Urban Spatial Development: A Real Options Approach

Urban Spatial Development: A Real Options ApproachTan Lee*Department of International Business,Yuan Ze UniversityJyh-bang Jou*Graduate Institute of National Development,National Taiwan University*Correspondence to: Jyh-bang Jou, Graduate Institute of National Development, College of SocialSciences, National Taiwan University, No. 1 Roosevelt Rd. Sec. 4, Taipei 106, Taiwan, R.O.C.Tel: (886-2) 33663331Fax: (886-2) 23679684E-mail: jbjou@ntu.edu.tw1

Urban Spatial Development: A Real Options ApproachAbstractThis article investigates urban spatial development in a real options frameworkwhere a landowner irreversibly develops property in an uncertain environment.Landthat is located far away from the central business district will be developed later, but ona larger scale regardless of whether uncertainty arises or not.However, landdevelopment will not continuously move outwards from the central business districtunder poor demand or supply conditions. It is also found that a landowner willdevelop property later, but more densely if (i) uncertainty becomes greater; (ii) thereturns to undeveloped land are higher; and (iii) the development costs are expected togrow more slowly.Keywords: Development Density, Real Options, Spatial DevelopmentJEL Classification: G13, R52.2

I. IntroductionThe traditional literature on urban spatial development (e.g., Turnbull, 1988) hasoffered insights regarding the pattern of urban development.By assuming that theenvironment in the real estate market is non-stochastic over time, it predicts thatdevelopment continuously moves either outward from or inward towards the centralbusiness district (henceforth CBD) of a city.Consequently, it precludes thepossibility that development may be temporally halted by adverse economic conditions.This article applies the real options model so as to incorporate “uncertainty” in theanalysis, which includes the “certainty” case as a polar case.The traditional literature that offers the “certainty” model also proceeds from thestatic, to the myopic expectation, and finally to the perfect foresight model.Thestatic long-run equilibrium model, including Alonso (1964), Mills (1967), and Muth(1969), uses the comparative-statics approach, and predicts that residential densitydeclines monotonically with greater distance to the CBD.The prediction portrayedby this kind of model, however, is restricted (Wheaton, 1982).The myopicexpectation model such as Anas (1978) assumes that urban development is anincremental process in which development occurs over time in successive zones fromthe CBD outwards, and that the development of each zone proceeds under myopicforesight. This kind of model produces predictions that are quite different from the3

static model.For example, Anas shows that if population grows over time, residentialdensity increases with greater commuting distance to the CBD.One key assumptionof the model is that capital investment in developing properties is costlessly reversible.By contrast, the perfect foresight model assumes that capital investment in propertydevelopment is irreversible due to land use restrictions, and thus a developer must waitfor a state of nature that is good enough to develop property.Fujita (1982),McFarlane (1999), and Wheaton (1982) make important contributions to this kind ofmodel, and Turnbull (1988) provides a generalized model in which multiplesimultaneous development site locations can arise.On the other hand, the real options literature (see, e.g., Dixit and Pindyck, 1994)that investigates the timing and density decisions of property development typicallyabstracts from the spatial factor. For example, Titman (1985) uses a binominal modeland demonstrates that uncertainty will delay property development.Clark and Reed(1989) allow a landowner to sequentially choose the timing and density ofdevelopment, while Williams (1991) allows a landowner to make these two decisionssimultaneously.In Capozza and Li (1994), a landowner chooses the timing ofdevelopment and the capital intensity (defined as capital over land) simultaneously.After solving the optimization problem, they set a pricing equation, which indicatesthat rents are an increasing function of distance from the edge of the urban area.4The

key difference between our article and theirs is that we explicitly place the spatialfactor into a landowner’s optimization problem, and thus this factor plays a central rolein our analysis.1The remainder of this article is organized as follows.basic model.Section II presents theSection III solves choices regarding the timing and density ofdevelopment of a landowner for both the certainty and the uncertainty case.SectionIV shows the comparative-statics results regarding how the spatial factor, together withvarious other exogenous forces, affects the timing and the density of developmentchosen by a landowner.Section V concludes by offering suggestions for futureresearch.II. The ModelWe construct a model that extends both the “certainty” model as in Turnbull (1988)and McFarlane (1999), and the real options model as in Williams (1991).Supposethat at date t 0 a risk-neutral landowner has a parcel of vacant land that isnormalized at one unit.At any time where t 0 , the landowner is able to developvacant land on a scale equal to Q , and thus also at a density equal to Q .1We assumeA minot difference it that their paper assumes that rents follow an arithmatic Browian motion, while weassume that both rents and the development costs follow joint geometric Brownian motions.5

