will be equal to the number of newly developed

properties, Q(t), and hence the rental rate determined by demand takes the form

RX tðÞ½¼cX tðÞ1

bQtðÞ;ð7Þ

where c¼a

b. The parameter, b, provides a measure of price elasticity.

Specifically, when the market has only the Leader’s supply Q

, the Leader can

enjoy a rental rate of

RX tðÞ½¼cX tðÞ1

bQL:ð8Þ

When the Follower decides to enter the market by building Q

units of rental space,

the increased supply will drive down the rental rate to

RX tðÞ½¼cX tðÞ1

bQ;ð9Þ

where Q¼QLþQF

. The Follower will experience only one demand function 9,

while the Leader will have both demand functions, first Eq. 8and subsequently Eq. 9.

Thus, the Leader’s strategy embeds that of the Follower. This chain of interactive

decisions constitutes a game between the two developers in equilibrium.

With standard backward induction, we first solve for the Follower’s market

equilibrium construction strategy and then work backward to determine that of the

Leader. We redefine the payoff V(X) in Eq. 5as F(X) for the Follower, and L(X) for

the Leader.

To solve Eq. 5for the Follower, F(X) must satisfy the following boundary

conditions (see Dixit & Pindyck, 1994, for more discussions)

F0ðÞ¼0;ð10aÞ



¼ermðÞdcX e

F1

rmðÞ

I;ð10bÞ

F0Xe



¼ermðÞdc

rmðÞ

;ð10cÞ

where Xe

F, the ‘trigger point,’is the level of demand shock with which the Follower

should start construction (the superscript eis used to denote the demand trigger

under market clearance; owill be used in the case of overbuilding in the next

section). The first boundary condition is the absorbing barrier specifying that the

development option will remain zero forever once the rental rate hits zero. The value-

matching condition (10b) simply states the payoff at the time of exercising the option.

The last is the smooth-pasting condition that ensures an optimal value for F(X)atXe

(also known as the ‘high-contact’condition by Merton, 1973).

and Q

do not need to be identical. For example, Wang and Zhou (2006) allow for developers with

uneven production capacity.

164 R.N. Lai et al.

Proposition 1 In equilibrium with a market clearance condition, the optimal

demand shocks that trigger developments are

L¼1

bQLþermðÞdrmðÞI



;ð11Þ

F¼b

cb1ðÞ

bQþermðÞdrmðÞI



;ð12Þ

where b¼1

2m

s2þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

s21



2þ2r

q, and Xe

L<Xe

Proof See Appendix.▪

Proposition 1 indicates that, in equilibrium with a market clearance condition,

when the demand shock is lower than Xe

L, no developer will invest. When the

demand shock reaches Xe

L, one will develop at once (and become the Leader), while

the other will wait until Xe

Fis reached (and become the Follower). Notice that since

the equilibrium payoffs will be the same for the Leader and the Follower, both

developers are indifferent between these two positions. (In other words, one

developer can be a Leader by chance, while the other will be the Follower.) Should

the demand shock reach Xe

Fbefore any construction occurs, both developers will

start building simultaneously and become joint developers. (The minimal time lag

between the developments, if any, can be neglected). In other words, different

market equilibria can be attained when the incremental demand shock reaches

different thresholds.

Causes for Asymmetric Development Strategies

It is easy to see that both trigger points in Eqs. 11 and 12 are positively affected by

new supply levels (Q

and Q, respectively), and the associated construction costs I.

The interpretation is simple. Developers need to wait for higher demand shocks that

can clear a market when there is a bigger supply, and/or that can generate acceptable

revenue to break even when the construction is more costly (any shock beyond the

trigger point level will bring her/him positive net income).

The optimal demand shocks in Eqs. 11 and 12 show that the Follower’s strategy

to begin construction (when the demand shock Xe

Fis realized) differs from that of the

Leader both in the level of new supply (that is, Qversus Q

) and in the term b

b1ðÞ

.In

fact, by rewriting Eq. 11 into Xe

L¼QL

bc þeðrmÞdrmðÞI

c, we can interpret the first

component as the demand shock that clears new supply, and the second component

as the demand shock that makes the developer break even (which can be called the

“reservation shock”). We can also rewrite Eq. 12 into two components in a similar

way.

Since the term b

b1ðÞ

>1, the Follower will wait longer than the Leader before

starting the development. When the Follower chooses the exercise time, the Leader

Stickiness of Rental Rates and Developers' Option Exercise Strategies 165

is already in the market. As such, the Follower does not need to worry about the

possible preemption of the rival. Hence, the Follower’s decision is simply a standard

option pricing problem that maximizes her/his own option value, and is affected by b

in a similar way as other real options. Transforming Eq. 12 into Xe

F¼1

c11

ðÞ

bQþerμðÞδrμðÞI



, we see that bnegatively affects Xe

F. In other words, a high

b, by affecting the development option value, leads to an early exercise.

In contrast, when the Leader chooses the exercise time, she/he has to consider the

Follower’s optimal exercise strategy. As shown in the Appendix,Xe

Lis solved by

making the Leader indifferent between being a Leader and being a Follower, rather

than by maximizing her/his own option value. Correspondingly, the Leader’sdecision

is based on the relative values of the Leader and the Follower at the moment the

Leader begins construction. With the option value that takes the functional form

LXðÞ¼BX bfor the Leader (versus FXðÞ¼AX bfor the Follower),

the relative

values depend on Aand B, but not the exponent b. The difference in these option

values stems from the construction sequence asymmetry of the two developers, which

contributes to their behavioral disparities in some aspects of the real estate market, as

we will show in Corollary 1 below (and later on in Corollary 2 when the market

clearance condition is absent).

Impact of External Conditions on Strategies with Market Clearance

In this subsection, we first estimate when the development triggering shocks will arrive

so that the developers will start their developments, and then analyze the impacts of

exogenous factors on the development timing of each developer and their development

time lag.

With Dynkin’s Equation (see Øksendal, 1998), the expected times when the

Leader and the Follower start constructing the rental units from a current date t

are



¼ln Xe



ln X t 0

ðÞðÞ

μ1

2σ2and Et



¼ln Xe



ln X t 0

ðÞðÞ

μ1

2σ2;ð13Þ

respectively. Correspondingly, the expected time lag between the two developments is

Ete



¼In Xe



ln Xe



μ1

2σ2:ð14Þ

The comparative static analyses on Et



,Et



and Ete



are reported in Corollary 1.

Corollary 1 In equilibrium with a market clearance condition, the time to start

construction is increasing in market uncertainty, while decreasing in demand

elasticity for both developers. That is,

@Et



@σ>0;@Et



@b<0;@Et



@σ>0;and @Et



@b<0:

As shown in the Appendix, the exponents in both developers’option value functions, βs, are the same.

166 R.N. Lai et al.

However, the magnitudes of the effects differ across developers, and the time lag

between their developments is increasing in market uncertainty, while decreasing in

demand elasticity. That is,

@Ete



@s>0;and @Ete



@b<0:

Proof See Appendix.▪

Corollary 1 indicates that the Leader and the Follower react consistently to the

changes of market uncertainty σand demand elasticity b, but to different extents due

to their asymmetric positions in the sequence of construction.

It is well known that uncertainty increases the real option values. Similarly, our

results, @Et

ðÞ

@σ>0, @Et

ðÞ

@σ>0, and @Ete

ðÞ

@σ>0, suggest that the development options are

more valuable with higher demand uncertainty. Developers will be more cautious

and willing to wait longer to start their developments. The mean exercise time lag

between the Leader and the Follower will also be longer.

Contradicting our analytical results, Grenadier’s(1996) numerical result shows

that the median exercise time lag between the Leader and the Follower should be

shorter when demand is more volatile. Grenadier’s result is surprising and counter-

intuitive because it suggests that developers will accelerate their development

activities when they are less certain about the future demand. Although it is possible

that the difference between Grenadier’s and our results could be due to the fact that we

use different parametric values, we believe that such a difference could be attributed to

the use of mean time (our model) versus median time (Grenadier’s model).

