- Email: [email protected]
Urban residential development timing
Regional Science and Urban Economics 11 (1981) 239-254, North.Holland
URBAN RESIDENTIAL DEVELOPMENT TIMING* David E. MILLS University of Virginia, Charlottesville, VA 22901, USA Received May 1980, final version received November 1980 This paper presents a dynamic model of the residential land development process with hetelogeneous housing capital. Its properties under a regime of competitive developers are examined for three informational scenarios: perfect foresight, myopia and imperfect foresight. A special case is presented where different types of housing construction are halted sequentially as the land supply diminishes rather than simultaneously at the moment it is finally exhausted.
1. Introdeetion The dynamic character of urban residential development has been subjected to rigorous economic analysis only within the last decade. Following the early work of Shoup (1970), contributions by Anas (1978), Arnott and Lewis (1979), Arnott (1980), Fujita (1976) and Mills (1978 and 1981), among others have focussed on various aspects of the intertemporal develvpment process. This paper seeks to extend that literature by taking up the timit:g of residential development and the evolution of the housing stock in a city where the land market is competitive, the demand for housing is growing and the supply of land is fixed. A dynamic model of the development process with heterogeneous housing capital is introduced in section 2. Its various properties under a regime of competitive developers with perfect foresight are discussed in section 3. A special case is presented where different types of housing construction are halted sequentially by competitors as the land supply diminishes rather than simultaneously at the moment it is finally exhausted. When the growth trend of housing demand is not monotonic but subject to intermittent contractions, competitors anticipate them and change the strategic timing of construction accordingly. In section 4 we turn to a regime of imperfect foresight. In this *This paper reports work that was supported by Grant no. H-5036 RG from the U.S. Department of Housing and Urban Development. The substance and findings of that work are dedicated to the U.S. public. The author is solely responsible for the accuracy of the statements and interpretations contained in this paper and such interpretations do not necessarily reflect the views of the U.S. Government. The author would like to thank Richard Arnott, Blaine Roberts and a referee for helpful comments on an earlier version.
0166-0462/81/0000--(O00/$02.50 © North-Holland Publishing Company
240
D.E. Mills, Urban residential development timing
connection, properties of the development process with both myopic and speculative decision-making are considered. Results are summarized in section 5. 2. A model of the development process To set the stage, consider a city with L units of homogeneous land for future residential development. The homogeneity assumption suppresses both Iocational and topographical differences among specific parcels. The assumption that land is fixed in supply accords with several actual situations" two are cities situated in urban areas and surrounded by other cities and natural boundaries like coastlines, and cities where comprehensive planning has restricted residential development to specific parcels. Initially all the land is in an unimproved state, bl,t as the demand for housing expands over time, it is developed residentially. The housing stock consists of dwelling units of n distinct types. The supply of housing at time t is indicated by the vector x(t)--(xl(t),xz(t)...xn(t)). Elements of this vector can be non-integers, indicating the divisibility of dwelling units. A fundamental difference among housing types is the amount of land they require per unit. These quantities are indicated by the vector =1~1.~2 .... , ~,,). There are three kinds of agents who together determine the course of development in the city. The first are tenants, whose collective demand for the ith type of housing at t is exogenous to the model and represented by the lcontinuously differentiable) inverse demand function"
f~(x(t),t),
i = I , 2 ....,n.
Since all dwelling units are in some sense substitutes for each other, we assume --<0 ?x~ -
for all i,j.
The rental rates on housing, characterized by the vector R(t)=(Rl(t), gz(t) ..... R,lt)), adjust instantaneously to changes in demand or supply. Thus.
Ri(t)=fi(x(t),t)t),
i= 1,2,...,n
(1)
holds at every moment. The second kind of agents in the model are firms in the competitive building industry, l'his industry has a constant-returns-to-scale technology,
D.E. Mills, Urban residential development timing
241
so the construction cost of a particular type of dwelling unit does not vary with the overall level of construction activity. These costs (net of land) for the n types of dwelling units are indicated by the vector c = (cz,c2,...,c,). For analytical convenience, construction is assumed to be instantaneous. The final kind of agents and the principal decision-makers in the model are landowners. Landownership is assumed to be sufficiently diffuse that no one has market power. As long as land remains unimproved, the owner's rate of return on it is R, per unit. The development decision concerning a parcel of land (of any size) involves determining a type of housing to be built and a time at which to undertake construction. Because dwelling units are durable and demolition costs are non-negligible, development is considered to be irreversible; previously developed land cannot be returned to an unimproved state or converted to another type of development. Landowners execute development decisions by contracting with firms in the competitive building industry at prices equal to construction costs. Then they either become landlords in the rental housing market or sell their property to other agents who become landlords. For purposes of exploring the timing of development decisions, it makes no difference whether this last transaction occurs, or if it does, when. All agents are assumed to have perfect information about R~, c and current values of R(t). The implications of whether this clairvoyance extends to the future are explored below. The process of land development in the city is fully characterized by x(t), for all t~O, or equivalently, by x(O), ~(t), for all t=>O. This vector time-path must satisly all previous assumptions, so I x(O)-O,
~(t)>=O, • x(t)<-L,
(2a)
t>-O,
(2b)
t>O.
(2c)
3. Perfect foresight In this section, we consider the development process that occurs when landowners have (in addition to perfect current information) perfect foresight concerning R(t) for all t>O. This is a strong assumption since R(t) is determined endogenously in the model. Recalling (1), it implicitly requires 1Another interpretation can be placed on the constraint (2C). Suppose the binding constraint on development is not land supply, but a legislated maximum number of residential units in the city. This resembles the actual constraint that applies in a growing number of cities with socalled 'growth controls'. To adapt (2) to this circumstance, let ~t be the unit vector and L the legislated maximum number of dwelling units. All subsequent results apply.
D~. Mills, Urban residenzial development timing
242
that landowners have exhaustive information about the f~(.) functions and can accurately foresee x(t) throughout the future. The reasons for studying this case are two. First, it enables us to characterize the development process when the perfect-foresight assumption is approximately true - - that is when landowners have good information about future events. Since returns on developed land will be shown to depend crucially on one's ability to predict R(t) accurately, landowners have considerable incentive to collect information concerning future rental rates. Second, this case serves as a benchmark for comparison with the development process that occurs under other assumptions and informational environments. Two departures from the present assumption are considered in section 4 below.
3.1. General properties of a competitive equilibrium Since landownership is decentralized, the land-conversion process must have all the properties of a competitive equilibrium. To see what this means, note that every landowner has the same set of development options for a parcel of land (no matter whether it represents all or only part of his holdings). These options are infinite in number and identified by both a type and time of development: (i,t). If landowners have perfect foresight, a competitive equilibrium will not exist unless all options actually exercised are equally profitable ex post. No option will be voluntarily exercised when a more profitab!e one exists. To make this property of the development process more rigorous, note first that the discounted (to t = 0 ) profit to the owner of one unit of land left perpetually unimproved is R j r , where r is the universal discount rate. Clearly, competitive landowners will not develop land unless there is an option that generates profit per land unit in excess of this amount. If a landowner exercises option (i, t) on a unit of lan::l, then differential profits attributable to development are V~(t)/~i, where V,(t }= j e - " [ f i ( x ( r ) , r ) - a i R . ] d r - e - " c i for all i. t
Let the maximum value of these profits over all options be p = max { V/(t)/a,}, i.!
where # > 0 by assumption. Then a necessary and sufficient condition for competitive equilibrium is (.,~(t)). (V~(t)/a~-#)=O
for all i, t.
