A finite set of equilibria for the indeterminacy of linear rational expectations models
aa r X i v : . [ q -f i n . E C ] J u l A finite set of equilibria for theindeterminacy of linear rationalexpectations models
Jean-Bernard Chatelain ∗ and Kirsten Ralf † July 18, 2018
Abstract
This paper demonstrates the existence of a finite set of equilib-ria in the case of the indeterminacy of linear rational expectationsmodels. The number of equilibria corresponds to the number of waysto select n eigenvectors among a larger set of eigenvectors related tostable eigenvalues. A finite set of equilibria is a substitute to contin-uous (uncountable) sets of sunspots equilibria, when the number ofindependent eigenvectors for each stable eigenvalue is equal to one. JEL classification numbers : C60, C61, C62, E13, E60.
Keywords:
Linear rational expectations models, indeterminacy,multiple equilibria, Riccati equation, sunspots. ”Das kann als Riccatische gleichung des matrizenkalk¨uls ange-sehen werden.”
Radon (1928) p.190.
This paper demonstrates that there is a finite set of rational expectationsequilibria in the case of indeterminacy for linear rational expectations models,which is a substitute to uncountable (continuously infinite) sets of sunspots ∗ Paris School of Economics, Universit´e Paris I Pantheon Sorbonne, CES, Centred’Economie de la Sorbonne, 106-112 Boulevard de l’Hˆopital 75647 Paris Cedex 13. Email:[email protected] † ESCE International Business School, 10 rue Sextius Michel, 75015 Paris, Email:[email protected]. n of pre-determined variables is lower than the number s of eigenvalues below onein absolute values. In this case, the initial values of the number m of nonpre-determined ”forward” variables may be driven by continuous randomvariables of zero mean, independently and identically distributed over time(Gourieroux et al. (1982)).Besides this continuous infinity of sunspots equilibria, it is feasible to ex-tend the computation of saddlepath unique rational expectations equilibrium(Blanchard and Kahn (1980), Boucekkine and Le Van (1996)) to the case ofmultiple equilibria. These rational expectations equilibria are solutions of amatrix Riccati equation (Radon (1928), Le Van (1986), Abou-Kandil et al. (2003)). This paper demonstrates that there is a finite a number of equilib-ria, at most equal to s ! n !( s − n )! . This is the number of ways to choose n distincteigenvectors among a larger set of s eigenvectors related to eigenvalues withabsolute values below one, when there is only one independent eigenvectorfor each of these eigenvalues. Blanchard and Kahn (1980) consider a linear rational expectations model: (cid:18) k t +1 t q t +1 (cid:19) = (cid:18) A nn A nm A mn A mm (cid:19)| {z } A (cid:18) k t q t (cid:19) + γ z t (1)where k t is an ( n ×
1) vector of variables predetermined at t with initial con-ditions k given (shocks can straightforwardly be included into this vector); q is an ( m ×
1) vector of variables non-predetermined at t ; z is an ( k ×
1) vec-tor of exogenous variables; A is ( n + m ) × ( n + m ) matrix, γ is a ( n + m ) × k matrix, t q t is the agents expectations of q t +1 defined as follows: t q t +1 = E t ( q t +1 p Ω t ) . (2)2 t is the information set at date t (it includes past and current values of allendogenous variables and may include future values of exogenous variables).A predetermined variable is a function only of variables known at date t so that k t +1 = t k t +1 whatever the realization of the variables in Ω t +1 . A non-predetermined variable can be a function of any variable in Ω t +1 , sothat we can conclude that q t +1 = t q t +1 only if the realization of all variablesin Ω t +1 are equal to their expectations conditional on Ω t .Boundary conditions for the policy-maker’s first order conditions are thegiven initial conditions for predetermined variables k and Blanchard andKahn (1980) hypothesis ruling out the exponential growth of the expectationsof w = ( k , q , z ): ∀ t ∈ N , ∃ w t ∈ R k , ∃ θ t ∈ R , such that | E t ( w t +1 p Ω t ) | ≤ (1 + i ) θ t w t , ∀ i ∈ R + . (3) Definition:
