Urban Land Markets with Factor Substitution
In what follows here, we will elaborate on the analysis of land-market equilibrium
with factor substitution, as outlined in O’Sullivan’s text. Starting with a model in
Chapter 6 of the text, we derive the model’s endogenous variables from its
Factor substitution refers to firms’ ability to substitute capital for land at locations
where land rent is relatively high – rather than using fixed land and capital inputs
regardless of land rent. The factor-substitution model in the text considers firms
that occupy office buildings. However, a factor substitution model can apply to
any type of firm. In these notes, factor substitution can be switched on or off so
that its impact on endogenous variables can be isolated. A zoning law controlling
firms’ land input is the tool used to switch factor substitution on or off.
Firms in the textbook model sell information products to their customers.
Customers might be paying for financial advice or legal advice – two examples of
information products. Firms need data – such as data on financial markets or data
on matters involved in contract law – in order to develop saleable information
The location variable in the model is distance from the office land market’s
median location (the location at the centre of a circular land market). We assume
that the cost of obtaining required data increases with distance from the median
location. Required data is obtained at meetings, and “travel cost” (an exogenous
variable) measures the cost of attending these meetings.
We begin with Table 6-7 in the text. We will re-order the table so that exogenous
variables (given from outside the model) are grouped in columns 1-4; the
endogenous variables (determined in the model) will be grouped in columns 5-9.
The resulting table appears here as Table 1.
Table 1 includes values for variables on rows omitted in Table 6 -7: the rows
added here show all data for locations at 0, 4 and 6 blocks from t he median
location, while travel cost numbers have been added for locations at 2 and 3
blocks. The blank spaces for variables at 2 and 3 blocks are not filled in since
data at these locations will not be needed in the discussion below.
The numbers in Table 1 could be interpreted as applying to seven firms at the
seven locations indicated in Column 1. Alternatively, we could interpret Table 1
1In what follows here, hyphenated numbers for tables or figures (for example Table 6-7) refer to tables or
figures in the text. These numbers are the same in both the 6th and 7th editions of the text. Table or figure
numbers without a hyphen (for example Table 1) refer to tables or figures included in these notes. These
notes cannot be read without the text book at hand, since there will be numerous references to tables and
diagrams in the text that are not reproduced in the notes.
1 as applying to a single firm that moves from one location to another. Either way,
the firm or firms will compete with large numbers of identical firms. The
conditions for perfect competition are assumed to be met here, including zero
economic profit in equilibrium (firms make only the profit required to keep them in
business). The profit required to stay in business is included in the “other non-
land cost” variable.
Table 1: Expanded / Re-ordered Version of Table 6-7 in Text
1. Distance 2. Travel 3. Total 4. Other 5. Product’n 6. Bldg. 7. Total 8. Capital9. Bid
in blocks cost revenue non-land site height rent paid cost of rent per
( x ) cost (hectares) (floors) bldg. ha
0 $0 $500 $150 0.02 50 $70 $280 $3500
1 $36 $500 $150 0.04 25 $64 $250 $1600
2 $74 $500 $150
3 $114 $500 $150
4 $156 $500 $150 0.125 8 $54 $140 $432
5 $200 $500 $150 0.25 4 $50 $100 $200
6 $246 $500 $150 0.50 2 $29 $75 $58
Column 1 in Table 1 shows distance from the median location, measured in
“blocks”. A block is treated as a standard distance measure, comparable to
kilometres or miles. This variable measures distance from the median location in
any direction – whether east, south, west, north (or any other compass direction)
makes no difference. All variables in the model will have equal values at any
given x, regardless of direction.
Travel cost is shown in Column 2. Since travel cost is an exogenous variable,
any numbers could be assumed for Column 2. The assumption in the text is that
travel cost increases at an increasing rate moving away from the median
location. In the example of Table 1, a move from 0 blocks (the median location
itself) to one block away increases travel cost by $36 - $0 = $36 per day; a move
from 1 block to 2 blocks increases travel cost by $74 - $36 = $38 per day, and so
on. The text provides some of the numbers for travel cost, and in Table 1
numbers have been filled in for other locations.
