Optimal Monetary Policy 1. Additive Uncertainty Ulf S¨oderstr¨om Bocconi University and Sveriges Riksbank
[email protected] ulf.c.soderstrom.googlepages.com
Uppsala University, August 20, 2008
Optimal Monetary Policy
1. Additive Uncertainty
Introduction • Optimization-based models typically include expectations of future variables. Such forwardlooking variables depend on expectations and outcomes of all other variables and may jump after any shock. • Traditional control theory (from engineering) need to be adjusted. • Distinguish between: – Commitment: Policymaker makes plan at t = 0 for entire future; Credible, Expectations adjust; Not time-consistent – Discretion: Policymaker cannot make credible commitments, chooses sequentially; Reoptimizes every period, Takes expectations as given; Time-consistent – Also: Commitment to simple rule (Taylor rule) • Analytical solutions are available for very simple cases, in general must use numerical methods.
1
Optimal Monetary Policy
1. Additive Uncertainty
Agenda 1. Optimal policy in a simple New Keynesian model: Analytical solution 2. More general models: Numerical solution 2.1 Solving a linear RE model 2.2 A simple rule for monetary policy 2.3 Optimal policy with commitment 2.4 Optimal policy with discretion 2.5 Alternative approaches 3. Conclusions 4. Matlab application • Main references: Gal´ı (2008, Ch. 5), S¨oderlind (1999). • See also Dennis (2004, 2007), Anderson and Moore (1985), Gerali and Lippi (2006). • The Matlab application uses code from Paul S¨oderlind’s webpage at the University of St. Gallen (http://home.datacomm.ch/paulsoderlind/).
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Optimal Monetary Policy
1. Additive Uncertainty
1 Optimal policy in a simple New Keynesian model • Simple model with monopolistic competition and staggered prices: πt = βEtπt+1 + κxt + ut 1 xt = Etxt+1 − [it − Etπt+1] + gt σ
(1.1) (1.2)
where πt is inflation, xt is the output gap, it is the one-period nominal interest rate • Similar to old-style AS-AD model, but crucial role for expectations Solving forward yields π t = Et
∞ X
β k [κxt+k + ut+k ]
(1.3)
k=0 ∞ X
1 − x t = Et (it+k − πt+k+1) + gt+k σ k=0
3
(1.4)
Optimal Monetary Policy
1. Additive Uncertainty
Monetary policy • What determines it? – Simple instrument rule (e.g., Taylor, 1993): it = (1 − gi) [gπ πt + gxxt] + giit−1
(1.5)
– Targeting rule (optimal policy): minimize objective function min Et
∞ X k=0
β
k
2 πt+k
+
λx2t+k
(1.6)
subject to the model • Note: – x∗ = 0 (yt∗ = y¯t), so no average inflation bias – π ∗ = 0, normalization (steady-state) – Output equation can be disregarded (one-to-one relationship between it and xt)
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Optimal Monetary Policy
1. Additive Uncertainty
1.1 Optimal policy with discretion • Take expectations as given, find optimal “time-consistent” policy • No endogenous state variables, so series of one-period problems: min π 2 + λx2t + Ft π ,x t t
(1.7)
t
subject to πt = κxt + ft
(1.8)
where Ft = Et
∞ X k=1
β
k
2 πt+k
+
λx2t+k
,
ft = βEtπt+1 + ut
(1.9)
• Optimal targeting rule κ xt = − πt λ
(1.10)
Monetary policy “leans against the wind”: πt > 0 ⇒ xt < 0 More aggressive response if κ large (θ small) or λ small
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Optimal Monetary Policy
1. Additive Uncertainty
• Substitute optimal targeting rule into Phillips curve: κ2 βλ λ πt = βEtπt+1 − πt + ut = E π + ut t t+1 λ λ + κ2 λ + κ2
(1.11)
• Assume ut follows AR(1) process ut = ρut−1 + uˆt • Repeated substitution for Etπt+j gives the reduced-form equation for inflation: λ λρ βλ βλ + E π + u ut πt = t t+2 t λ + κ2 λ + κ2 λ + κ2 λ + κ2 = ... k k ∞ λ X βλρ βλρ u + lim E π = t t t+k 2 2 2 k→∞ λ + κ λ + κ k=0 λ + κ | {z }
=0
=
λ ut κ2 + λ(1 − βρ)
(1.12)
• Using the policy rule yields the reduced-form equation for the output gap: xt = −
κ ut κ2 + λ(1 − βρ)
(1.13)
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Optimal Monetary Policy
1. Additive Uncertainty
• Note that λ ρut κ2 + λ(1 − βρ) = ρπt
Etπt+1 =
(1.14)
Etxt+1 = ρxt
(1.15)
• To derive a rule for the interest rate, use the optimal targeting rule in the output equation: κρ 1 κ − πt = − Etπt+1 − [it − Etπt+1] + gt λ λ σ
(1.16)
and use πt = ρ−1Etπt+1 to obtain
κσ(1 − ρ) Et πt+1 + σgt 1 + it = λρ
(1.17)
or, equivalently,
κσ(1 − ρ) 1 + ρπt + σgt it = λρ λρ + κσ(1 − ρ) = 2 ut + σgt κ + λ(1 − βρ)
(1.18)
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Optimal Monetary Policy
1. Additive Uncertainty
Properties of the optimal policy with discretion • Time consistency : Policy same in every period. • Certainty equivalence: Policy independent of Var(ut), Var(gt). • Policy brings inflation back to target by moving the output gap in the opposite direction. • Output (demand) shocks do not create any trade-off, but are completely offset: dit/dgt = σ. Monetary policy affects output in the same period, no preference for interest rate stability. • Inflation (supply) shocks create trade-off. Monetary policy offsets supply shocks only by affecting the output gap, so these will typically not be offset completely (unless λ = 0). • Trade-off between inflation and output variability (not levels):
2
2
κ Var(ut ) (1.19) Var(xt) = κ2 + λ(1 − βρ)
λ Var(ut ), Var(πt) = κ2 + λ(1 − βρ) λ ↑ ⇒ Var(πt) ↑, but Var(xt) ↓
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Optimal Monetary Policy
1. Additive Uncertainty
Response to inflation shocks (a) Interest rate
(b) Inflation
1.2
2 Discretion
1
1.5
0.8 0.6
1
0.4
0.5
0.2 0
0 −0.2
0
5
10
15
20
15
20
−0.5
0
5
10
15
20
(c) Output gap 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 0
5
10
• Numerical example: β = 0.99, ρ = 0.5, σ = 1, κ = 0.17, σu2 = 0.1 • uˆt = 1 for t = 0 and zero afterwards • Inflation high, CB tightens policy, negative output gap, gradual return to target (ρ > 0).
