A Model of Development Timing
Commercial areas of a city, even those regarded as "built-up" or fully developed, typically
undergo continual redevelopment. Except for protected sites, no building is immune from
economic forces that could lead to its replacement. The timing of such development depends on
a number of factors including the current and expected levels of net income from the existing
building, the expected net income from a new building, the cost of construction, the time rate of
discount and the permitted density of a new building.
If there exists a reasonably well functioning market for commercial land, current property values
will reflect market views about the best time for development, which could be very distant or
very near, and the best density for the new building. Thus, given expectations, current property
values imply some path or paths of future property use. Density regulations can affect the
development of individual lots and therefore the path of overall development in ways that are not
always obvious. Thus a model of the effect of density on development is an important planning
tool and one that is necessary in order to understand the impact of density changes.
The model used here begins with a production function for floor space, measured as density, i.e.,
as floor space per unit of land area, D . D is a strictly concave function of the ratio of
construction capital to lot area, K . Thus D  f ( K ) with f '( K )  0, f ''( K )  0 . For our
purposes, this can be immediately specialized to the standard function used in this type of study,
D  K  , 0    1 .1 Construction capital as a function of density can therefore be written as
1
K  D .
Time enters explicitly into this model through the functions for construction cost and the expected
net income per unit of floor area. These two parameters are given as Ct and Pt respectively,
where " t " stands for time, with the present time represented by t  0 . The cost of construction
1

at some time t can be written as KCt , or as D Ct , and the net revenue from some building with
density D may be written as DPt .2
Consider some property, with or without an existing building, and suppose that it is planned to
develop on this site a new building with density D at some time T in the future. The new
1
One unit of capital with this production function produces one unit of density. Thus a unit of capital is a
one square foot or square metre of building per square foot or square metre of land when density is one.
Higher densities take more than a proportionate amount of capital. The price of capital, Ct , is the price of
one square foot of floor space with a density of one. Because capital requirements often change discretely
and not continuously when certain densities are reached, this production function does not capture all the
details of building construction but rather just the broad tendency for costs per unit of floor area to rise as
buildings are denser, quality held constant.
2
For any given building only one market price of floor space is introduced. Thus the possibility that the
market price of floor space may be higher on higher floors of some given building is not modelled
explicitly. However, this does not affect the results as long as building costs per square foot rise faster with
height than market floor-space price. This has to occur at some point or else optimally sized buildings with
no zoning restrictions would be indefinitely high.
1
building will then exist and earn revenue into the indefinite future, i.e., only a one-time
conversion to a new building is being modelled.
Given this, the present value at time T of the property may be written as the present value then of
the expected net income flow minus the cost of constructing the new building. Revenue from the
existing building, if any, does not enter into the valuation at T since the old use has ended.3 This
present value at T , VT , is therefore

1
VT   DP0emt e r (t T ) dt  D  C0enT , where " r " is the time rate of discount, " m " is the rate at
T
which floor space net rents are growing, P0 is the floor space net rent at time " 0 ", today, " n " is
the rate at which construction costs are rising, and C0 is the construction cost today per unit of
capital.
Brought back to the present, t  0 , the present value today of the property at conversion time
T is
(1)
VT e
1


mt  r ( t T )

   DP0e e
dt  D C0e nT  e rT .
T

 rT
Until conversion time T , the existing property is earning a net revenue of R (t ) per unit of lot
area at time t with t  T . The present value today of that stream of revenue is given as
T
(2)
V0   R(t )e  rt dt
R
0
The complete expression for the present value today of the property is given by the sum of
equations (1) and (2)
(3)
V0  V0  VT e
R
 rT
1


mt  r ( t T )