It should be noted that the movement of geometric Brownian motion is based on

percentage change. When applying this motion to a developer’s exercise strategy,

there will be two opposite forces. On one hand, high demand volatility increases the

probability that the demand level will hit the trigger point in the short run. In this

regard, higher demand volatility might shorten a developer’s exercise time. On the

other hand, in the long run, the demand level should reach zero when demand

volatility is sufficiently large (see Øksendal, 1998, for a formal proof of this

proposition). Intuitively, consider a case in which the demand level starts at 100

units. The demand level will go up 10% and move down 10% (in a random order) in

the next two periods. It is easy to see that the demand level will be lower than 100

units at the end of the second period. The demand level will be even lower when the

percentage change (the volatility) is larger than 10%. (If the variance is sufficiently

large and the growth rate is zero, the demand level might approach zero at a certain

point, and there will be no exercise in the future.) This indicates that it will be more

To the best of our knowledge, most papers that analyze option-exercise time issues use the mean.

Grenadier (1996) justifies the use of the median by pointing out that the average time is undefined when

μ<1

2σ2.However, in order to make the expectation infinite, the demand volatility must be extremely

large. When μ=0.02 (a parameter used in Grenadier’s simulation analysis), σmust be greater than 0.20 in

order for μ<1

2σ2.In other words, the standard deviation of the percentage change in demand must be

more than ten times the expected percentage change in demand to make the average time infinite. In most

situations, we should not need to worry about this possibility.

Stickiness of Rental Rates and Developers' Option Exercise Strategies 167

difficult for the demand level to hit the trigger point in the long run if it cannot be

reached in a short period.

Given these two opposite forces, it is clear that the use of mean and median could

result in opposite conclusions. Since the use of median neglects the long tail on the

right side of the distribution (the longer expected time), it is possible that high

demand volatility might decrease the median exercise time between the two

developers. However, because our model prediction is based on analytical results

while Grenadier’s result is based on simulations, we believe that our analysis based

on the mean exercise time should offer a more complete picture than that based on

the median exercise time.

Note that market volatility exerts a more significantly negative effect on the

Follower’s development than on the Leader’s, as indicated by @Ete

ðÞ

@σ>0. This is also

attributed to the asymmetry in the sequence of development. From Eqs. 11 and 12,

we can see that the parameter, b, which is negatively affected by volatility, appears

in the trigger point solution for the Follower but not for the Leader. A higher

volatility increases the value of exercise option while it decreases the value of

development (reflected by a lower bvalue), causing the Follower to further delay

her/his development and lengthen the time lag from the Leader.

In terms of demand elasticity b, the time of development is negatively affected, as

confirmed by our results that @Et

ðÞ

@b<0 and @Et

ðÞ

@b<0. With the market clearance

condition, the market rent has to drop to clear the market if there is an increase in

supply. A lower elasticity means that the rent has to drop relatively more than in case

of higher elasticity in order to clear the market. Therefore, developers will delay the

development exercise when demand elasticity is low. Note that waiting becomes

valuable if the market demand is extremely inelastic (or bis very small). Such effect

is more pronounced in the case of the Follower, which is why the time lag of the two

developments will be widened, as suggested by @Ete

ðÞ

@b<0. The intuition can be seen

from Eqs. A.7 and A.11 in the Appendix.@Xe

@b¼QL

b2cand @Xe

@b¼ bQ

b1ðÞb2cimply that

the Follower needs to clear more new supply than the Leader (Q¼QLþQFversus

). They also imply that the Follower needs to wait for a larger demand increase

(by a factor of b

ðb1Þ) than the Leader before exercising the development option. Both

factors magnify the development time gap between the Leader and the Follower.

Strategic Exercises with Overbuilding

The previous section presents the optimal development strategies under market

equilibrium in which the rental rates are always adjusted so that demand equals

supply. In reality, developers are often reluctant to reduce the rental rate to eliminate

all supply. This is rational if the total rental income can be maximized, albeit with

some vacancy. In this section, we develop a strategic game where oversupply occurs

when developers collude on a sticky rent, and provide explanations on why a

collusion outcome is optimal. As in Wang and Zhou (2000), there are two stages in

the game: the developers choose their optimal time to construct in the first stage, and

choose the optimal collusive rent in the second stage. With backward induction, we

solve for the collusive rent first, and then the optimal times for construction.

168 R.N. Lai et al.

Collusion in Rent

In this section, our model starts with two assumptions for the property rental market.

First, we assume that landlords are able to adjust the rental rate of properties in short

intervals. (This could be an accurate description for commercial properties that have

hundreds of tenants with different maturities.) In addition, we assume that the rent

information can be observed by market players in a short interval. (This assumption

is particularly true for large metropolitan areas where several information gathering

and rental agents are active in the market.) With these two assumptions, the

determination of rental rates in a given market can be viewed as an infinitely

repeated game with perfect information among players (landlords).

Under this framework, the well-known Folk Theorem indicates that the collusion

(or Pareto) outcome (rent level) will dominate the Nash equilibrium as long as all the

players are patient (see, for example, Abreu, Dutta & Smith, 1994, and Aumann,

1959,1960,1981). The intuition behind this theorem is simple. When players can

retaliate at their opponents’current predatory actions in their long-term interactions,

predatory behaviors will not be optimal. Related to this, Friedman (1971) shows that

a Pareto outcome can be supported in a perfect equilibrium of a repeated game if any

player’s deviation from Pareto will induce other players to revert to Nash

equilibrium for the remainder of the game.

Although games in the real world may not be repeated infinitely due to limited

life spans of the players, Fudenberg and Maskin (1986) point out that, from

anecdotal and experimental evidence, a cooperative strategy is a likely outcome as

long as there are a large (even though finite) number of repetitions. Numerous

studies also point out that a collusion is still the favorable outcome even if (1) the

information on the number of repetitions is slightly imperfect to players (see

Neyman, 1999), (2) when players have some “irrationality”because of incomplete

information about players’options or motivations (see Fudenberg & Maskin, 1986

and Kreps, Milgrom, Roberts & Wilson, 1982), or (3) when a slight departure from

the strict Nash equilibria is acceptable by a player who is not perfectly rational

(Radner, 1980). To sum up, the literature on the repeated games seems to indicate

that cooperation is the optimal outcome as long as players’actions can be observed

by other players in the market and players in the market can retaliate promptly if

other players deviate from the agreed strategies. It is fair to say that most property

rental markets we observe exhibit the two characteristics that will ensure a

cooperative game.

Wang and Zhou (2000) apply the Folk Theorem to study the sticky rental rate and

the overbuilding phenomenon. They use a two-stage game framework to prove that

it is optimal for developers to collude on rents as long as they are allowed to adjust

rental rates reasonably promptly (consistent with the framework of an infinite

repeated game) and if the discount rate used by the developers is not substantial (that

is, the present value of future penalties associated with the low perpetual Bertrand

payoff is significant, consistent with the “patience”presumption of the Folk

Theorem).

See Proposition 1 of Wang and Zhou (2000) for the formal proof of (and detailed conditions for) an

optimal sticky rental rate.

Stickiness of Rental Rates and Developers' Option Exercise Strategies 169

The intuition for this phenomenon is very simple. Once a collusive rent is set, one

developer will try to deviate from the collusive rent only if the developer can benefit

from this “cheating”strategy. However, every developer realizes that, if one

developer lowers the rent, all other developers will follow the strategy; and when all

other developers follow the strategy, the rent will drop to the level that the

occupancy rate for all properties in the market reaches to 100%. Since the rental rate

at the full occupancy is below that under the collusion, every developer will suffer.

This means that all developers will receive a lower rent (than the collusive rent) from

the moment on and into perpetuity.

Given this, before any developer decides to cheat, she/he has to compare her/his

one time gain by cheating (by reducing the rental rate, where a developer can

exhaust her/his supply and earn a one time better payoff than all other developers in

the market during a short period) with the potential loss when all other developers

retaliate (where all developers will earn a perpetual income that is much lower than

collusive income thereafter). When rents can be adjusted very conveniently, the

temporarily high profit that a developer earns from cheating will quickly evaporate,

while the present value of penalties (lower rents) will be overwhelming when

compared with the benefit of cheating. Given this, Wang and Zhou (2000) argue that

there is no incentive for a developer to cheat by deviating from the collusive rent and

it is possible for a property rental market to exhibit a sticky rental rate.

