equilibrium and stability in economic dynamics · · 2012-02-21international workshop on economic...

Equilibrium and Stability in EconomicDynamics

Alfredo Medio

Department of Statistics, University of Udine

International Workshop on Economic Dynamics,University of Siena, Collegio S. Chiara

18-19 December 2009

Alfredo Medio Equilibrium and Stability

Outline

Disequilibrium Dynamics

John Chipman (1965), economist

The real content of the equilibrium concept is to befound not so much in the state itself as in the laws ofchange which it implies; that is in the tendencies tomove towards it, away from it, or around it

Brian Skyrms (1992), philosopher, game theorist

The explanatory value of equilibrium depends onthe underlying dynamics. When the underlyingdynamics is taken seriously, it becomes apparent thatequilibrium is not the central explanatory concept

Outline

Canonical Dynamical Systems

1 Discrete Timext+1 = G(xt ;α) (1)

x ∈ Rn: state variables

G : U → Rm: “law of motion”

α ∈ Rm: given parameters

t ∈ Z+: time

2 Continuous Time

x = f (x ;α) (2)

x ≡ d/dtf : U → R

t ∈ R

Canonical Dynamical Systems

1 Discrete Timext+1 = G(xt ;α) (1)

x ∈ Rn: state variables

G : U → Rm: “law of motion”

α ∈ Rm: given parameters

t ∈ Z+: time

2 Continuous Time

x = f (x ;α) (2)

x ≡ d/dtf : U → R

t ∈ R

Definition

An orbit is a time–ordered collection of states of a dynamicalsystem starting from a given initial point. In continuous time, anorbit defines a curve {x(t)|t ∈ I ⊂ R}, in discrete–time it is asequence of discrete points (x0, G(x0), G2(x0), . . .)

Definition

equilibrium is a fixed point x of (1) or (2), such that G(x) = x(discrete case) , or f (x) = 0 (continuous case)

Definition

A set A ⊂ Rn is invariant if orbits never leave it under the action

of G (or f )

Definition

of G (or f )

Definition

of G (or f )

Stability: there are more than 50 definitions of stability,but...

Definition

A point, or a set, is weakly (Lyapunov) stable if orbits startingnear it stay near

Definition

A point, or a set, is strongly (asymptotically) stable if (i) it isweakly stable and (ii) orbits starting in a (not too small) basinconverge to it

Stability: there are more than 50 definitions of stability,but...

Definition

A point, or a set, is weakly (Lyapunov) stable if orbits startingnear it stay near

Definition

A point, or a set, is strongly (asymptotically) stable if (i) it isweakly stable and (ii) orbits starting in a (not too small) basinconverge to it

Strategy

locate equilibria (invariant sets) and investigate theirstability properties

study the behavior of typical orbits (in a topological ormetric sense)

study effects of changing parameters (bifurcation theory)

Strategy

Outline

Models of Disequilibrium Dynamics

Example

Multiplier–accelerator cycle models: e.g., Samuelson (1939),Kaldor (1940), Hicks (1950), Goodwin (1951), Kalecki (1954)

aggregate “stylized facts”, instead of “first principles”

lags and nonlinearities

adaptive, backward–looking adjusting mechanism

methods: stability analysis, phase–diagrams,Poincaré–Bendixson Theorem

result: conditions for persistent oscillations

Example

Tâtonnement: Walras (1926 [1874]), ..., Arrow and Hurwicz(1958) and Arrow, Block and Hurwicz(1959)

rational (mazimizing) agents; macroeconomic compatibility(markets clear); no “false trading”

adjustment: “law of demand and supply”

method: second method of Lyapunov

result: equilibrium strongly stable only under stringentassumptions (gross substitutability)

Example

The Abandonment of Disequilibrium Dynamics

Muth (1961), Hahn (1974), Prescott and Mehra (1980), Lucas(1996, Nobel Lecture)

the scientist’s approach (as in Chipman’s and Skyrms’squotes) is rejected: equilibrium is the central explanatoryconcept of economic analysis

rational expectations: only non–systematic mistakes areallowed; probability distributions on which agents’decisions are based are the same as those resulting fromthose decisions (in a deterministic context, this amounts to“perfect foresight”)

The Abandonment of Disequilibrium Dynamics

Muth (1961), Hahn (1974), Prescott and Mehra (1980), Lucas(1996, Nobel Lecture)

the scientist’s approach (as in Chipman’s and Skyrms’squotes) is rejected: equilibrium is the central explanatoryconcept of economic analysis

rational expectations: only non–systematic mistakes areallowed; probability distributions on which agents’decisions are based are the same as those resulting fromthose decisions (in a deterministic context, this amounts to“perfect foresight”)