Besides other sunspots equilibria (Gourieroux et al. [1982]),let us define a set of rational expectations solutions, which are such that nonpredetermined variables are a linear function of pre-determined variables,where the matrix N mn is to be found, and with bounded solutions for pre-determined variables, so that the eigenvalues λ i of the matrix A nn − A nm N mn are below one (”stable eigenvalues”): q t +1 = − N mn k t +1 (4) k t +1 = ( A nn − A nm N mn ) k t (5) λ ( A nn − A nm N mn ) = { λ i with | λ i | < , i ∈ { , ..., n }} (6) Proposition: A has s stable eigenvalues and n + m − s unstable eigen-values.Case 1. When ≤ s < n , the number of stable eigenvalues is strictly be-low the number of pre-determined variables, there is no rational expectationsequilibrium (Blanchard and Kahn (1980)).Case 2. When s = n , the number of stable eigenvalues is strictly equal tothe number of predetermined variables, there is a unique rational expectationsequilibrium (Blanchard and Kahn (1980)).Case 3. When n < s ≤ n + m , the number of rational expectationsequilibria defined above is given by the number of ways of selecting n inde-pendent (right column) eigenvectors (cid:18) P nn P mn (cid:19) among a larger set of inde-pendent eigenvectors related to stable eigenvalues. If P nn is invertible, they orresponds to the number of rational expectations equilibria determined byeach matrix N mn = − P mn P − nn :Case 3.1. Finite number of equilibria . If the number of independenteigenvectors (geometric multiplicity) of each stable eigenvalues of A is ex-actly one, the number of equilibria is given by s ! n ! s ! where the number of stableeigenvalues not counting their multiplicity is denoted s ≤ s . In particular,if all the stable eigenvalues of A are distinct, then the number of equilibriais s ! n ! s ! .Case 3.2. Uncountable number of equilibria . If there is at least onestable eigenvalue of A with its number of independent eigenvectors (geomet-ric multiplicity) which is at least equal to two, then, there always exists anuncountable number of equilibria. This condition for an uncountable numberof equilibria is distinct from e.g. Gourieroux et al. (1982). For example, for n = 1, m = 1, and with a unique stable eigenvalue λ with two independent column vectors P = ( P , P ) , there is an uncountablenumber of single eigenvectors P α = P + α P with α ∈ C leading to solutions N mn,α = − P mn,α P − nn,α . For n = 2, m = 1, including another eigenvalue λ with a multiplicity equal to one and an eigenvector denoted P , there is asingle case of n = 2 columns eigenvector ( P , P ) and an uncountable numberof n = 2 eigenvector matrix P α = ( P , P α ) with α ∈ C allowing to computesolutions N mn = − P mn P − nn (see a numerical example for n = 2, m = 2 inAbou-Kandil et al. (2003) p.25). Proof:
