Download Optimal Property Development

©David M. Nowlan 1999
Optimal Property Development with S = F(K,L)
I have shown how the urban land market works to allocate land to its value-maximizing use and
in so doing to determine the type of use and the size of building on a lot, and the timing of
development or redevelopment. In this note, the value-maximizing argument is formalized
through the use of a simple static model and the model is then applied to a discussion of the role
of land value in allocating land to alternative uses and to an analysis of the impact of property
taxes on property use.
The model begins by setting out a function that describes the amount of building floor area that
can be created through the application of various amounts of building or construction capital and
land.
Let L stand for the area of a lot on which a building is to be constructed; let K stand for the
amount of construction capital that is used in the building; and let S stand for the usable floor
area of the building that results from the application of L and K . So, we have defined a
production function
S  F ( K , L)
First, a few asides about this functional relationship. How K should be measured may be
puzzling. Some index of tons of concrete and steel and person-hours of workers, professional
services and all of the other inputs to the construction process would be the appropriate measure,
and indeed there is such an index – the money cost of construction. This weights all the
component parts by their market cost and is readily observable. The only drawback is that
changing relative prices of component parts will change the measure of capital input even if
nothing in fact has changed with respect to the construction process (general inflation or deflation
of all prices in equal proportion can be corrected for, so it is not a problem). In this model, the
problem of changing relative weights (read "prices") will simply have to be lived with as I
assume that the measure of capital will be the cost of construction. In fact, I would like to
maintain the fiction of there being an "real" measure of capital, K , and a price per unit of this
"real" capital which I will label Pk . The value of capital is then Pk K . If I then define my unit of
capital so that Pk  1 , then K effectively becomes the amount of "real" capital measured by the
construction cost.
Another aside: there are many different types of buildings – office buildings, apartment buildings,
industrial space and single-family residences. Presumably the production function linking capital
and land to floor-area output will be different for each. So, this simply function F applies to one
type of building. This is not a problem since the results we get from the model will be the same
for whatever type of building is assumed and whatever the corresponding production function is.
Similarly, even for buildings of the same type, there exists the possibility of having many
different levels of quality. We could handle this within the simple model by defining Q , say, as
a measure of quality, with Q  1 representing standard, class B quality. The production function
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could then be written S  F (
K
, L) . Class C buildings would then have Q  1 so that each
Q
dollar's worth of construction capital would produce more floor space of a lower quality.
Superior class A quality would be represented by Q  1 .
The introduction of a quality measure, Q , would allow the model to be used to solve for size and
quality at the same time, but the extra dimension would complicate the graphical exposition
without adding anything to our understanding. So I'll stick to the simple two-input function,
assuming that there is one such function for each different type and quality of building.
Back to the basic model. I will assume that the functional relationship between capital and land
input and floor-space output can be represented by a constant-returns to scale, Cobb-Douglas
production function. (The literature appears to justify this, as a first approximation.) Thus, the
functional relationship becomes
S  K  L1 , with   1 .
This may also be written in a form that relates the "density" of the building, defined as
ratio of capital to land,
get
K
. Thus if both sides of the above functional form are divided by L , we
L
F
IJ . Let S  s and K  k . Then we can write, s  k
G
HK L
L
S
K