Recursive Equilibrium Dynamics

Dynamics are not generated by off–equilibrium forces butare derived from equilibrium conditions

Agents’ maximizing behavior + clearing markets +intertemporal constraints generate the implicit “law ofmotion”, with or without uncertainty

disequilibrium analysis is disqualified becausedisequilibrium behavior is irrational and therefore is not a“proper economic behavior”

Example

I. Optimal growth

Ramsey problem, discrete time

solution orbits correspond to equilibria of the completemarket, Arrow–Debreu model

application of Bellman’s dynamic programming equationand the associate policy function may reduce the problemto a recursive model (in appropriate state variables)

result: “anything goes”; equilibrium orbits may tend to astationary equilibrium, to periodic or even chaotic sets

Example

I. Optimal growth

Example

I. Optimal growth

Example

I. Optimal growth

Example

II. Overlapping generations models

no complete markets and unrestricted participation; doubleinfinity of agents and commodities

implicit dynamics:

F (xt , xt+1) = 0 (3)

if F can be inverted w.r.t. xt+1, (3) → (1)

result: “anything goes” againproblems (Medio and Raines, 2007):

backward dynamics, ifF can only be inverted w.r.t. xt

initial conditions (are they arbitrary?)

Example

implicit dynamics:

F (xt , xt+1) = 0 (3)

Example

implicit dynamics:

F (xt , xt+1) = 0 (3)

Example

implicit dynamics:

F (xt , xt+1) = 0 (3)

Example

implicit dynamics:

F (xt , xt+1) = 0 (3)

Bounded Rationality

the perfect foresight/rational expectations hypothesisrequires some very demanding assumptions (e.g., a greatdeal of common knowledge, infinite and non–distortedmemory, a super–human calculating ability)

loosening one or the other of them yields non–equilibriumdynamics models, often in the form of iterated maps (withor without random elements)

Bounded Rationality

Example

forecasting methods (homogeneous or heterogeneous,e.g. Brock and Hommes (1997))

learning (“eductive”, trial–and–error, etc., e.g. Binmore(1990), Gale (2000))

dynamic games (fictitious games, evolutionary games, e.g.Skyrms (1992))

Random Difference Equations

Recursive equilibrium, or bounded rationality models +random perturbations with a Markov representation

two basic types: perturbations of fundamentals (tastes andtechnology) and perturbations of expectations (sunspots)

Basic Stochastic Equations

xt+1 = T (xt , εξt+1)

xt+1 = T (xt , εξt+1) = F (xt)︸︷︷︸

deterministic core

+ G(xt , εξt+1)︸︷︷︸

perturbation term

F (xt) = T (xt , 0) ∀t ≥ 0G(xt , 0) = 0 ∀x

xt+1 = T (xt , εξt+1)

xt+1 = T (xt , εξt+1) = F (xt)︸︷︷︸

deterministic core

+ G(xt , εξt+1)︸︷︷︸

perturbation term

F (xt) = T (xt , 0) ∀t ≥ 0G(xt , 0) = 0 ∀x

xt+1 = T (xt , εξt+1)

xt+1 = T (xt , εξt+1) = F (xt)︸︷︷︸

deterministic core

+ G(xt , εξt+1)︸︷︷︸

perturbation term

F (xt) = T (xt , 0) ∀t ≥ 0G(xt , 0) = 0 ∀x

xt+1 = T (xt , εξt+1)

xt+1 = T (xt , εξt+1) = F (xt)︸︷︷︸

deterministic core

+ G(xt , εξt+1)︸︷︷︸

perturbation term

F (xt) = T (xt , 0) ∀t ≥ 0G(xt , 0) = 0 ∀x

Dynamics of Probability Distributions

Transition Probability Kernel

Pε(x , A) =

χA[T (x , εξ)]νε(dεξ) x ∈ X , A ∈ B(X )

πt+1(A) = P[πt (A)], π ∈ M

P[π(A)] =

XP(x , A)π(dx)

π(A) =∫

P(x , A)π(dx) : invariant probability distribution

Pε(x , A) =

πt+1(A) = P[πt (A)], π ∈ M

P[π(A)] =

XP(x , A)π(dx)

π(A) =∫

Pε(x , A) =

πt+1(A) = P[πt (A)], π ∈ M

P[π(A)] =

XP(x , A)π(dx)

π(A) =∫

Stochastic Stability

A general result on stochastic stability, Medio (2004)

1 applications: perturbed optimal growth models; sunspotOLG models (with well–defined forward dynamics);(bounded rationality) models of dynamic matching andbargaining games, with a Markov representation