  R(t )e dt    DP0e e
dt  D C0e nT  e  rT
0
T

T
 rt
Equation (3) has two variables, density, D , and conversion time, T . Given expectations about
prices and costs and given the discount rate, the optimal conversion time and the optimal density
of the new building can be worked out. This may be done by differentiating equation (3) first
with respect to density and then with respect to timing and setting each partial derivative equal to
zero. This yields two first-order optimality conditions which can be used to solve for optimal
T and optimal D . If the density were constrained to some binding density, D , this could be
inserted in equation (3) in place of the variable D and optimal conversion timing at the allowed
density worked out.
For purposes of this report, it is more useful to move to a graphic illustration of the effect of
density and timing on property values. This allows us to look easily at the effect in particular of
changing a density constraint.
3
Demolition costs are ignored.
2
To do this, take equation (3) and solve the integral expressions. Simplifying things slightly by
assuming that the net revenue stream from the existing property, R  t  , is a constant4 R , this
yields
(4)
V0 
1
DP0 mT  rT
R
1  e rT  
e e  D  C0enT e  rT
r
r m
We can use equation (4) to simulate the present property value V0 given various assumptions
about density and conversion timing. To do this we need to put specific numbers on the various
parameters R, r , P0 , m, C0 , n and  .
By way of illustration, suppose the existing property is generating a net revenue stream per square
foot of land area of $60 a year. This might represent an older 5-times coverage building yielding
a net rent of $12 a square foot. Suppose further that today's market net rents for new buildings
are $25 a year per square foot of floor space and expected to rise at a rate of 4 per cent a year,
construction costs are $120 per square foot and expected to rise at 2.5 per cent a year, and  is
0.75. This is equivalent to the following:
R  60, r  0.1, P0  25, m  0.04, C0  120, n  0.025, and   0.75 .
If these numbers are put in to equation (4) and density allowed to vary from 0 to 60 times
coverage while conversion timing varies from 0 to 40 years from today, then a series of values
can be calculated using equation 4. These values represent the present value today of the
property under each pair of assumptions about conversion timing and density. The matrix of
values so calculated can best be displayed as a contour plot of present value, as in Figure 1.
Figure 1 should be read just like any topographical contour map. Each line shows an equal
valued combination of density and timing. Moving towards the point at the centre of the contours
is equivalent to going up hill, only now it is the present property value not the altitude that is
rising. Given the assumed parameters, the highest or optimal present property value is achieved
with a planned conversion to a new building with a density of 26-times coverage in nine years.
Those are the vertical and horizontal coordinates of the highest point on the contour plot. Given
the parameters, any other density or development timing entails a lower value today for the
property. I.e., any other pair of density and timing coordinates will put one on a lower contour
line.
Suppose that density regulations do not permit the optimal 26-times coverage building. Suppose
that the maximum allowable zoning is only, say, 10-times coverage. The density is thus set but
Figure 1 can still be used to determine the best time to convert to a 10-times coverage building.
Do this by finding the density of 10 on the vertical axis and then look horizontally across at this
level until the highest value contour is reached.
In this case, notice that even as one begins to move to the right from a time of "0", i.e., today, the
contour values start going down. In fact, at 10-times coverage the highest contour that can be
reached is at time 0. If 10-times coverage were the limit, the property would be ready today for
4
This does not affect the interpretation of the results, but it does simplify equation (4). It also might not be
an unreasonable assumption with respect to an older building unable to keep up with rising market rents
because of its growing obsolescence.
3
60
50
density
40
30
20
10
0
0
5
M
10
15
20
25
30
35
40
years from today
Figure 1
60
50
density
40
30
20
10
0
0
M
5
10
15
20
25
years from today
Figure 2
4
30
35
40
60
50
30
20
10
0
0
5
10
M
15
20
25
30
35
40
years from today
Figure 3
60
50
40
density
density
40
30
20
10
0
0
M
5
10
15
20
25
years from today
Figure 4
5
30
35
40
development. If the assumed numbers were in fact the numbers for this property and the density
restricted to 10, then the value-maximizing model says that the property is ripe for development
at that density.
From this, the effect of raising the allowable density can be determined. If allowable density is
raised to above 10-times coverage, then the optimal development time will be pushed back. If the
density restriction is either removed altogether or raised to anything above 25-times coverage
then, as we have seen, the optimal thing to do is to delay development for nine years and then
build a 26-times coverage building. Raising the density limitation has delayed the optimal
development timing. This is a common response even for properties that are ripe for
development: if the allowable density is raised it becomes worthwhile to wait for a period and
develop either at the new, higher allowable density or at an unrestricted optimal density. To go
ahead today with the larger coverage would entail building an inefficiently large building for
today's market; it is more profitable to let net rents rise for a period before building.
Suppose now that the revenue stream from some existing property is not $60 per square foot of
land area but only $30. This might represent 5-times coverage building in poor repair generating
only $6 per square foot per year in net revenue. With no change in the other parameters, the
contour map shown in Figure 2 is the result.
If in the situation depicted in Figure 2, the maximum allowable density was 10-times coverage
then, as in Figure 1, it would be optimal to proceed right away with development. However, if
the density restriction were eased, or eliminated altogether, the optimal development would still
occur very soon, in one or two years, with a density of just under 20-times coverage. With such a
low current net revenue stream, it is not worthwhile to wait any longer. This would be regarded
as a "soft" site both with respect to the optimal timing of development with the original density
