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

2. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 1 CONTENTS Introduction to dynamical systems Example of a model of supply and demand – residential real estate market in Croatia Conclusions

3. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 2 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

4. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 3 DEFINITIONS – DYNAMICAL SYSTEMS

5. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 4 EXAMPLE

6. 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 (1/2) x f(x) “Periodic” behavior for the value of the parameter p = 1.5

7. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 6 ORBITS OF THE PREDATOR-PREY MODEL (2/2) x f(x) “Chaotic” behavior for the value of the parameter p = 3.9

8. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 7 DEFINITION – ATTRACTORS AND BIFURCATIONS

9. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 8 BIFURCATION DIAGRAM OF THE PREDATOR – PREY MODEL Phase space X=[0,1] Parameter r Attractor of the dynamical system for each parameter, period doubling bifurcation

10. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 9 CONTENTS Introduction to dynamical systems Example of a model of supply and demand – residential real estate market in Croatia Conclusions

11. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 10 FACTS REGARDING THE RESIDENTIAL REAL ESTATE MARKET IN CROATIA Number of flats being put on the market in Zagreb 2002 2003 2004 2005 2006 3341 4627 4015 4771 6139 Currently more than 60,000 people look for an appartment Current oversupply of over 2000 flats Is the market working ? Source: CBRE

12. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 11 DECISION MAKING MODEL OF A TYPICAL DEVELOPER Sanitized investment plan of a leading European developer for a residential project in Zagreb

13. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 12 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

14. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 13 DECISION MAKING MODEL OF A TYPICAL RESIDENTIAL BUYER Income of the family: Example Factor 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

15. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 14 SUPPLY – DEMAND CURVE FOR RESIDENTIAL REAL ESTATE 0 5000 10000 Number of flats developed / year Price / sqm Euro Conceptual 1500 2000 2500 3000 Demand Supply (by developer group)

16. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 15 KEY IDEAS FOR MODELING DYNAMICAL SUPPLY AND DEMAND Variables: x n+1 = r x n (1 – x n ) l n – the price of the residential zoned land / sqm (Euro), 1 Jan of each year b n – 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 x n – 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

17. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 16 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

18. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 17 CONTENTS Introduction to dynamical systems Example of a model of supply and demand – residential real estate market in Croatia Conclusions

19. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 18 EXAMPLE – COMPLEX MODELING OF SUPPLY AND DEMAND Model 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

20. Siniša Slijepčević, Department of Mathematics Attractors and bifurcation of a model of supply and demand – 25 February 2008 19 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?