All variables with a time dimension (Columns 2, 3, 4, 7, 8, 9) are specified on a
per-day basis. The text indicates that an office firm’s revenue ($500) is per day,
so all other time-specific variables must be measured on a per-day basis. (The
time period could alternatively be a month, a year or any other time interval, as
long as it is specified consistently for all variables.)
The product that is sold for $500 per day is not identified in the text, but
identifying it is unnecessary. Whatever the product is, both the output level
produced by a firm and the output price are exogenous; price times quantity of
output is fixed at $500 per day per firm.
Not shown in Table 1 is another exogenous variable assumed to apply equally to
each firm. Each firm is assumed to require an office building with 10,000 sq.
metres (1 hectare) of floor area to produce its output. The firm could rent that
2 floor area from another firm producing office space for rent. However, in the
model here and in the text the firm constructs its own office building, renting land
and capital for that purpose. It then uses its office building to produce the
information product that sells for $500 per day.
Column 4 (“other non-land cost”) shows a firm’s cost of inputs other than travel
cost, capital cost and land cost. ONLC includ2s labour cost and the profit
required by the firm to stay in business. The assumption is that inputs included
in ONLC cannot be substituted for land or capital: their cost is a constant $150 /
day given exogenously.
Columns 5 and 6 in Table 1 are effectively the same variable. Column 5 shows
the land input in hectares rented by a firm at each location (x-value), and Column
6 shows that same land input divided into the firm’s fixed floor area (1 hectare).
Thus Column 6, the ratio of floor area to land area, is the reciprocal of Column 5.
For example firms at x = 5 rent 0.25 hectares of land on which they construct 1
hectare of floor area, so the floor area / land area ratio is 1 / 0.25 = 4. In these
notes we will refer to “land input” rather the equivalent term “production site” used
in the text.
We assume that a firm’s land input is limited to land under the building (no
additional land for landscaping, parking, etc.). We also assume that every floor
has the same area as the ground floor, so a building on 0.25 hectares of land has
four floors of 0.25 hectares each. Given these assumptions, the floor area / land
area ratio is also “building height” measured in numbers of floors. 3
The variable in Column 7 is a firm’s land cost. Land cost is a firm’s land input in
hectares (Column 5) times land rent per hectare (Column 9). For example at x =
5, a firm rents one quarter hectare of land at a rent of $200 / hectare. Multiplying
these numbers together yields land cost of $50. The text uses an equivalent term
for land cost: “total rent paid”.
In equilibrium, land cost can also be referred to as “willingness to pay” for land
(“WTP” – see text Table 6-4). Willingness to pay for land means the maximum
amount a firm is willing to pay – paying more would reduce profit below the level
required to keep the firm in business.
Other non-land cost may also include travel cost at x = 0. In Table 1 and in the text, travel cost at x = 0 is set
equal to zero. However, that is consistent with firms at x = 0 actually incurring travel cost. To achieve that
consistency, define travel cost as the difference between travel cost at each x and travel cost at x = 0. Travel
cost at x = 0 is then included with the other constants in ONLC.
3Note: “building height” is equivalent to the floor area / land area ratio only if that ratio ≥ 1.0. The minimum
building height is obviously one floor. For example with a floor area / land area ratio =
0.5 (1 hectare floor area on 2 hectares of land), we would either have a building one floor high covering half
the site or some taller building covering less than half the site. A building 0.5 floors high covering the entire
site is not an option.
3 When costs in Columns 2, 4 and 8 are subtracted from revenue (Column 3), the
residual amount is available for land cost and (potentially) for economic profit.
However, if any of this residual actually does go to economic profit, “wannabe”
firms will be attracted. These wannabe firms will need land to get into production.
With all land already occupied, there will be excess demand for land so rent
levels will be bid up until economic profit is zero. At that point, firms will pay land
cost = the maximum they are willing to pay for land.