restriction and with respect to development under a looser density restriction. With a higher
allowable density, development still might be delayed, as in Figure 1, but not by very much.
Rather than having lower existing net revenues, i.e., a lower R value, suppose these revenues are
substantially higher. Figure 3 is drawn on the assumption that R  120 with no other change in
the parameter values. This might represent a well maintained older building at 6-times coverage
earning net annual rents of $20 a square foot.
With unrestricted density, the best conversion time would not be expected to be about seventeen
years from today and the optimal conversion density would be at about 37-times coverage.
Suppose, however, that as before allowable density is only 10-times coverage. Going through the
same exercise as before, one sees that it would be better not to develop today to this restricted
density. The contours keep rising as a horizontal line at the 10 density level is imagined going
from 0 years to the right. The best time to convert to 10-times coverage would be 10 or 11 years
from now. The reason this is different from the previous situation is that it is more costly to give
up the current net revenue earnings from the existing building. If the density restriction is now
eliminated, development will be pushed back a number of years. This is not a soft property, one
that is ripe for current development under either the current density restriction or one that was less
restrictive.
Looking still at Figure 3, one can see that if a 10-times coverage density limitation were eased to
say 18- or 20-times coverage, the timing of development would not be greatly affected. A more
dense building would be built, but the optimal timing for an 18- or 20-times coverage building
would still be around ten years, maybe even slightly earlier. This possibility, that raising the
allowable density will encourage earlier development, may be seen more clearly if the initial
6
density restriction had been only, say, 5-times coverage. At that allowable density, the optimal
time for development would be about twenty years hence. If the allowable density was now
raised to somewhere between 10- and 20-times coverage, the optimal development time would be
brought forward to about ten years hence, as we have seen. If the allowable density had been 5times coverage but density restrictions were then eliminated, optimal development timing would
become seventeen years, not much different from the optimum with only 5-times coverage
allowed.
Clearly, with an increase in allowable density, development timing could be moved ahead or
delayed. This is one of the reasons why density controls are not very effective instruments for
controlling the path over time of floor space development.5
This analysis has not taken into account the possibility that developers will be uncertain about the
allowable density at some time in the future. Suppose that allowable density has been 10-times
coverage. The City then announces that allowable density will be raised to 30-times coverage, at
least for some properties. Every affected owner or developer believes that they will always be
able to build at least at 10-times coverage but they have considerable uncertainty about how long
a window of opportunity to build up to 30-times coverage will last.
The decision thus facing the developer may be illustrated by again turning to Figure 1. At
allowable densities up to 30-times coverage, the best course of action for the developer of the
Figure 1 property would be to wait nine years and develop at 26-times coverage if it was believed
that they would with certainty be allowed to develop at that density at that time. But if there is
uncertainty about the future ability to develop at a higher density, the owner would do better to
develop now, at about 18-times coverage, as can be seen from the diagram, rather than wait and
find that he or she has to develop at the original density limitation of 10-times coverage. A
combination of 18-times coverage and developing today puts the owner on a higher present value
contour the combination of 10-times coverage and developing in nine years. Neither, of course,
is as valuable as waiting nine years to develop at 26-times coverage. There is an incentive, in the
face of the uncertainty, to develop early at a inefficiently low density.6
The matter is not so clear in the situation illustrated in Figure 3. Under the same circumstances as
described in the above paragraphs, a developer deciding to go ahead today under a policy that
allowed up to 30-times coverage would, as in the situation illustrated in Figure 1, build to a
density of just under 20-times coverage. But now, developing today this coverage would put the
owner on about the same contour line as waiting ten or fifteen years and then finding that density
was again restricted to 10-times coverage. The owner of this property would probably prefer to
wait and take a chance that the higher allowable density would continue into the future.
One final matter may be illustrated through the use of contour plots. Suppose we have a set of
parameters exactly the same as those underlying Figure 1 except that P , the current net rent
available from new properties, is lowered from $25 per square foot to $20. This might occur
because a new supply of floor space has come on to the market driving down net rents. Figure 4
5
There is one circumstance under which raising the density limitation will always bring forward the
optimal time of development. This is when the expectation of rates of increase in net floor space revenues
is about the same as the expectation about rats of increase in construction costs. In terms of the symbols
used here, when m  n , raising or eliminating the density limitation will always result in earlier optimal
developments.
6
It is inefficient in the sense that the present value of the property is less than it would be with the optimal
combination of density and timing, i.e., 26-times coverage in nine years.
7
illustrates the effect on the present value contours. Recall that in Figure 1 the optimal density is
26-times coverage for a building to be built in nine years. Increased current competition in the
space market drives down P and leads, as in Figure 4 to a less dense building, at about 22-times
coverage, and a substantial delay in its development, with the optimal timing now being about
twenty years hence. This illustrates the way in which the market acts to govern the pace of floor
space development: increases in current supply lower current net rents and lead to some
developments being delayed, and supply shortages lead to higher current net rents with the result
that developments are brought forward.
8