Bifurcations and attractors of a model of supply and demand Siniša Slijepčević 22 February 2008 PMF – Deparment of Mathematics

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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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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DEFINITIONS – DYNAMICAL SYSTEMS

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EXAMPLE

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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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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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DEFINITION – ATTRACTORS AND BIFURCATIONS

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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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CONTENTS



• Conclusions

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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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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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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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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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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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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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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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CONTENTS



• Conclusions

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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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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?

Documents

Bifurcations and attractors of a model of supply and demand Siniša Slijepčević 22 February 2008 PMF – Deparment of Mathematics