Since the conditions specified (and arguments advocated) for a sticky rental rate

by Wang and Zhou (2000) seem to conform to at least some commercial markets we

can observe in the real world, our model in the section starts with the assumption

that there is an optimal rent level (and hence, an optimal occupancy rate) in the

market. To accommodate this assumption, we define an effective rental rate as the

product of the sticky rental rate and occupancy level (which is now less than unity), or

Effective Rental Rate ¼Occupancy Rental Rate per Unit of Rented Space:

Given this definition, we can replace the incremental demand function in Eq. 1with

D¼aX tðÞbRt;ð15Þ

where Rtis the sticky rental rate selected by the landlord/developer for each unit of

rental space at time t. This sticky rental rate is achieved to maximize the developers’

collusive revenue, rather than to clear the market.

With the absence of the market clearance condition, the existing space Mhas a

vacancy rate v. Correspondingly, the total market demand equals 1 vðÞMþe

D, and

we have a total occupancy rate of k¼1vðÞMþ~

MþQtðÞ (0≤k≤1). The pro rata effective

rental rate for each rental unit (vacant and occupied) at time tis therefore

RX tðÞ½¼kRt¼1vðÞMþaX tðÞbRt

MþQtðÞ Rt:ð16Þ

Given this, the total revenue for all developers in the market equals total units

supplied multiplied by the effective rental rate, that is

TR Rt



¼kMþQtðÞðÞ

Rt¼1vðÞMþaX tðÞbRt



Rt:ð17Þ

170 R.N. Lai et al.

It should be noted that, for simplicity, we do not explicitly distinguish between

the natural vacancy rate and the excess vacancy rate when discussing the vacancy

rate. However, the existence of the natural vacancy can be easily incorporated into

our model. We can follow Jud and Frew’s(1990) work and define the natural

vacancy rate as a level where landlords will not willingly adjust rents anymore.

With this definition and by assuming demand elasticity is infinite when the vacancy

rate reaches the natural vacancy level,

the effect of the natural vacancy rate can be

reflected in Eq. 17. Since developers will select the optimal rent (and hence the

optimal occupancy level) based on Eq. 17, the assumption that demand elasticity is

infinite when the vacancy rate reaches the natural vacancy level ensures that the

impact of natural vacancy rate on investment decision is implicitly considered in the

optimization process.

Following Wang and Zhou (2000), our model setup implies that the developers of

new units compete with existing vacant units for market demand.

At any given

level of demand shock, X(t), the optimal rigid rental rate, R*, that developers will

stick to is one that maximizes the total revenue function (17). That is

R*¼1

2b1vðÞMþaX tðÞ½:ð18Þ

Correspondingly, the optimal occupancy rate is

k*¼1vðÞMþaX tðÞ

2MþQtðÞ½

;ð19Þ

and the optimal effective rental rate in Eq. 16 becomes

R*XtðÞ½¼

1vðÞMþaX tðÞ

4bMþQtðÞ½:

4ð20Þ

When the Leader enters the market by supplying Q

, all developers should

optimally rent at RL*.

Then, when the Follower enters the market, the new rental

rate should be adjusted. The issue is what level of demand shock can actually

optimize all developers’total revenue and sustain equilibrium (that is, when the

strategic game holds). Certainly, the demand shock trigger points Xe

Land Xe

Ffound

in “Developers’Game with a Market Clearance Condition”in which the resulting

One of the causes of natural vacancy is the matching delay between supply and demand, as mentioned

in Read (1988).

At this flat end of the demand function, renting out any additional unit cannot increase the total rental

income any further because the costs of providing more services by landlords, such as searching and

leasing costs as well as loss of tenants, become very high. In fact, Gabriel and Nothaft (1988) posit that

landlords may strategically hold some vacant units for the purpose of maximizing net rental income, given

imperfect information and rigid lease agreements.

It should be noted that any demand function can be applied in place of the revenue equation as long as

the total revenue function thus generated is concave, and hence ensures a unique maximum.

We continue to follow the notion of Leader and Follower as in “Developers’Game with a Market

Clearance Condition.”

Stickiness of Rental Rates and Developers' Option Exercise Strategies 171

optimal rental rates and occupancy rates have not been considered are no longer

applicable. The new triggers are derived in the following subsection.

Overbuilding

Following the approach adopted in “Developers’Game with a Market Clearance

Condition,”we will deduce the optimal demand shocks for the Follower and the

Leader, Xo

Fand Xo

L, respectively, given the rigid rent in Eq. 18. These will lead to

market situations with the coexistence of an (optimal) vacancy rate, 1 –k*, and total

maximized revenues for all developers (whose supplies include previously vacant

space as well as new supplies).

Proposition 2 In the strategic game with overbuilding, the optimal demand shocks

that trigger developments are

L¼1

aﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4ermðÞdIrmðÞbMþQL

ðÞ

q1vðÞM



;ð21Þ

F¼1

ab2ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4bb2ðÞerμðÞδIrμðÞbMþQðÞþM21vðÞ

qb1ðÞ1vðÞM



ð22Þ

where b¼1

2m

s2þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

s21



2þ2r

q,and Xo

L<Xo

Proof See Appendix.▪

From Eqs. 21 and 22, it is easy to see that the two factors affecting Eqs. 11 and 12

still persist. First, the trigger points are positively affected by the new supply levels,

and Q, respectively. With more new supply, the developers will have incentives

to wait for a larger demand shock before exercising their options in order to repress

the vacancy rate. Second, the trigger points are positively affected by the investment

costs. The developers still need to wait for a larger demand shock to generate

acceptable revenue when the buildings are more costly to construct. Moreover,

similar to the market clearance situation, the construction sequence asymmetry still

exists. The trigger shock functions for the Leader and the Follower differ in both the

levels of new development supply (Qas versus Q

) and the functions of b.

A major difference between this model and the market-clearance model is that,

with rent collusion, the trigger points decrease with more existing demand, measured

by (1−v)M. Given a higher existing demand, from R*¼1

2b1vðÞMþaX tðÞ½in

Eq. 18, we know that the equilibrium collusion rents should be higher for any

demand shock, hence motivating earlier developments. A similar rationale also

applies to Eq. 22. However, despite the complexity of the trigger value formula in

Eq. 22 relative to Eq. 21, it is easy to see that both positive and negative effects from

existing demand coexist on Xo

F, making (1−v)Mless negatively associated with Xo

than with Xo

172 R.N. Lai et al.

Impact of External Conditions on Strategies with Overbuilding

In this subsection, we analyze the impacts of exogenous factors on the degree of

stickiness in the rental rates, and therefore the persistence of overbuilding. On this,

however, we will focus on two parameters: the demand elasticity band the existing

supply M, because the implications of the market uncertainty σare similar to those

reported in Corollary 1.

The demand elasticity is now of greater importance

because of its close relation with the rent stickiness. In addition, given the strategic

game with collusion, the existing supply which does not play a role in the strategic

game with market clearance, will become a crucial factor in the formation of rental

rates, the extent of overbuilding, and hence, the timing of developments. Our

analytical results derived using Proposition 2 is reported in Corollary 2.

Corollary 2 In the strategic game with overbuilding, the time for the Leader to start

her/his construction is increasing in demand elasticity and decreasing in existing

supply; and the corresponding sticky rent is decreasing in demand elasticity and

increasing in existing supply. That is,

@Et



@b>0;@Et



@M<0;@Ro

@b<0 and @Ro

@M>0

In contrast, the time for the Follower to start her/his construction is increasing in

demand elasticity and decreasing in existing supply; and the corresponding sticky

rent is decreasing in both demand elasticity and existing supply. That is,

@Et



@b>0;@Et



@M<0;@Ro

@b<0 and @Ro

@M<0

In addition, the time lag between their developments is decreasing in both demand

elasticity and existing supply. That is,

@Eto



@b<0 and @Eto



@M<0:

Proof See Appendix.▪

In terms of demand elasticity b, the insight from Corollary 2 is interesting and

rich. First, contradicting the results derived in “Developers’Game with a Market

Clearance Condition,”we find that @Et

ðÞ

@b>0 and @Et

ðÞ

@b>0. In other words, if

developers are allowed to collude on rents, they will accelerate their exercises

whenever demand elasticity is low. The intuition is simple. From Eq. 20, we see that

@R*

@b<0 for any demand shock. That is, a lower demand elasticity will increase the

effective rental rate and hence the development revenue. This is confirmed by @Ro

@b<

0 and @Ro

@b<0. In other words, a lower demand elasticity makes developments more

attractive, leading to earlier exercises of development options. Therefore, as opposed

See the Appendix for our proof on the derivatives of the developers’expected times to start construction

and their construction time lag with respect to market volatility, σ.