(3)
D.E. Mills, Urban residential development timing
243
If option (i, t) is exercised, then ~(t) > 0 and Vj(t)/0c~=/~ so that no forgone option is superior to it. If the option is forgone, then ~ ( t ) = 0 and V~(t)/a~<#. No option is exercised when a more profitable one exists, so all exercised options are equally profitable. Although (3) is necessary and sufficient for a competitive equilibrium in the land-development process, it does not imply that the process is unique. Because housing production is characterized by constant-returns-to-scale, there is an absence of strict cost-convexity in the model. For this reason there can be an infinite number of vector time-paths x(t), for t_>_0, that
satisfy (3). One implication of (3) worthy of note is that if ~(t)>O
for some i and all t ~ (tl,t2),
where t2 > t ~, then
R~(t)=aiR,+rci
for all
t6(tl,t2).
This is sho~n as follows. The hypothesis implies that
~(t)-O
for all
t6(t~,t2),
which frown differentiating V~(t) implies that
fi(x(t),t)-o~R,-rc~=O
for all
t~(tl,t2).
This with (1) produces the stated conclusion. Over any interval of time where continuous construction of some dwelling-unit type occurs, its rental rate is maintained at the 'opportunity cost' level: that level where revenues equal the sum of opportunity land-cost and amortized building cost. The effect of the land-supply constraint on the competitive development process depends on the time-dependence properties of the various f~ functions. If demand for all types of housing stops growing before L units of land are developed, then the constraint has no effect. In order for this to be an equilibrium, # must be zero so that owners of undeveloped land are no worse off than others. In other circumstances, the land supply is exhausted and # is positive. (A strong sufficient condition for all L units of land to be developed in finite time is that for some i and all t>__O, ~f~/~t> 6, a small positive constant.) Even where the land supply is exhausted in finite time, there is no reason to believe that all types of dwelling-unit construction are halted simultaneously. ~f the moment at which type i construction is finally stopped
244
D.E. Mills, Urban residential development timing
is defined as Ti=min{t:~i(z)=O
if
r>t},
then the moment at which the last unit of land is developed is: max {7"1, T2,..., T.}. All type i construction occurs within the time-interval Since (3) requires that every option actually exercised be equally profitable, the wide variation in their cash-flow characteristics is immaterial. In particular, the discounted profit on a type i dwelling unit built at some t < T~ must be the same as one one built at T~. This means no profit accrues on the earlier-constructed dwelling unit until T~. The discounted sum of revenue it produces between the moment of construction and T~ is exactly offset by the discounted sum of the amortize~ building cost and the opportunity cost of land. Thus for any (i,t), wllere ~i(t)>0, Ti
e-"[fi(x(r), r ) - a , R . - rci] dz=O. t
If there is a market for land or dwelling units (in addition to the market fol their services) in this world of perfect foresight, prices will reflect the full exp~cta,.io_-, of future profit. The price of a unit of unimproved land at t = 0 will be p + R./r. At any future date prior to its development, the price will be e"p+R./r. It increases with time unless p = 0 in which case it remains constant. The price of an existing type i dwelling unit (including the site it occupies) at t will be the discounted (to t) sum of future revenues: oo
Pi(t)= $ e-'~'-')f/(x(r), z)d¢. !
Utilizing (1), the rate at which it changes is
15i(t)= rPi(t ) - Ri(t ), and is positiv~ or negative depending on the relative magnitudes of terms on the right-hand side. Owing to the fact that it is a competitive equilibrium [as depicted in (3)], the development process under conditions of perfect foresight is efficient. Barring fiscal cr other externalities (all of which have been excluded from the analysis), there is no vector time-path x(t) that generates greater (discounted) aggregate econo~nic benefits. 2 ZThis is rigorously demonstrated in Mills (1978) for a similar model.
D.E. Mills, Urban residential development timing
245
3.2. A special case Further properties of the development process can be stated if housing demand satisfies two further assumptions. The first is that initial demand for every housing type is non-negligible. Specifically, assume
~(O,O)>a~R,+rc~ for al~ i. The second is that demand changes at every t in such a way that the vector
~(t) that solves [ fdOxj] .
) + bgfdet] = o
is strictly positive [that is every element ~ ( t ) > 0 ] for every t>0, where [c3fffdx~] and [dfffdt] are respectively fin n × n matrix and an n-vector with typical elements shown. Since all elements in this matrix are non-positive, a necessary condition for this assumption to hold is that demand for all types of housing be strictly increasing:
d~lOt > 0
for all i and t > O.
(4)
With the first of these assumptions, x(t) is discontinuous at t = 0 and the vector ~(0) has elements that are infinite. But after this initial building spurt, constrt~ction rates remain finite (as long as elements of [dfffc3t] do). With the second a:sumption, the existence of a vector x(t) for which ~i(t)>O
for all i and te(O, TJ
is assured, and from previous elucidation of (3), this means
Ri(t)=~R..+rci
for all i and t e (0, T~].
15)
Simultaneous construction of all housing types continues throughout the time-interval [0, min {Tt,..., T~}], and the rental rate on each type of dwelling unit remains at its opportunity-cost level. At min {Tl,..., T~} (at least) one type of construction is halted. From this moment forward, while other types of construction continue as before, the rental rate on the halted type(s) increases and profits accumulate: Ri(t ) > ~iR~ + rc~
for t > T~ for all i.
Subsequently and in the order of increasing T~s, other types of building are halted and profits begin to accumulate on them as well. Thus, the process is brought to a close in a sequence of cascading cessations. Many types of
246
D.Eo Mills, Urban residential development timing
construction (at most, all but one) will have been stopped before the city's supply of land is exhausted. This pattern of cascading cessations emphasizes the importance of foresight and strategic timing on the part of landowners. Once a type of building has been stopped, its rental rate rises above the opportunity-cost level. Owners of vacant land therefore exercise restraint in forgoing development opportunities with high immediate returns in favor of future options that are, in the f'mal analysis, more remunerative. Another consequence of this pattern is that rental rates on different housing types begin to rise above their opportunity-cost levels at different times. To a bystander, observer or tenant who is less well-informed than landowners about the housing-supply mechanism and future market conditions, it will appear irrational that construction stops and rentals rise sconer on one type of housing than another. It is important therefore to remind ourselves that under the assumptions posed this pattern is a property of the eOicient development process. The order of cessation among dwelling-unit types is something about which nothing ~:ery general can be said. It depends not only on the relative magnitude of the elements of [c~fJc~t], but also on the elements of [OfJc3x/l. An example of ~he kind of result that can be wrung from the model if further assumptions are made is the following: Suppose
~(x(t),t)-eehi(xi(t))
for all i,
where q
Integrating and substituting e*T'h(xi(T, ))= a~R. + rc~ into this expression yields
e-'r'(R.+rG/a~)ll
r q
!/=/~
for all i.
Since the last term on the left-hand side is a positive constant, this expression implies the desired result.
D.E. Mills, Urban residential development timing
247
that this implies T~> Tj if and only if Ck/ak> cj/oej. That is, if there is no cross-elasticity of demand among housing types, and the 'pure' secular trend in "demand is uniform across types, then the order of cessation will be that of increasing building costs per unit of land. High-rise apartment buildings will continue to be constructed after new single-family dwellings and garden-apartment buildings have been stopped. If the first assumption above - - that initial demand for every housing type is non-negligible- is dropped while the second is kept, then f~(O, O) < ~iR~ + rc~ for one or more i. In this case, the development process differs from (5) only in that ~(t)=O over an initial time-interval for those housing types with negligible initial demand. A pattern of staggered starts for different construction types emerges that resembles the subsequent pattern of cascading cessations. Nevertheless, the order is of starts and stops unrelated.
