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Development Effort in Speculative Real Estate Competitions
JOURNAL OF HOUSING ECONOMICS ARTICLE NO.
6, 1–15 (1997)
HE970201
Development Effort in Speculative Real Estate Competitions COLIN READ* University of Alaska–Fairbanks, Fairbanks, Alaska 99775 Received June 14, 1996
I model speculative development races when the time at which the development effort becomes profitable is uncertain. I first treat the decision of a speculator who has exclusive rights to the development prize. I then show that development races intensify development effort compared with the collectively optimal level of effort. Development races with identical speculative developers reduce aggregate profits and bring forward development time, whether or not joint development externalities are present. In such races, land use policies that restrict the number of permitted developers can enhance collective returns. 1997 Academic Press
I. INTRODUCTION
Until recently, land use planning has typically been used to ensure compatible use and to prohibit or limit certain ‘‘undersirable’’ uses. It can, however, also be employed to discourage speculative overbuilding, that is, excessive building due to uncertainty. Some recent examples of real estate overbuilding have led local planners to consider methods to prevent or discourage such overbuilding. But there are no formal economic models that predict overbuilding by speculators. I describe a competitive model of speculative development in real estate markets which results in overbuilding. The simple problem of optimal real estate development rates for a sole developer is well understood. Such a developer facing no competitors would choose to time development of a commercial property to coincide with the arrival of a mature market for the property. Williams (1991) explores such optimal real estate development timing and proposes the use of zoning as a policy tool to curb suboptimal development rates. However, he does not explicitly model the suboptimality which occurs when competing developers race to bring their developments to market by using development time as a strategic variable. * I thank the editor of this journal and an anonymous referee for their helpful comments. 1 1051-1377/97 $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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A search through the industrial organization literature yields two distinct strands of literature that could perhaps be adapted to treat the strategic development of land. The first class of models shows that early development can be used to strategically preclude subsequent development by competing speculators. For instance, Eaton and Lipsey (1981, 1980) introduce time as a strategic variable. By investing in nondecaying or slowly-decaying physical capital early, a firm can ‘‘stake out’’ its market to preclude entry of others and hence earn losses in the short run but monopoly profits in the long run once the market matures. In the development analogy, strategic premature development are balanced by exclusive profits later. Helsley and Strange (1994) explore market commitment of this sort in an urban context. Such models of market commitment do not adequately capture the uncertainty of development when competing developers separately determine an optimal development time. In an uncertain environment, two or more speculators could prematurely develop similarly situated parcels of land, perhaps resulting in long run losses for each speculator. So the question still remains—at a given time, what is the return to a developer who may face one or more simultaneous competitors, each vying for a market that will ‘‘arrive’’ at an uncertain date? It is this second literature that my paper addresses. Perhaps the most useful starting point for such an exploration is the patent race literature. In the seminal paper in this field, Loury (1979) modeled uncertain development as a race to capture rewards of innovation when research and development investment is a one shot game. Lee and Wilde (1980) repeat the exercise when research and development requires an ongoing investment until the innovation is realized. Stewart (1983) and Delbono and Denicolo (1991) expand the literature still further by allowing the winner a larger share of the prize but permitting the co-innovators to share in the consolation prize. These authors are primarily interested in the relationship between market structure and performance. I extend this literature to speculative development races and explore how the number of competitors affects individual and aggregate effort. I also model joint development externalities by acknowledging the interdependence of various developers’ investment on other developers’ ultimate success. In this context, a real estate development can be modeled as a collection of developers’ efforts. Each development produces a positive joint development externality by enhancing the attractiveness of the overall development, and speeds the date at which the entire development becomes viable. I argue that there may be some need to use zoning and land use planning to control speculative competition because such competition can be economically wasteful. As an example of jointness in production, consider a mall development complex such as Buckland Hills north of Hartford, Connecticut. This com-
DEVELOPMENT EFFORT IN COMPETITIONS
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plex is actually a series of small malls in close proximity to each other. If there are n developers of these mini-malls and each of these mini-malls has a chance of becoming the preeminent mini-mall once the market matures, how will speculators compete to bring the overall development to market? How will the level of development effort differ with the number of developers allowed to vie for the grand prize of the preeminent minimall? How will the level of effort change as the size of the consolation prizes change? Will such competition produce economic waste? It is these questions that I address in this paper. I find that a sole developer without fear of competition may use exclusive development rights to delay production. In the limit with a zero discount rate, the sole developer will commit trivial effort to a project if the future prize remains constant. Yet competition enhances effort, even under a zero discount rate. In effect, the threat of competition increases project urgency and simultaneously lowers individual and aggregate profits. While the level of land prices may also detract from gross profits and effectively limit the number of competitors in the marketplace, the market-determined level of entry does not necessarily support joint profits maximization and optimal effort among speculators. In Section II, I present the basic model of speculative development under uncertainty. I model development costs as increasing in effort, either as a one-shot investment or as an ongoing cost that is incurred until the market matures. In Section III I determine the solution for a single speculator facing no competitors. In Section IV I determine the symmetric equilibrium for multiple speculators, and I introduce joint development externalities between speculators in Section V. I discuss the policy ramifications in Section VI, and conclude in Section VII.
II. THE MODEL
Consider n identical profit maximizing developers, each indexed by i, with a zero discount rate, who invest C(li ) to purchase a development effort li . Developers invest in speculative ventures with the prospect that their investment will eventually become profitable at some uncertain time in the future. I define this time as the moment that the market matures. Their effort li represents the probability that their development effort will attract a mature market in a given period. Increased individual effort will thus reduce the time at which the development becomes profitable (when the market matures) and will also increase the probability that the individual speculator will win the development race and capture the grand prize of the preeminent element of the development. If there is equal probability that the market for the development can
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mature in any given period, then the uncertainty of the development is appropriately modeled as a Poisson process. This distribution is completely characterized by an arrival rate parameter li . 0, representing the Poisson probability that the development becomes viable in any given period, with the expected wait for a mature market to ‘‘arrive’’ given by 1/ li . The costs of development effort can be modeled as either a one-shot cost F (li) or as an ongoing development cost C(li ) until the market matures. Let all speculators vie to receive the ‘‘winning’’ prize in a development of fixed value p, which represents the sum of the flow of all net future revenues. The developer capturing the winning prize will receive a share d of the prize p, while the n other developers will split the remainder (1 2 d)p. Then, if there are a total of n 1 1 competitors vying for prizes of average size p/(n 1 1), one developer will capture the grand prize dp . p/(n 1 1), while the n other developers will each receive consolation prizes of size (1 2 d)p/n , p/(n 1 1).
