Michel Dimou
Professor in Urban and Regional Economics
4 many series of theoretical work:
▪ Land use according to the economic activity. The prices and costs of production are
fixed, the market is concentrated in a single point. The explanatory variable is distance
(transport costs).
▪ The location of a firm. The prices and costs of production are fixed, the market is
concentrated in a single point. The explanatory variable is distance (transport costs).
▪ The location of a firm according to its competitors. The prices and costs of production
depend upon the state of competition (free competition, oligopoly, duopoly) the market
is dispersed in a homogeneous plane. The explanatory variable is distance (transport
costs).
▪ The location of an economic agent in a generalized spatial equilibrium. The prices and
costs of production depend upon the state of competition (free competition, oligopoly,
duopoly) the market is dispersed in a homogeneous plane. The explanatory variable is
distance (transport costs).
2
The market and the exchanges are
concentrated in a single point. No
possible exchange beyond this point
Heterogenous land
The market and the exchanges are
randomly distributed. The population
is randomly distributed, the expected
density is the same everywhere.
Homogenous plaine
3
H. Von Thunen
4
Von Thunen (1826) The isolated State
(Der isolierte Staat)
▪ What do farmers produce around a
town (market)?
▪ Where do farmers produce
agricultural products?
An innovating analysis :
- The spatial dimension of economic
activity
- Transport costs govern the use of land
- First marginalist approach –
- The concept of bid rent
The main assomptions are:
▪ Land is rare.
▪ Land fertility is the same everywhere.
(homogeneity assumption). Land
productivity is the same.
▪ No geological specificity.
▪ Farming activity is in a stationary state.
▪ Each farmer has the knowledge to
produce and aims in maximising his
profit (rationality hypothesis).
▪ Price is determined by demand and
supply.
▪ Each farmer has access to transport.
5
▪ An environment of free competition
▪ Land is scarce
▪ The soil productivity is not the same
RICARDO’S
MODEL OF LAND
USE
Price
Rent
Production
cost
Land fertility (decreasing)
6
Cost/price
Total cost
C
B
Price
Rent
Transport cost
A
Production cost
D
The triangle ABC
corresponds to a
rent due to location
The soil’s productivity is the
same everywhere.
The land value depends
upon its distance to the
center
Distance to center
7
Total Cost = CP + CT = 10 + X
▪ Price is P depends on Demand
Price 20
▪ CP: production cost
▪ CT : transport cost
▪ R is surplus
Production cost 10
A single product case: wheat
▪ CP : 10
▪ P : 20,
▪ CT : 1 per Km.
▪ Surplus = P – (CP + CT*X) with X distance
.
▪ 0 when X = 10
▪ max (10) when X=0
Distance to center
0
10
Rent 10
0
10
Distance to center
8
9
Rent
▪ What happens if:
▪ the price grows?
0
Distance to center
▪ the cost of production falls?
▪ the transport cost falls?
▪ A sprawl of the farming area.
▪ The boundaries of the wilderness are pushed away from the
town
10
Rent 20
15
corn
▪ Two products: wheat and corn
10
▪ CP wheat: 10, CP corn: 15
▪ P wheat: 20, P corn: 30
wheat
0
5
7,5
10
▪ CT wheat: 1 per km, CT corn: 2 per km.
Distance to market
▪ Surplus wheat = P – (CP + CT*X)
▪ 0 when X = 10 ; max (10) when X= 0
▪ Surplus corn = P – (CP + CT*X)
▪ 0 when X = 7,5 ; max (15) when X= 0
▪ Resolution
▪ Pwh - (Cwh + X)= Pcorn - (Ccorn + 2X)
=> X = 5
11
Initial situation (1)
•
•
•
•
2 goods, wheat & corn.
CP wheat: 10, CP corn: 15
P wheat: 20, P corn: 30
CT wheat: 1 per km, CT corn: 2 per km.
First change (2)
•
•
•
•
2 goods, wheat & corn.
CP wheat: 10, CP corn: 15
P wheat: 20, P corn: 30
CT wheat: 1 per km, CT corn: 1 per km.
Second change (3)
•
•
•
•
2 goods, wheat & corn.
CP wheat: 10, CP corn: 15
P wheat: 20, P corn: 35
CT wheat: 1 per km, CT corn: 2 per km.