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Optimal Monetary Policy
1. Additive Uncertainty
Variance trade-off under discretion 1 Discretion 0.9
0.8
Output gap variance
0.7
0.6
0.5
0.4
0.3
0.2
• λ = 0.25
0.1
• 0
0
0.05
0.1
0.15
0.2 0.25 0.3 Inflation variance
• Vary λ ∈ [0, 10], calculate Var(πt), Var(xt)
10
0.35
0.4
λ = 0.5 0.45
0.5
Optimal Monetary Policy
1. Additive Uncertainty
1.2 Optimal policy with commitment • CB makes plan for entire future, credible, so private expectations adjust. CB “chooses” expectations to stabilize the economy. • Complete intertemporal optimization problem: min E π ,x t t
t
∞ X
β
k
k=0
2 πt+k
+
λx2t+k
(1.20)
subject to πt = βEtπt+1 + κxt + ut
(1.21)
• Lagrangian: L = Et
∞ X
β
k=0
k
2 πt+k
+
λx2t+k
+ φt+k [πt+k − βπt+k+1 − κxt+k − ut+k ] ,
where φt+k is the multiplier on the constraint for period t + k.
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(1.22)
Optimal Monetary Policy
1. Additive Uncertainty
• First-order conditions: 0 = 2β k πt+k + φt+k − βφt+k−1
(1.23)
0 = 2β k λxt+k − κφt+k
(1.24)
so κ xt+k − xt+k−1 = − πt+k , λ κ xt = − πt λ
k≥1
(1.25) (1.26)
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Optimal Monetary Policy
1. Additive Uncertainty
Properties of the optimal policy with commitment 1. History-dependence: relates ∆xt to πt (not xt). Not purely forward-looking. 2. ∆xt < 0 as long as πt > 0, foreseen by rational agents, so the initial effect on inflation is smaller than under discretion 3. Time inconsistency : Policy different at k = 0 and k > 0. Reoptimization gives different policy rule. Cf. optimality from a “timeless perspective”, Woodford (2003): Implement κ xt+k − xt+k−1 = − πt+k λ
(1.27)
in every period. As if optimized long time ago. 4. Certainty equivalence: Policy independent of Var(ut), Var(gt).
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Optimal Monetary Policy
1. Additive Uncertainty
Response to inflation shocks (a) Interest rate
(b) Inflation
1.2
2 Commitment Discretion
1
1.5
0.8 0.6
1
0.4
0.5
0.2 0
0 −0.2
0
5
10
15
20
15
20
−0.5
0
5
10
15
20
(c) Output gap 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 0
5
10
• Commitment: CB commits to future deflation, so smaller initial effect on inflation (expectations lower); smaller initial policy tightening; deeper recession • Commitment policy time inconsistent: deflation not optimal if policy reoptimized
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Optimal Monetary Policy
1. Additive Uncertainty
Variance trade-off under commitment and discretion 1 Commitment Discretion 0.9
0.8
Output gap variance
0.7
0.6
0.5
0.4
• λ = 0.25
0.3
0.2
• λ = 0.5
• λ = 0.25
0.1
• 0
0
0.05
0.1
0.15
0.2 0.25 0.3 Inflation variance
0.35
0.4
λ = 0.5 0.45
0.5
• More favorable trade-off under commitment • “Stabilization bias”: Output overstabilized under discretion, inflation too volatile
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Optimal Monetary Policy
1. Additive Uncertainty
1.3 Commitment to a simple rule: Determinacy • Simple Taylor rule: it = gπ πt + gxxt
(1.28)
• Commitment device: – Easy verify ex post – Potentially more efficient than discretionary policy • Can policy guarantee a unique, stable equilibrium? • Avoid “sunspot equilibria”: non-fundamental movements in inflation self-fulfilling
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Optimal Monetary Policy
1. Additive Uncertainty
• Substitute (1.28) into output equation, write on simple matrix form:
Etxt+1 Etπt+1
= A
xt πt
+ C
gt ut
,
(1.29)
where
A=
−1
−1
−1
1 + σ (gx + κβ ) σ (gπ − −κβ
−1
β
β −1)
−1
,
C=
1
−(βσ)−1
0
−1
β
(1.30)
• Blanchard and Kahn (1980), Proposition 1: Unique solution if the number of eigenvalues in A outside the unit circle (nδ ) = the number of forward-looking variables (n2) – nδ > n2: no stable solution (non-existence) – nδ < n2: infinite number of stable solutions (indeterminacy) • Here 2 forward-looking variables, so determinacy if both eigenvalues of A are outside the unit circle. • See also Gal´ı (2008, Ch. 4).