In these notes we will refer to “land cost” rather than the equivalent terms “total
rent paid” or “willingness to pay” – thus assuming land rent and land input are at
their equilibrium values.
Column 8 shows the firm’s capital input. While the capital used in buildings has
multiple components (concrete, glass, steel, elevators, air conditioners etc.), all
capital is lumped together in the model so that it can fit onto one axis in an
isoquant / isocost diagram – for example Figure 1 below, or Figure 6A-2 in the
text. Thus the capital input is measured in dollars rather than physical units: it is
the dollar amount per day required to rent the capital used in the firm’s building.
With capital measured in dollar units, its unit price is $1 by definition.
The text defines the term “building cost” as the total of a firm’s capital cost plus
land cost. However, in these notes we will refer to “capital cost plus land cost”
rather than using a separate label for this total.
Deriving endogenous variables from the exogenous variables
In this section we will use Figure 1 to derive endogenous variables graphically
from exogenous variables. Figure 1 is modeled on Figure 6A-2 in the text, with a
key modification. The modification is to extend the tangent isocost line shown in
6A-2 to the vertical axis, with intercept point Cishown in Figure 1.
Figure 1 then resembles Figure 6A-1 in the text. Although Figure 6A-1 illustrates
a different model – a model of residential land market equilibrium – the
relationship between exogenous and endogenous variables is similar in both
Figures 6A-1 and 6A-2. The vertical axis in both Figures 6A-1 and Figure 6A-2 is
measured in dollar units. As was noted above, Figure 6A-2’s vertical axis is a
firm’s capital input. Fig. 6A-1’s vertical axis is a resident’s monthly spending on
Given an axis measured in dollars, other variables measured in dollars can be
displayed on that axis. In Figure 6A-1, values for monthly income net of
commuting cost can be displayed. Point A is income net of commuting cost for a
resident with a 10 mile commute, assuming commuting cost t is $50 / month /
mile. All residents are assumed to have $2000 monthly income, so a resident
with (10 X $50) = $500 / month commuting cost will have $1500 / month available
to spend on housing and other goods. Because residents are assumed to spend
all of their income, $1500 / month will in fact be spent on housing and other
goods where commuting distance x is 10 miles.
4 Figure 1
0.25 Land Input (Hectares)
Still referring to Figure 6A-1 in the text, point A = $1500 is one point on the
equilibrium budget line for a resident at x = 10 miles. One other point is required
to determine the position of that line. Tangency point i on the equilibrium
indifference curve U is1the required point.
No line through A other than Ai can be the equilibrium budget line. A line steeper
than Ai will not allow residents to reach the equilibrium indifference curve U .
The slope of a budget line is equal to the price of housing, so a slope that is too 4
steep means the price of housing is temporarily above its equilibrium level. If
the price of housing at x = 10 miles will not allow residents to obtain the
equilibrium utility level U1, residents will not pay that price. They will offer a
lower price and move to another location if housing firms do not accept the offer.
Thus a budget line through A that is too steep will rotate counter-clockwise on
pivot point A until it coincides with the tangent line Ai.
A line through A flatter than Ai will allow residents to reach indifference curves
above U , If residents can reach Indifference curves above U , their utility would
be temporarily above the equilibrium level. Other residents would be attracted to
the location x = 10 miles, causing the budget line to rotate clockwise on pivot
point A, until it coincides with Ai. After that, residents can no longer reach higher
indifference curves than the equilibrium indifference curve U 1
Thus the tangent line Ai is the resident’s equilibrium budget line. As was noted
above, Its slope is the price of housing at x = 10 miles: $0.30 per square foot per
4A budget line’s slope (in absolute value) is the ratio of two prices: the price of the good on the horizontal
axis divided by the price of the good on the vertical axis. The latter price equals $1, so the budget-line slope
is $p / $1 = $p: the equilibrium housing price at x = 10 miles.