12
Situation 1 & 2
Rent 20
Situation 1
Surplus wheat = P – (CP + CT*X)
15
▪ 0 when X = 10 ; max (10) when X= 0
corn
10
Surplus corn = P – (CP + CT*X)
corn (2)
▪ 0 when X = 7,5 ; max (15) when X= 0
Situation 2
wheat
0
7,5
10
Surplus wheat = P – (CP + CT*X)
15
Distance to market
▪ 0 when X = 10 ; max (10) when X= 0
Surplus corn = P – (CP + CT*X)
urban sprawl & substitution
▪ 0 when X = 15 ; max (15) when X= 0
13
Situation 1 & 3
Rent 20
15
Situation 1
corn
10
Surplus wheat = P – (CP + CT*X)
▪ 0 when X = 10 ; max (10) when X= 0
corn (2)
Surplus corn = P – (CP + CT*X)
wheat
▪ 0 when X = 7,5 ; max (15) when X= 0
Situation 2
0
7,5
10
15
Distance to market
Surplus wheat = P – (CP + CT*X)
▪ 0 when X = 10 ; max (10) when X= 0
Substitution without urban sprawl (if Price climbs)
Surplus corn = P – (CP + CT*X)
▪ 0 when X = 10 ; max (20) when X= 0
14
In Von Thunen’s model land
is assumed to be
heterogeneous.
▪ The town already exists
▪ The town is built around
the city-center (the
market)
▪ When no more human
activity subsists, there is
only wilderness.
▪ The real estate price of
wilderness is null (0)
15
Von Thunen doesn’t model a separate transport
cost function.
In Von Thunen’s transport system it is necessary
to transport both the products and the energy
of transport (a char driven by an ox – which
consumes a part of the product for energy
purposes). This additional load is proportional
to the final product the ox char is transporting
and which is to be delivered to the market.
Y = ÿ(1 – ax)
▪ Y: final product delivered at the market place,
▪ ÿ: total product (final product + energy)
▪ a: part of the total load consumed by km for
transport
▪ x: the distance.
16
▪ According to Samuelson the load is getting less
heavy as the ox consumes a part of it, which means that
the ox moves faster.
▪ But in Von Thunen’s function of cost, the marginal cost
is always the same. Marginal cost is linear to distance.
17
Getting smaller, going faster….
The melting of the iceberg is exponential
because in every moment it is
proportional of the remaining load
Y = ÿe-ax
Y: final product delivered at the market place,
ÿ: total product (final product + energy)
a: part of the total load consumed by km for transport
x: the distance.
18
19
3 main concepts :
▪ The CBD
▪ The monocentric urban configuration
▪ The bid rent
Alonso, Muth, Fujita, Thisse.
What can people be paying Manhattan or
downtown Chicago rents for, if not to be
with other people?
R. Lucas, 1987, Nobel Prize in Economics.
20
21
Place du Chatelet
La Défense
Piccadilly Circus
The City
22
23
24
▪ The main goal of the model is to explain the empirical regularities that we observe in cities
▪ The main mechanism is the relationship between commuting costs, housing prices, and
housing consumption
▪ The main output is spatial equilibrium within the city where households feature identical
utility levels and firms identical profits across space
25
26
Land use in Paris
Housing prices in Berlin
27
▪ A pattern of declining density
radiating from the center.
▪ •Tall multi-family buildings tend
to be located near the city
center, while single-family
houses are at the fringe
▪ •Land and housing prices per
square meter/foot tend to be
high near the city center and
lower farther away
28
▪ Land rent (R) is the price for using
one unit of land, say a hectare, for
one unit of time, say a year.
▪ Land value (V) is the price of buying
one unit of land, again say a hectare
▪ •Land is an asset; like any asset its
price (=value) is the present value of
the benefits (=net rent) from owning
it.