2 state variables and random variables take values ingeneral space

3 result applies to any type of deterministic attracting set Λ(fixed point, periodic, chaotic)

Assumptions

1 regularity of maps and perturbations2 the deterministic map F possesses an asymptotically

stable attracting set Λ with basin B(Λ)

3 the perturbation is sufficiently small (ε small) w.r.t. the sizeof B(Λ) and the strength of attraction

Assumptions

Transitivity (indecomposability)

Three cases:1 Case 1. Λ is strongly indecomposable (F strongly

topologically transitive) (e.g. Λ is a fixed point or a mixingchaotic set)

2 Case 2. F is topologically transitive on Λ but not stronglyso (e.g., Λ is a periodic set or a non–mixing chaotic set)

3 Case 3. F is not topologically transitive on Λ (e.g., thereexist multiple attractors for the map F )

1 Case 1: for each ε the operator P convergesasymptotically to a unique, invariant distribution πε

2 Case 2: there exist a finite number d of invariant (possiblyidentical) distributions. The operator (P)d converges to aunique distribution depending on the initial conditions

3 Case 3: there exists a finite decomposition of the statespace X into absorbing sets. If we restrict the dynamics ofT to one or the other of these sets, we are back to case 1or 2

A bouquet of simple ideas

1 in the theory of competitive general equilibrium anormative notion of equilibrium prevails

2 in a descriptive sense, equilibrium loses much of itsplausibility without an account of the dynamics that bring itabout

3 mathematical stability theory retains a fundamentalimportance in the economist’s toolkit

4 in the presence of random perturbations, the dynamics of asystem are best described in terms of probabilitydistributions and their stochastic stability. This significantlyblurs the differences among the canonical types ofattractors (fixed points, periodic and chaotic sets)

References I

Arrow, K. J. and L. Hurwicz, 1958. On the stability of thecompetitive equilibrium I,Econometrica, 26, 522–52

Arrow, K. J., H. D. Block and L. Hurwicz, 1959. On the stabilityof the competitive equilibrium II,Econometrica27, 82–109

Binmore, K. 1990. Modelling Rational Players: Parts I and II, in:K. Binmore,Essays on the Foundations of Game Theory, Oxfordand Cambbridge, MA: Blackwell, 151–185, 186–231

Brock, W.A. and C.H. Hommes, 1997. A Rational Route toRandomness,Econometrica, 65(5), 1059-1096,

Chipman, John S. 1965. The Nature and Meaning of Equilibriumin Economic Theory, in: H. Townsend (ed.),Price Theory,Middlesex, England: Penguin Books, 341–372

Gale, D. 2000.Strategic Foundations of General Equilibrium,Cambridge: Cambridge University Press

References II

Goodwin, R.M., 1951. The nonlinear accelerator and thepersistence of business cycles.Econometrica, 19, 1–17.

Hahn, F. 1974. On the Notion of Equilibrium in Economics,Inaugural Lecture, Cambridge University, reprinted in F. Hahn,Equilibrium and Macroeconomics, Cambridge MA.: MIT Press,1984

Hicks, J. A., 1950.Contribution to the Theory of the Trade CycleOxford: Clarendon Press

Kaldor, N., 1940. A model of the trade cycle.Economic Journal,50, 78–92

Kalecki, M., 1954.Theory of economic dynamics: An essay oncyclical and long- run changes in capitalist economy, London:Allen and Unwin

References III

Medio, A. 2004. Invariant Probabaility Distributions inEconomic Models: A General Result,MacroeconomicDynamics, 8, 162–187

Medio, A. and B. Raines, 2007. Backward Dynamics inEconomics. The Inverse Limit Approach,Journal of EconomicDynamics and Control, 31 1633-1671

Muth, J., 1961. Rational expectations and the theory of pricemovements.Econometrica, 29, 315–335

Prescott, E.C. and R. Mehra, 1980. Recursive CompetitiveEquilibrium: The Case of Homogeneous Households,Econometrica48, No. 6, 1365–1379

Lucas, R.E., Jr. 1996. Nobel Lecture: Monetary Neutrality,TheJournal of Political Economy, 104, No. 4, 661–682

References IV

Samuelson, P. A., 1939. Interaction Between the MultiplierAnalysis and the Principle of Acceleration,Review of EconomicStatistics21, 75–78

Skyrms, B. 1992. Chaos and the Explanatory Significance ofEquilibrium: Strange Attractors in Evolutionary GameDynamics,PSA, Volume 2 374-394

Walras, L., 1926.Élements d’économie politique pure; ou,Théorie de la richesse sociale. Paris: Pichon et Durand-Auzias;Lausanne: Rouge

equilibrium and stability in economic dynamics · · 2012-02-21international workshop on economic...

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