Introductory Notes on Rectangular Land Market Model
These notes elaborate on a land-market model presented in OSullivans
textbook Urban Economics.
Building on the textbooks analysis, our focus will be on these key questions:
What is the impact on land rents if an endoge1ous output price replaces
the textbooks exogenous output price?
What is the impact on land rents and output price if the land supply is
expanded or contracted?
The rectangular model is introduced in the sectthn Bid-Rent Curves for tth
Manufacturing Sector (pages 122-124, 7 edition, or 102-104, 6 edition). In
that sections Figure 6-1, we see an equilibrium bid rent curve (= bid rent
function, abbreviated here as BRF). The BRF shows zero-economic-profit land
rents for manufacturing firms as a function of distance (x) from a highway. The
zero-economic-profit condition is met when firms earn only the profit required to
stay in business.
The textbook discussion continues in A General Equilibrium Model of a
Monocentric City (pages 192-194, 7 thedition, or 157-160, 6 edition). In Figure
7A-3 we again see an equilibrium BRF for manufacturing firms now labelled R b.
(the b being for businesses that manufacture a product).
Figure 7A-3 includes residential land as well as manufacturing and agricultural
land, but the discussion in these introductory notes will be limited to
manufacturing and agriculture. Thus Figure 7A-3s bid rent functions labelled
R r (initialr (streetcar)be disregarded for now. They are considered, along with
the upper-right (labour market) diagram, in separate notes entitled The General
Equilibrium Rectangular City Model. 2
In the lower part of Figure 7A-3 we see the manufacturing land area (marked D),
with its western boundary formed by a highway 3
1An endogenous is determined in the model; an exogenous price is given from outside the model.
2A land market could function without residential land if firms provide their employees with living
accommodation at the job site.
As illustrated in Figure 7A-3, this western boundary is labelled the centre, but we will assume
here it is a north-south highway as in Figure 6-1. The lower part of Figure 7A-3 shows a land-
market map (the land market as viewed from above). As has been noted, the area marked D
(demand for labour) is land occupied by manufacturing firms firms that generate demand for
labour. The area S (supply of labour) is land occupied by housing firms (firms housing residents Some manufacturing firms are located directly on the highway, and so have zero
freight cost to access the highway. Others ship their product to the highway from
interior locations, along local roads. Freight cost is defined here as the cost to
reach the highway along local roads. These roads run east and west, so distance
from the highway (x) is measured to the east.
The model includes a variable called non-land production cost (NLPC), which is
the same for all firms regardless of location Included in this variable is the cost of
freight incurred after a firms vehicles have accessed the highway an equal
amount for all firms whether located at x = 0 or an interior location.
The rectangular land market has northern and southern boundaries, shown as
lines perpendicular to the highway. In the textbooks Fig. 7A-3 they are assumed
to be 1 mile apart. However, in these notes we will let the rectangular land
markets north-south dimension be an exogenous variable y (in kilometres).
We will assume that northern and southern boundaries are imposed on the
market by zoning regulations. Zoning regulations specify which land uses are
legal at various locations. We will assume that manufacturing is legal only
between the northern and southern boundaries, and only east of the highway.
Thus manufacturing land is limited to the rectangular area bounded by the
highway along its western edge, and bounded by zoning lines along its the
northern and southern edges. 4
Agriculture is legal not only in the rectangular area but everywhere else in the
model as well. Agriculture is assumed to be the only legal land use outside
northern and southern zoning boundaries, and the only legal land use on the
other side of the highway.
With the upper left part of Figure 7A-3 modified to delete residential bid rent
functions, we are left with just two bid rent functions: manufacturing BRF (R )
and agricultural BRF (R ). a , tha zero-economic-profit BRF for farms, is
horizontal reflecting an assumption that farm costs do not vary with location.
who supply labour); this residential land area is ignored in these introductory notes, so
manufacturing land will abut agricultural land. Agricultural land is marked A in Figure 7A-3. In the
textbook, distance x from the western boundary is in miles. Miles are replaced with kilometres in
these notes to simplify calculations combining area and linear measurement.
4 Using zoning boundaries to confine manufacturing land to a permitted zone allows the land
supply at each x to be expanded or contracted: the local town or city council can widen or narrow
the zone by amending its zoning law, making y a policy variable. The impact of changes to y will
be considered later in these notes. The manufacturing land supplys eastern boundary is not
specified by zoning but is determined in the land market. The eastern boundary of manufacturing
land is where manufacturing firms freight cost has increased to the point they can no longer
outbid farms for land,
2 The intersection of R and R is at x kilometres from the highway. Figure 1 (at
b a a 5
the end of these notes) shows the bid rent functions and map diagram.
The manufacturing BRF shown in Figure 6-1 has an equation R(x) = $60 10x.
This is evident from the straight-line BRFs intercept on the $ axis (where x = 0)
at $60/hectare/day, and its intercept on the x-axis (Where R = 0) at x = 6. More
generally, the manufacturing BRF equation is derived from a firms zero-
economic-profit equilibrium condition: total revenue = total cost, where total cost
includes the profit required to stay in business: 6
P b = NLPC + LR(x) + t b x. so
R(x) = (P b NLPC t b x) / L. [Equation (1)]
Total revenue Pb is price times output (a firms output is assumed fixed at
b units per day);
NLPC is a firms non-land production cost: a fixed amount per day that
includes labour and capital costs, cost of freight after accessing the
highway, intermediate inputs (such as tires) and profit required to stay in
L is a firms land input, assumed to be a fixed area of land in hectares.
R (x) is equilibrium land rent per hectare per day, as a function of x, so
L R(x) is a firms equilibrium land cost as a function of x;
t is a firms freight cost per unit output per kilometre distance from the
highway. Thus t b x is freight cost per day for a firm x kilometres from the
The numbers in the textbooks example are as follows:
P is fixed exogenously at $50 per bicycle, b at 5 bicycles per day, so a
firms total revenue = $250 per day at all locations;
NLPC is $130 / day so a firms non-land production cost is $130 / day at
L = 2 hectares, a fixed land input per firm at all locations;
t = $4 / unit output / kilometre so $4 (5) x is a firms freight cost per day.
Substituting these numbers into Equation (1) gives us:
R (x) = (250 130 20x) / 2 = 60 10x.
5Figure citations with no dash, for example Figure 1, refer to diagrams at the end of these notes.
Citations with a dash, for example Figure 6-1, refer to diagrams in the textbook.
6Profit required to stay in business is included in the NLPC variable discussed above. Economic
profit is profit in excess of profit required to stay in business. Our model here being perfectly
competitive, economic profit exists only in temporary disequilibrium.