III. THE SINGLE DEVELOPER CASE
In the absence of competition, a sole developer with exclusive rights to the entire development maximizes profits by choosing an optimal development effort li . If a developer with a zero discount rate will eventually secure a prize of present value p by investing in arbitrarily little effort, either on a one-shot or ongoing basis, he will commit only a trivial amount of effort. Net profits monotonically decrease in additional effort. To provide some urgency in development, let us assume in this section that the prize p depreciates in size at some rate r. Then the developer maximizes the following expression with respect to the level of effort li fi 5
E
y
0
2
p 3 Pr(market arrives at t)e2rt dt
E HE (effort costs to time t)dtJ y
t
0
0
3 Pr(market arrives at t)dt 2 one-shot effort costs. If the market arrives according to a Poisson process, then the profit function becomes fi 5
E
y
0
pli e2(li1r)t dt 2
E HE C(l ) dtJ le y
t
0
0
i
5 pli /(li 1 r) 2 C(li )/ li 2 F (li ).
2lit
dt 2 F (li ),
DEVELOPMENT EFFORT IN COMPETITIONS
5
The profit maximizing sole developer will then choose the level of effort l to maximize net profits max pli /(li 1 r) 2 C(li )/ li 2 F (li ). li
We will treat two cases of this problem. We first consider the case where all development effort is one-shot, and then consider the case when development costs are ongoing until the market matures. The One-Shot Development Cost Case In the one shot cost case, the developer’s first-order condition is dfi /dli 5 0 5 pr /(li 1 r)2 2 F 9(li ), ⇒ pr 5 F 9(li )(li 1 r)2,
(1)
with the second-order condition satisfied if costs are increasing at an increasing rate d 2fi /dl 2i 5 22pr /(li 1 r)3 2 F 0(li ) , 0. Taking the differential of the first-order expression gives
r dp 1 [p 2 2(li 1 r)F 9] dr 5 (li 1 r)[(li 1 r)F 0 1 2F 9] dli . We can substitute in the results from the first-order condition to determine the relationship between effort and exogenous variables
r dp 1 p[(li 2 r)/(li 1 r)] dr 5 [(li 1 r)[(li 1 r)F 0 1 2F 9] dli . If costs increase at an increasing rate, we see that the sole developer will increase his level of effort if the development prize p rises. However, more rapid depreciation of the prize will spur additional effort only if the equilibrium effort level is sufficiently large or the depreciation rate is sufficiently small (such that li 2 r . 0)). In such a case, the optimal effort level li brings the project to fruition at a rate greater than the depreciation in the value of the prize.
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The Ongoing Development Cost Case Repeating the exercise for development costs that are ongoing each period, we maximize the expression max pli /(li 1 r) 2 C(li )/ li , li
yielding a similar first-order condition dfi /dli 5 0 5 pr /(li 1 r)2 2 (C9 2 C/ li )/ li ⇒ pr 5 (C9 2 C/ li )(li 1 r)2 / li .
(2)
This expression implicitly defines an equilibrium level of effort l*. We find that the second-order condition is satisfied if costs are sufficiently increasing at an increasing rate d 2fi /dl 2i 5 22pr /(li 1 r)3 2 (C0 2 2C9/ li 1 2C/ l 2i )/ li . We can write the first-order condition (2) in terms of marginal effort costs MC and average effort costs AC dfi /dli 5 0 ⇒ pr 5 (MC 2 AC)(li 1 r)2 / li .
(2a)
Taking the differential of the first order expression for ongoing development costs gives
r dp/p 1 [ p 2 2(MC 2 AC)(li 1 r)/ li ] dr 5 d[(MC 2 AC)(li 1 r)2 / li ]/dli , which can be evaluated at the solution to the first-order condition
r dp/p 1 p[(li 2 r)/(li 1 r)] dr 5 (MC 2 AC)(li 1 r)(li 2 r)/ l 2i 1 ((li 1 r)2 / li ) d(MC 2 AC)/dli . Since marginal effort costs exceed average costs in equilibrium, d(MC 2 AC)/ dli . 0. As in the earlier one-shot development cost case, if equilibrium effort li exceeds the depreciation rate r, a rise in the value p of the project or a rise in the prize depreciation rate r increases development effort. The sole developer will commit development efforts to a property only if the development prize depreciates over time. Otherwise, there is no
DEVELOPMENT EFFORT IN COMPETITIONS
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urgency to increase development effort because the market prize will eventually arrive. In the next section, we find that competition creates such an urgency.
IV. THE SOLUTION WITH MULTIPLE DEVELOPERS
In this section, I consider the case when n 1 1 identical speculators compete for the development prize. Interestingly, while the previous section suggested that some sort of external project urgency is required for development to take place, I will show in this section that competition between developers is sufficient to spur development, even if there is a zero discount rate and no depreciation of the development prize. I further specify effort costs that are increasing in development effort according to a simple power function of development effort. These development effort costs can be one shot, as characterized by F (l) 5 bl11f, or ongoing, according to C(l) 5 al11f, where the parameter f . 21 represents diseconomies of scale. If f is less than zero, costs are increasing at a decreasing rate while f greater than zero signifies costs that are increasing at an increasing rate. In this case of multiple developers, each developer contributes to the collective rate at which the overall development will become profitable. Then, the market arrival rate is (li 1 nli ). Once the market matures, there are two possible states of nature—either a given developer receives the grand prize, represented as a larger than average share d of the prize p (i.e., 1 . d . 1/(n 1 1)), or shares in the residual prize (1 2 d)p with the n other ‘‘losing’’ developers. The probability that each of n developers will receive the grand prize pd is equal to li /(li 1 nli ). Then the profit function becomes fi 5
E
y
0
2
p(dli 1 ((1 2 d)/n)nl2i )e2(l11nl2i) dt
E HE al y
t
0
0
11f i
J
dt (li 1 nl2i )e2(li1nl2i) dt 2 bl 1i 1f ,
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which reduces to the developer’s maximization problem max [p(dli 1 ((1 2 d)/n)nl2i ) 2 al 1i 1f]/(li 1 nl2i ) 2 bl 1i 1f. li
(3)
The noncooperative developer will choose a level of effort li to maximize the above expression for expected profits, given the fixed strategy l2i of n other developers. In the symmetric equilibrium, each competitor facing an identical profit function determines an identical level of effort l. The One-Shot Development Cost Case Consider first the case when all development costs occur at time t 5 0. Then, the first-order condition is pd /(li 1 nl2i ) 2 [p(dli 1 ((1 2 d)/n)nl2i )]/(li 1 nl2i )2 2 b(1 1 f)l if 5 0. (4) As before, the second-order condition for a global maximum is satisfied 22pl2i(d(n 1 1) 2 1)/(li 1 nl2i )3 2 b(1 1 f)fl fi 21 , 0. Solving the first-order condition for the symmetric equilibrium gives pd /(n 1 1)l 2 pl /((n 1 1)l)2 2 b(1 1 f)lf 5 0.