that structures are completely irreversible once put into place. 2The cost ofdevelopment is assumed to be equal to (see, e.g., Quigg, 1993; Williams, 1991)C (Q, x1 (t )) Q η x1 (t ),(1)where the constant cost of scale η 1 (which indicates that development on a largerscale is more costly), and x1 (t ) is a disturbance term which captures supply shockssuch as unexpected changes in weather or labor market conditions.Following Turnbull (1988, 2005b), we assume that land rent during time t forthe plot of developed property per unit is given byR (Q, D, x 2 (t )) Q b 1 D a x 2 (t ), 2 b 1, a 0 .(2)In equation (2), the term x 2 (t ) denotes the macroeconomic shock from the demandside, and D is the distance from the CBD, which is the only characteristic thatdistinguishes parcels of land in a city that is monocentric. Equation (2) indicates thatthe rent to developed property per unit is increasing concave in structural density (i.e. R / Q 0 , and 2 R / Q 2 0 ), and decreasing in the distance from the CBD (i.e. R / D 0 ).2This indicates that renters are willing to pay less per unit of developedMcFarlane (1999) argues that investment on land development will be fully irreversible if demolitioncosts are extremely high. Similarly, Riddiough (1997) suggests that irreversibility is a reasonableassumption with real estate in which the physical asset is long-lived and switching costs to alternativeuses are quite high.Turnbull (2005a) argues that the irreversibility assumption may not be realistic, butprovides analytically tractable solutions.6

property as it is developed more densely or the distance from the CBD is greater.Both the supply shock, x1 (t ) , and the demand shock, x 2 (t ) , follow jointgeometric Brownian motions given bydxi (t ) α i xi (t )dt σ i xi (t )dΩ i (t ),where i 1, 2 .(3)Each variable x i (t ) has a constant expected rate of growth α i and aconstant variance of the growth rate σ i 2 , and each dΩ i (t ) is an increment to astandard Wiener process, with E{dΩ i (t )} 0 and E{d Ωi (t )}2 dt .Furthermore,E (dΩ 1 (t )dΩ 2 (t )) r12 σ 1 σ 2 dt , where 1 r12 1 .We assume that the risk-less rate of interest ρ is constant per unit of time andthat vacant land per unit earns an agricultural rent, γ x 2 (t ) , which is invariable acrossspace.We further assume that γ 0 such that a landowner has no incentive toabandon vacant land. In other words, the landowner does not have the option valueof abandonment.For tractability, we also abstract from both the time-to-buildproblem that usually occurs in the real estate industry (see, e.g., Bar-Ilan and Strange,1996; Capozza and Li, 1994; Grenadier, 2000), and the redevelopment problemaddressed in Williams (1997).Consequently, in what follows, the landowner willsimultaneously choose the date and the scale of development once and for all.33We also assume that all lots are simultaneously developed and are finished instantly.Theseassumptions are usually adopted in the real options literature (see, e.g., Capozza and Li, 1994; Childs, et7

III. Choices of the Date and the Density of Development1. The Certainty Case4In what follows, without risk of confusion, we will write x1 (t ) x1 and x2 (t ) x2 .The parcel of vacant land belonging to a landowner has the expected value given by Et [ t γx2 (τ)e ρ ( τ t ) dτ T Q b D a x2 (τ)e ρ ( τ t ) dτ Q η x1 (T )e ρ (T t ) ] .T(4)Equation (4) indicates that the expected present value of returns to the unit plot of landat a particular location D in the urban area is the sum of the expected present value ofagricultural rents received until time T , plus the expected present value of land rentbeginning at the time of development, less the expected present value of thedeveloping costs.Equivalently, the expected value of vacant land can be rewritten as γx2 Et T (Q b D a γ ) x2 (τ)e ρ ( τ t ) dτ Q η x1 (T )e ρ ( T t ) ,(ρ α 2 )where it is required that ρ α 2 .(4a)The first term of equation (4a) is the expectedpresent value of agricultural rents received from t until infinity, i.e. Et t γx2 (τ)e ρ ( τ t ) dτ γx2.(ρ α 2 )Under the certainty case,(5)x1 (T ) x1e α (T t )1andx2 (τ) x2 e α τ2since σ1 σ 2 0 .al., 1996; Williams, 1991). We thus neither allow lots to be developed sequentially nor allow thedevelopment of real estate as a sequential investment (see, e.g., Bar-Ilan and Strange, 1998).4The analysis of the “certainty” model can be found in several studies in the real options literature.See, for example, Dixit and Pindyck (1994, pp. 138-139) and Majd and Pindyck (1989).8

Substituting this into (4a) yields the objective function of the developer as given by γx2 max W (T , Q) , T ,Q (ρ α 2 ) where W (T , Q) (4b)(Q b D a γ ) x2 e ( ρ α )(T t ) Q η x1e ( ρ α )(T t ) .(ρ α 2 )2(5)1Intuitively, under the certainty case, a landowner needs to decide either todevelop immediately at the current time t or to delay development.The formeroccurs if the landowner expects the development costs to grow at a rate that is greaterthan or equal to the growth rate of the rent for developed property, i.e. α 1 α 2 .Begin by assuming that development is made immediately, i.e. T t , and later wewill show that the condition for this to be true is indeed α 1 α 2 .Substitute T tinto equation (4a) yields the expected value of vacant land as given byQ b D α x2 Q η x1 .(ρ α 2 )(4c)If the value given by equation (4c) is greater than zero, then vacant land will bedeveloped immediately. Assuming this condition holds, we can derive the choice ofdevelopment scale, denoted by Q * , by differentiating equation (4c) with respect toQ , and then setting it equal to zero.This yieldsb 1bQ* D a x2η 1 ηQ * x1 0.(ρ α 2 )(6)Solving equation (6) yieldsQ* (1bx2( η b )).η(ρ α 2 ) D a x1(7)9