Stickiness of Rental Rates and Developers' Option Exercise Strategies 173

to the market clearance condition in “Developers’Game with a Market Clearance

Condition”where market rent has to drop to clear excess supply, when rent collusion

is possible, developers can adjust rent (and correspondingly the vacancy rate) to a

level that optimizes total revenue.

It should also be noted that the demand elasticity exerts different degrees of

influence on the two developers. The result, @Eto

ðÞ

@b<0, indicates that when demand

becomes less elastic, the Leader will speed up construction at a faster pace than the

Follower. As a consequence, the time lag of the two developments will be enlarged.

The mathematical interpretation of this result is as follows. Comparing @Xo

@band @Xo

@bin

Eqs. A.20 and A.26 in the Appendix, we see that the existing demand (1 −v)Mis

embedded within the effect of demand elasticity on the Follower’s trigger point. As

discussed earlier, the existing demand has a positive effect on Xo

Fbut not on Xo

L.Asa

result, the sensitivity of the Follower’s trigger point to demand elasticity is reduced,

leading to @Xo

@b<@Xo

@b, and hence @Eto

ðÞ

@b<0.

Our result can help us understand better the market structure for different types of

property markets. In property markets with high demand elasticity (such as office

and retail), the construction decisions should not differ much whether developers

collude or not. In other words, if we assume that owners of class A office buildings

and super-regional malls find it easier to collude on rents than the owners of class C

office buildings and neighborhood malls, we would expect that the overall vacancy

rates between the high quality and low quality properties in these two types of

markets would not differ much.

On the other hand, the story could be different for property markets with a low

demand elasticity (such as hotel and storage). Given this, if we assume that

developers can collude on rents for better constructed hotels and storage facilities,

but not for mediocre hotels and storage facilities, holding everything else constant,

we would expect to see more developments on the high end of these two property

markets than the low end of the markets. We suggest that an empirical validation on

this proposition is warranted.

In terms of existing supply M, the results also generate important implications.

First, the derivations of @Ro

@Mand @Ro

@Min Eqs. A.23 and A.29 in the Appendix show that

the rents can be affected by Min two opposite ways. On one hand, holding the

existing vacancy rate constant, a bigger Mcorresponds to a bigger existing demand,

thus leading to a higher rent. On the other hand, a bigger Minduces earlier exercises,

hence reduces the equilibrium demand shock triggers, and leads to lower rents. The

result @Ro

@M>0 indicates that the former effect dominates the latter for the Leader, while

@Ro

@M<0 indicates the latter effect dominates the former for the Follower.

Moreover, we have @Et

ðÞ

@M<0 and @Et

ðÞ

@M<0. When there are already many existing

units in the market, there is an incentive for developers to increase the speed of

development. A possible explanation is that the total vacancy rate in the market (as a

weighted average of the vacancy rate of the existing space and the vacancy rate of

the new space) will be less sensitive to the vacancy rate of the new space if there is a

large amount of existing supply. (Under this circumstance, the weight of the new

supply will be smaller when compared with the weight of the existing supply). In

other words, developers have more incentives to build as long as the consequence of

the overbuilding can be shared by many other developers in the market. This finding

might shed light on the persistent excess vacancies in property markets, a puzzle in

174 R.N. Lai et al.

real estate research for decades. Our prediction is also consistent with the prediction

from Wang and Zhou (2000)’s static model in that developers might be able increase

their profits by offering more supplies if the existing supplies are large enough.

The result that @Eto

ðÞ

@M<0 indicates that the effect of existing supply on new

development is stronger for the Follower than for the Leader. That is, the Follower

will speed up development more actively than the Leader. This result is intuitively

simple. Since new supplies share the market with existing capacity, after entering the

market, the Leader’s new supply becomes existing inventory when the Follower

makes a decision. With a larger existing inventory, it is not surprising to see that the

speed of exercise is faster for the Follower than for the Leader.

The implication of this finding on real estate markets is interesting. Holding other

things constant, it implies that developers will have more incentives to build when

the market size is large, but not if the market size is small. If this implication is

correct, we should observe that large markets would constantly exhibit a high level

of excess vacancy. This should be valid because developers will have the incentives

to capture all the development opportunities if others will share the consequence

with them when things go bad. On the other hand, developers in a market with small

inventory will be more careful in their construction decisions because they will have

to bear the consequences on their own. Given this, we should observe that in large

markets, the vacancy rates will be consistently higher than their natural vacancy

rates. In the contrast, for a small market, there is a chance that the vacancy rate will

be low or around its natural vacancy rate. This is a proposition that can be

empirically tested.

Market Clearance versus Overbuilding

In this section, we compare the construction strategies under the market clearance

condition to that in a market with sticky rents. Intuitively, we expect developments

to begin earlier if there are sticky rents, which is confirmed by the following

corollary.

Corollary 3 In equilibrium, a development under a sticky rental rate condition

starts earlier than under a market clearance condition, as shown by

<Et

and Et



<Et



;

and the time disparity is enlarged with an already abundant existing supply or a low

demand elasticity, as shown by

@Et



Et



@M>0;@Et



Et



@M>0;@Et



Et



@b<0 and

@Et



Et



@b<0:

Stickiness of Rental Rates and Developers' Option Exercise Strategies 175

Proof See Appendix.▪

The results Et

<Et

and Et



<Et

in Corollary 3 confirm our

expectation that when developers collude on rents (which simultaneously optimizing

their profits), they are able to start the construction earlier than if they have to

ascertain full occupancy. This is true because the rental level at which the developers

collude provides the highest total revenue to them when compared with the rental

levels at other occupancy levels.

Corollary 3 also indicates that the construction time disparity is negatively

affected by the demand elasticity b. This is true because the trigger shocks are

decreasing in bunder the market clearance condition, but are increasing in bunder

the rent collusion condition. This outcome is not surprising. Any demand loss due to

an increase in the sticky rent is less significant when demand is less elastic. Given

this, the cost of collusion is reduced, while the incentive to build early increases.

Furthermore, the results that the disparities in construction are positively influenced

by the existing space Mconfirm our earlier findings. A larger existing supply is more

likely to encourage property developments when developers can collude on rents.

The mathematical interpretation is simple. The trigger shocks are decreasing in M

under the rent collusion condition, but are unaffected by Munder the market

clearance condition.

Conclusion

The persistence of overbuilding in real estate market cycles is rationalized in our real

options model where developers do not reduce rent to reach a full occupancy level. In

particular, we derive the optimal rental rate at which developers collude, the optimal

occupancy rate, and the critical values of demand shocks that trigger developments in

equilibrium. Our sticky-rent based real options model performs better than the

traditional market-clearance based real options models (such as Grenadier, 1996, and

Wang & Zhou, 2006) in explaining the overbuilding puzzle. Our real-options based

model also performs better than the static sticky rent model (by Wang & Zhou,

2000) in its ability to characterize dynamic developments and developers’behaviors.

Our conclusion on demand volatility contradicts that reported by Grenadier (1996).

We find that a high demand volatility actually delays the speed of developments

(rather than increases the speed of developments). We also find that, in cooperative

strategic games where developers collude on rental rates, a high demand elasticity

will discourage development. This finding contradicts the result reported in market-

clearance models such as Wang and Zhou (2006). In addition, a large existing supply

makes the real estate market insensitive to the new supply, which in turn increases

the speed of developments.