3.3. Discontinuous development If we depart from the second of the assumptions that define this case, holdin$ in place the first, the development process becomes more di~cult to describe. 3ecause (4) is a necessary condition for the assumption in question, a sufficient condition for departing from it is that demand contracts over some interval of time for one or more of the housing types. The most prominent examples of such periods are associated with macroeconomic recessions and the building cycle. We consider here only the least complicated departure from (4) that is possible. The results suggest the kind of changes that might be expected from more complicated departures since the latter are not taken up explicitly. Let there be one housing type k such that
dfkldt
0
for
~t=t~,t2
,
(6)
~.t q~[tl,t2] where 0 < t l
D.E. Mil/s. Urban residential development timing
248
it is clear that with (6) the competitive development process cannot proceed as it did before beyond time tl. Suppose, however, that (5) holds for every i ¢ k, and that
Rt(t)=~hR.+rch
for O < t < t t .
Then, because of (6) and because other (substitute) types of construction continue, Rh(t) begins at tt to fall below ~tR.+rch and landowners c e a s e building type k dwelling units. They do not begin again until Rk(t) rises to its previous level, beyond which time (5) holds for type k as well. Clearly, this does not occur until after t2. One consequence of all this is that throughout the period when type k construction is stalled, loss s accumulate on every previously-constructed type k dwelling unit. Furthermore, these losses are not offset by previous or subsequent gains. This means landowners who build type k housing after t2 make greater discounted profits (per-unitof-land) than those who build them before tt. This is a clear violation of (3), and so cannot be an accurate description of the competitive development process. From this we conclude that landowners with perfect foresight halt the construction of type k dwelling units before t~. In fact, they will discontinue it sufficiently far in advance that profits accumulate on existing type k dwelling units to offset the losses incurred after t t. Thus, the competitive developmet.:t process differs from (5) in that type k construction is stalled between to .rod th, where
O
e - n i l k(x(t ), t ) - ~tR. + rct] dt = 0.
(7)
la
This equation assures that discounted profits on type k dwelling units built after t b are the same as on those built before ta. The rental and construction rates for type k dwelling units must therefore satisfy
t=t {
R~(t) ~ ~kR.+rck
for
(0, ta) and
tE
[ta, tt,]
and
it(t)
(tb, Tk)} .
tt
>= 0
D.E. Mills. Urban residential development timing
249
If t,, should be 0, then type k construction does not begin in the first place until tb. The importance of foresight and strategic timing on the part of landowners is even more apparent here than in the previous special case. Laadowners must not only give attention to the question of when construction of a particular type is to be finally stopped, but also to the question of when it should be temporarily stopped to avoid net losses during a contraction. When building is stopped in anticipation of a coatraction, rental rates rise above, then fall below, and finally return again to their opportunity-cost level. Once again, the housing-market observer who is less well-informed than landowners will regard such discontinuities in development as irrational (or at least so whenever rental rates exceed their opportunity-cost level). Before proceeding, it might be well to summarize the main characteristics of perfect-foresight residential development.
(i)
Since no development option is exercised when a more profitable one exists, all options actually exercised are equally profitable. As a consequence, net profits do not begin to accumulate on dwelling units of the ith type until T~, the moment in which the last one is built. (ii) For the special case of 3.2, all construction rates are strictly positive throughout (0, min {TI,..., T~}). Not all types of dwelling-unit construction are halted simultaneously, but rather follow a pattern of cascading cessations. Rental rates on dwelling units remain at their ooportunity cost level for as long as construction of their respective typ~s continues; profits do not begin to accumulate on type i dwelling units "mtil T~. (iii) If demand for some type of housing contracts before all dwelling units of that type have been built, landowners will anticipate the contraction, halt construction in advance so profits can accumulate and offset foreseen losses. During the stall the rental rate first rises above, then falls below before finally returning to the opportunity-cost level. !
4. Imperfect foresight The assumption that landowners are endowed with perfect foresight about rental rates is not, in a purely descriptive sense, accurate. It is therefore important to consider how the development process might change if foresight is imperfect. We begin that task in this section by looking at the case where landowners are myopic - - where they make current decisions using current information only and disregard completely whatever information they have about the future? 4This is essentially the assumption attributed to decision-makers by Anas (1978).
250
D.E. Mills, Urban residential development timing
The purpose in taking up this case is not that it is more realistic than perfect foresight. Indeed it is probably less realistic given landowners' considerable incentive to acquire good information about the future. Instead, the purpose is to define a spectrum of outcomes that arise when landowners have imperfect foresight. These two cases m perfect foresight and myopia m lie at opposite ends of this spectrum and their comparison allows several conclusions about the nature of intermediate outcomes, all without constructing an elaborate expectational model. Before proceeding, we impose one further assumption on the model of section 2. In addition to the non-positivity assumption, the elements of [?f~/~xj] are henceforth assumed to satisfy
I f,/vx, I> max {I ~J Of,lO sl , I}
for all i,j.
(8)
J
This means there is no way to develop a parcel of vacant land to reduce Ri as much as if it were developed as type i housing. 4.1. Myopic behavior
The effect of landowner decisions based solely on current market information is easiest to see in the special case of 3.2. The process is the same here as there throughout the time-interval [0,min {Tl,..., Tn}). Construction of all types proceeds at rates that maintain rental rates at their opportunitycost level. At min{T~ .... , T,}, landowners with perfect foresight stop building one (or more) type of dwelling unit even though demand for this type of housing continues to grow and vacant land remains in the city. This is because additional building of that type would produce less profit than other alternatives. If landowners are myopic, they forgo this kind of consideration and plunge ahead with building at a rate that keeps the rental rate at its opportunity-cost level. This is true not only for the first dwelling-unit type which prescient landowners would halt, but for the others too. In fact, building of all types will continue as long as any vacant land remains. Similarly, rental rates will remain at their opportunity-cost level as long as vacant land remains. Thus R,(t)=~iR.+rc i
for te(O, T ' )
for all i,
(9)
where T = is defined as the moment when the land supply is exhausted: Tm= min {t :~t . x(t )= L}.
it is clear that T m lies somewhere strictly between the maximum and
D.E. Mills, Urban residential development timing
251
minimum elements of (7~,..., T,), since otherwise a . x ( T ' ) would be either greater or less than L. Thus, relative to perfect foresight, some kinds of construction are continued too long and others are stopped too soon because the land supply is exhausted. Too many dwelling units of some types are built and too few of others. Because of this, some development options exercised are more profitable ex post than others. To see this, note first from (9) that no profits accumulate on any dwelling unit until T". This means that profits on dwelling units of the same type are, here as before, equal regardless of the time of development:
V,(T')
---cti
for any t ~ (0, T m) for any i.