(4a)
which can be rearranged to produce
l* 5 [(p/b)(d 2 1/(n 1 1))/((1 1 f)(n 1 1))]1/(11f).
(5)
The level of effort l* is positive for the full range of any feasible scale economies (i.e., f . 21) if d . 1/(n 1 1). This condition is ensured since the share going to the winner must not be less than the average share 1/(n 1 1) going to all developers. As intuition would suggest, the firstorder condition shows that the level of effort rises if the share d going to the winner rises, the level of diseconomies f rises, the value of the prize p rises, or the cost parameter b falls. The Ongoing Development Cost Case In the case of a development characterized by ongoing development costs that each developer must incur until the market matures, the firstorder condition becomes
DEVELOPMENT EFFORT IN COMPETITIONS
(pdli 2 a(1 1 f)l if)/(li 1 nl2i ) 2 (pdli 1 (1 2 d)(l2i ) 2 al 1i 1f)/(li 1 nl2i )2 5 0.
9 (6)
Rearranging the first-order condition for the symmetric equilibrium gives p((n 1 1)d 2 1) 2 a(n 1 f(n 1 1))lf 5 0,
(6a)
which can be solved for the optimal development effort l*
l* 5 [(p/a)(d(n 1 1) 2 1)/(n 1 (n 1 1)f)]1/ f.
(7)
In the ongoing development cost case, the level of effort l* is positive if f . 2n/(n 1 1). Such is the case if marginal costs are increasing. Our intuition is again confirmed. Effort shows the same qualitative sensitivity as before to changes in exogenous parameters. The level of effort rises as the share parameter d rises, the level of diseconomies f rises, the value of the prize p rises, or the cost parameter a falls. Competition and Effort An interesting exploration is the response of development efforts and profits to increasing numbers of developers at a given project. Substituting the equilibrium level of effort into the symmetric-firm profit function fˆ * gives fˆ * 5 (p 2 alf)/(n 1 1) 2 bl11f,
(8)
ˆ * for n 1 1 competing developers are given by while aggregate profits P ˆ * 5 (n 1 1) fˆ *. P
(9)
The One-Shot Development Cost Case Substituting the equilibrium level of effort for the one-shot development cost case into the symmetric-firm profit function gives
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fˆ * 5 (p/(n 1 1))[1 2 (d(n 1 1) 2 1)/((1 1 f)(n 1 1)2)] 5 (p/(1 1 f))[(1 1 f 2 d)/(n 1 1) 1 1/(n 1 1)2], ˆ * 5 (p/(1 1 f))[(1 1 f 2 d) 1 1/(n 1 1)]. P
(10) (11)
We may differentiate individual and aggregate profits to determine the implications of increased competition on profits dfˆ */dn 5 2(p/(1 1 f))[(1 1 f 2 d)/(n 1 1)2 1 2/(n 1 1)3] , 0, ˆ */dn 5 2(p/(1 1 f))(1/(n 1 1)2) , 0. dP Developers’ individual and aggregate profits unambiguously fall with increased competition. Turning to the level of effort in the one-shot development cost case, we can differentiate the optimal effort level
l* 5 [(p/b)(d 2 1/(n 1 1))/((1 1 f)(n 1 1))]1/(11f)
(5)
to give dl*/dn 5 (pl*11f /b)[1/((1 1 f)(n 1 1)3) 2 (d 2 1/(n 1 1))/((1 1 f)(n 1 1))2] 5 (pl*11f /b)[(1 2 d(n 1 1) 1 1)/((1 1 f)(n 1 1))3] 5 (pl*11f /b)[(2 2 d(n 1 1))/((1 1 f)2(n 1 1))3] The level of effort depends on the size of the grand prize. Effort is increasing in the number of competitors if the winner’s share d is more than twice the size of the average share 1/(n 1 1). For prizes less grand, effort is decreasing as competition increases. The Ongoing Development Cost Case Given the first-order condition for the level of development effort, we can calculate equilibrium profits. Substituting the equilibrium level of effort from (7)
l* 5 [(p/a)(d(n 1 1) 2 1)/(n 1 (n 1 1)f)]1/ f
(7)
DEVELOPMENT EFFORT IN COMPETITIONS
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for the ongoing development cost case into the symmetric-firm profit function fˆ * gives fˆ * 5 p[1 2 (d(n 1 1) 2 1)/(n 1 (n 1 1)f)])/(n 1 1) 5 p[(n 1 1)(1 1 f 2 d)]/[(n 1 (n 1 1)f)(n 1 1)] 5 p(1 1 f 2 d)/(f 1 n(1 1 f)), ˆ * 5 (n 1 1)p(1 1 f 2 d)/(f 1 n(1 1 f)). P
(12) (13)
Inspection of the expressions for the level of development activity yields the following sensitivity of effort to changes in the number of competitors dl*/dn 5 (1/ f)l*(12f)[d /(n 1 (n 1 1)f) 2 (d(n 1 1) 2 1)(1 1 f)/((n 1 (n 1 1)f)2] 5 (1/ f)l*(12f)[(d(n 1 (n 1 1)f) 2 (1 1 f)(d(n 1 1) 2 1)/((n 1 (n 1 1)f)2] 5 (1/ f)l*(12f)[(2d 1 1 1 f)/((n 1 (n 1 1)f)2] . 0. Comparing the numerator of this expression with the numerator of the profits expression in (12), we see that development effort strictly increases with an increase in the number of competitors. But, by differentiating the level of profits fˆ * in a symmetric equilibrium, we see that individual profits fall with an increase in the number of competitors dfˆ */dn 5 2(1 1 f)fˆ */(f 1 n(1 1 f)) , 0, as do aggregate profits with increased competition ˆ */dn 5 fˆ * 2 (n 1 1)(1 1 f)fˆ */(f 1 n(1 1 f)) dP 5 fˆ *(1 2 (n 1 1)(1 1 f)/(f 1 n(1 1 f)) 5 2fˆ */(f 1 n(1 1 f)) , 0. Again, since development effort is increasing in the level of competition, average development time and individual and collective profits are decreasing with an increase in the number of competitors. Speculative competition increases effort and decreases profits, both individually and in the aggregate. While increased uncertainty tends to reduce effort in some economic models, increased competition and its concomitant
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increase in development uncertainty is found to increase effort here. I next consider the effect of joint externalities in speculative development.