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Optimal Monetary Policy
1. Additive Uncertainty
• Woodford (2003, Ch. 4): determinacy iff 1. det A > 1 2. det A − trA > −1 3. det A + trA > −1 • Here: 1" gx gπ κ # det A = 1+ + >1 β σ σ gx κ 1 trA = 1 + + + >0 σ βσ β
(1.31) (1.32)
• 1 and 3 satisfied automatically, 2 satisfied if gx κgπ # gx κ 1 1" 1+ + − 1+ + + > −1 β σ σ σ βσ β ⇔ 1−β gπ + gx > 1 κ
18
(1.33)
(1.34)
Optimal Monetary Policy
1. Additive Uncertainty
• Interpretation: 1 − β dπ = 1% permanently ⇒ dx = % permanently κ 1 − β Policy response: di = gπ + gx % > 1% κ
“Taylor principle”: change i more than one-for-one with π (else real interest rate ↓) • Else: sunspot-driven changes in inflation are self-fulfilling • Relevant in practice? Do central banks commit forever to a bad rule? Davig and Leeper (2007): Determinacy even with gπ < 1 if Prob(gπ > 1) > 0 in future. • Optimal policy with discretion (and commitment) satisfies Taylor principle:
κσ(1 − ρ) 1 + Et πt+1 + σgt it = λρ
(1.35)
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Optimal Monetary Policy
1. Additive Uncertainty
2 More general models: Numerical solutions • Define by – x1t an n1-vector of predetermined variables (including exogenous processes), with initial conditions x10 given, – x2t an n2-vector of forward-looking variables, without initial conditions, – εt an n1-vector of iid innovations with zero mean and covariance matrix Σε. • Then most (log) linear models can be written on the form
x1t+1 Etx2t+1
=A
x1t x2t
+ But +
εt+1 0
where the matrix A includes the parameters of the model. • Not very restrictive: any number of lags can be added to x1t. We can often solve also with non-singular matrix H in front of Etx2t+1. • To simplify notation, define the n-vector xt ≡ [x01t x02t]0, where n = n1 + n2.
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(2.1)
Optimal Monetary Policy
1. Additive Uncertainty
2.1 Solving a linear RE model • Solve the model without the instrument ut • We seek the non-explosive solution of (2.1), i.e., solutions with limj→∞ |Etxt+j | < ∞. • Take expectations at t:
Et
x1t+1 Etx2t+1
= A
x1t x2t
.
(2.2)
• The Schur decomposition of A in (2.2) is given by A = ZT Z H ,
(2.3)
where T is an upper triangular matrix with the eigenvalues of A on the diagonal, Z is a unitary matrix (Z −1 = Z H ; Z H Z = ZZ H = I), and Z H is the conjugate transpose of Z. • Reorder the rows in T, Z so that the nθ stable roots (the eigenvalues inside the unit circle) are first, and the nδ unstable roots are last.
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Optimal Monetary Policy
1. Additive Uncertainty
• Define the auxiliary variables
θt δt
H
≡Z
x1t x2t
,
i.e.,
x1t x2t
=Z
θt δt
,
(2.4)
where θt is related to the stable eigenvalues and δt to the unstable ones. • Then
Et
θt+1 δt+1
= Z H A
x1t
x2t
= T Z H
= T
using (2.2)
x1t x2t
using A = ZT Z H and Z H Z = I
θt δt
.
(2.5)
This is of the same form as (2.2), but is easier to solve as T is upper triangular and contains the stable eigenvalues in the upper-left block.
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Optimal Monetary Policy
1. Additive Uncertainty
• Partition T conformably with θt, δt:
E
t
θt+1 δt+1
=
Tθθ Tθδ 0 Tδδ
θt δt
.
(2.6)
• By construction, Tδδ includes the unstable eigenvalues. Therefore, to get a stable solution, we must have δt = 0 for all t, and thus Etθt+1 = Tθθ θt.
(2.7)
• Partition Z in (2.4) to get:
x1t x2t
=
Zkθ Zkδ Zλθ Zλδ
=
θt δt
Zkθ Zλδ
θt .
(2.8)
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Optimal Monetary Policy
1. Additive Uncertainty
• Since x10 is given, we can solve for the initial conditions θ0 as −1 θ0 = Zkθ x10,
(2.9)
if Zkθ is invertible. • Zkθ is (n1 × nθ ), so a necessary condition for Zkθ to be invertible is that nθ = n1, i.e., the number of stable roots is equal to the number of predetermined variables (the “saddle-point property” of Blanchard and Kahn, 1980). – nθ < n1 (nδ > n2): too few stable roots, no stable solution (non-existence). – nθ > n1 (nδ < n2): too many stable roots, infinite number of stable solutions (indeterminacy).
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Optimal Monetary Policy
1. Additive Uncertainty
Putting back the innovations • From (2.1) and (2.8), εt+1 = x1t+1 − Etx1t+1 = Zkθ [θt+1 − Etθt+1] .
(2.10)
• Solve for θt+1 and use Etθt+1 = Tθθ θt from (2.7): −1 θt+1 = Etθt+1 + Zkθ εt+1 −1 = Tθθ θt + Zkθ εt+1,
(2.11)
which together with δt = 0 and θ0 gives the solution for θt, δt. • To rewrite in terms of x1t, x2t, use (2.8) to get −1 −1 −1 Zkθ x1t+1 = Tθθ Zkθ x1t + Zkθ εt+1.
(2.12)
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Optimal Monetary Policy
1. Additive Uncertainty
Solution • The solution for x1t+1 then is −1 x1t+1 = Zkθ Tθθ Zkθ x1t + εt+1
≡ M x1t + εt+1,
(2.13)
i.e., a VAR(1) process. • Likewise, using (2.8) the solution for x2t is x2t = Zλθ θt −1 = Zλθ Zkθ x1t
≡ N x1t,
(2.14)
i.e., a linear function of the predetermined variables. • The key step in solving the model is the Schur decomposition. This is readily available in many software packages.
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Optimal Monetary Policy
1. Additive Uncertainty
Impulse response functions • From the solution (2.13)–(2.14), it is straightforward to trace the effects on the economy of a shock at time t, i.e., the impulse response function. • For the predetermined variables, the expected effects of a shock εt on future variables are x1t = εt Etx1t+1 = M εt Etx1t+2 = M 2εt ...
(2.15)
Etx1t+j = M j εt. • The effects on the forward-looking variables are Etx2t+j = N Etx1t+j = N M j εt .
(2.16)
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Optimal Monetary Policy
1. Additive Uncertainty
Unconditional variances • From (2.13)–(2.14), the unconditional covariance matrices of x1t and x2t satisfy Σx1 = M Σx1M 0 + Σε,
(2.17)
Σx2 = N Σx1N 0.
