Introductory Notes on Rectangular Land Market Model
These notes elaborate on a land-market model presented in O’Sullivan’s
textbook “Urban Economics”.
Building on the textbook’s analysis, our focus will be on these key questions:
• What is the impact on land rents if an endogenous output price replaces
the textbook’s exogenous output price? 1
• 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 section “Bid-Rent Curves for the
Manufacturing Sector” (pages 128-30, 8 edition, or 122-124, 7 edition, or 102-
104, 6 edition). In that section’s 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 Gethral Equilibrium Model ofth
Monocentric City” (pages 198-201, 8 edition, or 192-194, 7 edition, 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-3’s bid rent functions labelled
R r (initiar (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”.
In the lower part of Figure 7A-3 we see the manufacturing land area (marked D),
with its western boundary formed by a highway
An endogenous price is determined in the model; an exogenous price is given from outside the
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 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
includes all costs that are independent of location. One of these costs is the cost
of freight incurred after a firm’s vehicles have accessed the highway – an equal
amount for all firms and thus independent of location.
The rectangular land market has northern and southern boundaries, shown as
lines perpendicular to the highway. In the textbook’s Fig. 7A-3 they are assumed
to be 1 mile apart. However, in these notes we will let the rectangular land
market’s 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 R , ahe zero-economic-profit BRF for farms, is horizontal –
reflecting an assumption that farm costs do not vary with location. The
labour. The area S (supply of labour) is land occupied by housing firms (firms housing residents
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 supply’s 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 intersection of R and R is at x kilometres from the highway. Figure 1
b a a
(diagrams 5re at the end of these notes) shows the bid rent functions and map
The manufacturing BRF shown in the text book’s Figure 6-1 has an equation R(x)
= $60 – 10x. This is evident from the straight-line BRF’s 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 firm’s
zero-economic-profit equilibrium condition: total revenue = total cost: 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 firm’s 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 firm’s 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 firm’s equilibrium land cost as a function of x;
• t is a firm’s 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 textbook’s example are as follows:
• P is fixed exogenously at $50 per bicycle, b at 5 bicycles per day, so a
firm’s total revenue = $250 per day at all locations;
• NLPC is $130 / day so a firm’s 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 firm’s 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.
3 Now assume that R is $2a / hectare per day and solve for the intersection of R a
and R (x). At the intersection, R (x) = R = 20a= 60 - 10x so = x a 4 km. Tais
equilibrium is illustrated in Figure 1.
How does the land market arrive at zero-economic-profit rents?
Land rents temporarily below the zero-economic-profit level – by definition –
allow firms to earn economic profit. While existing firms would like this to
continue indefinitely, “wannabe” firms act as spoilers.
Wannabe firms are currently outside the model, but learn about any location
where economic profit is earned. A wannabe firm can appropriate an existing
firm’s economic profit by taking over the land the existing firm is renting. To take
over the land, it offers to outbid the existing firm. The land owner then gives the
existing firm a choice: either match the competing bid or leave. Competing bids
only stop going up when rent is at the zero-economic-profit level.
Land rents above the zero-economic-profit level are also unsustainable. If an
existing firm’s profit is too low to keep it in business it will lower its rent bid. We
assume that all firms are alike, so no other firm can outbid the existing firm. The
land owner then has to accept the zero-economic-profit bid.
As is indicated in the note to Figure 1, manufacturing land area is a rectangle x
kilometres by y kilometres. Its area measures y x sq. km =a100 y x hectares. a
Output per hectare is b / L, since each firm produces b output units per day on L
hectares. Thus total output produced on t9e manufacturing land area (B, in
bicycles per day) is 100 y x a (b / L). .
Given that B = 100 y x (b a L), it follows that.
xa = [L / (100 b y)] B [Equation (2)].
7Land owners in the model have no market power. Their role is simply to rent their land to the
highest bidder. We assume they accept a manufacturing firm’s bid if it equals or exceeds
agricultural rent, and will supply all of their land at any rent greater than or equal to zero – that is,
each land owner’s supply function is zero-elastic (vertical).
1 sq. km = 1,000,000 sq. m and 1 hectare = 10,000 sq. m, so 1 sq. km = 100 hectares. All land
rents in these notes and in relevant sections of the text are per hectare, so the area unit does not
need to be specified in every reference to land rent. Also in these notes all revenue and cost
numbers are per day so a time unit does not need to be specified in every reference to costs or
revenues. Thus for example, land rent (R) = $40 means $40 per hectare per day.
9We assume that land area required for local roads is small enough to ignore, so output is
produced on all land in the manufacturing land area.
4 We can use this equation as a starting point in deriving the supply function for
output. One point on that supply function (see note to Figure 1) is P = $50, B =
2000. Obtaining an equation for the supply function will allow output supplied to
be calculated for any price level.
The output supply function of a competitive industry is its marginal cost (MC)
function. Marginal cost is the addition to total cost when output increases by one
unit / day. With the model considered here, the additional cost is calculated by
adding one firm producing b units / day; the resulting addition to cost is then
divided by b to arrive at marginal cost for one unit.
Since all land between the highway and x is alaeady occupied, the marginal firm
locates at x a Its freight cost is t bax , or a x per unit. Thus t a is the freight
component of marginal cost.
An additional firm’s b units of output also require expenditure on non-land
production inputs, so NLPC / b is the non-land production cost component of
marginal cost. This component of marginal cost is constant, regardless of the
distance between x ana the highway.
An additional firm also requires L hectares of land. Given that firm’s location at
the agricultural boundary (x a), its land cost will be L Ra. Thus the land
component of marginal cost is L R / b.aThis component of MC is also constant
regardless of distance between x and the highway.
Adding the three components of marginal cost (NLPC, land and freight) we
MC = NLPC / b + L R a/b + t xa. [Equation (3)]
Substituting Equation (2) into Equation (3) we obtain
MC = NLPC / b + L R a/ b + [t L / (100by)] B.
Thus the supply function is
P = MC = NLPC/b + L R a/b + [tL / (100by)] B. [Equation (4)]
Figure 2 illustrates the equilibrium supply / demand intersection when P is set
exogenously at P* (P* = $50 in the example considered here), and the supply
function S has the parameter values indicated in footnote 10, so B 1= 2000
bicycles / day. P i0 Figure 2 is the constant term of marginal cost [NLPC / b + (L
R a) / b] = $34 in the example considered here.
With the numbers assumed above, the supply function will be
P = 130/5 + (2 x 20) / 5 + [(4 x 2) / (100 x 5 x 2)] B = 34 + 0.008B.
5 Also illustrated in Figure 2 is equilibrium with the supply function S’