~i
Profits per unit-of-land are not the same, however, across all dwelling unit types. V~(T")/0q must (generally) be greater than p, the profits that would occur on al~ parcels under perfect foresight, for some types i and less for others. In particular, profits are greater than p for those types where T~> T", and less for those types where T~< T". It will suffice to prove only the first half of this result. Consider some type k of housing for which Tk> T s. Under perfect foresight, this type of building would persist longer than under myopia. Because of this, xk(T") under myopia is less than xk(Tk) under perfect foresight. Together with (8), this assures us that fk(x(t),t) is greater under myopia than under perfect foresight for all t > Tm. This means that ~(T")/ah ~'nder myopia is greater than Vk(T~)/ak under perfect foresight which equals #. The implications of myopia are even more extensive if we return to the case in 3.3 where demand contractions occur. Myopic landowners will not at to or any other moment previous to t l stop construction of type k dwelling units. The contraction will not be anticipated. At tl, Rk(t) begins to fall due to the contraction and type k construction is at last stopped. But it is too late for owners of type k housing to accumulate profits to offset losses that begin at tl. Since new type k construction will begin once Rk(t) rises again to its opportunity.cost level (and providing that this occurs before T m when the land supply is exhausted), the ex post profit on type k housing built before t l will be less than that on those built later. By failing to anticipate the contraction and strategically halt construction before it arrives, myopic landowners cause profits earned on type k housing to depend on the time of development, s SAn example of failure to foresee such a contraction L,; provided in the U.S. recession of 19741975. Homebuilder inventories were at very high levels when this housing :lump arrived. Had it been properly anticipated, builders would have slowed down their activity in advance and would not have had to bear considerable inventory holding-costs during the period. At this writing, homebuiider anticipation of the 1980 contraction seems to have been better anticipated.
252
D.E. Mills, Urban residential development timing
It was noted earlier that the perfect-foresight development process is efficient; no alternative one creates greater land-generated benefits. Because the myopic development process violates (3), it is inefficient. 6 While some development options are more profitable than others ex post, the sum of land-generated benefits throughout the city must be less. Greater benefits would be achieved by withdrawing land from certain of the (less profitable) options and reassigning it to others (more profitable). This, of course, comes as no surprise. If foresight is costless (and we have disregarded the possibility that it could be achieved by short-sighted landowners at a price), there are bound to be gains associated with acquiring and using it. 4.2. Uncertainty and speculation
If landowners have perfect foresight, the development process is characterized in the special case of 3.2 by cascading cessations of the various types, and all exercised development options, regardless of type or time, are equally profitable ex post. If landowners are myopic, all types of development are stopped simultaneously when the land supply is exhausted, but some exerci~d options are more profitable than others. These differences are wholly attributable to whether landowners have and use perfect future information, or only current information. Suppose instead that landowners make current decisions using imperfect information about the future. If the demand-generating mechanism that drives the development process is equally well-understood by all landowners, they will share a common (objective) probability distribution over futur,e outcomes. If they have different informational or forecasting endowments, their expectations will be subjective and various. In the first ease, there will be unanimity among landowners at every moment about the set of development options that should be exercised. In the second, there will generally be a variance of opinion. 7 For purposes here it makes no difference which of these expectational scenarios applies. Either leads to speculative behavior. Consider only the special case of 3.2. A principal difference between the development process in this environment and both the previous ones is that more, pertinent information becomes available to landowners as time passes. Because of this, 6This of course assumes away the extraordinary case where T~= T", i'or all i. "The second case - diverse expectations - may be more important l~ere than in some other markets (e.g., commodities and currency) where speculation is important. This is because these other markets are recurring ones where 'na0ural selection' principles can operate to narrow the range of opinion among speculators over time. The land-development process described here is more a once-for-all kind of allocation problem. There is no mechanism to assure that less skillful ,,peculators are sorted out before irrevocable decisions are made, Even so, arbitrage among ,~ould-be landowners before or at t = 0 in the model might be expected to narrow the range of opinion among eventual decision-makers.
D.E. Mills, Urban residential development timing
253
some types of construction may be stopped while vacant land remains only to be restarted later due to revised expectations about the future. The possibility of such intermittent construction stops and starts means there is no condition here analagous to (5) or (9). All that can be said is that whenever type i construction occurs, its rental rate is at the opportunity-cost level: Ri(t)=oqR,+rci
ff ~i(t)>0.
In spite of these intermittent construction stops, it is unlikely that the process will culminate in simultaneous final stops of all development types when vacant land disappears as under myopia. Just as under perfect foresight, some types are likely to be finally stopped before others. We denote these stopping points as (TT,..., T~). They will generally differ from (T~,..., T,) because landowner foresight is imperfect and ex post mistakes are probable. Only very little can be said by way of comparing these stopping points. If there is at least one housing type - - call it k - - where T~,< Tk, then there must be at least one other type - - call it j - - for which T~> T~. Otherwise, vacant land would remain at max { T~,..., T~,}. Related to the event that ex post mistakes occur is the result that not all exercised options are equally profitable. This occurs whenever an intermittent stop allows profit to accumulate on earlier dwelling units of a type but not later ones. It also occurs when T7 4= T~ for all or several i. Because of the possibility of intermittent stops and starts before T~', it is not possible to infer as much here as under myopia about the profitability of type i development from the rCationship between T~' and T~. All that can be said without further qualification is that if T~' < T~, profits per-unit-of-land for all landowners who fortuitously choose type i housing exceed It. Since stopping points are nonoptimal, and some development options are more profitable ex post than others, the development process is inefficient. The sum of land-generated benefits throughout the city is less than it co'aid be if landowners had perfect foresight, s 5. Conclusion Once we step back from the perfect foresight assumption, some or all of the development process characteristics listed at the end of section 2 will fail to hold. If landowners are myopic, it is most unlikely that all types of development
sit does not follow from this that the development process generated by speculating landowners is not as efficient as it can be. it probably i,~ given their incentive to acquire and use whatever information they can obtain about future market conditions. RSUE
D
254
D,E. Mills, Urbaq residential development timing
will be equally profitable e x post; there will be both winners and losers in retrospect. Related to this is the fact that myopic landowners do not follow the pattern of cascading cessations of the development types as the land supply diminishes. Also, myopic landowners do not anticipate demand contractions and strategically halt development in advance. Because of this, some dwelling units of a type produce more profit than others due only to their time of construction. When the future is uncertain and landowners are not rigidly myopic bm speculative, the development process has some of the non-uniformity characteristics of both the perfect-foresight and myopia models. Stopping points for lhe various types of development are different as under perfect foresight, and e x post profits vary among development options as under :nyopia. These two extreme models define a spectrum of outcomes along which must lie the actual outcome produced by competitive, speculating oevelopers. Finally, it might be added that the results presented in this paper carry over to the case where L is infinite, or so large that the land supply is never a binding constraint. For this case T~ and T~, for all i, and T" are all infinite. Every feature of perfect-foresight, myopic and speculative landowner behavior is preserved except strategic stopping. References Anas, A., 1978. Dynamics of urban residential growth, Journal of Urban Economics 5, 66-87, Arnott, R.J,. 1980, A simple urban growth model with durable housing, Regional Science and Urban Economics !0, 53 76. Arnott. R.J. and F.D. Lewis, 1979. The transitiozb of land to urban use, Journal of Political Economy 87. 161 170. I:ujlta, M., 1976, Spatial patterns of urban growth, Journal of Urban Economics 3, 299-341. Mills, D.E., 1978. Competition and the residential land allocation process, Quarterly Journal of Economics 92, 227-244. Mills. D.E.. 1981, Growth, speculation and sprawl in a monocentric city, Journal of Urban Economics, forthcoming. Shoup. DC.. 1970, The optimal timing of urban land development, Papers of the Regional Science Association 25. 33 44.