V. THE SOLUTION WITH MULTIPLE DEVELOPERS AND DEVELOPMENT EXTERNALITIES
Joint development externalities are an important and underexplored component of agglomeration economies in speculation. Joint developments profit from the success of each of the codevelopments because the overall development becomes more attractive. While I do not model such externalities as enhancing the value of the project, I do model the effects of an externality which act to speed the arrival of a market to an overall development as a consequence of marketing of associated subdevelopments. Let this development externality represent an increase in the market arrival rate for the ith developer by a proportion c of the efforts of each of the other n developers. Then the maximization of the profit function for the ith developer becomes max [p(1 1 nc)(dli 1 ((1 2 d)/n)nl2i ) li
2 al
11f ]/[(1 i
(15)
1 nc)(li 1 nl2i )] 2 bl
11f i
Again, I assume that the noncooperative developer will maximize the above expression for expected profits, given the fixed strategy l2i of n other developers. The One-Shot Development Cost Case Note first that the maximization exercise for the developer in the one shot development cost case is identical to the exercise (from Eq. (3)) without such an externality max [p(1 1 nc)(dli 1 (1 2 d)l2i )]/[(1 1 nc)(li 1 nl2i )] 2 bl 1i 1f li
5 max [p(dli 1 (1 2 d)l2i )]/(li 1 nl2i ) 2 bl li
(14)
11f . i
This is because the revenue expression is unchanged. While the development will mature earlier, since investment is sunk, there is no net profit gain. The results from the case of one-shot development costs in Section IV still apply.
DEVELOPMENT EFFORT IN COMPETITIONS
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The Ongoing Development Cost Case In the case of a development characterized by ongoing development costs that must be borne until the market matures, any reduction in the time the project becomes viable as a consequence of joint development externalities allows a developer to economize on the ongoing flow of development costs. The maximization exercise for a developer in the variable cost development case is max [p(1 1 nc)(dli 1 (1 2 d)l2i ) 2 al 1i 1f]/[(1 1 nc)(li 1 nl2i )]. li
(16)
Under positive development externalities, the symmetric equilibrium firstorder condition becomes 0 5 [(p(1 1 nc)d 2 a(1 1 f)l if)(1 1 n) 2 (p(1 1 nc) 2 al if)]/[p(1 1 nc)l 2 al 1i 1f)(1 1 n)l] 5 (p(1 1 nc)(d(1 1 n) 2 1) 2 al if((1 1 f)(1 1 n) 2 1)), which can be solved at the symmetric equilibrium to give the optimal level of development effort
l** 5 [(p(1 1 nc)/a)(d(n 1 1) 2 1)/(n 1 (n 1 1)f)]1/ f.
(17)
With positive development externalities, this level of effort is strictly larger (by a factor (1 1 nc)1/ f) than in their absence. While the other comparative statics results remain unchanged, we can see by comparing (7) to (17) that an increase in externalities c increased development effort. We can now explore the effect of increased competition of development effort when such positive externalities exist dl**/dn 5 d(1 1 nc)1/ fl*/dn 5 (1/ f)(1 1 nc)(12f)/ f cl* 1 (1 1 nc) dl*/dn. We find that, if there are joint development externalities, effort levels are increasing with additional competition if marginal costs are increasing. Inserting the equilibrium effort level into the symmetric equilibrium profit function gives
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fˆ ** 5 p/(1 1 n) 2 alf /(1 1 nc)(1 1 n)] 5 p/(1 1 n) 2 p(1 1 nc)(d(n 1 1) 2 1)/[(n 1(n 1 1)f)(1 1 nc)(1 1 n)] 5 p/(1 1 n)[1 2 (d(n 1 1) 2 1)/(n 1 (n 1 1)f)] 5 fˆ *.
While development externalities increase effort, profits do not rise. This is because the decrease in development time is offset by the rise in development costs arising from speculative competition. VI. DISCUSSION
We have determined that greater development competition induces greater development effort but lower collective and individual speculative profits. While competition causes greater aggregate effort, it brings forward market maturity at a reducing rate. While we have not modeled the benefit of more rapid development, we have found that the intensity of effort arises solely because of a race to capture a fixed prize. Of course, with profits declining as more speculators enter the fray, the overall gross profit level must still cover local land costs. Land prices effectively provide a market cutoff to the number of speculative developments. Nonetheless, if there are sufficient speculative development surpluses over the next best use of land, there may be wide margin for excessive speculative development. It is perhaps in that event that land use planning should reduce duplication of competing enterprises through restrictions in the number of competitive speculating developers. Such caps on the number of competing projects can ensure greater individual and aggregate profits for those entering the market. This approach also argues for a speculation tax that is linked to the speed of development (to remove some of the urgency inherent in a development race) or a land tax that limits the number of speculators that will enter the development race. Indirect instruments to control effort can also be used. An increase in effort costs or taxation to decrease the size of the development prize reduces effort and hence moves the development race toward the optimal solution. VII. CONCLUSION
Speculative development is an analytically challenging problem. Elements such as uncertainty in the payoff date, the level of speculative profits earned by one developer at the expense of similarly situated developers, the
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optimal level of development effort, and externalities between substitutable and separately managed development projects have all been treated here in a stochastic model of speculative investment. The most interesting result is intuitive—greater development competition induces greater development effort. However, such increased development effort may be wasteful. For example, stochastic development competition spurs greater aggregate effort yet produces diminishing savings in time costs. This occurs even under a zero discount rate. I am currently working on extending this model in a number of ways. Consideration of the other side of the market is necessary to consider the welfare effects of this model. For instance, consumers of the development may benefit from shortened development time. In addition, some complementarity in competing development may imply that the development prize rises with entry. Finally, this model perhaps applies to other real estate issues that involve speculative effort over time to secure a prize. Most notable in this area would be real estate brokering where there exist similar economies and diseconomies, some mechanisms for prize sharing, and a Poisson type of matching process over time which while not of indeterminate length may perhaps be sufficiently long for the analogy here to stand.