(2.18)
• To solve the Lyapunov equation in (2.17), use the “vec” operator to write vec (Σx1) = vec (M Σx1M 0) + vec (Σε) = (M ⊗ M ) vec (Σx1) + vec (Σε) = (I − M ⊗ M )−1 vec (Σε) ,
(2.19)
where we have used vec(A + B) = vec(A) + vec(B), and vec(ABC) = (C 0 ⊗ A) vec(B), and where ⊗ is the Kronecker product. • Alternatively, if n1 is large, iterate on (2.17) until convergence, or use Matlab function to solve directly.
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Optimal Monetary Policy
1. Additive Uncertainty
Reintroducing monetary policy • Put back the instrument vector ut:
x1t+1 Etx2t+1
=A
x1t x2t
+ But +
εt+1 0
,
(2.20)
where we will typically have only one instrument, so nu = 1. • Simple rule: ut is set as a linear function of the variables in xt ut = F xt
(2.21)
• Optimal policy: ut is set to minimize a loss function subject to (2.20). The policymaker’s loss function is typically assumed to be quadratic in the variables of the model and the instrument: E0
∞ X t=0
β t [x0tQxt + 2x0tU ut + u0tRut] .
(2.22)
(Not the same Q as in the Schur decomposition!)
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Optimal Monetary Policy
1. Additive Uncertainty
2.2 A simple rule for monetary policy • Specify the policy instrument as a linear function of the variables in the model: ut = F xt .
(2.23)
• Using the rule in the model (2.20) we get
x1t+1 Etx2t+1
= A
x1t x2t
+ But +
= (A + BF )
x1t x2t
εt+1 0
+
εt+1 0
,
(2.24)
and the model can be solved using the method described above. • Alternatively, the interest rate rule can be included as one of the equations in the model (2.1), and the model is solved in the same way. • Note that we are implicitly assuming commitment to the rule (2.23), since the policy is expected to be followed forever.
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Optimal Monetary Policy
1. Additive Uncertainty
An optimized simple rule • Use the solution x2t = N x1t to write
x1t x
2t
=
ut
I N
F
I
N
x1t ≡ P x1t.
(2.25)
• Then we can write the objective function (2.22) in terms of x1t as E0
∞ X t=0
β
t
0 0 P x 1t
Q U 0
U R
∞ X
t=0
≡ E0 P x1t
β t [x01tW x1t] .
(2.26)
• Guessing a quadratic value function gives Jt = x01tV x1t + v = x01tW x1t + βEtJt+1.
31
(2.27)
Optimal Monetary Policy
1. Additive Uncertainty
• Substitute for EtJt+1 and use the solution x1t+1 = M x1t + εt+1: x01tV x1t + v = x01tW x1t + βEt [x01t+1V x1t+1 + v] = x01tW x1t + βx01tM 0V M x1t + βEt [ε0t+1V εt+1] + βv.
(2.28)
• Then V and v satisfy V = W + βM 0V M,
(2.29)
v = βEt [ε0t+1V εt+1] + βv.
(2.30)
• To find V , iterate on (2.29) until convergence, starting out from some symmetric positive definite V . To find v, solve (2.30) and recall the rule tr(ABC) = tr(BAC) = tr(CAB): v =
β β Et [ε0t+1V εt+1] = tr(V Σε). 1−β 1−β
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(2.31)
Optimal Monetary Policy
1. Additive Uncertainty
• The value of the objective function then is J0 = x010V x10 +
β tr(V Σε). 1−β
(2.32)
• To find the optimal coefficients in F use a non-linear optimization routine (e.g., fminsearch in Matlab) to minimize the objective function.
33
Optimal Monetary Policy
1. Additive Uncertainty
2.3 Optimal policy with commitment • At t = 0 the policymaker chooses a sequence for the instrument for the entire future • Perfectly credible, so private expectations adjust; CB “chooses” expectations. • In future periods the policymaker must honor past commitments, represented by Lagrange multipliers on the forward-looking variables (ρ2t). When optimizing (at t = 0) the policymaker ignores past commitments, so ρ20 = 0. Therefore the optimal policy is not time-consistent. • “Commitment in a timeless perspective” (Woodford, 2003): policymaker committed to the optimal policy long ago (at time t = −∞); ρ20 > 0. • The state of the economy is given by the predetermined variables x1t and ρ2t (with initial conditions), while x2t, ut and ρ1t are forward-looking variables (without initial conditions). • The optimal policy will be a linear function of the state (x1t, ρ2t). • To solve the model we set up a Lagrangian, derive the first-order conditions, and use the generalized Schur decomposition to follow almost the same steps as before.
34
Optimal Monetary Policy
1. Additive Uncertainty
• The policymaker solves min E0 {ut }
∞ X t=0
β t [x0tQxt + 2x0tU ut + u0tRut] ,
(2.33)
subject to xt+1 = Axt + But + ξt+1,
(2.34)
and x10 given, where ξt+1 ≡ [ε0t+1 (x2t+1 − Etx2t+1)0]0. • Set up the Lagrangian L0 = E0
∞ X t=0
β t [x0tQxt + 2x0tU ut + u0tRut + 2ρt+1 (Axt + But + ξt+1 − xt+1)] .
(2.35)
• The first-order conditions w.r.t. xt, ut, ρt+1 are 0 = βQxt + βU ut + βA0Etρt+1 − ρt,
(2.36)
0 = U 0xt + Rut + B 0Etρt+1,
(2.37)
0 = Axt + But + ξt+1 − xt+1.
(2.38)
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Optimal Monetary Policy
1. Additive Uncertainty
• Write the first-order conditions as
I 0
0
0 0 βA
0
0 0 −B 0
xt+1 ut+1 Etρt+1
=
A
B
0
−βQ −βU I U0
R
0
t
xt u
ρt
+
ξt+1 0 0
.