URBAN RESIDENTIAL DEVELOPMENT TIMING* David E. MILLS University of Virginia, Charlottesville, VA 22901, USA Received May 1980, final version received November 1980 This paper presents a dynamic model of the residential land development process with hetelogeneous housing capital. Its properties under a regime of competitive developers are examined for three informational scenarios: perfect foresight, myopia and imperfect foresight. A special case is presented where different types of housing construction are halted sequentially as the land supply diminishes rather than simultaneously at the moment it is finally exhausted.
1. Introdeetion The dynamic character of urban residential development has been subjected to rigorous economic analysis only within the last decade. Following the early work of Shoup (1970), contributions by Anas (1978), Arnott and Lewis (1979), Arnott (1980), Fujita (1976) and Mills (1978 and 1981), among others have focussed on various aspects of the intertemporal develvpment process. This paper seeks to extend that literature by taking up the timit:g of residential development and the evolution of the housing stock in a city where the land market is competitive, the demand for housing is growing and the supply of land is fixed. A dynamic model of the development process with heterogeneous housing capital is introduced in section 2. Its various properties under a regime of competitive developers with perfect foresight are discussed in section 3. A special case is presented where different types of housing construction are halted sequentially by competitors as the land supply diminishes rather than simultaneously at the moment it is finally exhausted. When the growth trend of housing demand is not monotonic but subject to intermittent contractions, competitors anticipate them and change the strategic timing of construction accordingly. In section 4 we turn to a regime of imperfect foresight. In this *This paper reports work that was supported by Grant no. H-5036 RG from the U.S. Department of Housing and Urban Development. The substance and findings of that work are dedicated to the U.S. public. The author is solely responsible for the accuracy of the statements and interpretations contained in this paper and such interpretations do not necessarily reflect the views of the U.S. Government. The author would like to thank Richard Arnott, Blaine Roberts and a referee for helpful comments on an earlier version.
0166-0462/81/0000--(O00/$02.50 © North-Holland Publishing Company
240
D.E. Mills, Urban residential development timing
connection, properties of the development process with both myopic and speculative decision-making are considered. Results are summarized in section 5. 2. A model of the development process To set the stage, consider a city with L units of homogeneous land for future residential development. The homogeneity assumption suppresses both Iocational and topographical differences among specific parcels. The assumption that land is fixed in supply accords with several actual situations" two are cities situated in urban areas and surrounded by other cities and natural boundaries like coastlines, and cities where comprehensive planning has restricted residential development to specific parcels. Initially all the land is in an unimproved state, bl,t as the demand for housing expands over time, it is developed residentially. The housing stock consists of dwelling units of n distinct types. The supply of housing at time t is indicated by the vector x(t)--(xl(t),xz(t)...xn(t)). Elements of this vector can be non-integers, indicating the divisibility of dwelling units. A fundamental difference among housing types is the amount of land they require per unit. These quantities are indicated by the vector =1~1.~2 .... , ~,,). There are three kinds of agents who together determine the course of development in the city. The first are tenants, whose collective demand for the ith type of housing at t is exogenous to the model and represented by the lcontinuously differentiable) inverse demand function"
f~(x(t),t),
i = I , 2 ....,n.
Since all dwelling units are in some sense substitutes for each other, we assume --<0 ?x~ -
for all i,j.
The rental rates on housing, characterized by the vector R(t)=(Rl(t), gz(t) ..... R,lt)), adjust instantaneously to changes in demand or supply. Thus.
Ri(t)=fi(x(t),t)t),
i= 1,2,...,n
(1)
holds at every moment. The second kind of agents in the model are firms in the competitive building industry, l'his industry has a constant-returns-to-scale technology,
D.E. Mills, Urban residential development timing
241
so the construction cost of a particular type of dwelling unit does not vary with the overall level of construction activity. These costs (net of land) for the n types of dwelling units are indicated by the vector c = (cz,c2,...,c,). For analytical convenience, construction is assumed to be instantaneous. The final kind of agents and the principal decision-makers in the model are landowners. Landownership is assumed to be sufficiently diffuse that no one has market power. As long as land remains unimproved, the owner's rate of return on it is R, per unit. The development decision concerning a parcel of land (of any size) involves determining a type of housing to be built and a time at which to undertake construction. Because dwelling units are durable and demolition costs are non-negligible, development is considered to be irreversible; previously developed land cannot be returned to an unimproved state or converted to another type of development. Landowners execute development decisions by contracting with firms in the competitive building industry at prices equal to construction costs. Then they either become landlords in the rental housing market or sell their property to other agents who become landlords. For purposes of exploring the timing of development decisions, it makes no difference whether this last transaction occurs, or if it does, when. All agents are assumed to have perfect information about R~, c and current values of R(t). The implications of whether this clairvoyance extends to the future are explored below. The process of land development in the city is fully characterized by x(t), for all t~O, or equivalently, by x(O), ~(t), for all t=>O. This vector time-path must satisly all previous assumptions, so I x(O)-O,
~(t)>=O, • x(t)<-L,
(2a)
t>-O,
(2b)
t>O.
(2c)
3. Perfect foresight In this section, we consider the development process that occurs when landowners have (in addition to perfect current information) perfect foresight concerning R(t) for all t>O. This is a strong assumption since R(t) is determined endogenously in the model. Recalling (1), it implicitly requires 1Another interpretation can be placed on the constraint (2C). Suppose the binding constraint on development is not land supply, but a legislated maximum number of residential units in the city. This resembles the actual constraint that applies in a growing number of cities with socalled 'growth controls'. To adapt (2) to this circumstance, let ~t be the unit vector and L the legislated maximum number of dwelling units. All subsequent results apply.
D~. Mills, Urban residenzial development timing
242
that landowners have exhaustive information about the f~(.) functions and can accurately foresee x(t) throughout the future. The reasons for studying this case are two. First, it enables us to characterize the development process when the perfect-foresight assumption is approximately true - - that is when landowners have good information about future events. Since returns on developed land will be shown to depend crucially on one's ability to predict R(t) accurately, landowners have considerable incentive to collect information concerning future rental rates. Second, this case serves as a benchmark for comparison with the development process that occurs under other assumptions and informational environments. Two departures from the present assumption are considered in section 4 below.
3.1. General properties of a competitive equilibrium Since landownership is decentralized, the land-conversion process must have all the properties of a competitive equilibrium. To see what this means, note that every landowner has the same set of development options for a parcel of land (no matter whether it represents all or only part of his holdings). These options are infinite in number and identified by both a type and time of development: (i,t). If landowners have perfect foresight, a competitive equilibrium will not exist unless all options actually exercised are equally profitable ex post. No option will be voluntarily exercised when a more profitab!e one exists. To make this property of the development process more rigorous, note first that the discounted (to t = 0 ) profit to the owner of one unit of land left perpetually unimproved is R j r , where r is the universal discount rate. Clearly, competitive landowners will not develop land unless there is an option that generates profit per land unit in excess of this amount. If a landowner exercises option (i, t) on a unit of lan::l, then differential profits attributable to development are V~(t)/~i, where V,(t }= j e - " [ f i ( x ( r ) , r ) - a i R . ] d r - e - " c i for all i. t
Let the maximum value of these profits over all options be p = max { V/(t)/a,}, i.!
where # > 0 by assumption. Then a necessary and sufficient condition for competitive equilibrium is (.,~(t)). (V~(t)/a~-#)=O
for all i, t.