REFERENCES DELBONO, F., AND DENICOLO, V. (1991). ‘‘Incentives to Innovate in a Cournot Oligopoly.’’ Quart. J. Econ. 56, 951–961. EATON, C. AND LIPSEY R. (1981). ‘‘Capital, Commitment, and Entry Equilibrium.’’ Bell J. Econ. 12(2), 593–604. EATON, C. AND LIPSEY, R. (1980). ‘‘Exit Barriers are Entry Barriers: The Durability of Capital as a Barrier to Entry.’’ Bell J. Econ. 11(2), 721–29. GOERING, G., AND C. READ (1993). ‘‘Spillovers in a Patent Race Model.’’ Unpublished discussion paper, 1993. HELSLEY, R., AND W. STRANGE (1994). ‘‘City Formation with Commitment.’’ Regional Sci. Urban Econ. 24(3), 373–390. KANE, E. (1994). ‘‘Has U.S. Overbuilding Affected Construction Activity Globally?’’ J. Housing Res. 5(2), 295–301. LEE, T. AND L. WILDE (1980). ‘‘Market Structure and Innovation: A Reformulation.’’ Quart. J. Econ. 44, 429–436. LOURY, G. (1979). ‘‘Market Structure and Innovation.’’ Quart. J. Econ. 43, 395–410. STEWART, M. (1983). ‘‘Non-Cooperative Oligopoly and Preemptive Innovation Without Winner-Take-All.’’ Quart. J. Econ. 48, 681–694. WILLIAMS, J. T. (1991). ‘‘Real Estate Development as an Option.’’ J. Real Estate Finance Econ. 4(2), 191–208.
6, 1–15 (1997)
HE970201
Development Effort in Speculative Real Estate Competitions COLIN READ* University of Alaska–Fairbanks, Fairbanks, Alaska 99775 Received June 14, 1996
I model speculative development races when the time at which the development effort becomes profitable is uncertain. I first treat the decision of a speculator who has exclusive rights to the development prize. I then show that development races intensify development effort compared with the collectively optimal level of effort. Development races with identical speculative developers reduce aggregate profits and bring forward development time, whether or not joint development externalities are present. In such races, land use policies that restrict the number of permitted developers can enhance collective returns. 1997 Academic Press
I. INTRODUCTION
Until recently, land use planning has typically been used to ensure compatible use and to prohibit or limit certain ‘‘undersirable’’ uses. It can, however, also be employed to discourage speculative overbuilding, that is, excessive building due to uncertainty. Some recent examples of real estate overbuilding have led local planners to consider methods to prevent or discourage such overbuilding. But there are no formal economic models that predict overbuilding by speculators. I describe a competitive model of speculative development in real estate markets which results in overbuilding. The simple problem of optimal real estate development rates for a sole developer is well understood. Such a developer facing no competitors would choose to time development of a commercial property to coincide with the arrival of a mature market for the property. Williams (1991) explores such optimal real estate development timing and proposes the use of zoning as a policy tool to curb suboptimal development rates. However, he does not explicitly model the suboptimality which occurs when competing developers race to bring their developments to market by using development time as a strategic variable. * I thank the editor of this journal and an anonymous referee for their helpful comments. 1 1051-1377/97 $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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A search through the industrial organization literature yields two distinct strands of literature that could perhaps be adapted to treat the strategic development of land. The first class of models shows that early development can be used to strategically preclude subsequent development by competing speculators. For instance, Eaton and Lipsey (1981, 1980) introduce time as a strategic variable. By investing in nondecaying or slowly-decaying physical capital early, a firm can ‘‘stake out’’ its market to preclude entry of others and hence earn losses in the short run but monopoly profits in the long run once the market matures. In the development analogy, strategic premature development are balanced by exclusive profits later. Helsley and Strange (1994) explore market commitment of this sort in an urban context. Such models of market commitment do not adequately capture the uncertainty of development when competing developers separately determine an optimal development time. In an uncertain environment, two or more speculators could prematurely develop similarly situated parcels of land, perhaps resulting in long run losses for each speculator. So the question still remains—at a given time, what is the return to a developer who may face one or more simultaneous competitors, each vying for a market that will ‘‘arrive’’ at an uncertain date? It is this second literature that my paper addresses. Perhaps the most useful starting point for such an exploration is the patent race literature. In the seminal paper in this field, Loury (1979) modeled uncertain development as a race to capture rewards of innovation when research and development investment is a one shot game. Lee and Wilde (1980) repeat the exercise when research and development requires an ongoing investment until the innovation is realized. Stewart (1983) and Delbono and Denicolo (1991) expand the literature still further by allowing the winner a larger share of the prize but permitting the co-innovators to share in the consolation prize. These authors are primarily interested in the relationship between market structure and performance. I extend this literature to speculative development races and explore how the number of competitors affects individual and aggregate effort. I also model joint development externalities by acknowledging the interdependence of various developers’ investment on other developers’ ultimate success. In this context, a real estate development can be modeled as a collection of developers’ efforts. Each development produces a positive joint development externality by enhancing the attractiveness of the overall development, and speeds the date at which the entire development becomes viable. I argue that there may be some need to use zoning and land use planning to control speculative competition because such competition can be economically wasteful. As an example of jointness in production, consider a mall development complex such as Buckland Hills north of Hartford, Connecticut. This com-
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plex is actually a series of small malls in close proximity to each other. If there are n developers of these mini-malls and each of these mini-malls has a chance of becoming the preeminent mini-mall once the market matures, how will speculators compete to bring the overall development to market? How will the level of development effort differ with the number of developers allowed to vie for the grand prize of the preeminent minimall? How will the level of effort change as the size of the consolation prizes change? Will such competition produce economic waste? It is these questions that I address in this paper. I find that a sole developer without fear of competition may use exclusive development rights to delay production. In the limit with a zero discount rate, the sole developer will commit trivial effort to a project if the future prize remains constant. Yet competition enhances effort, even under a zero discount rate. In effect, the threat of competition increases project urgency and simultaneously lowers individual and aggregate profits. While the level of land prices may also detract from gross profits and effectively limit the number of competitors in the marketplace, the market-determined level of entry does not necessarily support joint profits maximization and optimal effort among speculators. In Section II, I present the basic model of speculative development under uncertainty. I model development costs as increasing in effort, either as a one-shot investment or as an ongoing cost that is incurred until the market matures. In Section III I determine the solution for a single speculator facing no competitors. In Section IV I determine the symmetric equilibrium for multiple speculators, and I introduce joint development externalities between speculators in Section V. I discuss the policy ramifications in Section VI, and conclude in Section VII.