(2.39)
• Reorder the matrices, placing the predetermined variables first, and take expectations:
GEt
kt+1 λt+1
= D
kt λt
,
(2.40)
where
kt ≡
x1t ρ2t
,
λt ≡
x2t u
t
,
(2.41)
ρ1t
so kt collects the n predetermined variables and λt the n + nu forward-looking variables. • This model is a generalization of the initial model in (2.2), since G 6= I. Solving the model follows similar steps as before, but using the generalized Schur decomposition (Klein, 2000; Sims, 2000). See the Appendix.
36
Optimal Monetary Policy
1. Additive Uncertainty
• As before, the solution is a VAR(1) in the predetermined variables and a linear relationship between the forward-looking and predetermined variables:
x1t+1 ρ2t+1
x2t u
= M
x1t ρ2t
+
εt+1 0
,
(2.42)
t
= N
ρ1t
x1t ρ2t
.
(2.43)
• In particular, the optimal rule for ut is given by c
ut = F
x1t ρ2t
,
(2.44)
where Fc is the submatrix given by rows (n2 + 1 : n2 + nu) of N .
37
Optimal Monetary Policy
1. Additive Uncertainty
Properties of the optimal policy with commitment • The optimal rule under commitment depends on the predetermined variables and the multipliers on the forward-looking variables: c
ut = F
x1t ρ2t
.
(2.45)
The presence of the “promise-keeping” multipliers is due to previous commitments. • History-dependence: The multipliers ρ2t can be written as ρ2t = M21x1t−1 + M22ρ2t−1 = M21
∞ X j=1
j−1 M22 x1t−j .
(2.46)
Thus, the optimal policy at t depends on the entire history of x1t. • Time-inconsistency: As ρ20 = 0, the optimal policy is different at t = 0 and at t > 0. Thus, optimal policy is not time-consistent. • Certainty equivalence: M and N depend on A, B, Q, U, R, β, but are independent of Σε. Thus, optimal policy is the same as in a non-stochastic economy.
38
Optimal Monetary Policy
1. Additive Uncertainty
Commitment in a timeless perspective • Reference: Woodford (2003). • Suppose the policymaker acts as if it had committed to the optimal policy long ago (at t = −∞). • Then ρ20 > 0, so policy is the same in every period (time-consistency). • Svensson and Woodford (2004): To implement, modify loss function to min E0
∞ X t=0
β t [x0tQxt + 2x0tU ut + u0tRut] + β −1ρ020 [x20 − E−1x20] ,
(2.47)
where ρ20 are the multipliers from the optimization problem in period t = 0. Then solve for optimal policy with discretion.
39
Optimal Monetary Policy
1. Additive Uncertainty
2.4 Optimal policy with discretion • Under discretion the policymaker is unable to commit to future policies and therefore does not honor past commitments. Instead the policymaker reoptimizes in each period, and we seek the optimal time-consistent policy. • The policymaker takes expectations as given, leading to a Nash equilibrium solution. • The state of the economy is given by the predetermined variables in x1t. Therefore the optimal rule and the forward-looking variables will follow ut = F x1t,
(2.48)
x2t = N x1t,
(2.49)
for some F and N . • No closed-form solution exists, and the properties of the solution algorithm are unknown, but they tend to work fine.
40
Optimal Monetary Policy
1. Additive Uncertainty
• To find the optimal rule in period t the policymaker solves min E0 {ut }
∞ X t=0
β t [x0tQxt + 2x0tU ut + u0tRut] ,
(2.50)
subject to
x1t+1 Etx2t+1
= Axt + But +
εt+1 0
,
(2.51)
and x10 given. • Since we have a linear-quadratic problem, we guess that the value function in t is a quadratic function of the state: Jt = x01tVtx1t + vt.
(2.52)
• Then the Bellman equation is x01tVtx1t + vt = min {x0tQxt + 2x0tU ut + u0tRut + βEt [x01t+1Vt+1x1t+1 + vt+1]} . u t
41
(2.53)
Optimal Monetary Policy
1. Additive Uncertainty
Rewriting the problem • The Bellman equation includes x2t which are endogenous and depend on expectations of all variables in the model. • However, using the guess Etx2t+1 = Nt+1Etx1t+1
(2.54)
we can rewrite the model in terms of only x1t+1. • Partition A, B in (2.51). Then we can combine with (2.54) to obtain Etx2t+1 = A21x1t + A22x2t + B2ut = Nt+1Etx1t+1 = Nt+1 [A11x1t + A12x2t + B1ut] .
42
(2.55)
Optimal Monetary Policy
1. Additive Uncertainty
• Then we can solve for x2t as x2t = Dtx1t + Gtut,
(2.56)
where Dt ≡ [A22 − Nt+1A12]−1 [Nt+1A11 − A21] ,
(2.57)
Gt ≡ [A22 − Nt+1A12]−1 [Nt+1B1 − B2] .
(2.58)
• Combine with (2.51) to write x1t+1 as x1t+1 = A11x1t + A12x2t + B1ut + εt+1 = A∗t x1t + Bt∗ut + εt+1,
(2.59)
where A∗t ≡ A11 + A12Dt,
(2.60)
Bt∗ ≡ B1 + A12Gt.
(2.61)
43
Optimal Monetary Policy
1. Additive Uncertainty
• Partition Q and U and write the loss function for period t as x0tQxt + 2x0tU ut + u0tRut
(2.62)
= x01tQ11x1t + x01tQ12x2t + x02tQ21x1t + x02tQ22x2t + 2 [x01tU1 + x02tU2] ut + u0tRut. • Using (2.56) and (2.59) we can write x0tQxt + 2x0tU ut + u0tRut = x01tQ∗t x1t + 2x01tUt∗ut + u0tRt∗ut,
(2.63)
where Q∗t ≡ Q11 + Q12Dt + Dt0 Q21 + Dt0 Q22Dt,
(2.64)
Ut∗ ≡ Q12Gt + Dt0 Q22Gt + U1 + Dt0 U2,
(2.65)
Rt∗ ≡ R + G0tQ22Gt + G0tU2 + U20 Gt.
