Aug 20, 2012 - âMailing address: Goethe University, House of Finance, Grueneburgplatz 1, 60323 Frankfurt am Main; Emai
Online Appendix to Optimal Monetary and Fiscal Policy with a Zero Bound on Nominal Interest Rates Sebastian Schmidt∗ Goethe University Frankfurt August 20, 2012
Contents A The model
2
A.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
A.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
A.3 The government budget constraint . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
A.4 The efficient equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
A.5 Aggregate private sector behavior in the sticky-price economy . . . . . . . . . . . .
6
B Linear-quadratic approximation to expected utility of the representative household C Numerical algorithm
7 10
C.1 Policy function approximation under discretion . . . . . . . . . . . . . . . . . . . .
10
C.2 Policy function approximation under commitment . . . . . . . . . . . . . . . . . . .
12
D Equivalent consumption loss
15
∗ Mailing address: Goethe University, House of Finance, Grueneburgplatz 1, 60323 Frankfurt am Main; Email:
[email protected]; Phone: +49(0)69/79833811.
1
References
A
16
The model
The economy is represented by a small-scale New Keynesian rational expectations model. The private sector consists of households and firms and the public sector is formed by the government. Model inhabitants interact in bond, labor and goods markets. I begin with a description of the optimization problems faced by the private sector and then turn to the public sector.
A.1
Households
The preferences of the representative household are defined over a composite private consumption good Ct , labor nt (i) for all types i ∈ [0, 1], and a composite public consumption good Gt provided by the fiscal authority. Expected lifetime utility of the household reads1
E0
∞ X t=0
β
t
U (Ct , ξt ) −
Z
1
v (nt (i) , ξt ) di + g (Gt , ξt ) , 0
(1)
where ξt represents a vector of preference shocks and β ∈ (0, 1). For each possible value of ξ, U (·, ξ) and g (·, ξ) are increasing, concave functions, and v (·, ξ) is an increasing, convex function. Both, the private and the public composite consumption good, consist of a continuum of differentiated goods
Ct = Gt =
Z Z
1
Ct (i)
θt −1 θt
di
0 1
Gt (i)
θt −1 θt
t θ θ−1
di
0
t
t θ θ−1 t
(2) ,
(3)
where θt > 1, Ct (i) denotes private consumption of good i and Gt (i) denotes public consumption of good i. Expenditure minimization results in the following household demand function for consumption good i Ct (i) = 1
Pt (i) Pt
−θt
Ct ,
I am considering a cashless limiting economy in the sense of Woodford (2003).
2
(4)
where Pt (i) denotes the price of good i and Pt is the price index defined as
Pt =
Z
1
Pt (i)
1−θt
0
di
1 1−θ
t
.
(5)
The household enters each period t with one-period government bond holdings Bt−1 paying her Bt−1 /Pt units in terms of the composite consumption good. She supplies nt (i) units of labor to R1 firm i and earns total labor income 0 Wt (i) nt (i) di, where Wt (i) denotes the nominal wage rate
for labor of type i. The household owns the goods-producing firms, and therefore receives dividend payments Ψt . Lump-sum taxes Tt are paid to the fiscal authority. Also, the household buys Bt oneperiod nominal government bonds at price
1 Rt ,
where Rt ≥ 1 is the riskless gross nominal interest
rate. She consumes Ct units of the private consumption good, purchased at price Pt , and Gt units of the public consumption good. The representative household chooses Ct , nt (i) , ∀i ∈ [0, 1] , and Bt to maximize expected lifetime utility (1) subject to a sequence of budget constraints Bt ≤ Pt Ct + Rt for all t, where Pt Ct =
R1 0
Z
1
Wt (i) nt (i) di + Bt−1 − Tt + Ψt ,
Pt (i) Ct (i) di. Finally, aggregate labor supply is given by
Nt =
A.2
(6)
0
Z
1
nt (i) di.
(7)
0
Firms
Consumption goods are produced by a continuum of firms of measure one acting under monopolistic competition. Each firm i ∈ [0, 1] possesses a production technology
Yt (i) = At nt (i) ,
3
(8)
where At > 0 denotes a common technology shock. Total demand for good i consists of household demand and government demand
Yt (i) = Ct (i) + Gt (i) ,
(9)
where government demand for good i is given by
Gt (i) =
Pt (i) Pt
−θt
Gt .
