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Chapter 3

Contents

I Overview 1

II Solution of Deterministic Models 1

II.1 Finite-Horizon Models . . . 1 II.2 Infinite Horizon Models . . . 3

III Solution of Stochastic Models 4

III.1 An Illustrative Example . . . 4 III.2 The Algorithm in General . . . 5

IV Further Applications 6

IV.1 The Benchmark Model . . . 6 IV.2 A Small Open Economy . . . 7

I Overview

• A solution for the deterministic finite-horizon Ramsey model: a system of non-linear equa- tions.

• A solution for the deterministic infinite-horizon Ramsey model: an infinite number of un- knowns.

• A solution for the stochastic infinite-horizon Ramsey model: “deterministic extended path.”

II Solution of Deterministic Models

II.1 Finite-Horizon Models

The Model. max

{C^t,Kt+1}^T_t=0 T

∑

t=0

β^tC

1−η t

1 − η

s.t. Kt+1 = K_t^α− Ct+ (1 − δ)Kt, K0 : given,

KT = 0,

1

where β ∈ (0, 1), η > 0, α ∈ (0, 1), and δ ∈ [0, 1]. The F.O.C.s can be written as a system of T non-linear equations in the T unknown capital stocks K1, . . ., KT.

0 = [(1 − δ)Kt+ K_t^α− Kt+1]^−η− β[(1 − δ)Kt+1+ K_t+1^α − Kt+2

]−η

(1 − δ + αK_t+1^α−1), t= 0, 1, . . . , T − 1,

K0 : given, KT = 0,

Non-Linear Equation Solvers.

• The common structure of the algorithms is the iterative scheme: x_s+1 = xs+ µ∆xs

• An initial guess of the solution x0

• A direction of change ∆x

• A step length µ

• Find xs so that f (xs) ≃ 0 or xs+1− xs ≃ 0 Solutions.

II.2 Infinite Horizon Models

The Model.

{Ct,Kmaxt+1}^∞_t=0

∞

∑

t=0

β^tC

1−η t

1 − η

s.t. Kt+1 = K_t^α− Ct+ (1 − δ)Kt, K₀ : given.

Kt is approaching to the steady state level K^∗.

K^∗ =[ 1 − β(1 − δ) αβ

]_α−1¹

We cannot use non-linear equation solvers in this case. However, in which T is large enough that KT+1 = K^∗, we can use non-linear equation solvers again. Shooting algorithm can also be applied to solve this problem.

Solutions.

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III Solution of Stochastic Models

III.1 An Illustrative Example

{Ct,Kmaxt+1}^∞_t=0^E0

[ _∞

∑

t=0

β^tC

1−η t

1 − η ]

s.t. K_t+1 = ZtK_t^α− Ct+ (1 − δ)Kt, ln Zt= ρ ln Z_t−1+ ǫt,

K₀ : given,

where ρ ∈ [0, 1) and ǫt∼ N (0, σ²). The F.O.C.s can be written as follows. 0 = [(1 − δ)Kt+ K_t^α− Kt+1]^−η− βEt

{[(1 − δ)Kt+1+ K_t+1^α − Kt+2

]−η

(1 − δ + αK_t+1^α−1)^}, t= 0, 1, . . . ,

K₀ : given,

and transversality condition. Under the assumption of the rational expectations, the expected future path of Zt is:

{Z0^{, Z}1, . . .} = {Zt}_t=0^∞ =^{Z₀^ρt}∞

t=0^.

Then we can obtain an approximation of this path from the solution of the system of T non-linear equations

0 =^[(1 − δ)Kt+ Z₀^ρtK_t^α− Kt+1

]−η

− β^[(1 − δ)Kt+1+ Z₀^ρt+1K_t+1^α − Kt+2

]−η

(1 − δ + αK_t+1^α−1), t= 0, 1, . . . , T − 1,

K₀ : given, KT+1 = K^∗,

III.2 The Algorithm in General

Notation.

• The system of stochastic difference equations that governs this model is generally described as

0 = Et[gⁱ(xt, yt, zt, x_t+1, y_t+1, z_t+1)] , z_t= Πzt−1+ σΩǫt,

i= 1, 2, . . . , n(x) + n(y), t= 0, 1, . . . ,

where ǫt ∼ N(0n(z)^{, I}n(z))

• xt ∈ R^n(x): the vector of state variables with x₀.

