Models of macroeconomic and financial systems with hysteresis.

Transcription

1 Models of macroeconomic and financial systems with hysteresis. Hugh McNamara Resonance Oscillations and Stability of Non-smooth Dynamical Systems Imperial College London June 2009 H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

2 Hysteresis Definitions of hysteresis The term hysteresis was proposed by Sir Alfred Ewing(1885) after identifying a lag between input and output in magnetic experiments. Further work in magnetism led to a number of models describing hysteresis effects, a key instance being the model first proposed by Ferenc Preisach(1935) which today bears his name. Output Input H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

3 Hysteresis Definitions of hysteresis Independently of these efforts, hysteresis was identified in the wetting and drying of soils by Haines(1930), after work a dispute between himself and Fisher(1926). The Independent Domain Model of soil-moisture hysteresis was proposed by Néel(1942), Everett(1952) and others independently. The Independent Domain model is equivalent in many ways to the Preisach model. Soil Particle Air Water H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

4 Hysteresis Definitions of hysteresis A more complete mathematical framework for hysteresis was developed by a group working at the Moscow Control Institute under Mark Krasnosel skii. Complex hysteresis nonlinearities are identified as connections of elementary hysterons, which in turn are simple functional relationships exposed to positive feedbacks(krasnosel skii & Pokrovskii 1989). R α₁,β₁ µ₁ x(t) R α₂,β₂ R α₃,β₃ R α N,βN µ₂ µ N µ₃ y(t) H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

5 Hysteresis Definitions of hysteresis Under this framework the Preisach model is a parallel connection of non-ideal relays simple step functions with feedback. Emergent properties and identification theorems could be established based on this framework. on 2 Threshold values off H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

6 Hysteresis Definitions of hysteresis One important identification is of systems with a wiping out property, or return-point memory. Isaak Mayergoyz(1991) identified such systems as being of Preisach-type (a second property distinguishes the Preisach model itself). The memory of such a system consists of the non-dominated extrema of the input. Once a past extremum is exceeded(dominated), it is forgotten. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

7 Hysteresis Definitions of hysteresis A further definition of hysteresis was suggested by Augusto Visitin (1994), Martin Brokate & Jürgen Sprechels(1996), who identify hysteresis as operators on function spaces which possess rate-independent memory. A feature of rate independent memory is that the only features remembered will be shocks, extrema of the input. Closing Value /07/ /10/ /01/ /04/2007 Date H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

8 Hysteresis Definitions of hysteresis A further definition of hysteresis arises from applications in mechanical engineering. In this case the identification is with dissipative phenomena which persist in the adiabatic limit(a very accessible description was an article by Bernstein in IEEE Control Systems Magazine Oct. 2007). This is consistent with the rate-independence property mentioned before. All of the above definitions of hysteresis imply dissipative behaviour. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

9 Hysteresis Definitions of hysteresis To summarise, there are a number of definitions of hysteresis, often complementary or equivalent. Systems where changes in output lag the causal changes in input. Collections of hysterons elementary feedback operators. Rate-independent operators on function spaces. Systems which are dissipative in the adiabatic limit. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

10 Macroeconomic hysteresis Macroeconomic Macroeconomics deals with the description of large-scale aggregate quantities such as output(gdp), inflation and unemployment. The mainstream macroeconomic framework inherited ideas from the so-called neo-classical revolution in the late 19th century, when many ideas from the physical sciences were introduced to economics. Many macroeconomic models involve representative agents, who make decisions with perfect rationality and information, and a large proportion involve linear relationships and assume that economic systems operate at equilibrium. In recent years many of these assumptions have come into question. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

11 Macroeconomic hysteresis Macroeconomic hysteresis Since the 70s, hysteresis has begun to be mentioned in an economic context, specifically in the labour market(phelps, 1972 and Sachs, 1986) and foreign trade trade(kemp & Wan, 1974 and Baldwin & Krugman, 1989). The evidence and theoretical support for hysteresis is somewhat varied, and includes features from a number of the senses of hysteresis mentioned before. The phenomena of boom-blessings and recession-curses support the persistence of the effects of shocks. A specific example was the failure of the mainstream models to account for the persistent high level of unemployment in Europe during the 1970s. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