The impacts of sticky rental rates on investment decisions are not confined to real

estate markets. In fact, the models derived in this study can be applied to other types

of markets. Specifically, our model demonstrates that the investment decisions of

any industry with few suppliers (where collusion on the output price can occur more

easily) could differ from other industries where the prices of products adjust

promptly to the level of new supply.

176 R.N. Lai et al.

Acknowledgements We thank João Duque, John Erickson, David Geltner, John Glascock, Robert Van

Order, and participants of the 2005 Cambridge-Maastricht Symposium for their helpful comments. The

authors are solely responsible for any remaining errors in the paper.

Appendix

Proof of Proposition 1 We follow the standard procedure by guessing that the

Follower’s solution to Eq. 5takes the form FXðÞ¼AX b. Equating FX



condition 10b and F0Xe



¼bAX



b1to condition 10c, we derive the value of the

development opportunity at any level of demand shock as

FXðÞ¼

erμðÞδ

rμðÞ

cX e

F1



I



if X<Xe

erμðÞδ

rμðÞ

cX 1



Iif XXe

:ðA:1Þ

where

F¼b

cb1ðÞ

bQþermðÞdrmðÞI



is the trigger point that prompts the Follower to start construction. Substituting the

differentials of FXðÞ¼AX binto Eq. 6, we have

2s2b2þm1

2s2



br¼0;

which generates

b¼1

2m

s2þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

s21



þ2r

To obtain the Leader’s strategy, notice that she/he enjoys perpetual rental earning of

L1XðÞ¼

ermðÞd

rmðÞ

cX 1

bQL



IðA:2Þ

before the Follower offers any rental units. By the time the Follower supplies rental

units at some time unknown to the Leader because Xe

Fis reached, the Leader will

lose the higher rent of RX tðÞ½¼cX tðÞ1

bQLand instead earn a diluted perpetual

rent of RX



¼cX e

F1

bQfrom then on (recall that Q¼QLþQF). The change in

rent, L

(X), will behave similarly as in Eq. 6, that is,

2s2X2d2L2

dX2þmXdL2

dX rL2¼0;

and the change in perpetual rental flow of

ermðÞd

rmðÞ

cX e

F1

bQcX tðÞ1

bQL



Stickiness of Rental Rates and Developers' Option Exercise Strategies 177

becomes the boundary condition to the partial differential equation. With similar

approach adopted for Eq. A.1, the solution becomes

L2XðÞ¼

ermðÞd

rmðÞ

cX e

F1

bQcX 1

bQL





:ðA:3Þ

Combining both rental values in Eqs. A.2 and A.3, the value of the construction

opportunity of the Leader at any demand shock level follows

LXðÞ¼

erμðÞδ

rμðÞ cX 1

bQL



1X





þcX e

F1







Iif X<Xe

erμðÞδ

rμðÞ

cX 1



Iif XXe

:ðA:4Þ

Note that the second equation is identical to that of the Follower in Eq. A.1, which

indicates simultaneous developments.

The best time for the Leader to start construction can be determined as follows. In

market equilibrium, a developer is indifferent to building now or building later (that

is, being a Leader or a Follower) if and only if the values of her/his construction

opportunity at both times are identical to her/his counterpart, that is, L(X)=F(X).

Otherwise, both developers will try to preempt the market in order to occupy the

market share as early as possible. Following this reasoning, there exists an Xe

0;Xe



at which one developer would start construction and become the Leader.

This is obtained by setting LX



¼FX



, which generates

L¼1

bQLþermðÞdrmðÞI



:ðA:5Þ

Given that b

b1>1 is always true, it is easy to see that Xe

L<Xe

F.▪

Proof of Corollary 1 From Eq. 11 we know that

@Xe

@σ¼0;and ðA:6Þ

@Xe

@b¼QL

b2c<0:ðA:7Þ

Hence, from Eq. 13, we derive

@Et



@σ¼σln X e



ln X t0

ðÞðÞ



μ1

2σ2



2>0;and ðA:8Þ

@Et



@b¼1

Lμ1

2σ2



@Xe

@b<0:ðA:9Þ

178 R.N. Lai et al.

In addition, from Eq. 12, we have

@Xe

@s¼ Xe

bb1ðÞ

@s>0;ðA:10Þ

where @b

@s¼2m

s31ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þ

m2rmðÞ

s21



þ2r

5<0;and,

@Xe

@b¼ bQ

b1ðÞb2c<0;

ðA:11Þ

Hence, from Eq. 13, we derive

@Et



@σ¼σln X e



ln X t0

ðÞðÞ



μ1

2σ2



2þ1

Fμ1

2σ2



@Xe

@σ>0;and ðA:12Þ

@Et



@b¼1

Fμ1

2σ2



@Xe

@b<0:ðA:13Þ

Finally, from Eq. 14, we have

@Ete



@σ¼

@Xe

@b@b

@σ

Fμ1

2σ2



þσln Xe



ln Xe



μ1

2σ2



2>0;ðA:14Þ

given @b

@σ<0and @Xe

@b<0, which can be easily seen from Xe

F¼1

c11

ðÞ

½1

bQþerμðÞδrμðÞI,and

@Ete



@b¼ berμðÞδrμðÞIQF

b2c2b1ðÞμ1

2σ2



FXe

<0:ðA:15Þ

▪

Proof of Proposition 2 To obtain the optimal strategies of the developers, first notice

that the optimal effective rental rate for the Follower from expression (20)is

Stickiness of Rental Rates and Developers' Option Exercise Strategies 179

RF*¼1vðÞMþaX o

½

4bMþQðÞ

, where Q¼QLþQF. With the same approach in Proposition 1,

the boundary conditions in deducing optimal Xo

Fare modified to



¼ermðÞd1vðÞMþaX o



4brmðÞMþQðÞ

I;ðA:16aÞ

F0Xo



¼ermðÞda1vðÞMþaX o



2brmðÞMþQðÞ

:ðA:16bÞ

With a guess that the Follower’s solution to Eq. 5takes the form FXðÞ¼AX band

subject to conditions A.16a and A.16b, the value of the development opportunity

becomes

FXðÞ¼

erμðÞδ1vðÞMþaX o



4brμðÞMþQðÞ

I



if X<Xo

erμðÞδ1vðÞMþaX½

4brμðÞMþQðÞ

Iif XXo

ðA:17Þ

and the corresponding optimal trigger point Xo

Fis

F¼1

ab2ðÞ

4bb2ðÞerμðÞδIrμðÞbMþQðÞþM21vðÞ

qb1ðÞ1vðÞM



Substituting into Eq. 6for FXðÞ¼AX b, we find that bis identical to that in the case

of market clearance situation (b>2 is a necessary condition for the equation to be

valid).

To obtain the optimal strategy of the Leader, notice from Eq. 18 that before the

Follower enters the market, the Leader (and all other current participants in the

market) should be requesting an optimal rigid rate of RL*¼1

2b1vðÞMþaX o



, so that

the corresponding effective rental rate is RL*¼kL*RL*¼1vðÞMþaXo

½

4bMþQL

ðÞ

. Following the

approach adopted in “Developers’Game with a Market Clearance Condition,”the

Leader’s profit function is L

where

L1¼ermðÞd

rmðÞ

1vðÞMþaX o



4bMþQL

ðÞ

I;

and

L2¼ermðÞd

rmðÞ

1vðÞMþaX o



4bMþQðÞ

1vðÞMþaX½

4bMþQL

ðÞ



(again Q¼QLþQF), when the Follower has entered the market. Note that the term

in the parentheses of L

must be negative because the profit without the Follower

must be higher than that with her/his existence. That is, there must be incentive for

180 R.N. Lai et al.

developers to enter the market first to become the Leader. There exists a unique

demand shock Xo

Lthat results in developers being indifferent to being Leader or

Follower. This is obtained by setting LX



¼FX



, which generates

L¼1

aﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4ermðÞdIrmðÞbMþQL

ðÞ

q1vðÞM



:ðA:18Þ

To show the existence of Xo

L20;Xo



, we make use of the fact that C−D>0if

C2D2¼CþDðÞCDðÞ>0givenC> 0 and D> 0 for some arbitrary constants

Cand D. In other words, combining Xo

Land Xo

Fand rearranging, we have



2Xo



2¼1

a2b2ðÞ

2"2M1vðÞb1ðÞ

"ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

M21vðÞ

qþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

b2ðÞ

4GL

b1ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

bb2ðÞGþM21vðÞ

q##þbGb2ðÞGL

½

b2ðÞ;

where G¼4ermðÞdIrmðÞbMþQðÞand GL¼4ermðÞdIrmðÞbMþQL

ðÞ.Itis

obvious that G>G

, which implies the last term on the right-hand side is positive.

