# linearization - Introduction to Linearization Methods...

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Introduction to Linearization Methods Jonathan Heathcote August 28th 2003 1. Solving systems of stochastic linear di f erence equations (see Farmer’s book) 1.1. Example of a f rst order stochastic di f erence equation. x t +1 = b + ax t + ε t +1 ε is a random variable. Conditions for stationary distribution of x (i) | a | < 1 (ii) ε drawn from an invariant probability distribution Solution to the model is a sequence of probability distributions { G t ( ·| x 0 ) } t =1 1.2. Simple application: a stochastic version of the Solow growth model. Equations describing the model: Y t = z t F ( K t ,N t ) (1) N t = γ t N K t +1 =(1 δ ) K t + Y t C t (2) C t s ) Y t (3) z t +1 ρ )+ ρz t + ε t +1 ε ˜ N (0 2 ) (4) z 0 ,K 0 0 given 1. Rewrite in terms of stationary variables. Divide 1, 2 and 3 by N t , and for any variable x let x t = X t N t y t = z t F ( k t , 1) = z t f ( k t ) (assuming F is homogenous of degree one) γk t +1 δ ) k t + y t c t c t s ) y t 1

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Introduction to Linearization Methods 2 2. Compute the (non-stochastic) steady state y = z f ( k ) z =(1 ρ )+ ρz γk δ ) k + y c c s ) y 3. Linearize around the steady state (take a f rst order Taylor series expansion) y t y = z f 0 ( k )( k t k f ( k )( z t z ) z t +1 z = ρ ( z t z ε t +1 ε ˜ N (0 2 ) γ ( k t +1 k )=(1 δ k t k )+( y t y ) ( c t c ) c t c s )( y t y ) 1.3. Review of some useful linear algebra. De f nition of eigenvalues and eigenvectors Consider an n × n matrix A that maps R n into R n . The roots of A are the n solutions λ 1 ...λ n to the equation AY = λY where each element of λ is a scalar and each Y is an eigenvector. There will gen- erally be n di f erent eigenvalues λ 1 ...λ n and n di f erent eigenvectors Y 1 ...Y n ,e a ch corresponding to a particular eigenvalue. Suppose n =2 . Then the two eigenvectors Y a and Y b are straight lines through the origin in the two dimensional Cartesian plane ( Y 1 ,Y 2 ) such that if the initial state of the system Y t =( y 1 ,t ,y 2 ,t ) lies on one the eigenvectors (say Y a ) then the state next period is (by de f nition) given by Y t +1 = AY t = λ a Y t Now, AY = λY implies ( A λI ) Y =0 . This has a non-zero solution only if ( A λI ) is singular or has a zero determinant (one column of A λI is a scalar multiple of the other). Thus the eigenvalues can be computed by solving the polynomial | A λI |
Introduction to Linearization Methods 3 Of course, there is a gauss command that will return all the eigenvectors and eigenvalues of any matrix, so we do not really need to worry about how to solve for them. The command is eigv. The eigenvalues of a system determine the set of initial conditions, if any, under which the system is stable (see below). For example, suppose in a two dimensional system, one eigenvalue is greater than one in absolute value, and one is less than one.

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## This note was uploaded on 11/10/2011 for the course ECON 601 at Cornell University (Engineering School).

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linearization - Introduction to Linearization Methods...

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