29
Housing is measured in units of housing
services = q
▪ q= quality-adjusted square meters
V= the value of a housing unit = the
present value of the rental flow
▪ Depends on housing characteristics
▪ For now, we assume that floor space is the
only characteristic
So, with a long lifetime T, for housing:
p= the price (rent) per unit of q per year
r= rent for a housing unit = pq
▪ If the unit is a rental apartment, r= contract
rent
▪ If the unit is owner-occupied, r is not
observed
30
1. All jobs are in the city center (CBD)
2.The city has a dense network of radial roads
3.The city contains identical households or
consumers or workers
▪ Same income/wage (y) and preferences (will
be relaxed later)
4.The residents consume only two goods:
housing (q) and a composite good (c)
▪ The price of the composite good is the same
everywhere (equal to 1)
Commuting costs
1. The per-kilometer cost of commuting is t, so
a resident living at distance x from the CBD
incurs a commuting cost tx
▪ •Commuting has only a monetary cost
▪ •Later we will introduce the opportunity cost of
time used in commuting
2. This leaves y –tx for expenditure on housing
and the composite good (= disposable income)
▪ •Disposable income decreases as x increases
▪ Land and the housing that sits on it are
allocated to the highest bidder at each location
31
Housing consumption
A housing unit or a dwelling has a variety of
characteristics
▪ Floor space, yard size, construction quality, age,
number of rooms, amenities
We assume that dwellings differ only in size
▪ q represents square meters and p is measured as
rental price per square meter
The consumer’s budget constraint is
y –tx= pq+ c
Consumer’s analysis
Consumers want to maximize the utility
(welfare) they get from consuming housing
and the common good, while taking into
account their budget constraint
The consumer chooses the c and q to
maximize utility U(c, q) subject to the
budget constraint at each distance x
▪ Location “choice” enters the problem only
through commuting costs
▪ Dwellings differ only with respect to size.
The expenditure on common good and housing is
equal to disposable income after commuting costs
32
Everyone wants to live right next to the
CBD, but everyone can’t live in the same
location
This equilibrium can hold only if price per
housing per square meter falls as distance
increases
Since higher commuting costs mean that
disposable income falls as x increases, some
offsetting benefit must be present to keep
utility from falling
Lower p at more distant locations serves as
a compensating differential
The price of the composite good is the same
everywhere, and doesn’t play a compensating
role
33
▪ The consumer chooses the point
where the indifference curve is
tangent to the budget line (c1, q1)
▪ •This is the highest possible
indifference curve that the
consumer can reach within the
budget constraint
34
35
What magnitude must the price of housing p1 be at
distance x1 to ensure that the suburban consumer reach
the same indifference curve as the central-city consumer?
▪ Prices per m² are higher in central-city, p0> p1
▪ The suburban resident consumes more housing space
(q1> q0) and less common good (c1< c0) than the
central-city resident
▪ This means that dwelling size q increases as distance x
from the CBD increases
The model’s two main predictions are
1. Price per square meter p of housing falls as distance x
increases
2. Size of the dwellings q increases when x increases
36
How does the total rent (pq) for a small centralcity dwelling compare to the total rent of a
larger suburban house?
▪ Since p falls with x while q increases, the
product pq could either increase or decrease.
It depends on the consumer’s preferences or
the shape of the indifference curve
▪ The price curve is convex if housing
increases with x. Consumers substitute
cheaper housing for common goods, so
housing prices don’t have to decline as
quickly to compensate consumers.
37
The assumptions
 =V − A −F
with p = profit, V = sales (income), A =
other costs, F = real estate costs.
and V = V(Kq) where:
K = distance to the centre and q = size of
the plant.
A = A(V, K, q)
F = P(K)q, where P(K) = real estate price at a
distance K from city center
Two equations
The location equation
dV dA dV dA
dP
−

−
+q
=0
dK dV dK dK
dK
The land consumption equation
dV dA dV dA dP
−

−
−
=0
dq dV dq dq dq
38
surface
surface
q2
q
q
q1
U1
U1
K
distance
K1
K
K2
distance
39
40
The German theory of location
(W. Roscher, A. Schäffle, W. Launhardt)
The Fermat/Toricelli Point: in a triangle,
which is the point that minimizes the
aggregated distance to these points?
Describes how/where to find the optimal
location for a:
▪ manufacturing firm. Its purpose it to place
a firm in a location
▪ where the final cost of transportation,
labor, and materials are minimized.
41
Transfer-oriented firm: transport cost is
the dominant factor in the location
decision
➢ The firm chooses the location that
minimizes total transport costs
Two types of cost:
- Procurement cost is the cost of
transporting raw materials from the
input source to the production facility
- Distribution cost is the cost of
transporting the firm’s output from the
production facility to the market
▪ Four assumptions:
1.Single transferable output. The firm produces a fixed
quantity of a single product, which is transported from
the production facility to a market M
2.Single transferable input. The firm may use several
inputs, but only one input is transported from an input
source, F, to the production facility. All other inputs are
ubiquitous.