Thus, as long as we confirm that ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

M21vðÞ

qþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

b2ðÞ

4GL

b1ðÞ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

bb2ðÞGþM21vðÞ



is positive, we

can show that Xo



2Xo



2>0, which leads to Xo

F>Xo

L. We again apply the same

concept that C−D>0ifC2D2¼CþDðÞCDðÞ>0givenC> 0 and D>0,

and obtain

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

M21vðÞ

qþﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

b2ðÞ

4GL

b1ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

bb2ðÞGþM21vðÞ



¼b2ðÞM2"ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

b2ðÞ

216erμðÞδrμðÞb1vðÞ

M1þQL



b1ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

b24erμðÞδrμðÞbI

M1þQ



s#þb2ðÞ

4GL

b1ðÞ

2>0;

and this is true given that I(M+Q

)(inG

) is far greater than I

M1þQL



and

M1þQ



(in the big parenthesis). This means that Xo



2Xo



2>0, and hence

that Xo

F>Xo

Comparing the two demand shock formulae, we see that one main difference lies

on the presence of bin Xo

Fbut not Xo

L. It is therefore worth to examine the effect of b

on Xo

F. Taking the derivative, we have

@Xo

@b¼ Xo

b2ðÞ

M1vðÞ

ab2ðÞ

1b1ðÞ

1vðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

b2ðÞbM2

BþM2



21vðÞ

A;ðA:19Þ

Stickiness of Rental Rates and Developers' Option Exercise Strategies 181

where B¼4be rmðÞdrmðÞbI M þQðÞ. Given r,μand vfar smaller than 1, and

hence M2

B¼1

4be rμðÞδrμðÞbI

M1þQ

ðÞ

will be very large, the second term in the big parenthesis

will be very small and we have @Xo

@b<0. ▪

Proof of Corollary 2 From Eq. 21, we have

@Xo

@b¼1

ab ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ermðÞdIrmðÞbMþQL

ðÞ



>0;ðA:20Þ

and given r,μand vfar smaller than 1, we have

@Xo

@M¼1

aﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ermðÞdrmðÞbI

MþQL

ðÞ

s1vðÞ

<0;ðA:21Þ

as long as M+Q

are larger than I, which is usually true. Then, from Eq. 18, we have

@Ro

@b¼ 1

2b2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

erμðÞδIrμðÞbMþQL

ðÞ



<0;ðA:22Þ

@Ro

@M¼1

MþQL

ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ermðÞdIrmðÞbMþQL

ðÞ



>0:ðA:23Þ

Furthermore, we have

@Xo

@σ¼0;ðA:24Þ

@Ro

@s¼a

@Xo

@s¼0;ðA:25Þ

Then, from definition (13), we can derive

@Et



@b¼1

Lμ1

2σ2



@Xo

@b>0;

@Et



@M¼1

Lμ1

2σ2



@Xo

@M<0;

and

@Et



@σ¼σln X o



ln X t 0

ðÞðÞ



μ1

2σ2



2þ1

Lμ1

2σ2



@Xo

@σ>0:

182 R.N. Lai et al.

In the Follower’s case, for simplicity, we assume Y¼2berμðÞδIrμðÞbMþQðÞ

and Z¼1

1vðÞ

2bb2ðÞermðÞdIrmðÞb



, and from Eqs. 22 and 18,

@Xo

@b¼Y

ab ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2b2ðÞYþ1vðÞ

2M2

q>0;ðA:26Þ

@Ro

@b¼1

2b2b2ðÞ

b1vðÞ

Yb2ðÞþ1vðÞ

2M2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2b2ðÞYþ1vðÞ

2M2

A<0; ðA:27Þ

In addition, we have

@Xo

@M¼1vðÞ

ab2ðÞ

Zþ1vðÞM

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

21vðÞZMþQðÞþ1vðÞ

2M2

qb1ðÞ

A<0;ðA:28Þ

which is proved as follows. As the sign of @Xo

@Mdepends only on the sign of the terms

in the big parentheses given b> 2 and r,μand vfar smaller than 1, we rewrite these

terms as Zþ1vðÞM½b1ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

21vðÞZMþQðÞþ1vðÞ

2M2



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

21vðÞZMþQðÞþ1vðÞ

2M2

p. Applying the concept that C−D<0

if C2D2¼CþDðÞCDðÞ<0 given C> 0 and D> 0 for some arbitrary

constants Cand D, we have

Zþ1vðÞMðÞ

2b1ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

21vðÞZMþQðÞþ1vðÞ

2M2



¼1vðÞMþZðÞ

2b1ðÞ

21vðÞ

2M2þ2Z1vðÞMþQðÞ



¼1vðÞ

2M2þ2Z1vðÞM



1b1ðÞ



þZ2b1ðÞ

22Z1vðÞQ



<0;

given that b>2andZ<2(b−1)(1 −ν)Q(due to the small r,μand v). This result

implies that Zþ1vðÞM½b1ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

21vðÞZMþQðÞþ1vðÞ

2M2

q<0and hence

@Xo

@M<0.

Also, from Eq. 18, we have

@Ro

@M¼1vðÞ

2bb2ðÞ

Zþ1vðÞM

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

21vðÞZMþQðÞþ1vðÞ

2M2

q1

A<0;ðA:29Þ

which can be proved with a similar approach that shows

Zþ1vðÞM<ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

21vðÞZMþQðÞþ1vðÞ

2M2

Stickiness of Rental Rates and Developers' Option Exercise Strategies 183

Then, from definition (13), we can derive

@Et



@b¼1

Fm1

2s2



@Xo

@b>0;

and

@Et



@M¼1

Fm1

2s2



@Xo

@M<0:

To facilitate discussion on the time lag between the two developments, we assume

that A¼ermðÞdrmðÞbI. Note that given r,μand vfar smaller than 1, while M

very large and larger than I, it must be that A<<M.

From Eqs. 21 and 22, we have

FXo

L¼1vðÞMþb2ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4AMþQL

ðÞ



þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4b2ðÞbAMþQðÞþ1vðÞ

2M2

ab2ðÞ :

Correspondingly,

@Xo

FXo



@b¼A

ab ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MþQL

rþ2b

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4b2ðÞbA

MþQðÞ

þ1vðÞ

1þQ



<0;

ðA:30Þ

because A<<M+Q

makes the first (negative) term significantly dominate the second

(positive) term in the outermost parentheses on the right-hand side.

Then from the definition in Eq. 14, we have

@Eto



@b¼1

m1

2s2



FXo

@Xo

@b@Xo



FþXo

LXo



@Xo



<0;ðA:31Þ

given @Xo

@b@Xo

@b¼@Xo

FXo

ðÞ

@b<0 (see result A.30), Xo

LXo

F<0 and @Xo

@b>0:In

addition,

@Xo

FXo



@M¼1

ab2ðÞ

1vðÞþ 2b2ðÞbA

Mþ1vðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4b2ðÞbA

M1þQ



þ1vðÞ

qb2ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MþQL

ðA:32Þ

184 R.N. Lai et al.

To study the sign of @Xo

FXo

ðÞ

@M, notice that the term in the large parenthesis on the

right hand side of the equation

1vðÞþ 2b2ðÞbA

Mþ1vðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4b2ðÞbA

M1þQ



þ1vðÞ

qb2ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MþQL

<1vðÞþ

2b2ðÞbA

Mþ1vðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2b2ðÞbA

Mþ1vðÞ

qb2ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MþQL

¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2b2ðÞfb

Mþ1vðÞ



1vþb2ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MþQL

With the same concept that C−D<0ifC2D2¼CþDðÞCDðÞ<0, and

given A<<M, the last expression can be proved to be negative, which implies that

@Xo

FXo

ðÞ

@M<0. Since from Eq. 14 Eto



and Xo

FXo



in general move in the same

directions as Mchanges, we have @Eto

ðÞ

@M<0.