(10)
Price-setting is described by the Calvo (1983) framework. Each period a fraction 1 − α of randomly chosen firms reoptimizes its price to maximize discounted profits while the remaining fraction α ∈ (0, 1) keeps its price unchanged. The maximization problem of a firm i that resets its price in period t reads max Pt (i)
∞ X
Et Qt,t+j αj Yt+j (i) [Pt (i) − M Ct+j (i)] ,
j=0
subject to (4), (9) and (10), where Qt,t+j = β j
UC (Ct+j )/Pt+j UC (Ct )/Pt
is the stochastic discount factor. The
has been used to substitute for solution to the cost minimization problem M Ct (i) = (1 − τ ) WAt (i) t the nominal wage, where M Ct (i) represent nominal marginal costs of firm i and τ denotes a wage subsidy paid by the government. This wage subsidy is set such that the distortions from monopolistic competition are eliminated in the non-stochastic steady state.
A.3
The government budget constraint
The government budget constraint reads
Pt Gt + Bt−1 + τ
where Pt Gt =
R1 0
Z
1
Wt (i) nt (i) di = 0
Bt + Tt , Rt
Pt (i) Gt (i) di. Without loss of generality, I assume that the supply of government
bonds is zero for all t. Hence, the budget constraint simplifies to
Pt Gt + τ
Z
1
Wt (i) nt (i) di = Tt . 0
4
A.4
The efficient equilibrium
Let the endogenous variables in the efficient equilibrium be denoted by a star superscript. The benevolent social planner maximizes the representative household’s expected lifetime utility (1) subject to (2), (3), (6), (7), (8), and (9). The optimality conditions read
Ct∗ (i) = Ct∗
(11)
G∗t (i) = G∗t
(12)
n∗t (i) = Nt∗
(13)
∗ , ξt+1 UC (Ct∗ , ξt ) = βRt∗ Et UC Ct+1
vn (Nt∗ , ξt ) UC (Ct∗ , ξt )
= At
(14) (15)
UC (Yt∗ − G∗t , ξt ) = gG (G∗t , ξt ) ,
(16)
for all i ∈ [0, 1], where Rt∗ = 1 + rt∗ denotes the ex-ante gross real interest rate in the efficient equilibrium. Using the resource constraint and log-linearizing (14) h i ∗ ˆ t∗ = rt∗ − r∗ = 1 Et Yˆt+1 ˆ ∗t+1 − γˆt+1 − Yˆt∗ − G ˆ ∗t − γˆt , R −G σ
(17)
where variables with a hat are expressed in percentage deviations from the non-stochastic steady (C,0) Y and state (government spending is expressed as a share of steady state output), σ −1 ≡ − UUCC C (C,0) U
(C,0)
Cξ γˆt ≡ − UCC (C,0)Y ξt .
Using the resource constraint and the production technology, and log-linearizing (15) yields ˆ ∗t + Γˆ Yˆt∗ = ΞAˆt + ΓG γt + (1 − Γ) υˆt , where Ξ ≡
1+η , σ −1 +η
Γ≡
σ −1 , σ −1 +η
η≡
vnn (n,0) vn (n,0) Y
v
(18)
(n,0)
, and υˆt ≡ − vnnnξ(n,0)Y ξt .