3.Fixed-factor proportions. The firm produces its fixed
quantity with fixed amounts of each input. No factor
substitution
4.Fixed prices. The firm is so small that it does not
affect the prices of its input or its output
➢ The only cost that varies across space is transport
cost
42
A Single establishment
M2
▪ 2 inputs - 1 output
d2
▪ A fixed location of the raw materials/1st input,
the labor market/2nd input and the final market
K
▪ Distance is Euclidian. Transport costs are
directly proportional to weight
▪ The profit of the firm is:
 = p3m3 − ( p1 + t1d1 )m1 − ( p2 + t2 d 2 )m2 − t3d3m3
d3
d1
M1
M3
3
Min Transport cost =  mi ti d i
i =1
43
A
1
2
3
3
2
1
B
3
2
1
C
For 1 $, how much distance can the
firm travel if carrying raw materials
(A), labor force (B), or the final
product (C)?
44
M3
Final Market
The material index (I) is
the ratio of the weights of
the intermediate products
(raw materials) to the final
product.
Output Transport Cost
M1
steel
Input Transport Cost
M2
plastic
M1
steel
M3
Final Market
M2
plastic
•
If I >1, the firm locates
close to the raw
materials source.
•
If I < 1, the firm locates
close to the final
market.
45
A Firm producing hockey bats
Monetary weight input = mi  ti = 10  $1 = $10
Monetary weight output = mo to = 3  $2 = $6
46
PC = mi  ti  x = 10  1  x
DC = mo  to  ( xM − x ) = 3  2  ( xM − x )
Total transport cost
(procurement cost plus
distribution cost is
minimized at the input
source (the forest)
because the monetary
weight of the input (($10)
exceeds the monetary
weight of the output ($6)
47
A bottling firm of beveradges
Monetary weight input = mi  ti = 1  $1 = $1
Monetary weight output = mo  to = 4  $1 = $4
48
Total transport cost is
minimized at the final
market because the
monetary weight of the
output (($4) exceeds the
monetary weight of the
input ($1)
PC = mi  ti  x = 1  1  x
DC = mo  to  ( xM − x ) = 4  1  ( xM − x )
49
THE MOSES
LOCATION
PRODUCTION
MODEL
- The firm can substitute in favour of the cheaper inputs
- The distance from the factory to the market, d3, is fixed
- The firm chooses a location along the arc IJ
50
Looks at the price ratio
between inputs
m1
Can locate anywhere within
specific distance from
output market between L & J
Envelope budget
constraint
The choice is then the
combination of inputs
L
Output
Isoquant
J
This allows the
development of an
envelope budget constraint
q2
m2
51
input 1
A’
Q
A
J
h
O
Variable-factor proportions.
L
Budget
Constraint in L1
c
The firm produces its fixed
quantity with variables
amounts of each input.
Budget constraint in L2
B’
B
Case 1: Minimizing the cost of production
❖ 1 firm, 1 product,
❖ A fixed volume of final product Q,
input 2
Factor substitution
Prices of inputs are different
according to location.
Location choice: a different
combination of inputs.
❖ Different input prices in each locality.
52
input 1
A’
A
Q2 Q3
Q1
kj
2
g t
B’
B
Case 2: Maximizing the quantity of production
❖ 1 firm, 1 product,
❖ Different input prices in each locality.
input 2
Variable-factor
proportions.
The firm produces its fixed
quantity with variables
amounts of each input.
Factor substitution
Prices of inputs are
different according to
location.
Location choice: a different
combination of inputs.
53
54
▪ Land is homogeneous
▪ People are randomly
distributed
▪ The expected density is the
same in every point of the land
55
Hotelling Model…
Price
p1 + t.x
p1 + t.x
V
V
t
z=0
Price
x1
t
p1
1/2
Shop 1
x1
z=1
p1 + t.x1 = V, so x1 = (V – p1)/t
56
Hotelling Model…
Price
p1 + t.x
p1 + t.x
Price
V
V
p1
p2
z=0
x2
x1
1/2
x1
x2
z=1
Shop 1
57
Hotelling Model…
◼
Suppose that all consumers are to be served at price p.