12 Macroeconomic hysteresis Macroeconomic hysteresis A number of unique challenges are faced when applying hysteresis to macroeconomic systems. To begin with, identifying which quantities are inputs (if any) is not a trivial question. In most cases a complicated relationship between a number of quantities must exist. In order to proceed, we first look at replicating certain stylised facts which are easily explained in a hysteresis framework. Among these are the phenomena of heterostasis, whereby the equilibrium level of a certain quantity is altered by a temporary stimulus. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

13 Applying hysteresis to economic systems Aims of the model The main stylised fact which we would like to explain include the persistent impact of a temporary stimulus. We would also like to avoid the use of representative agents responding smoothly and rationally to changes. However, rate-independence is not easily identified in the relationships of interest. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

14 Applying hysteresis to economic systems Fundamental assumptions Our model begins by examining how an individual economic agent (for example a firm) responds to changes in the economic situation. The agent can either be active, producing output(normalised to unity), or inactive. This decision is made based on whether the agent believes that the price of their output is high enough to cover the costs of producing it. However, some of these costs are sunk, they cannot be recovered by reversing the decision to begin production(examples include the purchase of plant machinery, training of staff, marketing efforts, etc.). For this reason the firm will not enter until both the ordinary costs of production and the sunk costs are covered by the price. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

15 Applying hysteresis to economic systems Fundamental assumptions Once active, the firm will continue to produce output even if the price falls below the level at which the entry decision was made. This behaviour is that of a non-ideal relay, one of the hysterons of the Krasnosel skii-pokrovskii framework. on 2 Threshold values off H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

16 Applying hysteresis to economic systems Parallel with hydrology As mentioned earlier, the physics of moisture in the soil was one of the earliest applications of hysteresis. In recent years this has advanced substantially(see recent article in IEEE Control Systems Magazine). The level of moisture in a quantity of soil depends on the amount of inflow in a hysteretic manner, however this relationship is not rate-independent. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

17 Applying hysteresis to economic systems Parallel with hydrology In soil moisture a quantity is introduced which represents the equilibrium level of wetting corresponding to a steady state of the moisture content. It turns out that this quantity has a physical significance, it is called the matric potential and can be expressed in units of energy. As a first approximation, the rate of increase of moisture content is proportional to the difference between the amount of wetting and this matric potential. ẋ(t)= k(w(t) M(t)). H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

18 Applying hysteresis to economic systems Back to economics In economics, a high level of activity will cause an increase in output, and a low level will cause a decrease. There should therefore be an intermediate level at which output will remain constant. We can quantify this as a similar differential equation as for the soil-moisture case: ẋ(t)=k(i(t) y(t)) where y(t) is this equilibrium level. A suitable relationship between x and y will close the system. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

19 Applying hysteresis to economic systems Closing the system The relationship between x and y is assumed to be rate-independent. This assumption allows for the desired persistence effects to manifest, but it is stressed that the relationship between x and I(which are observable quantities) is not rate-independent. This can be written as x( )=Gy( ), where G is a suitably defined rate-independent operator. As described before, there is some good economic justifications for using a collection of non-ideal relay type hysterons to define this operator. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

20 Applying hysteresis to economic systems Preisach example If it is assumed that the individual agents act independently, the Preisach model may be used to close the system, and this type of operator-differential equation has some convenient properties which admit proofs of existence and uniqueness and numerical implementations. Output x(t) Input I(t) Equilibrium rate y(t) H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

21 Applying hysteresis to economic systems More realistic candidates It is more realistic to consider situations in which agents are not independent. This will also incorporate herding effects. A network of dependencies between the agents describes the interaction. u v w z w k H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

22 Applying hysteresis to economic systems Preisach model with feedback A mean field approximation of this network of interactions may be a candidate for use. This would be the Preisach model with feedback shown below. Adding feedback to the input-output behaviour of the Preisach model will introduce some interesting behaviour. Most of the time, the input-output relationship will be continuous, however at certain moments a strong avalanche will occur, and the output will jump. u v w z w k H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23

23 Applying hysteresis to economic systems Other alternatives Another option is to consider a system with multiple interacting inputs. In this case I(t) and y(t) will be vector valued. In this case, one option for relationship x( )=Gy( ) is a variation of the Mayergoyz vector Preisach-type model. H. McNamara (U.C.C.) Hysteresis in Economics 24 June / 23