Let us consider a matrix N mn such that: (cid:18) k N ,t q N ,t (cid:19) = (cid:18) I n nm − N mn I m (cid:19) (cid:18) k t q t (cid:19) with T = (cid:18) I n nm − N mn I m (cid:19) and T − = (cid:18) I n nm N mn I m (cid:19) (7)So that: (cid:18) k N ,t +1 q N ,t +1 (cid:19) = (cid:18) I n nm N mn I m (cid:19) (cid:18) A nn A nm A mn A mm (cid:19) (cid:18) I n nm − N mn I m (cid:19) (cid:18) k N ,t q N ,t (cid:19)(cid:18) k N ,t +1 q N ,t +1 (cid:19) = (cid:18) A nn − A nm N mn A nm g ( N mn ) A mm + N mn A nm (cid:19) (cid:18) k N ,t q N ,t (cid:19) with (8) g ( N mn ) = A mn + A mm N mn − N mn A mm − N mn A nm N mn = mn (9) g ( N mn ) = mn = ∂ N mn /∂t is a matrix equation including a constant,two linear terms and a quadratic term N mn A nm N mn , which Radon (1928)4enoted as matrix Riccati extension of scalar Riccati differential equations.If N mn is a solution with constant coefficients of g ( N mn ) = mn , then thecharacteristic polynomial of matrix A is the product of two characteristicpolynomials, as det ( T ) = 1 = det ( T − ):det ( A − λ I n + m ) = det ( A nn − A nm N mn − λ I n ) · det ( A mm + N mn A nm − λ I m ) = 0(10)Each solution N mn of g ( N mn ) = mn corresponds to a particular par-tition of the eigenvalues of the matrix A since its eigenvalues are exactlythe eigenvalues of A nn − A nm N mn (with n eigenvalues counting multiplicity)and A mm + N mn A nm (with m eigenvalues counting multiplicity). A Jor-dan canonical transformation J of the A matrix with P a matrix of righteigenvectors is: (cid:18) A nn A nm A mn A mm (cid:19) (cid:18) P nn P nm P mn P mm (cid:19) = (cid:18) P nn P nm P mn P mm (cid:19) (cid:18) J n nm mn J m (cid:19) (11)where J nn is a n × n Jordan matrix with the eigenvalues of A nn − A nm N mn and J mm is a m × m Jordan matrix with the eigenvalues of A mm + N mn A nm .One has: (cid:18) A nn − A nm N mn A nm mn A mm + N mn A nm (cid:19) (cid:18) I n nm − N mn I m (cid:19) (cid:18) P nn P nm P mn P mm (cid:19) = (cid:18) I n nm − N mn I m (cid:19) (cid:18) P nn P nm P mn P mm (cid:19) (cid:18) J n nm mn J m (cid:19) (12)which implies: (cid:18) ( A nn − A nm N mn ) P nn + A nm ( P mn − N mn P nn ) ∗ ( A mm + N mn A nm ) ( P mn − N mn P nn ) ∗ (cid:19) = (cid:18) P nn J nn ∗ ( P mn − N mn P nn ) J nn ∗ (cid:19) (13)Because the eigenvalues of A mm + N mn A nm are not the eigenvalues of J nn , then ( P mn − N mn P nn ) cannot stack eigenvectors (each of them distinctfrom the zero vector by definition) of A mm + N mn A nm . Then, the secondequality for block matrices ( i = 2 , j = 1) is valid:( A mm + N mn A nm ) ( P mn − N mn P nn ) = ( P mn − N mn P nn ) J nn (14)only and only if P mn − N mn P nn = . Then, if P nn is invertible, one findsthe solutions N mn = − P mn P − nn . n stable eigenvalues, and compute N mn using a set of n columneigenvectors (cid:18) P nn P mn (cid:19) related to these stable eigenvalues. The number ofrational expectations equilibria is then given by the number of ways of se-lecting n independent (right column) eigenvectors (cid:18) P nn P mn (cid:19) related to thestable eigenvalues s ≥ n .Finally, the first equality for block matrices ( i = 1 , j = 1) becomes:( A nn − A nm N mn ) P nn = P nn J nn (15)Hence, the matrix P nn is an eigenvectors matrix of the matrix A nn − A nm N mn . Q.E.D.
A similar demonstration with transpose matrices holds for left row eigen-vectors (cid:18) Q mn Q mm (cid:19) with Q = P − chosen among a set of s > n row eigenvec-tors related to stable eigenvalues. If Q mm is invertible, one finds the solutions N mn = − P mn P − nn = Q − mm Q mn . A finite set of rational expectations equilibria (when the number of indepen-dent eigenvectors for each stable eigenvalue is equal to one) exists at eachperiod. For a chosen equilibrium with a given set of eigenvectors at a givenperiod to be find again on the following periods, one needs to assume that theeconomic agents select the same set of eigenvectors at each period. In thiscase, economic agents shape their rational expectations following the sameprocedure at each period in a time-consistent manner.
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