Finally,

@Eto



@σ¼σln Xo



ln Xo



μ1

2σ2



2þ

@Xo

@σXo

@Xo

@σ

μ1

2σ2



FXo

>0;ðA:33Þ

because @Xo

@s>0and@Xo

@s¼0. ▪

Proof of Corollary 3 Again, to facilitate discussion, we assume that A¼erμðÞδ

rμðÞbI, and we have A<<Mgiven r,μand vfar smaller than 1 while Mvery large

and larger than I.

Following Eqs. 11 and 21, we have

LXo

L¼1

aM1vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

M1þQL



þQLþA

>0;

given A<<M.

FXo

F¼

b2ðÞbQþAðÞþb1ðÞ

21vðÞM



b1ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4b2ðÞbAMþQðÞþ1vðÞ

2M2



ab2ðÞb1ðÞ >0;

which can be proven using similar approach as before. That is, by b2ðÞbQþAðÞþð

b1ðÞ

21vðÞMÞ2b1ðÞ

2ðﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4b2ðÞbAMþQðÞþ1vðÞ

2M2

qÞ2

>0:These results in-

Stickiness of Rental Rates and Developers' Option Exercise Strategies 185

dicate that Xe

LXo

L>1, and Xe

FXo

F>1. Then with Eq. 13, we can derive



Et

¼ln Xe



ln Xo



μ1

2σ2¼ln Xe

LXo



μ1

2σ2>0;

and



Et



¼ln Xe



ln Xo



μ1

2σ2¼ln Xe

FXo



μ1

2σ2>0:

Given A

Mvery small, we can also derive

@Xe

LXo



@b¼1

ab ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

AMþQL

ðÞ

p1þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MþQL

<0;ðA:34Þ

and

@Xe

LXo



@M¼1

a1vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MþQL

>0;ðA:35Þ

@Xe

FXo



@b¼1

ab b1ðÞ

bA12b1ðÞMþQðÞ

4b2ðÞbAMþQðÞþ1vðÞ

2M2

A<0;ðA:36Þ

because it can be shown that 2b1ðÞMþQðÞ

4b2ðÞbAMþQðÞþ1vðÞ

2M2

q>0;

and

@Xe

FXo



@M¼1

ab2ðÞ

b1ðÞ1vðÞ

2b2ðÞbA

M2þ1vðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4b2ðÞbA

M1þQ



þ1vðÞ

A>0:

ðA:37Þ

Again from Eq. 13,Et



Et

and Xe

LXo

Lshould in general move in the

same directions when bor Mchanges, and so do Et



Et



and Xe

FXo

F.▪

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Intercity Rent Differentials in the U.S. Housing Market 2000: Understanding Rent Variations as a Sociological Phenomenon

Article

Full-text available

Oct 2009
J URBAN AFF

This study extends the intercity rent differentials investigation by Gilderbloom and Appelbaum (1988) in relatively independent housing markets to see how it can be replicated using U.S. census data from the year 2000 against the 1970 and 1980 models with the addition of several new variables to measure its impact on intercity rents. We find that region, race, and climate no longer explain rent differentials in 2000 as it did in the 1980 research, while affirming that a large percentage of old houses and small mom-and-pop landlords causes rents to fall. We find that both the cost of homeownership and the level of household income remain critical factors in explaining the level of median rent across cities. We also find a strong correlation between cities with extensive anti-war activity in the late 1960s and same sex households having higher rents, although more research needs to be done before we argue a causal relationship. We contend that sociology needs to be put back into the equation in order to understand how rents vary from city to city. Our explanation of rent variations adds a social dimension that most other researches miss. We also show how the amount of explanatory power is increased significantly by adding in a sociological dimension. The factors that determine median rent levels across cities have been a topic of much debate and intense research in the literature on urban housing markets and urban public policy.

International Real Estate Review

Article

Full-text available

Dec 2018

Kwabena Mintah

Studies have demonstrated the potential of real options analysis (ROA) in property development decision-making. However, practitioners have yet to accept, adopt and integrate ROA in property development decision-making in Australia. This paper therefore investigates how Australian residential property developers manage uncertainties and risks, examines flexibility as a risk management tool, and evaluates the receptiveness and acceptance of ROA for decision making. Data are collected through face-to-face semi-structured interviews with twelve participants, and analysed by using thematic analysis. The results indicate that a discount rate is insufficient for managing uncertainties and risks; rather, contingency is used. Receptiveness and acceptance of the RO theory are mixed due to lack of unanimity among responses. Some participants are positive about flexibility, while others are dismissive. Beyond quantitative ROA models, the findings suggest that practitioners are receptive to ROA, but concerns remain over adoption. Flexibility cases executed by some participants in practice indicate that practitioners are subconsciously using ROA. Therefore, it is possible that acceptance and adoption could be achieved in the future. Evidence of the use of contingency as a risk management tool challenges the long-held notions of risk-return relationships in property development and investment. This is initial evidence of qualitative research on ROA in practice within Australian property developments.

International Real Estate Review

Article

Dec 2011

This paper develops a stylized model to provide a rational explanation for the boom-and-bust price movement pattern that we frequently observe in the real world. Our stylized model indicates that there are three conditions to form a boom-and- bust price pattern in a community: a move-in of high income residents, wide income gap between new and existing residents, and supply process that leads to an inventory buildup. It seems that, based on these three conditions, China is more likely to experience a boom-and-bust price movement pattern than a developed country with a more mature and less vibrant economy.

Real Options Analysis in Residential Property Development Decision-Making in Australia: Perspectives of Executives

Article

Full-text available

Apr 2021

Kwabena Mintah

Unbundling curbside parking costs from housing prices

Article

Jan 2019
J ECON GEOGR

This article empirically shows that at least some of the costs of curbside parking spaces are capitalized in housing prices even though these parking spaces are not formally bundled with housing units. Making use of Istanbul’s transition from free and informal curbside parking to paid and formal, we find that housing prices decreased substantially in response to city operated curbside parking, whereas rents remained statistically the same or, at best, decreased only very slightly.

Contemporary neighborhood housing dynamics in a mid-sized US city: The policy consequences of mismeasuring the dependent variable

Article

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Mar 2015

This paper analyzes median assessed residential property values using three different operationalizations of a tract-level dependent variable for a mid-sized US city: Louisville, Kentucky. We estimated the impacts of accessibility, socio-demographics, and housing characteristics as well as three policy interventions (HOPE VI, historic preservation districts, and university-community partnerships) on median assessed values and changes to them over the 2000-2006 housing bubble. Our interpretation of models employing the three different operationalizations leads to different conclusions about neighborhood health and the efficacy of policy programs. Conventional operationalizations employed by advocates, such as looking at medians or raw dollar changes in median values, are likely to find that policy interventions have less of an impact compared to measuring recent percent changes in property values. Thus, we provide a methodological contribution that shows that percent changes should accompany traditional analysis in capturing the effects of contemporary policy interventions. Mismeasuring neighborhood housing markets has played a role in prematurely concluding that targeted policy programs in neighborhoods are ineffective. Based on our analysis, we invite academics and policy-makers to rethink contemporary neighborhood housing dynamics.

Quantifying and Explaining Stickiness in Housing Rents: A Turkish Case Study with Micro-Level Data

Article

Sep 2014

Using a national panel of housing units, this paper documents that the rate of nominal rigidity in housing rents is high in Turkey between 2008 and 2011. We find that, on average, 31.5% of the rents did not change from year to year in nominal terms. We then ask if the incidence of nominal rigidity depends on the turnover status of the housing unit. We show that 35.4% of the nonturnover units had rigid rents, while for only 17.1% of the turnover units rents did not change. We also present evidence that grid pricing is associated with more than half of the nominal rigidity in housing rents in our sample. The household- and individual-level determinants of the nominal rigidity in rents and turnover status are also investigated using the micro-level details available in our dataset. We document that, relative to the low-income tenants, high-income tenants are less likely to have rigid rents and they are also less likely to change units frequently. This finding suggests that search and moving costs impose frictions that amplify the opportunity costs of high-income tenants; thus, they are more likely to agree on reasonable rent increases for the purpose of saving time and reducing emotional stress.

