Supplement to Chapter 3
C.5 Regulator for Discretion Problem
This section shows how to derive the regulator problem (3.31) with state transition (3.32) from Section 3.2. For convenience, the regulator is reproduced here
X′tV∗Xt+ v∗ = min
Ut
X′tQ0Xt+ 2X′tN0Ut+ Ut′R0Ut+ βEtX′t+1V0Xt+1+ v0
(3.31)
s.t. Xt+1 = A0Xt+ B0Ut+ Dwt+1 (3.32)
Xt≡
Xt
Xt|t−1
Public beliefs about current and future policies are taken as given. They are represented by F0 = [F10 F20] and G0 as in (3.26) and (3.29). As will be seen regulator matrices with superscript “0” depend on F0 and G0. Future policies of the policymaker are also taken as given and represented by the value function Vt+10 . Since the problem is linear quadratic,
the value function can be taken to be linear quadratic, too, (Bertsekas 2005):
Vt+10 = X′t+1V0Xt+1+ v0 (C.9)
The definition of the discretion problem in Section 3.2, takes the matrix V0 and the scalar v0 as given. For the policy improvement algorithm, they are however calculated to be consistent with continuing the policy F0 and the beliefs G0 forever. This is shown at the end of this section.
The derivation proceeds by using the temporary equilibrium (3.30) and the Kalman filter (3.27) to substitute Yt|t out of the loss function (3.25) and transition equation (3.21) for Xt. The Kalman filter also yields the transition equation for Xt|t−1. The Kalman filter also depends on a prior belief about observables Zt|t−1 = CxXt|t−1+ CuUt|t−1 and thus on a prior belief on policy. To simplify the regulator, it is assumed that this belief is consistent with F0 (as it will be in equilibrium), such that
Zt|t−1 = (Cx+ Cu(F10+ F20))
| {z }
Cˆ
Xt|t−1
The Kalman update can then be written as
Xt|t = KCxXt+ (I − K ˆC)Xt|t−1+ KCuUt
Together with the temporary equilibrium (3.30) this yields
Yt|t = G0xXt+ ˆG0xXt|t−1+ G0uUt
with G0x = (Gx+ GuF10)KCx
Gˆ0x = (Gx+ GuF10)(I − K ˆC) + GuF20 G0u = (Gx+ GuF10)KCu
Loss Function
The loss function (3.25) can then be rewritten in terms of the regulator’s states and control using
where Q0, N0 and R0 conformably partition the above quadratic form as:
H0′QH0 =
Likewise, the state transitions for Xt and Xt|t−1 can be derived as
Xt+1 = (Axx+ AxyG0x)Xt+ AxyGˆ0xXt|t−1 + (AxyG0u+ Bx)Ut+ Dwt+1 Xt+1|t = AxxKCxXt+
Axx(I − K ˆC) + (AxyGu+ By)F20
Xt|t−1 + AxxKCuUt
where Axx = AxxAxy(Gx+ GuF10) + ByF10
The matrices A0, B0 and D in (3.32) are thus given by:
Value Function consistent with F0 and G0
The policy improvement algorithm described at the end of Section 3.2 uses a continuation value consistent with carrying out the policy F0 forever. The continuation value is linear quadratic in Xt as in (C.9). It is computed from the closed loop representation of the regulator obtained by plugging the policy F0 into (3.31) and (3.32). V0 then solves the Lyapunov equation
V0 =
Q0+ N0F0+ F∗′R0F0
+ β(A0+ B0F0)′V0(A0+ B0F0)
The equation has a unique solution if the matrix in curly braces is positive definite and if the closed loop transition matrix (A0+ B0F0) has all eigenvalues inside the unit circle.
The former is assured by the form of the original loss function3 and the latter holds if a stationary equilibrium exists.4
The scalar v0 can be computed from:
v0 = β
1 − βtr (V0DD′)
where tr is the trace operator. Notice that optimal policies are certainty equivalent (for
3Please recall that Q in (3.25) is assumed to be positive definite.
4Efficient methods for solving Lyapunov equations are available for example via the LAPACK routines encoded in MATLAB or by using the doubling algorithms of Anderson et al. (1995).
given F0) and do not depend on v0. Details behind these derivations can be found in Sargent and Ljungqvist (2004) or Svenssson (2007). Unconditionally expected losses are computed as
E(Vt0) = tr (V0EXtX′t) + v0
where EXtX′t solves another Lyapunov equation:
EXtX′t= (A0+ B0F0) (EXtX′t) (A0+ B0F0)′+ DD′
C.6 Additional Results
This appendix contains supplementary results: Figure C.1 compares impulse responses under hidden and symmetric information for the simple NK model, this collects results from Figure 3.1 and 3.4 on one figure. Figure C.2 compares impulse responses under full information measure with time-zero public expectations for same model. A comparison with the commitment policy under full information is given in Figure C.3.
Figure C.4 documents the dependence of the policymaker’s value function on the extent of credibility problems, measured by σb. Figure C.5 supplements the sensitivity analysis of Figure 3.6 for fε/fτ with the corresponding picture for ¯σ2 = (fε/fτ)2σε.
Figure C.1: Hidden Information versus Symmetric Information
Note: Impulse responses to structural shocks in the simple NK model. Straight lines show optimal policy under hidden information and are identical to those in Figure 3.4. Dashed lines show optimal policy under symmetric information and are identical to those in Figure 3.1.
Figure C.2: Public Beliefs versus Actual Policies
Note: Impulse responses in simple NK model with optimal policy. Straight lines show structural responses under full information measure. (Identical to those in Figure 3.4.) Dashed lines show publicly expected response paths as of time zero.
Figure C.3: Discretion with Hidden Information versus Commitment under Symmetric
Note: Impulse responses in simple NK model with optimal policy. Straight lines show responses under discretion policy with hidden information, identical to those in Figure 3.4.) Dashed lines show commitment outcomes under symmetric information.
Figure C.4: Transparency and Welfare
0 0.5 1 1.5 2 2.5 3
Note: Unconditionally expected value function of policymaker E(Vt) of model with belief shocks (Sec-tion 3.4) when varying the volatility of belief shocks, σb. Other parameters at baseline values of Table 3.1.
178SUPPLEMENTTOCHAPTE
Figure C.5: Sensitivity of Effective Noise ¯σ2
(a) Perspective 1 (b) Perspective 2
Note: Sensitivity of ¯σ2= (fε/fτ)2σεto changes in αx and σε, keeping other parameters at their baseline values (Table 3.1).
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