(3)
D.E. Mills, Urban residential development timing
243
If option (i, t) is exercised, then ~(t) > 0 and Vj(t)/0c~=/~ so that no forgone option is superior to it. If the option is forgone, then ~ ( t ) = 0 and V~(t)/a~<#. No option is exercised when a more profitable one exists, so all exercised options are equally profitable. Although (3) is necessary and sufficient for a competitive equilibrium in the land-development process, it does not imply that the process is unique. Because housing production is characterized by constant-returns-to-scale, there is an absence of strict cost-convexity in the model. For this reason there can be an infinite number of vector time-paths x(t), for t_>_0, that
satisfy (3). One implication of (3) worthy of note is that if ~(t)>O
for some i and all t ~ (tl,t2),
where t2 > t ~, then
R~(t)=aiR,+rci
for all
t6(tl,t2).
This is sho~n as follows. The hypothesis implies that
~(t)-O
for all
t6(t~,t2),
which frown differentiating V~(t) implies that
fi(x(t),t)-o~R,-rc~=O
for all
t~(tl,t2).
This with (1) produces the stated conclusion. Over any interval of time where continuous construction of some dwelling-unit type occurs, its rental rate is maintained at the 'opportunity cost' level: that level where revenues equal the sum of opportunity land-cost and amortized building cost. The effect of the land-supply constraint on the competitive development process depends on the time-dependence properties of the various f~ functions. If demand for all types of housing stops growing before L units of land are developed, then the constraint has no effect. In order for this to be an equilibrium, # must be zero so that owners of undeveloped land are no worse off than others. In other circumstances, the land supply is exhausted and # is positive. (A strong sufficient condition for all L units of land to be developed in finite time is that for some i and all t>__O, ~f~/~t> 6, a small positive constant.) Even where the land supply is exhausted in finite time, there is no reason to believe that all types of dwelling-unit construction are halted simultaneously. ~f the moment at which type i construction is finally stopped
244
D.E. Mills, Urban residential development timing
is defined as Ti=min{t:~i(z)=O
if
r>t},
then the moment at which the last unit of land is developed is: max {7"1, T2,..., T.}. All type i construction occurs within the time-interval Since (3) requires that every option actually exercised be equally profitable, the wide variation in their cash-flow characteristics is immaterial. In particular, the discounted profit on a type i dwelling unit built at some t < T~ must be the same as one one built at T~. This means no profit accrues on the earlier-constructed dwelling unit until T~. The discounted sum of revenue it produces between the moment of construction and T~ is exactly offset by the discounted sum of the amortize~ building cost and the opportunity cost of land. Thus for any (i,t), wllere ~i(t)>0, Ti
e-"[fi(x(r), r ) - a , R . - rci] dz=O. t
If there is a market for land or dwelling units (in addition to the market fol their services) in this world of perfect foresight, prices will reflect the full exp~cta,.io_-, of future profit. The price of a unit of unimproved land at t = 0 will be p + R./r. At any future date prior to its development, the price will be e"p+R./r. It increases with time unless p = 0 in which case it remains constant. The price of an existing type i dwelling unit (including the site it occupies) at t will be the discounted (to t) sum of future revenues: oo
Pi(t)= $ e-'~'-')f/(x(r), z)d¢. !
Utilizing (1), the rate at which it changes is
15i(t)= rPi(t ) - Ri(t ), and is positiv~ or negative depending on the relative magnitudes of terms on the right-hand side. Owing to the fact that it is a competitive equilibrium [as depicted in (3)], the development process under conditions of perfect foresight is efficient. Barring fiscal cr other externalities (all of which have been excluded from the analysis), there is no vector time-path x(t) that generates greater (discounted) aggregate econo~nic benefits. 2 ZThis is rigorously demonstrated in Mills (1978) for a similar model.
D.E. Mills, Urban residential development timing
245
3.2. A special case Further properties of the development process can be stated if housing demand satisfies two further assumptions. The first is that initial demand for every housing type is non-negligible. Specifically, assume
~(O,O)>a~R,+rc~ for al~ i. The second is that demand changes at every t in such a way that the vector
~(t) that solves [ fdOxj] .
) + bgfdet] = o
is strictly positive [that is every element ~ ( t ) > 0 ] for every t>0, where [c3fffdx~] and [dfffdt] are respectively fin n × n matrix and an n-vector with typical elements shown. Since all elements in this matrix are non-positive, a necessary condition for this assumption to hold is that demand for all types of housing be strictly increasing:
d~lOt > 0
for all i and t > O.
(4)
With the first of these assumptions, x(t) is discontinuous at t = 0 and the vector ~(0) has elements that are infinite. But after this initial building spurt, constrt~ction rates remain finite (as long as elements of [dfffc3t] do). With the second a:sumption, the existence of a vector x(t) for which ~i(t)>O
for all i and te(O, TJ
is assured, and from previous elucidation of (3), this means
Ri(t)=~R..+rci
for all i and t e (0, T~].
15)
Simultaneous construction of all housing types continues throughout the time-interval [0, min {Tt,..., T~}], and the rental rate on each type of dwelling unit remains at its opportunity-cost level. At min {Tl,..., T~} (at least) one type of construction is halted. From this moment forward, while other types of construction continue as before, the rental rate on the halted type(s) increases and profits accumulate: Ri(t ) > ~iR~ + rc~
for t > T~ for all i.
Subsequently and in the order of increasing T~s, other types of building are halted and profits begin to accumulate on them as well. Thus, the process is brought to a close in a sequence of cascading cessations. Many types of
246
D.Eo Mills, Urban residential development timing
construction (at most, all but one) will have been stopped before the city's supply of land is exhausted. This pattern of cascading cessations emphasizes the importance of foresight and strategic timing on the part of landowners. Once a type of building has been stopped, its rental rate rises above the opportunity-cost level. Owners of vacant land therefore exercise restraint in forgoing development opportunities with high immediate returns in favor of future options that are, in the f'mal analysis, more remunerative. Another consequence of this pattern is that rental rates on different housing types begin to rise above their opportunity-cost levels at different times. To a bystander, observer or tenant who is less well-informed than landowners about the housing-supply mechanism and future market conditions, it will appear irrational that construction stops and rentals rise sconer on one type of housing than another. It is important therefore to remind ourselves that under the assumptions posed this pattern is a property of the eOicient development process. The order of cessation among dwelling-unit types is something about which nothing ~:ery general can be said. It depends not only on the relative magnitude of the elements of [c~fJc~t], but also on the elements of [OfJc3x/l. An example of ~he kind of result that can be wrung from the model if further assumptions are made is the following: Suppose
~(x(t),t)-eehi(xi(t))
for all i,
where q
Integrating and substituting e*T'h(xi(T, ))= a~R. + rc~ into this expression yields
e-'r'(R.+rG/a~)ll
r q
!/=/~
for all i.
Since the last term on the left-hand side is a positive constant, this expression implies the desired result.
D.E. Mills, Urban residential development timing
247
that this implies T~> Tj if and only if Ck/ak> cj/oej. That is, if there is no cross-elasticity of demand among housing types, and the 'pure' secular trend in "demand is uniform across types, then the order of cessation will be that of increasing building costs per unit of land. High-rise apartment buildings will continue to be constructed after new single-family dwellings and garden-apartment buildings have been stopped. If the first assumption above - - that initial demand for every housing type is non-negligible- is dropped while the second is kept, then f~(O, O) < ~iR~ + rc~ for one or more i. In this case, the development process differs from (5) only in that ~(t)=O over an initial time-interval for those housing types with negligible initial demand. A pattern of staggered starts for different construction types emerges that resembles the subsequent pattern of cascading cessations. Nevertheless, the order is of starts and stops unrelated.