❑
The highest price is that charged to the consumers at the ends of the
market.
❑
Their transport costs are t/2 : since they travel ½ mile to the shop
❑
So they pay p + t/2 which must be no greater than V.
❑
So p = V – t/2.
◼
Suppose that marginal costs are c per unit.
◼
Suppose also that a shop has set-up costs of F.
◼
t


Then profit is  = N  V − − c  − F
2 

58
Hotelling Model…
Price
p1 + t.x
p2 + t.x
p1 + t.x
V
p3 + 1/2t
Price
p2 + t.x
V
p1
p2
p3
z=0
x3
x2
x1
1/2
Shop 1
x1
x2
z=1
x3
59
The only stable equilibrium
is not optimal neither for
each prisoner nor for the
collectivity (both prisoners).
A non cooperative game
60
• A beach
• A random distribution
of people
• 2 ice cream shops
• Same ice cream
• Same price
• No ccoperation
• Minimize distance from
the ice cream shop
61
•
A beach
•
A random distribution of people
•
2 ice cream shops
•
Same ice cream
•
Same price
•
No ccoperation
•
Minimize distance from the ice
cream shop
62
25%
25%
25%
25%
63
Transport costs
Production cost
Distance
A
B
▪ Transport cost
64
Transport Costs
Production Costs
Distance
A
B
Agglomeration under certain circumstances
65
Hotelling Model…
p1 + txm = p2 + t(1 - xm) 2txm = p2 - p1 + t
xm(p1, p2) = (p2 - p1 + t)/2t
There are N consumers in total
So demand to firm 1 is D1 = N(p2 - p1 + t)/2t
Price
Price
p2
p1
xm
Shop 1
Shop 2
66
Hotelling Model…
Profit to firm 1 is 1 = (p1 - c)D1 = N(p1 - c)(p2 - p1 + t)/2t
1 = N(p2p1 - p12 + tp1 + cp1 - cp2 -ct)/2t
Differentiate with respect to p1
1/ p1 =
N
(p2 - 2p1 + t + c) = 0
2t
p*1 = (p2 + t + c)/2
What about firm 2? By symmetry, it has a
similar best response function.
p*2 = (p1 + t + c)/2
EC 3322 (Industrial Organization I)
Yohanes E. Riyanto
67
Hotelling Model…
Finding the Bertrand-Nash Eq.:
p*1 = (p2 + t + c)/2
p2
R1
p*2 = (p1 + t + c)/2
2p*2 = p1 + t + c
R2
= p2/2 + 3(t + c)/2 c + t
 p*2 = t + c
(c + t)/2
 p*1 = t + c
Profit per unit to each
(c + t)/2 c + t
firm is t
Aggregate profit to each firm is Nt/2
p1
68
Hotelling Model…
Price
Price
p*2 = t+c
p*1 = t+c
xm = (p2 - p1 + t)/2t
xm =1/2
Shop 1
Shop 2
69
Strategic Complements and Substitutes…
❑
❑
Suppose firm 2’s costs increase
This causes Firm 2’s Cournot best
response function to fall
◼
❑
aggressive
response by
firm 1
q2
at any output for firm 1 firm 2
now wants to produce less
Firm 1
Cournot
Firm 1’s output increases and firm
2’s falls
Firm 2
q1
❑
Firm 2’s Bertrand best response
function rises
◼
❑
at any price for firm 1 firm 2
now wants to raise its price
firm 1’s price increases as does
firm 2’s
p2
passive
response by
firm 1
Firm 1
Firm 2
Bertrand
p1
70
Transport Costs
Production Costs
Distance
A
B
▪ High Transport costs prevent concentration
▪ Low Transport Costs prevent concentration.
71
Production cost B
Production cost A
Distance
A
B
A different strategy
▪ A seeks agglomeration
▪ B flies away
Distance prevent from competition
72
Transport costs
Production costs
Distance
A
B
Assymety of the transport cost function
73
Production cost B
Production cost A
Distance
A
B
Local market effects.
74
W. Christaller: the role of distance
A. Lösch: the role of people welfare
75
The main asumptions:
▪ There is an isotropic plain of flat land throughout the plain.
