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Economics Dynamics Problems 217

# Economics Dynamics Problems 217 - CHAPTER 5 Discrete...

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CHAPTER 5 Discrete systems of equations 5.1 Introduction In chapter 3 we considered linear difference equations for a single variable, such as x t = 2 x t 1 , x t = 4 x t 1 + 4 x t 2 , x t = ax t 1 + b Each of these equations is linear and autonomous. But suppose we are interested in such systems as the following: (i) x t = ax t 1 + by t 1 y t = cx t 1 + dy t 1 (ii) x t = 4 x t 1 + 2 y t = − 2 y t 1 3 x t 1 + 3 (iii) x t = 2 x t 1 + 3 y t 1 + 4 z t 1 y t = 0 . 5 x t 1 z t = 0 . 7 y t 1 All these are examples of systems of linear autonomous equations of the first order. As in previous chapters, we shall here consider only autonomous equations (i.e. independent of the variable t ), but we shall also largely restrict ourselves to linear systems. If all the equations in the system are linear and homogeneous, then we have a linear homogeneous system. If the system is a set of linear equations and at least one equation is nonhomogeneous, then we have a linear nonhomogeneous
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