3.3. Discontinuous development If we depart from the second of the assumptions that define this case, holdin$ in place the first, the development process becomes more di~cult to describe. 3ecause (4) is a necessary condition for the assumption in question, a sufficient condition for departing from it is that demand contracts over some interval of time for one or more of the housing types. The most prominent examples of such periods are associated with macroeconomic recessions and the building cycle. We consider here only the least complicated departure from (4) that is possible. The results suggest the kind of changes that might be expected from more complicated departures since the latter are not taken up explicitly. Let there be one housing type k such that
dfkldt
0
for
~t=t~,t2
,
(6)
~.t q~[tl,t2] where 0 < t l
D.E. Mil/s. Urban residential development timing
248
it is clear that with (6) the competitive development process cannot proceed as it did before beyond time tl. Suppose, however, that (5) holds for every i ¢ k, and that
Rt(t)=~hR.+rch
for O < t < t t .
Then, because of (6) and because other (substitute) types of construction continue, Rh(t) begins at tt to fall below ~tR.+rch and landowners c e a s e building type k dwelling units. They do not begin again until Rk(t) rises to its previous level, beyond which time (5) holds for type k as well. Clearly, this does not occur until after t2. One consequence of all this is that throughout the period when type k construction is stalled, loss s accumulate on every previously-constructed type k dwelling unit. Furthermore, these losses are not offset by previous or subsequent gains. This means landowners who build type k housing after t2 make greater discounted profits (per-unitof-land) than those who build them before tt. This is a clear violation of (3), and so cannot be an accurate description of the competitive development process. From this we conclude that landowners with perfect foresight halt the construction of type k dwelling units before t~. In fact, they will discontinue it sufficiently far in advance that profits accumulate on existing type k dwelling units to offset the losses incurred after t t. Thus, the competitive developmet.:t process differs from (5) in that type k construction is stalled between to .rod th, where
O
e - n i l k(x(t ), t ) - ~tR. + rct] dt = 0.
(7)
la
This equation assures that discounted profits on type k dwelling units built after t b are the same as on those built before ta. The rental and construction rates for type k dwelling units must therefore satisfy
t=t {
R~(t) ~ ~kR.+rck
for
(0, ta) and
tE
[ta, tt,]
and
it(t)
(tb, Tk)} .
tt
>= 0
D.E. Mills. Urban residential development timing
249
If t,, should be 0, then type k construction does not begin in the first place until tb. The importance of foresight and strategic timing on the part of landowners is even more apparent here than in the previous special case. Laadowners must not only give attention to the question of when construction of a particular type is to be finally stopped, but also to the question of when it should be temporarily stopped to avoid net losses during a contraction. When building is stopped in anticipation of a coatraction, rental rates rise above, then fall below, and finally return again to their opportunity-cost level. Once again, the housing-market observer who is less well-informed than landowners will regard such discontinuities in development as irrational (or at least so whenever rental rates exceed their opportunity-cost level). Before proceeding, it might be well to summarize the main characteristics of perfect-foresight residential development.
(i)
Since no development option is exercised when a more profitable one exists, all options actually exercised are equally profitable. As a consequence, net profits do not begin to accumulate on dwelling units of the ith type until T~, the moment in which the last one is built. (ii) For the special case of 3.2, all construction rates are strictly positive throughout (0, min {TI,..., T~}). Not all types of dwelling-unit construction are halted simultaneously, but rather follow a pattern of cascading cessations. Rental rates on dwelling units remain at their ooportunity cost level for as long as construction of their respective typ~s continues; profits do not begin to accumulate on type i dwelling units "mtil T~. (iii) If demand for some type of housing contracts before all dwelling units of that type have been built, landowners will anticipate the contraction, halt construction in advance so profits can accumulate and offset foreseen losses. During the stall the rental rate first rises above, then falls below before finally returning to the opportunity-cost level. !
4. Imperfect foresight The assumption that landowners are endowed with perfect foresight about rental rates is not, in a purely descriptive sense, accurate. It is therefore important to consider how the development process might change if foresight is imperfect. We begin that task in this section by looking at the case where landowners are myopic - - where they make current decisions using current information only and disregard completely whatever information they have about the future? 4This is essentially the assumption attributed to decision-makers by Anas (1978).
250
D.E. Mills, Urban residential development timing
The purpose in taking up this case is not that it is more realistic than perfect foresight. Indeed it is probably less realistic given landowners' considerable incentive to acquire good information about the future. Instead, the purpose is to define a spectrum of outcomes that arise when landowners have imperfect foresight. These two cases m perfect foresight and myopia m lie at opposite ends of this spectrum and their comparison allows several conclusions about the nature of intermediate outcomes, all without constructing an elaborate expectational model. Before proceeding, we impose one further assumption on the model of section 2. In addition to the non-positivity assumption, the elements of [?f~/~xj] are henceforth assumed to satisfy
I f,/vx, I> max {I ~J Of,lO sl , I}
for all i,j.
(8)
J
This means there is no way to develop a parcel of vacant land to reduce Ri as much as if it were developed as type i housing. 4.1. Myopic behavior
The effect of landowner decisions based solely on current market information is easiest to see in the special case of 3.2. The process is the same here as there throughout the time-interval [0,min {Tl,..., Tn}). Construction of all types proceeds at rates that maintain rental rates at their opportunitycost level. At min{T~ .... , T,}, landowners with perfect foresight stop building one (or more) type of dwelling unit even though demand for this type of housing continues to grow and vacant land remains in the city. This is because additional building of that type would produce less profit than other alternatives. If landowners are myopic, they forgo this kind of consideration and plunge ahead with building at a rate that keeps the rental rate at its opportunity-cost level. This is true not only for the first dwelling-unit type which prescient landowners would halt, but for the others too. In fact, building of all types will continue as long as any vacant land remains. Similarly, rental rates will remain at their opportunity-cost level as long as vacant land remains. Thus R,(t)=~iR.+rc i
for te(O, T ' )
for all i,
(9)
where T = is defined as the moment when the land supply is exhausted: Tm= min {t :~t . x(t )= L}.
it is clear that T m lies somewhere strictly between the maximum and
D.E. Mills, Urban residential development timing
251
minimum elements of (7~,..., T,), since otherwise a . x ( T ' ) would be either greater or less than L. Thus, relative to perfect foresight, some kinds of construction are continued too long and others are stopped too soon because the land supply is exhausted. Too many dwelling units of some types are built and too few of others. Because of this, some development options exercised are more profitable ex post than others. To see this, note first from (9) that no profits accumulate on any dwelling unit until T". This means that profits on dwelling units of the same type are, here as before, equal regardless of the time of development:
V,(T')
---cti
for any t ~ (0, T m) for any i.