(2.66)
44
Optimal Monetary Policy
1. Additive Uncertainty
• Thus the Bellman equation in terms of x1t is x01tVtx1t + vt = min {x01tQ∗t x1t + 2x01tUt∗ut + u0tRt∗ut u
(2.67)
t
+
βEt (A∗t x1t
+
Bt∗ut
+ εt+1)
0
Vt+1 (A∗t x1t
+
Bt∗ut
+ εt+1) + vt+1 .
• The nu first-order conditions are Ut∗0x1t + Rt∗ut + βBt∗0Vt+1A∗t x1t + βBt∗0Vt+1Bt∗ut = 0.
(2.68)
• Rearranging gives the decision rule in t: ut = Ftx1t,
(2.69)
where Ft ≡ −
Rt∗
+
∗0 ∗ −1 βBt Vt+1Bt Ut∗0
+
βBt∗0Vt+1A∗t
45
.
(2.70)
Optimal Monetary Policy
1. Additive Uncertainty
• Combine with the Bellman equation (2.67): x01tVtx1t + vt = x01tQ∗t x1t + 2x01tUt∗Ftx1t + x01tFt0Rt∗Ftx1t +
βEt (A∗t x1t
+
Bt∗Ftx1t
+ εt+1)
0
Vt+1 (A∗t x1t
+
= x01t (Q∗t + 2Ut∗Ft + Ft0Rt∗Ft) x1t + βx01t (A∗t +
Bt∗Ftx1t + εt+1) + vt+1 Bt∗Ft)0 Vt+1 (A∗t + Bt∗Ft) x1t
+ βEt [ε0t+1Vt+1εt+1 + vt+1] .
(2.71)
• Thus, Vt and vt satisfy Vt = Q∗t + 2Ut∗Ft + Ft0Rt∗Ft + β (A∗t + Bt∗Ft)0 Vt+1 (A∗t + Bt∗Ft) ,
(2.72)
vt = βEt [ε0t+1Vt+1εt+1 + vt+1] .
(2.73)
46
Optimal Monetary Policy
1. Additive Uncertainty
The time-invariant policy • To obtain the time-invariant policy, start from some symmetric positive definite Vt and some Nt and iterate on the equations for Dt, Gt in (2.57)–(2.58), A∗t , Bt∗ in (2.60)–(2.61), Q∗t , Ut∗, Rt∗ in (2.64)–(2.66), Ft in (2.70) and Vt in (2.72). • This gives the stationary solution ut = F x1t,
(2.74)
x2t = (D + GF ) x1t ≡ N x1t,
(2.75)
x1t+1 = (A11 + A12N + B1F ) x1t + εt+1 ≡ M x1t + εt+1,
(2.76)
so again the solution for x1t is a VAR(1), and the guesses for ut and x2t are confirmed. • The value of the loss function is J0 = x010V x10 +
β tr(V Σε). 1−β
(2.77)
47
Optimal Monetary Policy
1. Additive Uncertainty
Properties of the optimal policy with discretion • The optimal rule under discretion depends on the predetermined variables: ut = Fdx1t.
(2.78)
Thus, the optimal policy is not history-dependent, but depends only on the current values of the state variables. • Time-consistency: The optimal policy is the same in every period. Thus, optimal policy is time-consistent. • Certainty equivalence: F, V, M and N depend on A, B, Q, U, R, β, but are independent of Σε. Thus, optimal policy is the same as in a non-stochastic economy. • Suboptimality: The optimal policy with discretion gives a worse outcome than with commitment. This is true also without an overly ambitious output/unemployment target and an inflation bias (as in Kydland and Prescott, 1977). This is due to the inefficient response to shocks (no history dependence), and is sometimes called a “stabilization bias.” How large is the cost of discretionary policy is an empirical issue; see Dennis and S¨oderstr¨om (2006).
48
Optimal Monetary Policy
1. Additive Uncertainty
2.5 Alternative approaches • Although the standard form used here is very flexible, it cannot accomodate all possible models. For example, Rudebusch (2002) uses the model πt = µπ Et−1π¯ t+3 + (1 − µπ ) yt = µy Et−1yt+1 + (1 − µy )
4 X
απj πt−j + αy yt−1 + εt,
(2.79)
βyj yt−j − βr [rt−1 − r∗] + ηt,
(2.80)
j=1 2 X
j=1
rt−1 = µr [Et−1¯ıt+3 − Et−1π¯ t+4] + (1 − µr ) [¯ıt−1 − π¯ t−1] , where ¯ıt = 1/4
P3
j=0 it−j
and π¯ t = 1/4
P3
j=0 πt−j
(2.81)
are the average yearly interest rate and
inflation rate. • In this model the output gap depends on expectations of the interest rate three periods ahead, Et−1it+3, and this equation is not easily rewritten to fit into the standard framework.
49
Optimal Monetary Policy
1. Additive Uncertainty
• In such cases, there are alternative frameworks available: – Dennis (2004, 2007) develops solution algorithms based on the structural form A0xt = A1xt−1 + A2Etxt+1 + A3ut + A4Etut+1 + A5vt
(2.82)
and shows how to calculate optimal policy under discretion and precommitment and how to solve for a given simple rule and calculate optimized rules. – The Anderson-Moore (AIM) algorithm, developed at the Federal Reserve Board, is commonly used to solve models with a simple rule for monetary policy. See Anderson and Moore (1985) and Zagaglia (2005). In general, the model is written on the form J X j=0
Gj xt−j +
K X
Hk Etxt+k = εt,
(2.83)
k=1
where one of the equations corresponds to the monetary policy rule. – These frameworks are more flexible than the standard one, and can handle future expected instruments in a simple way. Also, there is no need to explicitly distinguish between predetermined and forward-looking variables. However, the optimization routines are probably less efficient and less reliable than the standard routines.
50
Optimal Monetary Policy
1. Additive Uncertainty
3 Optimal policy with additive shocks: Conclusions • Optimal policy with commitment: – History dependent – Time consistent – Certainty equivalent • Optimal policy with discretion: – Time consistent – Not history dependent – Suboptimal – Certainty equivalent • Optimized simple rule: – Not certainty equivalent – Commitment to a rule, may dominate optimal policy with discretion
51
Optimal Monetary Policy
1. Additive Uncertainty
Certainty equivalence • Stochastic properties have no impact on optimal policy, same as in non-stochastic economy. • Applies if linear model, quadratic objectives, only additive uncertainty, unrestricted optimal policy. • Optimized simple rule: CE does not apply. • Data uncertainty: CE still applies wrt optimal estimates of unobservable variables. • Multiplicative uncertainty: CE does not apply.
