Linearizing the Hyperinflation Model

Transcription

1 Linearizing the Hyperinflation Model Klaus Adam George Evans Seppo Honkapohja March 7, 003 Abstract This is a technical appendix to Are Stationary Hyperinflation Paths Learnable? by the same authors which derives the linearization coefficients for the hyperinflation model at the high steady state. Introduction Consider the hyperinflation model with linear asset demand and money supply m d t ψ E t Π t ωith > ψ > 0 m s t m t Π t v t Setting m s t m d t and imposing market clearing in t, oneobtains which implies ψ E t Π t ψ E t Π t Π t v t ψ E t Π t Π t () ψ E t Π t g v t Below we linearize this model around the high inflation steady state and determine the linearization coefficients α, β,andβ 0 of Steady states x t α β E t x t β 0 E t x t u t The non-stochastic steady states of () satisfy Π ( ψ Π g) ( ψ Π)ψ Π ( g ψ ) Π 0

2 The are two steady states Π l ψ q( ψ g ψ gψ ψ ) Π h ψ ψ q( g ψ gψ ψ ) where the high inflation steady state Π h is the one around which we linearize. 3 Calculating the linearization coefficients at Π h Π h ψ ψ p (ψ 0 g ψ gψ ψ ) 3. The parameter α: α ψ Π ψ Π g ψ ψ ψ p ) ( g ψ gψ ψ ψ ψ ψ p ) ( g ψ gψ ψ g ψ p ( g ψ gψ ψ ) g ψ p ( g ψ gψ ψ ) () 3. The parameter β 0 : ψ β 0 ψ Π g ψ ψ ( ψ ψ p (ψ 0 g ψ gψ ψ ) ) g ψ ψ p (ψ 0 g ψ gψ ψ ) (3) 3.3 The parameter β : The denominator : β ( ψ Π) ψ ( ψ Π g)

3 ( ψ Π g) ( ψ ψ 4 ψ The numerator: ψ q( g ψ gψ ψ ) g) q( g ψ gψ ψ ) ( ψ Π) ψ ψ ψ ψ As a result ψ q( g ψ gψ ψ ) ψ q( g ψ gψ ψ ) ψ β ( ψ Π) ψ ( ψ Π g) ψ 0 ψ p ) ( g ψ gψ ψ 4 ψ p (ψ 0 g ψ gψ ψ ) ψ p ) ( g ψ gψ ψ ψ ψ p ( g ψ gψ ψ ) (4) ψ 4 Express coefficients in terms of ω ψ ξ g g max and Now, rewrite the linearization coefficients by dividing numerator and denominator by : α ψ p ( g ψ gψ ψ ) g ψ p ( g ψ gψ ψ Ã s ) ψ g ψ! gψ ψ Ã s g ψ g! ψ gψ ψ 3

4 β ψ p ) ( g ψ gψ ψ ψ ψ p ( g ψ gψ ψ ) ψ Ã Ã ψ ψ s g ψ! gψ ψ s g! ψ gψ ψ and r β 0 ψ p 0 ψ (ψ 0 g ψ gψ ψ ) β ψ p ( g ψ gψ ψ Ã s ) ψ g ψ! gψ ψ Ã s ψ g ψ! gψ ψ The coefficients now depend on ω ψ / g/ only. As argued in the text, the maximum deficit is p g max ψ p Defining ξ g/g max we have s g ξ( ψ ψ ) Using this, we can express the coefficients as functions of ξ and ω, whichisthe result used in the Mathematica Notebook: α ξ(ω r ω)ω ξ(ω ω)ω(ξ(ω ω)) ωξ(ω ω)(ω) ξ(ω r ω)ω ξ(ω ω)ω(ξ(ω ω)) ωξ(ω ω)(ω) 4

5 β ω r ξ(ω r ω)ω ξ(ω ω)ω(ξ(ω ω)) ωξ(ω ω)(ω) r ξ(ω ω)ω(ξ(ω ω)) ωξ(ω ω)(ω) ξ(ω ω)ω ξ(ω r ω)ω ξ(ω ω)ω(ξ(ω ω)) ωξ(ω ω)(ω) ξ(ω r ω)ω ξ(ω ω)ω(ξ(ω ω)) ωξ(ω ω)(ω) 5 The linearization for g 0 5. The constant term α: From equation (): α ψ Π ψ Π d ψ p ( g ψ gψ ψ ) ψ p ( g ψ gψ ψ ) (5) As g 0 numerator and denominator converge to zero. Therefore, we must apply De L Hopital. The derivative of the numerator: ψ p ) ( g ψ gψ ψ g p ( g ψ gψ ψ ) ( g ψ ) Set g 0and evaluate this expression using > ψ : p ( g ψ gψ ψ ) ( g ψ ) r (ψ ) ψ ( ψ ) The derivative of the denominator: ψ p ) ( g ψ gψ ψ g p ( g ψ gψ ψ ) ( g ψ ) 5

6 Set g 0and evaluate using > ψ : p ( g ψ gψ ψ ) ( g ψ ) r (ψ ψ ) ψ ( ψ ) As a result, the constant converges for g 0 to lim α lim ψ Π h (g) g 0 g 0 ψ Π h (g) g ψ ψ ψ ψ 5. The parameter β 0 on E t Π t : We have ψ β 0 ψ Π g from equation (3). Since money is valued, the denominator is larger than zero. Therefore, lim g 0 β The parameter β on E t Π t : From (4) we have β ( ψ Π) ψ ( ψ Π g) Since both denominator and numerator converge to zero, we must apply De L Hopital again. From the calculations following (5), we know I know that the derivative of the numerator converges to ψ ψ 6

7 The denominator is given by ( ψ Π g) ( ψ ψ 4 ψ Taking the derivative w.r.t. g delivers 4 Ã gψ q (ψ ψ q( g ψ gψ ψ ) g) q( g ψ gψ ψ ) 0 gψ g gψ ψ) g ψ p ) ( g ψ gψ ψ! q (ψ 0 gψ g gψ ψ) (ψ 0 g ψ ) Setting g 0and evaluating this expression delivers that it is equal to zero. This implies that ψ lim β ψ g The ratio r β 0 β as g 0 The limit for r β 0 β is given by lim r lim β 0 g 0 g 0 β ψ ψ Πg g 0 ( ψ Π)ψ ( ψ Πg) lim ψ Π lim g 0 ψ Π lim ψ Π g g 0 ψ Π ψ where the last line follows from the calculations in section 5.. 7