Macroeconomics Theory II

Transcription

1 Macroeconomics Theory II Economies as ynamic Systems Francesco Franco Nova SBE February 11, 2014 Francesco Franco Macroeconomics Theory II 1/18

2 First-order The variable z t follows a first-order di erence equation z t = az t 1 + m t (1) where m t is an exogenous function of time. A solution of this equation is to express z t as a function of the current, future, and lagged values of m t as well as some intial value of z 0.Try backward substitution Francesco Franco Macroeconomics Theory II 2/18

3 The stable case: a < 1 efine the lag operator L by Lx t = x t 1 hence the lag operator maps a variable to its value in the previous period. We can write equation (1) as (1 al) z t = m t Francesco Franco Macroeconomics Theory II 3/18

4 The stable case: a < 1 efine the inverse of the (1 al) as (1 al) 1 = 1 + al + a 2 L an operator that converges because a < 1. Obviously (1 al) 1 (1 al) =1 Francesco Franco Macroeconomics Theory II 4/18

5 The stable case: a < 1 Now write z t =(1 al) 1 (1 al) z t =(1 al) 1 m t with (1 al) 1 m t = m t + am t 1 + a 2 m t A solution is therefore tÿ z t = a t s m s s= Œ Francesco Franco Macroeconomics Theory II 5/18

6 The stable case: a < 1 It is important to notice that it is not the only solution. If b 0 is a constant then adding the term b 0 a t yields another solution. Therefore we refer to z t = tÿ s= Œ a t s m s + b 0 a t as the general solution Francesco Franco Macroeconomics Theory II 6/18

7 The stable case: a < 1 The arbitrary constant b 0 is determined by an initial condition on the variable z t. Suppose we know z 0 then the general solution is correct only if 0ÿ z 0 = a s m s + b 0 holds, that is, if s= Œ b 0 = z 0 0ÿ s= Œ a s m s Francesco Franco Macroeconomics Theory II 7/18

8 The stable case: a < 1 Substitute in the general solution corresponding to the initial value z t = tÿ a t s m s + z 0 a t s=1 gives the particular equation Francesco Franco Macroeconomics Theory II 8/18

9 The unstable case: a > 1 Unstable roots usually govern the behavior of forward-looking economic variables, such as equity prices. efine the lead operator Fx t = x t+1 forward by one period our equation z t+1 = az t + m t+1 and isolate z t z t = 1 a z t+1 1 a m t+1 Francesco Franco Macroeconomics Theory II 9/18

10 The unstable case: a > 1 Now apply our forward operator: 31 1 a F 4 z t = 1 a Fm t and given - 1 a - < 1 we can define the inverse operator 11 1 a F 2 1 Francesco Franco Macroeconomics Theory II 10/18

11 The unstable case: a > 1 As in the stable case a solution is therefore z t = a F a F z t = a F 1 a Fm t z t = Œÿ s=t a 4 s t m s Francesco Franco Macroeconomics Theory II 11/18

12 The unstable case: a > 1 As before we find a general solution by adding the term b 0 a t z t = Œÿ s=t a 4 s t m s + b 0 a t now you can determine b 0 using an initial condition on z 0,however any choice for b 0 other than b 0 = 0 would lead to z t exploding. Under the assumption that agents are not willing to participate in an unstable economy (see Shiller, 1978), the markets will determine z 0 such that b 0 = 0. Francesco Franco Macroeconomics Theory II 12/18

13 First-order vector systems In our models we have systems of variables. If we interpret z as a vector and the parameter a as a conformable matrix we can apply the same method. The only question is to determine which variables are driven by unstable dynamics and which variables by stable dynamics. Consider the system C z1t z 2t = A C z1t 1 z 2t 1 + C m1t m 2t C a11 a where A = 12 is non singular. Now we have to a 21 a 22 transform the system into an unstable and stable part. Francesco Franco Macroeconomics Theory II 13/18

14 First-order vector systems Any square matrix can be decomposed into AE = E C e1 e where E = 2 is a matrix containing the eigenvectors and 1 1 C 1 0 = is a matrix containing the eigenvalues of A. 0 2 Thus A = E E 1 Francesco Franco Macroeconomics Theory II 14/18

15 First-order vector systems Now use the decomposition C z1t z 2t = E E 1 C z1t 1 z 2t 1 + C m1t m 2t and premultiply by E 1 E 1 C z1t z 2t = E 1 C z1t 1 z 2t 1 + E 1 C m1t m 2t Francesco Franco Macroeconomics Theory II 15/18

16 First-order vector systems efine the transformed vectors z Õ t = E 1 z t and m Õ t = E 1 m t. The last matrix equation becomes C z Õ 1t z Õ 2t = C z Õ 1t 1 z Õ 2t 1 + C m Õ 1t m Õ 2t where because is diagonal we have expressed the system in terms of two variables with non interacting dynamics. Each of these can be solved separately using methods described above and the solution of the original variables can be recovered by applying the reverse transformation z t = Ez Õ t markets. Francesco Franco Macroeconomics Theory II 16/18

17 Under Rational expectations The Solution of Linear i erence Models under Rational Expectations by Olivier Jean Blanchard and Charles M. Kahn Update: SOLVING LINEAR RATIONAL EXPECTATIONS MOELS by Chris sims Francesco Franco Macroeconomics Theory II 17/18

18 Readings M. Obstfeld and K. Rogo, Foundations of International Macroeconomics (MIT Press, 1996). Supplement C, Solving Systems of Linear i erence Equations. The Solution of Linear i erence Models under Rational Expectations Author(s): Olivier Jean Blanchard and Charles M. Kahn Source: Econometrica, Vol. 48, No. 5 (Jul., 1980), pp *Chris Sims: Francesco Franco Macroeconomics Theory II 18/18