52
Optimal Monetary Policy
1. Additive Uncertainty
4 Matlab application: Optimal monetary policy with additive shocks • Consider the “hybrid” New Keynesian model πt = απt−1 + (1 − α)βEtπt+1 + κ [yt − y¯t] + vtπ , 1 yt = δyt−1 + (1 − δ)Etyt+1 − [it − Etπt+1] + vty , σ π π π vt = ρπ vt−1 + εt ,
(4.2)
y vty = ρy vt−1 + εyt ,
(4.4)
y¯t = ρy¯y¯t−1 + εyt¯,
(4.5)
(4.1)
(4.3)
where y¯t is potential output. • Generalization of simple model; there α = δ = 0, and y¯t = 0 for all t. • Microfoundations for inertia: habits in consumption, indexation or rules of thumb in price setting, adaptive expectations, . . . • System of second-order difference equations, cannot be solved analytically. Need numerical methods.
53
Optimal Monetary Policy
1. Additive Uncertainty
Setting up the model • Output and inflation depend on current expectations of future values of all variables in the model, and are thus free to adjust in response to any shock in the model. These are therefore forward-looking variables. • The two shocks and potential output, on the other hand, depend only on past values and on exogenous disturbances, thus these are predetermined. • We also need to add lags of inflation and output to write model on first-order form. These are endogenous state variables that are predetermined at t. And to allow for an interest rate smoothing objective, we add the lagged interest rate, which is also predetermined at t. • Thus, we define
x1t ≡
t t−1 t−1
vtπ vty y¯ π y
,
x2t ≡
πt yt
,
u t ≡ it ,
it−1
54
εt ≡
επt εyt εyt¯ 0 0 0
.
Optimal Monetary Policy
1. Additive Uncertainty
• To write the model on the required form, define
A0 ≡
1 0 0 0 0 0
0
0
0 1 0 0 0 0
0
0
0 0 1 0 0 0
0
0
0 0 0 1 0 0
0
0
0 0 0 0 1 0
0
0
0 0 0 0 0 1
0
0
0 0 0 0 0 0 (1 − α)β 0 0 0 0 0 0
0 (1 − δ)
1/σ
B1 ≡
#0
"
0 0 0 0 0 1 0 1/σ
;
Σε ≡
55
; A1 ≡
σπ 0
ρπ
0
0
0
0 0 0
0
0
ρy
0
0
0 0 0
0
0
0 ρy¯
0
0 0 0
0
0
0
0
0
0 0 1
0
0
0
0
0
0 0 0
1
0
0
0
0
0 0 0
0
−1 0
κ −α 0 0 1 −κ
0 −1 0
0 0 0 0
0 σy 0 0 0 0 0
0 σy¯ 0 0 0
0
0
0 0 0 0
0
0
0 0 0 0
0
0
0 0 0 0
0
−δ 0 0
1
;
Optimal Monetary Policy
1. Additive Uncertainty
• Then we can write the model as
A
0
x1t+1 Etx2t+1
1
=A
x1t x2t
+ B1ut +
εt+1 0
,
(4.6)
and premultiplying by A−1 0 we obtain the standard form
x1t+1 Etx2t+1
= A
x1t x2t
+ But +
εt+1 0
56
(4.7)
Optimal Monetary Policy
1. Additive Uncertainty
Modeling monetary policy • Assume that the central bank loss function is ∞ X
min E0
β
t
t=0
πt2
2
+ λy (yt − y¯t) + λi (it − it−1)
2
(4.8)
• To write the loss function on the required form, define the vector of target variables as
Yt ≡
πt yt − y¯t it − it−1
= Cxxt + Cuut,
(4.9)
where
0 0 0 0 0 0 1 0
0
Cx ≡ 0 0 −1 0 0 0 0 1 ; Cu ≡ 0 0 0 0 0 0 −1 0 0
57
1
Optimal Monetary Policy
1. Additive Uncertainty
• Then the period loss function is Lt = Yt0ZYt = (Cxxt + Cuut)0 Z (Cxxt + Cuut) = x0tCx0 ZCxxt + 2xtCx0 ZCuut + u0tCu0 ZCuut = x0tQxt + 2x0tU ut + u0tRut,
(4.10)
where
1 0
0
Z ≡ 0 λy 0 0 0 λi
,
Q ≡ Cx0 ZCx,
U ≡ Cx0 ZCu; R ≡ Cu0 ZCu.
(4.11)
• Alternatively, a Taylor rule can be implemented as it = (1 − gi) [gπ πt + gy (yt − y¯t)] + giit−1 = F xt,
(4.12)
where "
#
F ≡ 0 0 −(1 − gi)gy 0 0 gi (1 − gi)gπ (1 − gi)gy .
58
Optimal Monetary Policy
1. Additive Uncertainty
Numerical exercises • “Structural” parameter values: α = 0.5, β = 0.99, κ = 0.17, δ = 0.5, σ = 1 • Shock parameters: ρπ = ρy = ρy¯ = 0.5 √ σπ = σy = σy¯ = 0.1 • Policy preferences: λy = 0.5, λi = 0.1 • Policy rule parameters: gπ = 2, gy = 0.8, gi = 0.8
59
Optimal Monetary Policy
1. Additive Uncertainty
• The Matlab script OptimalPolicy.m shows how to 1. Set up the model 2. Solve for a given simple rule, and calculate unconditional variances 3. Derive optimal monetary policy with commitment and discretion 4. Optimize a simple rule 5. Calculate impulse responses • This script uses Paul S¨oderlind’s Matlab routines, described in S¨oderlind (1999), “Solution and Estimation of RE Macromodels with Optimal Policy,” available from his website at http://home.datacomm.ch/paulsoderlind/.
