© David M. Nowlan 1999
The Simple Economics of Urban Land
The basic unit of urban land is the lot. This is a legally defined and registered piece of land and
as such may, in market economies, be owned or traded. It is the proprietary land unit: you can
buy or sell one or two or three land units, but not one-half a land unit. If you want to sell part of a
lot that you own, you must first go through a legal process of dividing your land into two new
registered units. If this is allowed, then you can sell one of the two new units.
A lot may be empty or it may contain buildings or other structures. I will generally use the word
"property" to mean a lot whether empty or built upon. If the distinction between vacant and builtupon lots is important, I will be specific and say "vacant land" or "land with a building."
Structures on land are sometimes called "improvements" and a lot with a structure is then called
an "improved property."
Capitalization and the Value of Property
When we think of the value of a property, we generally think of its asset or capital value, a price
that a buyer and seller can agree upon. In the case of property consisting of land or land-plusbuildings, this asset value is related to expectations of future flows of benefits, or of costs and
benefits, that will be associated with the property. This relationship between the expected future
flow values and the present capital value is of key importance; the process of deriving a present
capital value from the future flows is called "capitalization."
Let's take as an example a property consisting of a commercial building standing on its registered
lot. Think of time as consisting of a sequence of discrete periods called years and suppose that
this property is expected to yield a certain net income or net rent in each future year. Obviously,
in most cases expectations of earnings will depend on a complicated set of expectations about not
just tenant rents but also about costs that the owner may have to bear, or bear in part -maintenance, improvements and taxes, perhaps. To make the example simple, I will use
expectations about just the net income or rental flow, the income after all costs have been netted
out.
Suppose our expectation is that the net rent in the first year will be R1 , in the second year R2 and
so on. Suppose further that each year's net rent accrues to the owner at the end of the year.1 Each
year's net rent will contribute something to the present capital value. The amount of money, R1 ,
that accrues at the end of the first year will be worth to the owner today something less than
R1 because it isn't received for a year: it will have to be discounted using a discount factor that is
based on prevailing earnings of other assets that the owner could be investing in instead of this
particular property.
1
When we work with discrete time, a choice has to be made whether to assume the flow value
accrues at the beginning of the year or the end of the year. It makes a slight difference to the
result, as we shall see. If continuous time is used, this choice is not necessary.
Suppose that the alternative earnings or interest rate is 5% a year, i.e., 0.05 per year. Then the
1
1

. The
1  0.05 105
.
R
contribution of this first year's earnings is R1 multiplied by the discount factor, or 1 . In
105
.
1
general, if the interest rate is i per year,2 the discount factor per year will be
. Thus, the
1 i
R2
contribution to present asset value of the net revenue at the end of year two would be
(1  i ) 2
R3
and the contribution of year three's net revenue would be
.
(1  i)3
discount factor that must be applied to the first year's net income is
Continuing with this line of reasoning, the present asset value is equal to the sum of all of the
discounted net revenues, from year 1 through to some indefinitely distant future, the infinite
future. This is called the capitalization of the revenue flow.
Let V0 stand for the present value at the beginning of year 1. Then what has just been said is that
V0 
R1
R2
R3
Rn


...
...
2
3
(1  i) (1  i) (1  i)
(1  i) n
To make things a little simpler, suppose that the expected net revenue each year is constant so
that R1  R2  R3  Rn  R . Then
V0 
R
R
R


...
2
(1  i) (1  i) (1  i)3
With i  0 , the right-hand-side of this expression is the sum of a convergent geometric series3
with a value of
R
. Thus, given the assumptions of a constant annual net revenue and a fixed
i
annual discount or interest rate, we end up with a simple relationship between the annual net
2
This example and the ones that follow assume that the discount rate does not change from year
to year. If it did, we could label each year's rate i1 , i2 , and so on so that the two year discount
factor would be
3
R2
.
(1  i1 )(1  i2 )
Such a series may be represented in general by the expression S  c  cx  cx 2  cx 3  cx 4 ... ,
with 0  x  1 . The value of the sum is then
equivalent to the constant term c while
c
R
. In the expression in the text,
is
1 x
1 i
1
is equivalent to the multiplier x .
1 i
2
revenue flow and the present capital value of the property:4
V0 
R
.
i
Without the assumptions of a constant R and a constant i , there would be a more complex
relationship between the revenue flows and the capital value, but this fundamental capitalization
relationship can sometimes be used to help simplify these more complicated relationships.
Suppose, for example, that the property in question was a vacant lot and that what was
contemplated was the construction in the first year of a building costing C dollars. This would be
followed in year two and beyond by a constant net revenue flow of R dollars per year. Suppose
again that the cost and the annual net revenues all accrue at the end of each year. In this case, the
present capital value would be
V0 
C
R
R
R



...
2
3
(1  i) (1  i) (1  i) (1  i)4
This can be written a little differently as
V0 
F
IJL
R
R

G
M
(1  i) (1  i)
H KN
C
1

(1  i)
(1  i)
2

O
P
Q
R
...
(1  i)3
The term in the square brackets may be thought of as the present value at the end of year one or
the beginning of year two (i.e., after the construction cost has been incurred). From the basic
capitalization formula, we know that this is equal to
R
. So the present value today of the vacant
i
lot is
V0 
1 IF R I
F
G
H1  i J
KG
HC  i J
K.
A Fundamental Theorem of Capital
Using the concept of revenue capitalization, we can work out a theorem that tells us about the
expected change in capital value over time.
Suppose we start with a property that has an expected flow annually of future net revenue. As
before, each year's net earnings, which accrue at the end of the year, may be labelled R1 , R2 , and
so on. The present value of the property is given by
4
The relationship is not quite as neat if we assume that the net revenue flows accrue at the
beginning of each time period rather than the end. With this change, present value is
R2
R3
R4