• yt ∈ R^n(y): the vector of control and co-state variables (sometimes these are so called “jump variables”).

• zt∈ R^n(z): the vector of purely exogenous variables which are logarithmic. The Algorithm.

Deterministic Extended Path: Simulation of the stochastic DGE model

✓ ✏

Step 1. Initialize: Let p denote the number of periods to consider and (x0, z0) the initial state of the model.

Step 1.1 Draw a sequence of shocks {ǫt}^p_t=0.

Step 1.2 Compute {zt}^p_t=1 from zt = Πzt−1+ σΩǫt. Step 1.3 Choose large T .

Step 2. For t = 0, 1, . . . , p repeat the following steps to obtain decisions for period t: Step 2.1 Given zt, compute {zt}^p_t=1 from zt+s= Π^sz_t.

Step 2.2 Solve the system of T (n(x) + n(y)) equations by using non-linear equations solver or shooting method,

0 = Et[gⁱ(x_t+s, y_t+s, z_t+s, x_t+s+1, y_t+s+1, z_t+s+1)] , i= 1, 2, . . . , n(x) + n(y),

s= 0, 1, . . . , T − 1, x^∗ = xt+T.

Keep xt+1 and yt.

Step 2.3 Use xt+1 as starting value for next period t + 1.

✒ ✑

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IV Further Applications

IV.1 The Benchmark Model

{Ct,Nmaxt,Kt+1}^∞_t=0^E0

[ _∞

∑

t=0

β^tC

1−η

t ^{(1 − Nt)θ(1−η)}

1 − η

]

s.t. Kt+1 = ZtK_t^α(AtNt)^1−α+ Ct+ (1 − δ)Kt, A_t+1 = aAt, (a ≥ 1)

ln Zt= ρ ln Zt−1+ ǫt, K₀ : given.

When we define the detrended variables as kt ≡ _A^K_tN^t_t and ct ≡ _A^C_tN^t_t, we obtain the F.O.C.s as follows.

0 =c^−η_t (1 − Nt)^θ(1−Nt)− λt,

0 =θc^1−η_t (1 − Nt)^{θ(1−η)−1}− (1 − α)λtZtN_t^−αk_t^α, 0 =akt+1− (1 − δ)kt+ ct− ZtN_t^1−αk_t^α,

0 =λt− βa^−ηEtλ_t+1(1 − δ + αZ_t+1N_t+1^1−αk_t+1^α−1).

• xt ≡ kt

• yt ≡ (ct, Nt, λt)

• T = 150. Therefore, a system contains 600 unknown variables.

However, we can eliminate ct and λt by using ct = ^1−α_θ (1 − Nt)ZtN_t^−αk_t^α. Then, we obtain 2T equations as follows.

0 =Z_t^ρsN_t+s^1−αk_t+s^α + (1 − δ)k^a_t+skt+s+1− ^{1 − α}

θ ^{(1 − Nt+s)Z}

ρ^s

t ^Nt+s^−αkt+s^{α ,}

0 =

(Z_t^ρs+1N_t+s+1^1−α k_t+s+1^α + (1 − δ)kt+s+1− akt+s+2

Z_t^ρsN_t+s^1−αk_t+s^α + (1 − δ)kt+s− akt+s+1

)η

( 1 − Nt+s

1 − Nt+s+1

)θ(1−η)

,

− βa^−η(1 − δ + αZ_t^ρs+1N_t+s+1^1−α k^α−1_t+s+1), s =0, . . . , T − 1,

kT =kT+1 ^{= k∗.}

IV.2 A Small Open Economy

• Since a small open economy model has often many points of steady state, the log-linearized approach in chapter 2 is not suitable to solve the model. The deterministic extended path approach can address this problem. (Omitted)

• To use the log-linearized approach in chapter 2, Schmit-Grohe and Uribe (2003) “Closing small open economy models” shows several approach to stabilize a small open economy model.

http://public.econ.duke.edu/~grohe/research/closing_jie.pdf

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