bifurcations and attractors of a model of supply and demand siniša slijepčević 22 february 2008...
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Bifurcations and attractors of a model of supply and demand
Siniša Slijepčević
22 February 2008PMF – Deparment of Mathematics
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
CONTENTS
• Introduction to dynamical systems
• Example of a model of supply and demand – residential real estate market in Croatia
• Conclusions
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
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
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
DEFINITIONS – DYNAMICAL SYSTEMS
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
EXAMPLE
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
ORBITS OF THE PREDATOR-PREY MODEL (1/2)
x
f(x)
“Periodic” behavior for the value of the parameter p = 1.5
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
ORBITS OF THE PREDATOR-PREY MODEL (2/2)
x
f(x)
“Chaotic” behavior for the value of the parameter p = 3.9
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
DEFINITION – ATTRACTORS AND BIFURCATIONS
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
BIFURCATION DIAGRAM OF THE PREDATOR – PREY MODEL
Phase spaceX=[0,1]
Parameter r
Attractor of the dynamical system for each parameter, period doubling bifurcation
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
CONTENTS
• Introduction to dynamical systems
• Example of a model of supply and demand – residential real estate market in Croatia
• Conclusions
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
FACTS REGARDING THE RESIDENTIAL REAL ESTATE MARKET IN CROATIA
Number of flats being put on the market in Zagreb
2002 2003 2004 2005 2006
33414627
40154771
6139• Currently more than 60,000 people look for an appartment
• Current oversupply of over 2000 flats
• Is the market working ?
Source: CBRE
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
DECISION MAKING MODEL OF A TYPICAL DEVELOPERSanitized investment plan of a leading European developer for a residential project in Zagreb
Income
Retail &Residential sales incomePrice / m2 inc. VAT
Price net of VAT Sales Total
Apartments 300 23.000 Sq.m. € 2.700 SqM 2.328€ € 53.534.000
Parking & Storage
underground Parking Spaces for Seller 300 units € 14.000 each 12.069€ € 3.621.000
above ground 100 units € 4.600 each 3.966€ € 397.000
storage 2.000 Sq.m. € 2.350 each 2.026€ € 4.052.000
TOTAL SALES € 61.604.000
Costs €/sqm Costs TotalSite Acquisition Costs
Land acquisition 12.000 Sqm 1.000 € 12.000.000
purchase tax and fees 5% € 600.000 € 12.600.000 22,3%
Residential building costs
Apartments 35.000 Sq.m. @ Sqm € 700 € 24.500.000
Basement 11.000 Sq.m. @ Sqm € 350 € 3.850.000 57,3%
Roads and on site parking 3.045 Sq.m. @ Sqm € 60 € 183.000
Green areas 1.218 Sq.m. @ Sqm € 30 € 37.000
Apartments communal charges 35.000 Sq.m. @ Sqm € 90 € 3.150.000
underground communal charges 11.000 Sq.m. @ Sqm € 60 € 660.000
€ 32.380.000
Soft costs
Design from construction costs 4,0% € 1.295.000
Site management from construction costs 2,5% € 810.000
G&A from construction costs 2,5% € 810.000
marketing from sales 2,5% € 1.540.100
Contingency from construction costs 5,0% € 1.619.000 € 6.074.100,00
Finance
Interest during construction 6,5% € 51.054.100 € 4.977.775
Loan cost 1,0% € 510.541 € 5.488.316 9,7%
TOTAL COSTS € 56.542.416 89,3%
Development Yield on costs 9,0%
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
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
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
DECISION MAKING MODEL OF A TYPICAL RESIDENTIAL BUYER
Income of the family:
ExampleFactor
Disposable income:
Required sqm:
Loan (number of years):
Max price / sqm:
12,000 kn
25 % of the income
60 sqm
30 years
2,300 Euro / sqm
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
SUPPLY – DEMAND CURVE FOR RESIDENTIAL REAL ESTATE
0 5000 10000
Number of flats developed / year
Price / sqmEuro
Conceptual
1500
2000
2500
3000
DemandSupply
(by developer group)
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
KEY IDEAS FOR MODELING DYNAMICAL SUPPLY AND DEMAND
Variables:
xn+1 = r xn (1 – xn)
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
• Assume growth of income distribution proportional to GDP growth; i.e. constant in the model
xn – the price of the residential real estate / sqm (Euro), 1 Jan of each year
i.e. the “normalized” price of the
residential real estate behaves accordingly to a predator – prey
model
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
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 ~ 2.71Attractor: stable growth
2004: r ~ 3.62Attractor: Period 4
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
CONTENTS
• Introduction to dynamical systems
• Example of a model of supply and demand – residential real estate market in Croatia
• Conclusions
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
EXAMPLE – COMPLEX MODELING OF SUPPLY AND DEMANDModel of energy supply and demand in two regions in China
Source: Mei Sun, Lixin Tian, Ying Fu; Chaos, Solitons, Fractals 32 (2007)
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
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Siniša Slijepčević, Department of MathematicsAttractors and bifurcation of a model of supply and demand – 25 February 2008
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?