II. THE MODEL
Consider n identical profit maximizing developers, each indexed by i, with a zero discount rate, who invest C(li ) to purchase a development effort li . Developers invest in speculative ventures with the prospect that their investment will eventually become profitable at some uncertain time in the future. I define this time as the moment that the market matures. Their effort li represents the probability that their development effort will attract a mature market in a given period. Increased individual effort will thus reduce the time at which the development becomes profitable (when the market matures) and will also increase the probability that the individual speculator will win the development race and capture the grand prize of the preeminent element of the development. If there is equal probability that the market for the development can
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mature in any given period, then the uncertainty of the development is appropriately modeled as a Poisson process. This distribution is completely characterized by an arrival rate parameter li . 0, representing the Poisson probability that the development becomes viable in any given period, with the expected wait for a mature market to ‘‘arrive’’ given by 1/ li . The costs of development effort can be modeled as either a one-shot cost F (li) or as an ongoing development cost C(li ) until the market matures. Let all speculators vie to receive the ‘‘winning’’ prize in a development of fixed value p, which represents the sum of the flow of all net future revenues. The developer capturing the winning prize will receive a share d of the prize p, while the n other developers will split the remainder (1 2 d)p. Then, if there are a total of n 1 1 competitors vying for prizes of average size p/(n 1 1), one developer will capture the grand prize dp . p/(n 1 1), while the n other developers will each receive consolation prizes of size (1 2 d)p/n , p/(n 1 1).
III. THE SINGLE DEVELOPER CASE
In the absence of competition, a sole developer with exclusive rights to the entire development maximizes profits by choosing an optimal development effort li . If a developer with a zero discount rate will eventually secure a prize of present value p by investing in arbitrarily little effort, either on a one-shot or ongoing basis, he will commit only a trivial amount of effort. Net profits monotonically decrease in additional effort. To provide some urgency in development, let us assume in this section that the prize p depreciates in size at some rate r. Then the developer maximizes the following expression with respect to the level of effort li fi 5
E
y
0
2
p 3 Pr(market arrives at t)e2rt dt
E HE (effort costs to time t)dtJ y
t
0
0
3 Pr(market arrives at t)dt 2 one-shot effort costs. If the market arrives according to a Poisson process, then the profit function becomes fi 5
E
y
0
pli e2(li1r)t dt 2
E HE C(l ) dtJ le y
t
0
0
i
5 pli /(li 1 r) 2 C(li )/ li 2 F (li ).
2lit
dt 2 F (li ),
DEVELOPMENT EFFORT IN COMPETITIONS
5
The profit maximizing sole developer will then choose the level of effort l to maximize net profits max pli /(li 1 r) 2 C(li )/ li 2 F (li ). li
We will treat two cases of this problem. We first consider the case where all development effort is one-shot, and then consider the case when development costs are ongoing until the market matures. The One-Shot Development Cost Case In the one shot cost case, the developer’s first-order condition is dfi /dli 5 0 5 pr /(li 1 r)2 2 F 9(li ), ⇒ pr 5 F 9(li )(li 1 r)2,
(1)
with the second-order condition satisfied if costs are increasing at an increasing rate d 2fi /dl 2i 5 22pr /(li 1 r)3 2 F 0(li ) , 0. Taking the differential of the first-order expression gives
r dp 1 [p 2 2(li 1 r)F 9] dr 5 (li 1 r)[(li 1 r)F 0 1 2F 9] dli . We can substitute in the results from the first-order condition to determine the relationship between effort and exogenous variables
r dp 1 p[(li 2 r)/(li 1 r)] dr 5 [(li 1 r)[(li 1 r)F 0 1 2F 9] dli . If costs increase at an increasing rate, we see that the sole developer will increase his level of effort if the development prize p rises. However, more rapid depreciation of the prize will spur additional effort only if the equilibrium effort level is sufficiently large or the depreciation rate is sufficiently small (such that li 2 r . 0)). In such a case, the optimal effort level li brings the project to fruition at a rate greater than the depreciation in the value of the prize.
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The Ongoing Development Cost Case Repeating the exercise for development costs that are ongoing each period, we maximize the expression max pli /(li 1 r) 2 C(li )/ li , li
yielding a similar first-order condition dfi /dli 5 0 5 pr /(li 1 r)2 2 (C9 2 C/ li )/ li ⇒ pr 5 (C9 2 C/ li )(li 1 r)2 / li .
(2)
This expression implicitly defines an equilibrium level of effort l*. We find that the second-order condition is satisfied if costs are sufficiently increasing at an increasing rate d 2fi /dl 2i 5 22pr /(li 1 r)3 2 (C0 2 2C9/ li 1 2C/ l 2i )/ li . We can write the first-order condition (2) in terms of marginal effort costs MC and average effort costs AC dfi /dli 5 0 ⇒ pr 5 (MC 2 AC)(li 1 r)2 / li .
(2a)
Taking the differential of the first order expression for ongoing development costs gives
r dp/p 1 [ p 2 2(MC 2 AC)(li 1 r)/ li ] dr 5 d[(MC 2 AC)(li 1 r)2 / li ]/dli , which can be evaluated at the solution to the first-order condition
r dp/p 1 p[(li 2 r)/(li 1 r)] dr 5 (MC 2 AC)(li 1 r)(li 2 r)/ l 2i 1 ((li 1 r)2 / li ) d(MC 2 AC)/dli . Since marginal effort costs exceed average costs in equilibrium, d(MC 2 AC)/ dli . 0. As in the earlier one-shot development cost case, if equilibrium effort li exceeds the depreciation rate r, a rise in the value p of the project or a rise in the prize depreciation rate r increases development effort. The sole developer will commit development efforts to a property only if the development prize depreciates over time. Otherwise, there is no
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urgency to increase development effort because the market prize will eventually arrive. In the next section, we find that competition creates such an urgency.
IV. THE SOLUTION WITH MULTIPLE DEVELOPERS
In this section, I consider the case when n 1 1 identical speculators compete for the development prize. Interestingly, while the previous section suggested that some sort of external project urgency is required for development to take place, I will show in this section that competition between developers is sufficient to spur development, even if there is a zero discount rate and no depreciation of the development prize. I further specify effort costs that are increasing in development effort according to a simple power function of development effort. These development effort costs can be one shot, as characterized by F (l) 5 bl11f, or ongoing, according to C(l) 5 al11f, where the parameter f . 21 represents diseconomies of scale. If f is less than zero, costs are increasing at a decreasing rate while f greater than zero signifies costs that are increasing at an increasing rate. In this case of multiple developers, each developer contributes to the collective rate at which the overall development will become profitable. Then, the market arrival rate is (li 1 nli ). Once the market matures, there are two possible states of nature—either a given developer receives the grand prize, represented as a larger than average share d of the prize p (i.e., 1 . d . 1/(n 1 1)), or shares in the residual prize (1 2 d)p with the n other ‘‘losing’’ developers. The probability that each of n developers will receive the grand prize pd is equal to li /(li 1 nli ). Then the profit function becomes fi 5
E
y
0
2
p(dli 1 ((1 2 d)/n)nl2i )e2(l11nl2i) dt
E HE al y
t
0
0
11f i
J
dt (li 1 nl2i )e2(li1nl2i) dt 2 bl 1i 1f ,
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which reduces to the developer’s maximization problem max [p(dli 1 ((1 2 d)/n)nl2i ) 2 al 1i 1f]/(li 1 nl2i ) 2 bl 1i 1f. li
(3)
The noncooperative developer will choose a level of effort li to maximize the above expression for expected profits, given the fixed strategy l2i of n other developers. In the symmetric equilibrium, each competitor facing an identical profit function determines an identical level of effort l. The One-Shot Development Cost Case Consider first the case when all development costs occur at time t 5 0. Then, the first-order condition is pd /(li 1 nl2i ) 2 [p(dli 1 ((1 2 d)/n)nl2i )]/(li 1 nl2i )2 2 b(1 1 f)l if 5 0. (4) As before, the second-order condition for a global maximum is satisfied 22pl2i(d(n 1 1) 2 1)/(li 1 nl2i )3 2 b(1 1 f)fl fi 21 , 0. Solving the first-order condition for the symmetric equilibrium gives pd /(n 1 1)l 2 pl /((n 1 1)l)2 2 b(1 1 f)lf 5 0.