5 month as shown in Figure 6A-1. The coordinates of the tangency point i are G
(non housing goods consumption) = $1200 / month; and housing consumption h
= 1000 square feet. The variables in this paragraph are equilibrium values for the
endogenous variables at x = 10 miles. The exogenous variables from which
these values are derived are income (w) = $2000 per month; commuting cost (t) =
$50 per mile per month; and utility level U = U . 1
Returning to the isoquant / isocost diagram (Figure 1), derivation of equilibrium
endogenous variables at any given location (for example x = 5 blocks) would
proceed via these steps:
Since the zero economic profit condition requires equality of total revenue and
total cost for each firm, total revenue minus other non-land cost minus travel
cost = (capital cost plus land cost). Using the relevant values from Table 1, we
have $500 (total revenue) minus $150 (other non-land cost) minus $200 (travel
cost) = $150 = total of capital cost plus land cost. $150 is then the C-axis
intercept of the equilibrium isocost line at x = 5 blocks. This point is represented
in Figure 1 as C = $i50. Firms at x = 5 must spend $150 per day on capital
and land to build an office building with 1 hectare of floor area while earning
zero economic profit.
C is one point on the equilibrium isocost line facing firms at x = 5. One other
point is required to determine the position of the equilibrium isocost line. The
required point is tangency point m, on the isoquant for an office building with 1
hectare floor area (labeled the S = 1 isoquant). This proposition is demonstrated
via steps similar to those already taken with the residential model discussed
The slope of an isocost line (in absolute value) is land rent [R (x)] divided by the
price of capital ($1 by definition). For example if an isocost line has slope =
minus $200, land rent is $200 / hectare / day.
If the x = 5 isocost line through C iis temporarily steeper than Cm, firmi will not be
able to reach the isoquant required for a 1-hectare office building. No firm will pay
a land rent so high that it cannot earn the profit required to stay in business while
also occupying the required 1 hectare of floor area. In that case land owners will
have to reduce rents. That will make the isocost line through C iless steep,
rotating counter-clockwise on pivot point C until it coincides with Cm.
If the x = 5 isocost line through C is timporarily flatter than Cm, firms iill be able
to reach the S = 1 isoquant while paying less than $150 / day, thus earning
economic profit. In this case wannabe firms will be attracted by the economic
profit. Requiring land to get into production, and with all land already occupied,
the wannabe firms will bid up land rent. This will rotate the isocost line clockwise
5An isocost line through A that is temporarily flatter ihan Cm will cut through the S = 1 isoquant. Points on
the isoquant under the isocost line will be on lower isocost lines, so firms can (temporarily) pay for their
required 1-hectare office building and have money left over.
6 on the pivot point C uniil it coincides with Cm. i
Thus the tangent line Cm is ihe equilibrium isocost line for firms at x = 5. Its slope
(in absolute value) is $200 / hectare. Thus $200 per hectare is the equilibrium
land rent entry in Column 9 of Table 1. The C-axis coordinate of point m is the
Column 8 entry ($100). The L-axis coordinate of point m is the Column 5 entry
(0.25 hectares). Multiplying the 0.25 hectare land input by the $200 / hectare land
rent yields the $50 land cost entry (column 7). The 0.25 hectare land input divided
into one hectare yields the building height entry in Column 6: building height = 1 /
0.25 = 4 floors.
Similar derivations of endogenous variables can be carried out at other locations
in the model. At any given value of x, subtract other non-land cost and travel cost
from total revenue. Plot the result of this subtraction, which will be different at
each x (since travel cost is different at each x), on the vertical axis of the isoquant
diagram to obtain C(x) i the equilibrium isocost line’s vertical-axis intercept at the
given location. Finally, draw the tangent from C ix) to the S = 1 isoquant. The
tangency point and tangent slope will yield equilibrium values for all endogenous
variables at the given value of x.
At central locations where travel cost is relatively low (for example at x = 1),
C ix) will be relatively high up on the C axis. On the x = 1 line of Table 1, C will i
equal [$500 (total revenue) minus $150 (other non-land cost) minus $36 (travel
cost)] = $314. A tangent to the S = 1 isoquant from C = $314 will be relatively
steep (i.e. land rent will be relatively high). Als