Is Intrinsic Energy Efficiency Reflected in the Pricing of UK Office Leases?

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Oct 2012

The aim of this research is to investigate whether the intrinsic energy efficiency rating of an office building has a significant impact on its rental value. Using a sample of 817 office buildings with Energy Performance Certificate (EPCs), we investigate whether a pricing differential can be identified depending on the energy rating. While previous analyses of this topic have typically relied on appraisal-based and/or asking rent data, our dataset contains actual contract rents as well as information on lease terms. The results indicate a significant rental premium for energy efficient buildings. However, we find that this premium appears to be mainly driven by the youngest cohort of state-of-the-art energy efficient buildings. We also find that tenants of more energy efficient buildings tend to pay a lower service charge but this link appears to be rather weak and limited to newer buildings. Hence, we argue that the information contained in the EPC is still not fully taken into account in the UK commercial property market with the possible exception of both the highest and the lowest EPC ratings.

Pricing of Presale Contracts with Macroeconomic Factors and Stochastic Basis Risk

Article

Oct 2020

This study analyzes the stochastic basis risk (presale price minus spot housing price) of presale housing from the viewpoint of forward pricing. We employ the modified Brownian bridge process to deal with the time-varying volatility and price convergence of presale and existing houses. The finding reveals an asymmetric effect of macroeconomic factors on the presale price in different market conditions. Such results might reflect the investors’ behaviors of overreaction and/or over-pessimism. In addition, the influence of macroeconomic variables is greater on the basis price than on the prices of existing houses, especially in a depressed housing market. Our results emphasize the importance of considering basis risk and the influence of macroeconomic factors when pricing presale houses to manage housing risk.

Micro-Level Stickiness as a Threat to Client Centric Knowledge Transfer among ICT Firms in Malaysian Technology Parks

Chapter

Jan 2012

The study in this chapter was carried out to ascertain whether the national ICT initiative on facilitating knowledge transfer among ICT firms by providing dedicated technology parks throughout Malaysia was client centric. It examines the issue from a micro level perspective of stickiness of knowledge transfer, which includes transfer mechanisms, types of transfer, knowledge barriers, and transfer contexts. The study solicited the opinions of competent informants (i.e., policy makers and competent ICT leaders) through face-to-face interviews in three technology parks located in northern and central Peninsular Malaysia (Kulim High Technology Park; MSC, Cyberjaya; and Selangor Science Park). The results confirm the presence of stickiness in terms of mechanisms specified in the literature.

The Theory of Rational Option Pricing

Article

Full-text available

Mar 1973
Bell J Econ Manag Sci

Robert C. Merton

Author begins by deducing a set of restrictions on option pricing formulas from the assumption that investors prefer more to less. These restrictions are necessary conditions for a formula to be consistent with a rational pricing theory. Attention is given to the problems created when dividends are paid on the underlying common stock and when the terms of the option contract can be changed explicitly by a change in exercise price or implicitly by a shift in the investment or capital structure policy of the firm. Since the deduced restrictions are not sufficient to uniquely determine an option pricing formula, additional assumptions are introduced to examine and extend the seminal Black-Scholes theory of option pricing. Explicit formulas for pricing both call and put options as well as for warrants and the new ″down-and-out″ option are derived. Other results.

Acceptable points in General Cooperative $n$-person Games

Chapter

Full-text available

Jan 1959

Robert John Aumann

Analyzing Convertible Bonds

Article

Full-text available

Nov 1980
J FINANC QUANT ANAL

Investment under Uncertainty

Book

Dec 1994

Universitext

Article

Jan 1998

Bernt Øksendal

The main new feature of the fifth edition is the addition of a new chapter, Chapter 12, on applications to mathematical finance. I found it natural to include this material as another major application of stochastic analysis, in view of the amazing development in this field during the last 10-20 years. Moreover, the close contact between the theoretical achievements and the applications in this area is striking. For example, today very few firms (if any) trade with options without consulting the Black & Scholes formula! The first 11 chapters of the book are not much changed from the previous edition, but I have continued my efforts to improve the presentation through out and correct errors and misprints. Some new exercises have been added. Moreover, to facilitate the use of the book each chapter has been divided into subsections. If one doesn't want (or doesn't have time) to cover all the chapters, then one can compose a course by choosing subsections from the chapters. The chart below indicates what material depends on which sections. Chapter 6 Chapter IO Chapter 12 For example, to cover the first two sections of the new chapter 12 it is recom mended that one (at least) covers Chapters 1-5, Chapter 7 and Section 8.6. VIII Chapter 10, and hence Section 9.1, are necessary additional background for Section 12.3, in particular for the subsection on American options.

The Strategic Exercise of Options: Development Cascades and Overbuilding in Real Estate Markets

Article

Dec 1996

Steven Grenadier

This article develops an equilibrium framework for strategic option exercise games. I focus on a particular example: the timing of real estate development. An analysis of the equilibrium exercise policies of developers provides insights into the forces that shape market behavior. The model isolates the factors that make some markets prone to bursts of concentrated development. The model also provides an explanation for why some markets may experience building booms in the face of declining demand and property values. While such behavior is often regarded as irrational overbuilding, the model provides a rational foundation for such exercise patterns.

Inestment Under Uncertainty

Book

Dec 1994

How should firms decide whether and when to invest in new capital equipment, additions to their workforce, or the development of new products? Why have traditional economic models of investment failed to explain the behavior of investment spending in the United States and other countries? In this book, Avinash Dixit and Robert Pindyck provide the first detailed exposition of a new theoretical approach to the capital investment decisions of firms, stressing the irreversibility of most investment decisions, and the ongoing uncertainty of the economic environment in which these decisions are made. In so doing, they answer important questions about investment decisions and the behavior of investment spending. This new approach to investment recognizes the option value of waiting for better (but never complete) information. It exploits an analogy with the theory of options in financial markets, which permits a much richer dynamic framework than was possible with the traditional theory of investment. The authors present the new theory in a clear and systematic way, and consolidate, synthesize, and extend the various strands of research that have come out of the theory. Their book shows the importance of the theory for understanding investment behavior of firms; develops the implications of this theory for industry dynamics and for government policy concerning investment; and shows how the theory can be applied to specific industries and to a wide variety of business problems.

Rental Housing Markets and the Natural Vacancy Rate

Article

Dec 1988
R Estate Econ

This paper employs new census vacancy rate data to analyze the price-adjustment mechanism for rental housing. The study extends previous research on this topic, which provided conflicting evidence concerning the traditional theory of rental housing market adjustment (see Smith [10], [11]; DeLeeuw and Ekanem [2]; Eubank and Sirmans [4]; and Rosen and Smith [8]). Cross-section and time-series data are pooled to estimate natural vacancy rates for sixteen United States cities for the 1981–85 period. The analysis further explores the determinants of variation in natural vacancy rates across those metropolitan areas.

The Valuation of Options for Alternative Stochastic Processes

Article

Mar 1976
J FINANC ECON

This paper examines the structure of option valuation problems and develops a new technique for their solution. It also introduces several jump and diffusion processes which have not been used in previous models. The technique is applied to these processes to find explicit option valuation formulas, and solutions to some previously unsolved problems involving the pricing of securities with payouts and potential bankruptcy.

Collusive Behavior in Non-Cooperative Epsilon-Equilibria of Oligopolies with Long But Finite Lives

Article

Apr 1980

Roy Radner

In a game of a finite number of repetitions of a Cournot-type model of an industry, if firms are satisfied to get close to (but not necessarily achieve) their optimal responses to other firms' sequential strategies, then in the resulting noncooperative “equilibria” of the sequential market game, (1) if the lifetime of the industry is large compared to the number of firms, there are equilibria corresponding to any given duration of the cartel, whereas (2) if the number of firms is large compared to the industry's lifetime, all equilibria will be close (in some sense) to the competitive equilibrium.

Stickiness of Rental Rates and Developers’ Option Exercise Strategies

Abstract

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