(4a)
which can be rearranged to produce
l* 5 [(p/b)(d 2 1/(n 1 1))/((1 1 f)(n 1 1))]1/(11f).
(5)
The level of effort l* is positive for the full range of any feasible scale economies (i.e., f . 21) if d . 1/(n 1 1). This condition is ensured since the share going to the winner must not be less than the average share 1/(n 1 1) going to all developers. As intuition would suggest, the firstorder condition shows that the level of effort rises if the share d going to the winner rises, the level of diseconomies f rises, the value of the prize p rises, or the cost parameter b falls. The Ongoing Development Cost Case In the case of a development characterized by ongoing development costs that each developer must incur until the market matures, the firstorder condition becomes
DEVELOPMENT EFFORT IN COMPETITIONS
(pdli 2 a(1 1 f)l if)/(li 1 nl2i ) 2 (pdli 1 (1 2 d)(l2i ) 2 al 1i 1f)/(li 1 nl2i )2 5 0.
9 (6)
Rearranging the first-order condition for the symmetric equilibrium gives p((n 1 1)d 2 1) 2 a(n 1 f(n 1 1))lf 5 0,
(6a)
which can be solved for the optimal development effort l*
l* 5 [(p/a)(d(n 1 1) 2 1)/(n 1 (n 1 1)f)]1/ f.
(7)
In the ongoing development cost case, the level of effort l* is positive if f . 2n/(n 1 1). Such is the case if marginal costs are increasing. Our intuition is again confirmed. Effort shows the same qualitative sensitivity as before to changes in exogenous parameters. The level of effort rises as the share parameter d rises, the level of diseconomies f rises, the value of the prize p rises, or the cost parameter a falls. Competition and Effort An interesting exploration is the response of development efforts and profits to increasing numbers of developers at a given project. Substituting the equilibrium level of effort into the symmetric-firm profit function fˆ * gives fˆ * 5 (p 2 alf)/(n 1 1) 2 bl11f,
(8)
ˆ * for n 1 1 competing developers are given by while aggregate profits P ˆ * 5 (n 1 1) fˆ *. P
(9)
The One-Shot Development Cost Case Substituting the equilibrium level of effort for the one-shot development cost case into the symmetric-firm profit function gives
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fˆ * 5 (p/(n 1 1))[1 2 (d(n 1 1) 2 1)/((1 1 f)(n 1 1)2)] 5 (p/(1 1 f))[(1 1 f 2 d)/(n 1 1) 1 1/(n 1 1)2], ˆ * 5 (p/(1 1 f))[(1 1 f 2 d) 1 1/(n 1 1)]. P
(10) (11)
We may differentiate individual and aggregate profits to determine the implications of increased competition on profits dfˆ */dn 5 2(p/(1 1 f))[(1 1 f 2 d)/(n 1 1)2 1 2/(n 1 1)3] , 0, ˆ */dn 5 2(p/(1 1 f))(1/(n 1 1)2) , 0. dP Developers’ individual and aggregate profits unambiguously fall with increased competition. Turning to the level of effort in the one-shot development cost case, we can differentiate the optimal effort level
l* 5 [(p/b)(d 2 1/(n 1 1))/((1 1 f)(n 1 1))]1/(11f)
(5)
to give dl*/dn 5 (pl*11f /b)[1/((1 1 f)(n 1 1)3) 2 (d 2 1/(n 1 1))/((1 1 f)(n 1 1))2] 5 (pl*11f /b)[(1 2 d(n 1 1) 1 1)/((1 1 f)(n 1 1))3] 5 (pl*11f /b)[(2 2 d(n 1 1))/((1 1 f)2(n 1 1))3] The level of effort depends on the size of the grand prize. Effort is increasing in the number of competitors if the winner’s share d is more than twice the size of the average share 1/(n 1 1). For prizes less grand, effort is decreasing as competition increases. The Ongoing Development Cost Case Given the first-order condition for the level of development effort, we can calculate equilibrium profits. Substituting the equilibrium level of effort from (7)
l* 5 [(p/a)(d(n 1 1) 2 1)/(n 1 (n 1 1)f)]1/ f
(7)
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for the ongoing development cost case into the symmetric-firm profit function fˆ * gives fˆ * 5 p[1 2 (d(n 1 1) 2 1)/(n 1 (n 1 1)f)])/(n 1 1) 5 p[(n 1 1)(1 1 f 2 d)]/[(n 1 (n 1 1)f)(n 1 1)] 5 p(1 1 f 2 d)/(f 1 n(1 1 f)), ˆ * 5 (n 1 1)p(1 1 f 2 d)/(f 1 n(1 1 f)). P
(12) (13)
Inspection of the expressions for the level of development activity yields the following sensitivity of effort to changes in the number of competitors dl*/dn 5 (1/ f)l*(12f)[d /(n 1 (n 1 1)f) 2 (d(n 1 1) 2 1)(1 1 f)/((n 1 (n 1 1)f)2] 5 (1/ f)l*(12f)[(d(n 1 (n 1 1)f) 2 (1 1 f)(d(n 1 1) 2 1)/((n 1 (n 1 1)f)2] 5 (1/ f)l*(12f)[(2d 1 1 1 f)/((n 1 (n 1 1)f)2] . 0. Comparing the numerator of this expression with the numerator of the profits expression in (12), we see that development effort strictly increases with an increase in the number of competitors. But, by differentiating the level of profits fˆ * in a symmetric equilibrium, we see that individual profits fall with an increase in the number of competitors dfˆ */dn 5 2(1 1 f)fˆ */(f 1 n(1 1 f)) , 0, as do aggregate profits with increased competition ˆ */dn 5 fˆ * 2 (n 1 1)(1 1 f)fˆ */(f 1 n(1 1 f)) dP 5 fˆ *(1 2 (n 1 1)(1 1 f)/(f 1 n(1 1 f)) 5 2fˆ */(f 1 n(1 1 f)) , 0. Again, since development effort is increasing in the level of competition, average development time and individual and collective profits are decreasing with an increase in the number of competitors. Speculative competition increases effort and decreases profits, both individually and in the aggregate. While increased uncertainty tends to reduce effort in some economic models, increased competition and its concomitant
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increase in development uncertainty is found to increase effort here. I next consider the effect of joint externalities in speculative development.