▪ There is a homogenous preference among people since people will always purchase
goods from the nearest place possible?
▪ Consumers bear the burden of shipping in terms of cost.
▪ That people act economically rationally.
The cost of product is defined by:
▪ The cost of production (fixed).
▪ The cost of transport to get to the seller (depends on distance).
76
▪ A unique product, different sellers
distributed across space.
▪ The price at the shop is 5€,
consumer’s
gain
▪ Transport cost is 2€/km,
40€
▪ The limit range of the product is 40€
(maximum price that an economic agent
is ready to pay in order to buy this
product),
▪ Market area: a radius of 17.5 km.
0
0
17,5
distance
77
▪ The limit range depends upon 3 variables:
•
▪ The price
▪ The transport cost
▪ The consumer’s budget
consumer’s
gain
40€
▪
▪ Transport cost depends upon:
▪ The transport mode ;
▪ The transport frequency (daily, weekly, monthly);
0
0
17,5
distance
▪ The product’s cost of production.
▪ The consumer’s budget.
78
79
•
▪ Each type of good has its own limit
range (all producers behave the
same way)
consumer’s
gain
▪ Two different types of goods have
two different limit ranges. How do
externalities interfere
good 1
good 2
0
0
distance
80
•
▪ A market mechanism
consumer’s
gain
▪ Agglomeration
▪ Public administration analysis
Example
good 1
▪ Kinder garden
▪ Primary school
good 2
▪ Secondary schools
0
▪ Universities
0
distance
81
82
▪ US city more segregationist than the European
city (to be discussed)
▪ Two important parameters when studying the
US city:
▪ Land consumption– neighbors.
▪ « Tell me who is your neighbor, I will guess
where you live »
▪ US sociologists aim to understand economic,
social and ethnical segregation in the US cities.
83
▪ Park & Burgess – McKenzie: the geographical dimension of
the social structure of Chicago. People change their location
when climbing up the social ladder. Residential mobility.
▪ Hoyt takes into account the importance of transport
infrastructures. Public equipment tend to settle down
newcomers.
▪ Harris & Ullman: the importance of public equipment away
from the city-center. A city is built on the aggregation of
multiple neighborhoods.
84
▪ 4 principles:
▪ principle of location (close or far
from city center & transport cost).
▪ principle of agglomeration (scale
and scope external economies).
▪ principle of exclusion (scale and
scope negative external economies).
▪ principle of the best possible choice
(all locations are not free).
85
The spatial & social
organization of Chicago
86
•
The importance of having
an access to the city
center.
•
Combining sociology to
the monocentric city.
87
A functional
organization of the
monocentric city
88
The US city
La ville Européenne
The European
city
High income Households
Industrial activity
Median income Households
Merchant Activities
CBD
Median income Households
High income
Households
Students
Ethnic districts
Low income households
89
Europe: Rich people in city-center – poor people in
the suburbs
M.Freyssenet: not quite true ! How social
segregation in Paris changed.
▪ In the early 1800, city center is poor.
▪ In 1950, city-center is heterogeneous but the
suburbs are homogeneous (workers).
▪ In 1980, the city-center becomes homogeneous
(bourgeois) but the suburbs are heterogeneous
(newcomers, workers and median income
families).
before & after
Baron Hausmann’s Paris
90
91
▪ Imbert : in France, in the 1960s and
the 1970s poor people live in large
parts of the buildings in Paris’s city
center (ancien and decrepit). But this
changed within two decades.
▪ The historical value of buildings.
▪ City center is not only the economic
but also the cultural center.
▪ Distance is also calculated in time of
transport.
92
▪ Urban renovation: a tool to move
away the poor population out of
the city center.
▪ Higher local taxes & higher
housing prices.
▪ M.Anderson (1970), M.Castells
(1970), H.Nonn (1977) : the victims
of urban renovation.
▪
▪ In France : the suburbs represent
le « débordement de la ville ».
93
94
95
Micromotives and macrobehavior
▪ The US city was not a segregated
area
▪ It became segregated through a
double mechanism:
▪ The hazard
▪ People preferences
▪ Market mechanisms
96
97
-
-
Static or dynamic context
Number of parameters that enter the location choice
Type of land
98
•
Transport costs and housing prices
•
The importance of land use
•
The neighbors
•
We didn’t introduce amenities