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Optimal Monetary Policy
1. Additive Uncertainty
Appendix: The generalized Schur decomposition • References: Klein (2000) and Sims (2000). • The generalized Schur decomposition of D, G in (2.40) are G = QSZ H ,
(A.1)
D = QT Z H ,
(A.2)
where Q, Z are unitary matrices, and S, T are upper triangular, with the eigenvalues given by tii/sii. We then follow almost the same steps as above. • Reorder the rows in Q, S, T, Z so that the nθ stable roots are first, and the nδ unstable roots are last. • Define the auxiliary variables
θt δt
≡ Z H
kt λt
.
(A.3)
61
Optimal Monetary Policy
1. Additive Uncertainty
• Then
SE
t
θt+1 δt+1
H
= SZ E
t
kt+1 λt+1
kt+1
= QH GEt
λt+1
kt
λt
= T
using S = QH GZ and Z H Z = I
using (2.40)
kt
= QH D = T Z H
λt
using D = QT Z H and QH Q = I
θt δt
.
(A.4)
• Partition conformably with θt, δt:
Sθθ Sθδ 0
Sδδ
t
E
θt+1 δt+1
=
Tθθ Tθδ 0 Tδδ
θt δt
62
.
(A.5)
Optimal Monetary Policy
1. Additive Uncertainty
• Since the lower right block (Sδδ , Tδδ ) includes all unstable eigenvalues, we must have δt = 0 for all t to get a stable solution. Thus, −1 Tθθ θt. Etθt+1 = Sθθ
(A.6)
since Sθθ is invertible (det Sθθ = siiθθ and siiθθ 6= 0 for all i). Q
• Use (A.3) and partition Z:
kt λt
=
Zkθ Zkδ Zλθ Zλδ
=
θt δt
Zkθ Zλδ
θt .
(A.7)
• If Zkθ is invertible, we can solve for
−1 −1 θ0 = Zkθ k0 = Zkθ
x10 0
.
(A.8)
63
Optimal Monetary Policy
1. Additive Uncertainty
• From (2.34), x1t+1 − Etx1t+1 = εt+1, and note that ρ2t+1 − Etρ2t+1 = 0 (see Backus and Driffill, 1986). Thus, using (A.7) we obtain
εt+1 0
= kt+1 − Etkt+1 = Zkθ (θt+1 − Etθt+1) .
(A.9)
−1 • Use Etθt+1 = Sθθ Tθθ θt from (A.6) to get
−1 θt+1 = Etθt+1 + Zkθ
εt+1 0
−1 −1 = Sθθ Tθθ θt + Zkθ
εt+1 0
.
(A.10)
64
Optimal Monetary Policy
1. Additive Uncertainty
• Finally, use (A.7) twice to get kt+1 = Zkθ θt+1
−1 Tθθ θt + = Zkθ Sθθ
εt+1 0
−1 −1 = Zkθ Sθθ Tθθ Zkθ kt +
≡ M kt +
εt+1 0
εt+1 0
,
(A.11)
which together with x10 given and ρ20 = 0 determines the dynamics of kt. • As before, we use (A.7) to obtain λt = Zλθ θt −1 = Zλθ Zkθ kt
≡ N kt.
(A.12)
65
Optimal Monetary Policy
1. Additive Uncertainty
References Anderson, Gary and George Moore (1985), “A linear algebraic procedure for solving linear perfect foresight models,” Economics Letters, 17 (3), 247–252. Backus, David K. and John Driffill (1986), “The consistency of optimal policy in stochastic rational expectations models,” Discussion Paper No. 124, Centre for Economic Policy Research. Blanchard, Olivier J. and Charles M. Kahn (1980), “The solution of linear difference models under rational expectations,” Econometrica, 48 (5), 1305–1311. Davig, Troy and Eric M. Leeper (2007), “Generalizing the Taylor principle,” American Economic Review , 97 (3), 607–635. Dennis, Richard and Ulf S¨oderstr¨om (2006), “How important is precommitment for monetary policy?” Journal of Money, Credit, and Banking, 38 (4), 847–872. Dennis, Richard (2004), “Solving for optimal simple rules in rational expectations models,” Journal of Economic Dynamics and Control , 28 (8), 1635–1660. ——— (2007), “Optimal policy rules in rational-expectations models: New solution algorithms,” Macroeconomic Dynamics, 11 (1), 31–55. Gal´ı, Jordi (2008), Monetary Policy, Inflation, and the Business Cycle, Princeton University Press. Gerali, Andrea and Francesco Lippi (2006), “Solving dynamic linear-quadratic problems with forward-looking variables and imperfect information using Matlab,” Manuscript, Banca d’Italia. Klein, Paul (2000), “Using the generalized Schur form to solve a multivariate linear rational expectations model,” Journal of Economic Dynamics and Control , 24 (10), 1405–1423. Kydland, Finn E. and Edward C. Prescott (1977), “Rules rather than discretion: The inconsistency of optimal plans,” Journal of Political Economy, 85 (3), 473–491. 66
Optimal Monetary Policy
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Rudebusch, Glenn D. (2002), “Term structure evidence on interest rate smoothing and monetary policy inertia,” Journal of Monetary Economics, 49 (6), 1161–1187. Sims, Christopher A. (2000), “Solving linear rational expectations models,” Manuscript, Princeton University. S¨oderlind, Paul (1999), “Solution and estimation of RE macromodels with optimal policy,” European Economic Review Papers and Proceedings, 43 (4–6), 813–823. Svensson, Lars E. O. and Michael Woodford (2004), “Implementing optimal policy through inflation-forecast targeting,” in Ben S. Bernanke and Michael Woodford (eds.), The Inflation-Targeting Debate, The University of Chicago Press. Taylor, John B. (1993), “Discretion versus policy rules in practice,” Carnegie-Rochester Conference Series on Public Policy, 39, 195–214. Woodford, Michael (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. Zagaglia, Paolo (2005), “Solving rational-expectations models through the Anderson-Moore algorithm,” Computational Economics, 26 (1), 91–106.
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