V. THE SOLUTION WITH MULTIPLE DEVELOPERS AND DEVELOPMENT EXTERNALITIES
Joint development externalities are an important and underexplored component of agglomeration economies in speculation. Joint developments profit from the success of each of the codevelopments because the overall development becomes more attractive. While I do not model such externalities as enhancing the value of the project, I do model the effects of an externality which act to speed the arrival of a market to an overall development as a consequence of marketing of associated subdevelopments. Let this development externality represent an increase in the market arrival rate for the ith developer by a proportion c of the efforts of each of the other n developers. Then the maximization of the profit function for the ith developer becomes max [p(1 1 nc)(dli 1 ((1 2 d)/n)nl2i ) li
2 al
11f ]/[(1 i
(15)
1 nc)(li 1 nl2i )] 2 bl
11f i
Again, I assume that the noncooperative developer will maximize the above expression for expected profits, given the fixed strategy l2i of n other developers. The One-Shot Development Cost Case Note first that the maximization exercise for the developer in the one shot development cost case is identical to the exercise (from Eq. (3)) without such an externality max [p(1 1 nc)(dli 1 (1 2 d)l2i )]/[(1 1 nc)(li 1 nl2i )] 2 bl 1i 1f li
5 max [p(dli 1 (1 2 d)l2i )]/(li 1 nl2i ) 2 bl li
(14)
11f . i
This is because the revenue expression is unchanged. While the development will mature earlier, since investment is sunk, there is no net profit gain. The results from the case of one-shot development costs in Section IV still apply.
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The Ongoing Development Cost Case In the case of a development characterized by ongoing development costs that must be borne until the market matures, any reduction in the time the project becomes viable as a consequence of joint development externalities allows a developer to economize on the ongoing flow of development costs. The maximization exercise for a developer in the variable cost development case is max [p(1 1 nc)(dli 1 (1 2 d)l2i ) 2 al 1i 1f]/[(1 1 nc)(li 1 nl2i )]. li
(16)
Under positive development externalities, the symmetric equilibrium firstorder condition becomes 0 5 [(p(1 1 nc)d 2 a(1 1 f)l if)(1 1 n) 2 (p(1 1 nc) 2 al if)]/[p(1 1 nc)l 2 al 1i 1f)(1 1 n)l] 5 (p(1 1 nc)(d(1 1 n) 2 1) 2 al if((1 1 f)(1 1 n) 2 1)), which can be solved at the symmetric equilibrium to give the optimal level of development effort
l** 5 [(p(1 1 nc)/a)(d(n 1 1) 2 1)/(n 1 (n 1 1)f)]1/ f.
(17)
With positive development externalities, this level of effort is strictly larger (by a factor (1 1 nc)1/ f) than in their absence. While the other comparative statics results remain unchanged, we can see by comparing (7) to (17) that an increase in externalities c increased development effort. We can now explore the effect of increased competition of development effort when such positive externalities exist dl**/dn 5 d(1 1 nc)1/ fl*/dn 5 (1/ f)(1 1 nc)(12f)/ f cl* 1 (1 1 nc) dl*/dn. We find that, if there are joint development externalities, effort levels are increasing with additional competition if marginal costs are increasing. Inserting the equilibrium effort level into the symmetric equilibrium profit function gives
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fˆ ** 5 p/(1 1 n) 2 alf /(1 1 nc)(1 1 n)] 5 p/(1 1 n) 2 p(1 1 nc)(d(n 1 1) 2 1)/[(n 1(n 1 1)f)(1 1 nc)(1 1 n)] 5 p/(1 1 n)[1 2 (d(n 1 1) 2 1)/(n 1 (n 1 1)f)] 5 fˆ *.
While development externalities increase effort, profits do not rise. This is because the decrease in development time is offset by the rise in development costs arising from speculative competition. VI. DISCUSSION
We have determined that greater development competition induces greater development effort but lower collective and individual speculative profits. While competition causes greater aggregate effort, it brings forward market maturity at a reducing rate. While we have not modeled the benefit of more rapid development, we have found that the intensity of effort arises solely because of a race to capture a fixed prize. Of course, with profits declining as more speculators enter the fray, the overall gross profit level must still cover local land costs. Land prices effectively provide a market cutoff to the number of speculative developments. Nonetheless, if there are sufficient speculative development surpluses over the next best use of land, there may be wide margin for excessive speculative development. It is perhaps in that event that land use planning should reduce duplication of competing enterprises through restrictions in the number of competitive speculating developers. Such caps on the number of competing projects can ensure greater individual and aggregate profits for those entering the market. This approach also argues for a speculation tax that is linked to the speed of development (to remove some of the urgency inherent in a development race) or a land tax that limits the number of speculators that will enter the development race. Indirect instruments to control effort can also be used. An increase in effort costs or taxation to decrease the size of the development prize reduces effort and hence moves the development race toward the optimal solution. VII. CONCLUSION
Speculative development is an analytically challenging problem. Elements such as uncertainty in the payoff date, the level of speculative profits earned by one developer at the expense of similarly situated developers, the
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optimal level of development effort, and externalities between substitutable and separately managed development projects have all been treated here in a stochastic model of speculative investment. The most interesting result is intuitive—greater development competition induces greater development effort. However, such increased development effort may be wasteful. For example, stochastic development competition spurs greater aggregate effort yet produces diminishing savings in time costs. This occurs even under a zero discount rate. I am currently working on extending this model in a number of ways. Consideration of the other side of the market is necessary to consider the welfare effects of this model. For instance, consumers of the development may benefit from shortened development time. In addition, some complementarity in competing development may imply that the development prize rises with entry. Finally, this model perhaps applies to other real estate issues that involve speculative effort over time to secure a prize. Most notable in this area would be real estate brokering where there exist similar economies and diseconomies, some mechanisms for prize sharing, and a Poisson type of matching process over time which while not of indeterminate length may perhaps be sufficiently long for the analogy here to stand.
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