L
L
S
, to the
L


as the production function.
This is often a more convenient form to work with.
Zoning by-laws very frequently set maximum densities that are allowed for different types of
buildings in different parts of a city.
Our initial goal is to use this model to establish the optimal building size or building density
given the price of capital and given the expected net earnings per unit of floor area. As indicated
above, the price of a unit of capital will be denoted by Pk . The net annual earnings of a unit of
floor space will be called Ps . (By "net" I mean that the gross annual revenue has maintenance and
other costs to the owner deducted from it.)
Using the s  k  formulation, the decision variable is the intensity of capital, k , to use. The
construction cost per unit of land (remember that k is capital per unit of land) will be Pk k .
Since the net revenue is an annual flow, this construction cost should be converted to an annual
equivalent by multiplying by the rate of interest i . (It is as if the developer had borrowed the
money to pay for the construction and then pays it back in perpetuity at the annual interest rate.)
The annual capital cost, then, as a function of k is iPk k per unit of land.
Correspondingly, the net revenue per unit of land will be Ps s . The developer's goal is to
maximize the difference between the net revenue flow per unit of land and the annual capital cost
per unit of land. Denoting this difference by R , meaning the "residual" value, we can write
R  Ps s  iPk k . We can solve for a maximum R value by differentiating this expression with
respect to the decision variable, k , and set the result equal to zero – the standard first-order
condition for a maximum (or a minimum; but we know from the shapes of the revenue and cost
functions that the result will give us a unique maximum).
R
s
 Ps
 iPk  0 , or
k
k
iPk  Ps
s
.
k
s
is the marginal product of
k
s
S
capital (using the Cobb-Douglas production functions, you can show that
and
are the
k
K
s
same), and that Ps
is therefore the marginal value product of capital. So the optimality
k
Notice that iPk is the marginal annual cost of a unit of capital, that
condition shown above is the usual welfare-economics condition that the marginal cost of a factor
equals the marginal value product: MCk  MVPk .
The maximization of the residual value may also be shown graphically if we plot the cost of
capital, iPk k , and the net earnings after development, Ps s , against the capital intensity, k .
$/yr
Ps s *
Ps s
iPk k
k*
k
Where the slope of the two functions are equal, the residual value will be maximized. Sine the
slope of the Ps s function is Ps
s
and the slope of the iPk k function is iPk , this equality of
k
slopes is exactly the same value maximizing condition as we had worked out algebraically. The
optimum building intensity is labelled k * in the diagram; the optimal density is s * .
If the property before development was a vacant lot, and if building space, s , is produced by
capital and land under conditions of constant returns to scale, then the residual value (in the
diagram, the vertical distance between the Ps s curve and the iPk k line) at the optimal capital
intensity and building density has a special interpretation. It is equal to the marginal value
product of land.
To see this, apply Euler's equation to production function S  K  L1 to get
S
S
L
or S  K  MPK  L  MPL . Multiply both sides by Ps to get
K
L
Ps S  K  MVPK  L  MVPL . Dividing both sides by L and rearranging yields
MVPL  Ps s  k  MVPk . This holds true everywhere.
But at the optimum, MVPk  iPk . So this result says that MVPL  Ps s  iPk k , i.e., the marginal
SK
value product of land is equal to the residual when the building intensity is optimal. The market
maximizes the residual value of the property; this will become the market price. This price in
turn signals to the market the value of land, given the price of building capital, and so will
encourage land to be used efficiently.
The model may now be used to examine briefly some issues associated with property taxation,
which is the most common local tax in North America and in many of the members of the
Commonwealth.
Suppose that there is initially no property tax and the model has defined the optimal building
density, s * . Now suppose that a property tax is introduced and that it will be imposed at a rate
or percentage t on the value of the built property, Ps s . This is very like most urban property
taxes except that they are typically given as rates on or percentages of the asset or capital value of
the property. In our simple, static model, asset and annual values are related by the discount rate
i (to convert from an annual to a capital value, divide by i , and to convert from a capital to an
annual value, multiply by i ), so a tax on asset values can easily be converted to and treated as a
tax on annual values.
With a property tax in existence, the residual or land value accruing to a property owner will be
diminished by the amount of the tax. So, given as before the capital intensity, k , as the decision
variable, the residual value that the market will maximize is given by
R  Ps s  tPs s  iPk k .
Differentiating this with respect to k and setting the result equal to zero yields a new first-order
condition for maximum value:
iPk  (1  t ) Ps
s
. With 1  t  0 , this means that developers will build only up to a density
k
where the marginal value product of capital remains above its marginal cost. The building
density will be less than optimal; capital will be underused and land overused. Essentially what
has happened is that every unit of additional building size costs the developer tax dollars as well
as construction dollars, whereas without taxes only the construction dollars had to be taken into
account.
Based on this result, many economists argue that property taxes are inefficient because they lead
encourage buildings to be built at less than optimal densities. (Other taxes are of course also
inefficient.) However, under some circumstances property taxes may actually encourage efficient
building densities. To see this, suppose that urban public costs rise as new development occurs.
This obviously does happen. Now suppose that the urban public costs per unit of land, c , are
related to the capital intensity of density of the building, so that c  c( k ) , with
c
 0 . This
k
means that the capital cost function that we have been using, iPk k does not represent all of the
costs of developing a property. The true cost function that we should be comparing with the net
revenue is iPk k  c( k ) . The first order condition for a maximum residual value then becomes
s c

. In this case, the true social optimum building density is less than the k * we
k k
c
s
originally found; indeed if
happens to equal tPs
, then the standard property tax at rate t
k
k
iPk  Ps
yields a socially optimal building density.
Of course it would be extremely unlikely that
c
s
 tPs
exactly, but even so it is clear that if
k
k
public costs are a function of building size, standard property tax may move the outcome towards
and not away from the optimum.
There is another type of property tax that is sometimes proposed, most famously by an economist
named Henry George writing early in the 1900s. His proposal and that of modern-day
"Georgeists" is that a property tax can be an important source of government revenue but that it
should be levied on the value of the land component of the property only, and not the building
component.
This Henry George tax may be illustrated by again assuming that a tax now exists at an annual
rate t applied only to the residual value of the land. Effectively, a developer will want to develop
at a building density such that the following expression is maximized:
R  Ps s  t ( Ps s  iPk k )  iPk k .
The first-order condition for this to be maximized is now
iPk 
(1  t ) s
. The (1 t )' s cancel out, and the optimum condition becomes the original
Ps
(1  t ) k
condition. The introduction of a Henry George tax has not led to an inefficiently small building
density.
Again, the Henry George argument is quite popular among economists and many others, but it
does have some weaknesses. The following four qualifications should be borne in mind:
1) the argument that a greater building density may entail greater public costs per unit of land
may still apply, in which case neither the original condition nor the identical Henry George
condition for optimal development are correct;
2) identifying the value of land as distinct from the overall value of a property may not only be
difficult and in some cases is conceptually impossible (for further discussion of this point see an
earlier paper of mine for which I can give you a reference);
3) if the Henry George argument is correct, then the land-value tax must be applied to all land, or
distortions will remain: differential tax treatment of different types of property within urban areas
will lead to inefficient resource allocations, and differential land tyaxes between urban and rural
areas will lead to inefficient city sizes; and
4) the higher the Henry George tax, the lower the incentive for the market to allocate land
efficiently. In the limit, if the Henry George tax were at a rate of 1 or close to it, the whole of the
land value would be removed in taxes and all incentives to use land efficiently will have
disappeared.