Bifurcations and attractors of a
model of supply and demand
Siniša Slijepčević
22 February 2008
PMF – Deparment of Mathematics
CONTENTS
• Introduction to dynamical systems
• Example of a model of supply and demand –
residential real estate market in Croatia
• Conclusions
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
1
MOTIVATION
Theory of dynamical systems in economical modeling:
• Theory of dynamical systems is used to model and explain deterministic phenomena,
without elements of randomness
• The theory can model complex looking phenomena with relatively simple models
Key tricks
• Lots of tricks to deduce and explain
behavior of a model without solving it
explicitly
• Developed theory to understand changes
of behavior of a class of models, depending
on a parameter (attractors, bifurcations)
Typical phase portrait of a 2D model
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
2
DEFINITIONS – DYNAMICAL SYSTEMS
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
3
EXAMPLE
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
4
ORBITS OF THE PREDATOR-PREY MODEL (1/2)
“Periodic” behavior for the value of the parameter p = 1.5
f(x)
x
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
5
ORBITS OF THE PREDATOR-PREY MODEL (2/2)
“Chaotic” behavior for the value of the parameter p = 3.9
f(x)
x
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
6
DEFINITION – ATTRACTORS AND BIFURCATIONS
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
7
BIFURCATION DIAGRAM OF THE PREDATOR – PREY MODEL
Attractor of the dynamical system for each parameter, period doubling bifurcation
Phase space
X=[0,1]
Parameter r
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
8
CONTENTS
• Introduction to dynamical systems
• Example of a model of supply and demand –
residential real estate market in Croatia
• Conclusions
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
9
FACTS REGARDING THE RESIDENTIAL REAL ESTATE MARKET IN
CROATIA
Number of flats being put on the market in Zagreb
6139
4015
3341
4771
4627
• Currently more
than 60,000
people look for an
appartment
• Current
oversupply of over
2000 flats
• Is the market
working ?
2002
2003
2004
2005
2006
Source: CBRE
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
10
DECISION MAKING MODEL OF A TYPICAL DEVELOPER
Sanitized investment plan of a leading European developer for a residential project in Zagreb
Income
Retail &Residential sales income
Apartments
Parking & Storage
underground Parking Spaces for Seller
above ground
storage
300
23.000
Sq.m.
300
100
2.000
units
units
Sq.m.
Price / m2 inc.
VAT
€ 2.700
SqM
€ 14.000
€ 4.600
€ 2.350
each
each
each
Price net of
VAT
€
2.328
€
12.069
€
3.966
€
2.026
TOTAL SALES
€/sqm
Costs
Site Acquisition Costs
Land acquisition
purchase tax and fees
Residential building costs
Apartments
Basement
Roads and on site parking
Green areas
Apartments communal charges
underground communal charges
12.000
35.000
11.000
3.045
1.218
35.000
11.000
Sqm
Sq.m.
Sq.m.
Sq.m.
Sq.m.
Sq.m.
Sq.m.
Sqm
Sqm
Sqm
Sqm
Sqm
Sqm
Total
€ 3.621.000
€ 397.000
€ 4.052.000
€ 61.604.000
Costs
5%
€ 12.000.000
€ 600.000
€ 700
€ 350
€ 60
€ 30
€ 90
€ 60
€ 24.500.000
€ 3.850.000
€ 183.000
€ 37.000
€ 3.150.000
€ 660.000
1.000
@
@
@
@
@
@
Sales
€ 53.534.000
Total
€ 12.600.000
22,3%
57,3%
€ 32.380.000
Soft costs
Design
Site management
G&A
marketing
Contingency
from
from
from
from
from
Finance
Interest during construction
Loan cost
construction costs
construction costs
construction costs
sales
construction costs
6,5%
1,0%
4,0%
2,5%
2,5%
2,5%
5,0%
€ 51.054.100
€ 1.295.000
€ 810.000
€ 810.000
€ 1.540.100
€ 1.619.000
€ 6.074.100,00
€ 4.977.775
€ 510.541
€ 5.488.316
9,7%
€ 56.542.416
89,3%
TOTAL COSTS
Development Yield on costs
9,0%
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
11
KEY PARAMETERS IN THE DECISION MAKING PROCESS OF A TYPICAL
DEVELOPER TO BUILD A RESIDENTIAL BLOCK IN ZAGREB
• Sales price / sqm (analysis in practice based on the current sales
price)
• Cost of land / sqm
• Cost of construction / sqm
• Communal and water tax / sqm
• Cost to finance (i.e. interest rates; likely leverage)
Developers discriminated by the cost of construction and
cost to finance
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
12
DECISION MAKING MODEL OF A TYPICAL RESIDENTIAL BUYER
Factor
Example
Income of the family:
12,000 kn
Disposable income:
25 % of the income
Required sqm:
60 sqm
Loan (number of years):
30 years
Max price / sqm:
2,300 Euro / sqm
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
13
SUPPLY – DEMAND CURVE FOR RESIDENTIAL REAL ESTATE
Conceptual
Price / sqm
Euro
Supply
(by developer
group)
Demand
3000
2500
2000
1500
0
5000
10000
Number of flats developed / year
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
14
KEY IDEAS FOR MODELING DYNAMICAL SUPPLY AND DEMAND
xn – the price of the residential real
estate / sqm (Euro), 1 Jan of each year
Variables:
ln – the price of the residential zoned
land / sqm (Euro), 1 Jan of each year
bn – number of flats put on market in
each year (pre sales)
Parameter:
r – proportional to interest rates and
average construction cost / sqm
Key principles:
• Model everything in “nominal”,
normalized terms, i.e. net of nominal
GDP growth
xn+1 = r xn (1 – xn)
i.e. the “normalized”
price of the
residential real
estate behaves
accordingly to a
predator – prey
model
• Assume growth of income distribution
proportional to GDP growth; i.e.
constant in the model
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
15
BIFURCATION DIAGRAM FOR THE MODEL OF THE RESIDENTIAL REAL
ESTATE SUPPLY AND DEMAND IN TIME
Normalized price of
the residential real
estate / year
Parameter r
2004: r ~ 2.71
Attractor: stable growth
2004: r ~ 3.62
Attractor: Period 4
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
16
CONTENTS
• Introduction to dynamical systems
• Example of a model of supply and demand –
residential real estate market in Croatia
• Conclusions
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
17
EXAMPLE – COMPLEX MODELING OF SUPPLY AND DEMAND
Model of energy supply and demand in two regions in China
X(t) – Energy supply in the region A
Y(t) – Energy demand in the region B
Z(t) – Energy import from the region A to the
region B
• Lorenz – type chaotic
attractor
• Phenomenologically
equivalent behavior to a
much simpler predator –
prey model
Source: Mei Sun, Lixin Tian, Ying Fu; Chaos, Solitons, Fractals 32 (2007)
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
18
QUESTIONS FOR FURTHER ANALYSIS
• Does the model faithfully represent behavior of the real estate market in a
longer period of time in Croatia? (to be checked experimentally)
• Can it be implemented to other markets (e.g. the US)?
• Which policy is optimal to “regulate” the market, i.e. prevent the real
estate prices bifurcating into the chaotic region?
– Regulating supply (i.e. the POS – type policy?)
– Regulating demand (i.e. the loan interest subsidies for the first time
purchasers)?
– Regulating land prices; e.g. by putting Government owned or Municipal
land for sale or “right to build” for residential development, for
preferential prices?
Siniša Slijepčević, Department of Mathematics
Attractors and bifurcation of a model of supply and demand – 25 February 2008
19