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ECO101H1 Lecture Notes - Shortage, Timit, Concave Function

Course Code
Peter Tomlinson

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Urban Land Markets With Factor Substitution
April 2011
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
exogenous variables.
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 products.
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.1 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 the median
location, while travel cost numbers have been added for locations at 2 and 3
1 In 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.

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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.
Table 1: Expanded / Re-ordered Version of Table 6-7 in Text
in blocks
( x )
cost 3.Total
revenue 4.Other
rent paid 8.Capital
cost of
rent per
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
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
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.
Exogenous Variables
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.)

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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
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 includes labour cost and the profit
required by the firm to stay in business.2 The assumption is that inputs included
in ONLC cannot be substituted for land or capital: their cost is a constant $150 /
day given exogenously.
Endogenous Variables
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
2 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..
3 Note: “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.
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