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Bifurcations and attractors of a model of supply and demand

Explore the dynamics of a model of supply and demand in the residential real estate market in Croatia, using concepts from dynamical systems theory. Learn about attractors, bifurcations, and the behavior of the market over time.

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Bifurcations and attractors of a model of supply and demand

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

  2. CONTENTS • Introduction to dynamical systems • Example of a model of supply and demand – residential real estate market in Croatia • Conclusions

  3. 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. DEFINITIONS – DYNAMICAL SYSTEMS

  5. EXAMPLE

  6. ORBITS OF THE PREDATOR-PREY MODEL (1/2) “Periodic” behavior for the value of the parameter p = 1.5 f(x) x

  7. ORBITS OF THE PREDATOR-PREY MODEL (2/2) “Chaotic” behavior for the value of the parameter p = 3.9 f(x) x

  8. DEFINITION – ATTRACTORS AND BIFURCATIONS

  9. 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

  10. CONTENTS • Introduction to dynamical systems • Example of a model of supply and demand – residential real estate market in Croatia • Conclusions

  11. FACTS REGARDING THE RESIDENTIAL REAL ESTATE MARKET IN CROATIA • Number of flats being put on the market in Zagreb • Currently more than 60,000 people look for an appartment • Current oversupply of over 2000 flats • Is the market working ? 6139 4771 4015 4627 3341 2006 2005 2004 2003 2002 Source: CBRE

  12. DECISION MAKING MODEL OF A TYPICAL DEVELOPER • Sanitized investment plan of a leading European developer for a residential project in Zagreb

  13. 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. DECISION MAKING MODEL OF A TYPICAL RESIDENTIAL BUYER Example Factor 12,000 kn Income of the family: Disposable income: 25 % of the income 60 sqm Required sqm: Loan (number of years): 30 years Max price / sqm: 2,300 Euro / sqm

  15. SUPPLY – DEMAND CURVE FOR RESIDENTIAL REAL ESTATE • Conceptual Price / sqm Euro Supply (by developer group) Demand 3000 2500 2000 1500 10000 5000 0 Number of flats developed / year

  16. 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 xn+1 = r xn (1 – xn) bn – number of flats put on market in each year (pre sales) i.e. the “normalized” price of the residential real estate behaves accordingly to a predator – prey model r – proportional to interest rates and average construction cost / sqm Parameter: 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

  17. 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. CONTENTS • Introduction to dynamical systems • Example of a model of supply and demand – residential real estate market in Croatia • Conclusions

  19. 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)

  20. 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?

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