... , or, assuming that R1  R2  R3  Rn ...  R and
2
1  i (1  i) (1  i)3
R(1  i)
applying the convergent sum formula, V0 
.
i
V0  R1 
3
V0 
R1
R2
R3


...
2
1  i (1  i) (1  i)3
In general the annual net revenue flows will be different; some may be negative as well as
positive.
Now consider the value of this property one year later. R1 has been received, so the asset value
will be a function only of the net revenue flows from year two onwards:
V1 
R2
R3
R4


...
2
1  i (1  i) (1  i)3
If this last expression is multiplied through on both sides by
1
, we get
1 i
V1
R2
R3
R4



...
2
3
1  i (1  i) (1  i) (1  i)4
Subtracting this from the expression for V0 yields the following
V0 
V1
R
 1 which can be rearranged in several ways to show the relationship between the
1 i 1 i
annual discount rate, the year's net revenue and the one-year change in asset or capital value.
V1  V0  iV0  R1 or
V1  V0
R
V V
R
 i  1 or i  1 0  1 .
V0
V0
V0
V0
This is one of the most fundamental relationships in capital theory. Capital has to earn its
required interest, if expectations are met, either by appreciating, i.e., V1  V0  0 , or by earning a
return, R1 , on the initial investment, or by some combination of the two. Looking at the third of
the above expressions, we see that the sum of the proportionate appreciation plus the net revenue
expressed as a proportion of the initial capital value must equal the interest or discount rate.
Thus, if there is no expected net revenue in the first year, R1  0 , the property must rise in value.
Vacant land incurring no cost and entailing no revenue must appreciate over time simply as a
condition of market equilibrium with expectations consistent with reality. If the appropriate
interest rate is, say, 10 per cent annually, then the vacant land will approximately double in value
every seven years.
Continuous Time
In the analysis so far, time has proceeded in discrete units, units of a year for example. Things
happen either at the beginning of a time period or at the end, and the end of one period is exactly
the same time as the beginning of the next.
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Continuous time, by contrast, is a concept more in keeping with the way in which we usually
think about time, that it proceeds smoothly from one instant to the next. It also happens that
modelling time continuously often makes the mathematics easier.
Let us begin by recalling that in discrete time, one dollar at the beginning of a period invested at
an annual interest rate of i will yield at the end of the period (1  i )1 dollars. If the interest, still at
i
2
an annual rate of i , were calculated and paid half-yearly, then the one dollar would equal (1  )
i
2
i
2
i
2
after six months and (1  )(1  )  (1  ) 2 after a year.
If the interest on the invested dollar were paid m times a year, the dollar would amount to
i m
i
) after one year and (1  ) mt after t years. If m grows large without limit, then the
m
m
period of the interest payment and compounding grows small without limit. If m is "infinitely"
(1 
large, then the interest payment and compounding may be thought of as taking place
continuously, and we are then in the analytical world of continuous time and can write the value
of one dollar invested at i per year for t years as (1 
i mt
) .
m
m
This last expression may be simplified using the mathematical constant e , which is defined as
1
(1  )n . This equals, approximately, 2.7183.
n
n 
i 1
 . Then the expression for the value of a dollar
m n
after t years at an annual interest rate of i compounded continuously becomes
Proceeding with the simplification, let
it
1
1 

(1  )nit   (1  )n  . Using the definition of e , this in turn may be written as e it .
n
n 

n
n 
So, with compounding taking place continuously, a dollar invested at interest rate i per year for
t years will yield e it dollars. After ten years, the dollar would become e10i dollars.
Anyone who has been concerned with how frequently interest on their saving account is
calculated will realize that the more often the interest is accrued the larger the initial investment
will become. Thus, 20% a year interest on a dollar yields $1.20 after one year if the accrual
occurs only once a year, but it yields $e.2  $1.22 if the accrual is continuous.
Discounting, as before, is simply the inverse of compounding. $10 seven years from now
10
or 10e 7i . A net annual revenue of R (t ) being earned
e7i
t years from now (where t no longer has to be an integer) has a present value of R(t )e it .
discounted at a rate of i per year is
5
If we go back now to the simple case of a constant net revenue stream of R dollars per year
where R stays constant over time, then the present value of this net revenue stream which occurs

continuously for an indefinitely long period of time may be written as
 Re
 it
dt . Notice that
0
R is the rate per year at which the net revenue is continuously flowing just as correspondingly i is
the rate per year at which this flow is being continuously discounted.

1 it
R R
The value of the integral is
e R  0   . We end up with our familiar present value
i
i i
0
R
formula,
for a flow of net revenue of R per year.
i
The fundamental capital relationship may also be readily derived in a continuous-time model.
The annual flow of net revenue at any point in time t is given by R (t ) , no longer assumed to be


constant. The present value of this flow at any time s , Vs , is given by Vs  R(t )e i ( t s )dt .
s
Now we work out the rate at which Vs changes with a change in s :
  R (t )e
Vs
  R( s )   
s
s
s

i ( t s )


dt   R( s )  i  R(t )e  i ( t  s )dt   R ( s )  iVs . This result is
s
exactly the same capital relationship that we derived in discrete time.
6