~i
Profits per unit-of-land are not the same, however, across all dwelling unit types. V~(T")/0q must (generally) be greater than p, the profits that would occur on al~ parcels under perfect foresight, for some types i and less for others. In particular, profits are greater than p for those types where T~> T", and less for those types where T~< T". It will suffice to prove only the first half of this result. Consider some type k of housing for which Tk> T s. Under perfect foresight, this type of building would persist longer than under myopia. Because of this, xk(T") under myopia is less than xk(Tk) under perfect foresight. Together with (8), this assures us that fk(x(t),t) is greater under myopia than under perfect foresight for all t > Tm. This means that ~(T")/ah ~'nder myopia is greater than Vk(T~)/ak under perfect foresight which equals #. The implications of myopia are even more extensive if we return to the case in 3.3 where demand contractions occur. Myopic landowners will not at to or any other moment previous to t l stop construction of type k dwelling units. The contraction will not be anticipated. At tl, Rk(t) begins to fall due to the contraction and type k construction is at last stopped. But it is too late for owners of type k housing to accumulate profits to offset losses that begin at tl. Since new type k construction will begin once Rk(t) rises again to its opportunity.cost level (and providing that this occurs before T m when the land supply is exhausted), the ex post profit on type k housing built before t l will be less than that on those built later. By failing to anticipate the contraction and strategically halt construction before it arrives, myopic landowners cause profits earned on type k housing to depend on the time of development, s SAn example of failure to foresee such a contraction L,; provided in the U.S. recession of 19741975. Homebuilder inventories were at very high levels when this housing :lump arrived. Had it been properly anticipated, builders would have slowed down their activity in advance and would not have had to bear considerable inventory holding-costs during the period. At this writing, homebuiider anticipation of the 1980 contraction seems to have been better anticipated.
252
D.E. Mills, Urban residential development timing
It was noted earlier that the perfect-foresight development process is efficient; no alternative one creates greater land-generated benefits. Because the myopic development process violates (3), it is inefficient. 6 While some development options are more profitable than others ex post, the sum of land-generated benefits throughout the city must be less. Greater benefits would be achieved by withdrawing land from certain of the (less profitable) options and reassigning it to others (more profitable). This, of course, comes as no surprise. If foresight is costless (and we have disregarded the possibility that it could be achieved by short-sighted landowners at a price), there are bound to be gains associated with acquiring and using it. 4.2. Uncertainty and speculation
If landowners have perfect foresight, the development process is characterized in the special case of 3.2 by cascading cessations of the various types, and all exercised development options, regardless of type or time, are equally profitable ex post. If landowners are myopic, all types of development are stopped simultaneously when the land supply is exhausted, but some exerci~d options are more profitable than others. These differences are wholly attributable to whether landowners have and use perfect future information, or only current information. Suppose instead that landowners make current decisions using imperfect information about the future. If the demand-generating mechanism that drives the development process is equally well-understood by all landowners, they will share a common (objective) probability distribution over futur,e outcomes. If they have different informational or forecasting endowments, their expectations will be subjective and various. In the first ease, there will be unanimity among landowners at every moment about the set of development options that should be exercised. In the second, there will generally be a variance of opinion. 7 For purposes here it makes no difference which of these expectational scenarios applies. Either leads to speculative behavior. Consider only the special case of 3.2. A principal difference between the development process in this environment and both the previous ones is that more, pertinent information becomes available to landowners as time passes. Because of this, 6This of course assumes away the extraordinary case where T~= T", i'or all i. "The second case - diverse expectations - may be more important l~ere than in some other markets (e.g., commodities and currency) where speculation is important. This is because these other markets are recurring ones where 'na0ural selection' principles can operate to narrow the range of opinion among speculators over time. The land-development process described here is more a once-for-all kind of allocation problem. There is no mechanism to assure that less skillful ,,peculators are sorted out before irrevocable decisions are made, Even so, arbitrage among ,~ould-be landowners before or at t = 0 in the model might be expected to narrow the range of opinion among eventual decision-makers.
D.E. Mills, Urban residential development timing
253
some types of construction may be stopped while vacant land remains only to be restarted later due to revised expectations about the future. The possibility of such intermittent construction stops and starts means there is no condition here analagous to (5) or (9). All that can be said is that whenever type i construction occurs, its rental rate is at the opportunity-cost level: Ri(t)=oqR,+rci
ff ~i(t)>0.
In spite of these intermittent construction stops, it is unlikely that the process will culminate in simultaneous final stops of all development types when vacant land disappears as under myopia. Just as under perfect foresight, some types are likely to be finally stopped before others. We denote these stopping points as (TT,..., T~). They will generally differ from (T~,..., T,) because landowner foresight is imperfect and ex post mistakes are probable. Only very little can be said by way of comparing these stopping points. If there is at least one housing type - - call it k - - where T~,< Tk, then there must be at least one other type - - call it j - - for which T~> T~. Otherwise, vacant land would remain at max { T~,..., T~,}. Related to the event that ex post mistakes occur is the result that not all exercised options are equally profitable. This occurs whenever an intermittent stop allows profit to accumulate on earlier dwelling units of a type but not later ones. It also occurs when T7 4= T~ for all or several i. Because of the possibility of intermittent stops and starts before T~', it is not possible to infer as much here as under myopia about the profitability of type i development from the rCationship between T~' and T~. All that can be said without further qualification is that if T~' < T~, profits per-unit-of-land for all landowners who fortuitously choose type i housing exceed It. Since stopping points are nonoptimal, and some development options are more profitable ex post than others, the development process is inefficient. The sum of land-generated benefits throughout the city is less than it co'aid be if landowners had perfect foresight, s 5. Conclusion Once we step back from the perfect foresight assumption, some or all of the development process characteristics listed at the end of section 2 will fail to hold. If landowners are myopic, it is most unlikely that all types of development
sit does not follow from this that the development process generated by speculating landowners is not as efficient as it can be. it probably i,~ given their incentive to acquire and use whatever information they can obtain about future market conditions. RSUE
D
254
D,E. Mills, Urbaq residential development timing
will be equally profitable e x post; there will be both winners and losers in retrospect. Related to this is the fact that myopic landowners do not follow the pattern of cascading cessations of the development types as the land supply diminishes. Also, myopic landowners do not anticipate demand contractions and strategically halt development in advance. Because of this, some dwelling units of a type produce more profit than others due only to their time of construction. When the future is uncertain and landowners are not rigidly myopic bm speculative, the development process has some of the non-uniformity characteristics of both the perfect-foresight and myopia models. Stopping points for lhe various types of development are different as under perfect foresight, and e x post profits vary among development options as under :nyopia. These two extreme models define a spectrum of outcomes along which must lie the actual outcome produced by competitive, speculating oevelopers. Finally, it might be added that the results presented in this paper carry over to the case where L is infinite, or so large that the land supply is never a binding constraint. For this case T~ and T~, for all i, and T" are all infinite. Every feature of perfect-foresight, myopic and speculative landowner behavior is preserved except strategic stopping. References Anas, A., 1978. Dynamics of urban residential growth, Journal of Urban Economics 5, 66-87, Arnott, R.J,. 1980, A simple urban growth model with durable housing, Regional Science and Urban Economics !0, 53 76. Arnott. R.J. and F.D. Lewis, 1979. The transitiozb of land to urban use, Journal of Political Economy 87. 161 170. I:ujlta, M., 1976, Spatial patterns of urban growth, Journal of Urban Economics 3, 299-341. Mills, D.E., 1978. Competition and the residential land allocation process, Quarterly Journal of Economics 92, 227-244. Mills. D.E.. 1981, Growth, speculation and sprawl in a monocentric city, Journal of Urban Economics, forthcoming. Shoup. DC.. 1970, The optimal timing of urban land development, Papers of the Regional Science Association 25. 33 44.