Finally, log-linearizing (16) ∗ ˆ t = σ −1 Yˆt∗ + ω −1 ψˆt − σ −1 γˆt , σ −1 + ω −1 G
5
(19)
gGξ (G,0) (G,0) where ω −1 ≡ − ggGG Y and ψˆt ≡ − gGG (G,0)Y ξt . Combining, (17), (18) and (19), the efficient G (G,0)
real rate of interest, the efficient level of output and the optimal level of government spending in the efficient equilibrium can be expressed as a function of the exogenous disturbances i h 1 ΞEt ∆Aˆt+1 + (1 − Γ) Et ∆ˆ υt+1 − Et ∆ˆ γt+1 − Et ∆ψˆt+1 σ + ω (1 − Γ) −1 h σ i σ ΞAˆt + (1 − Γ) υˆt − (1 − Γ) γˆt + ψˆt = 1+ −Γ ω ω σ ˆ = (1 + ΛΓ) ΞAt + (1 + ΛΓ) (1 − Γ) υˆt + (1 − Λ (1 − Γ)) Γˆ γt + ΛΓ ψˆt , ω
ˆ t∗ = R ˆ ∗t G Yˆt∗
where ∆ is the first difference operator, and Λ ≡ 1 +
A.5
σ ω
−Γ
−1
.
Aggregate private sector behavior in the sticky-price economy
Solving the optimization problem of the representative household and firms in the sticky-price economy, imposing market clearing and aggregating, private sector behavior can be summarized by the following two log-linear approximations ˆ gap + βEt π π ˆt = κ Yˆtgap − ΓG ˆt+1 + ut t ˆ t − Et π ˆ gap − σ R ˆ gap + Et Yˆ gap − Et G + dt , ˆ Yˆtgap = G t+1 t t+1 t+1 ˆ t∗ . Here, Yˆ gap = Yˆt − Yˆt∗ denotes the output gap, and where ut ≡ − (θ−1)(σκ −1 +η) θˆt , and dt ≡ σ R t ˆ gap = G ˆt − G ˆ ∗t denotes the government spending gap expressed as a share of steady state output. G t Following Woodford (2003), the output gap is defined as the difference between the actual level of output and the level of output consistent with the efficient equilibrium.2 The government spending gap is defined as the difference between actual government spending and the optimal level of government spending in the efficient equilibrium. 2
The difference between the efficient level of output and the output level in the distorted flexible-price economy accrues to the markup shock − (θ−1) σκ−1 +η θˆt . ( )
6
B Linear-quadratic approximation to expected utility of the representative household The period-utility function of the representative household reads
Vt = U (Ct , ξt ) −
Z
1
v (nt (i) , ξt ) di + g (Gt , ξt ) ,
(20)
0
In order for the second-order Taylor series approximation to expected household utility to describe an accurate welfare ranking of alternative policies, we have to substitute out consumption and hours of work in (20) prior to the expansion, see Woodford (2003), chapter 6. Using the production function (8) and the market clearing condition for the goods market we get
Vt = U (Yt − Gt , ξt ) −
Z
1
v (yt (i) /At , ξt ) di + g (Gt , ξt ) .
(21)
0
The period utility function can then be approximated as follows. Beginning with the first expression ˜ t + 1 UCC Y˜ 2 + 1 UCC G ˜ 2t − UCC Y˜t G ˜ t + UCξ Y˜t ξt U (Yt − Gt , ξt ) ≈ UC Y˜t − UC G t 2 2 ˜ t ξt + t.i.p., −UCξ G
(22)
˜ t ≡ Xt − X, where variables without a time subscript denote non-stochastic steady state values, X for some variable Xt , and t.i.p. captures terms independent of monetary and fiscal policy. First- and second-order derivatives of the utility function are evaluated at the steady state, respectively. We then ˜ t using substitute out Y˜t and G 1 Y˜t ≈ Y Yˆt + Y Yˆt2 2 ˆ 2t , ˜t ≈ Y G ˆt + 1 Y G G 2
7
where government spending is expressed in terms of its share of steady state real GDP. Equation (22) then becomes n 2 ˆ t + 1 1 − σ −1 Yˆt2 − 1 1 + σ −1 G ˆ t + σ −1 Yˆt G ˆt U (Yt − Gt , ξt ) ≈ UC Y Yˆt − G 2 2 o ˆ t γˆt + t.i.p. (23) +σ −1 Yˆt − G Next, we approximate the term in (21) capturing disutility of labor 1 1 2 1 y 1 v (yt (i) /At , ξt ) ≈ vn y˜t (i) + vnn 2 y˜t (i) − vn 2 + vnn 3 y˜t (i) A˜t A 2 A A A 1 +vnξ y˜t (i) ξt + t.i.p. o nA 1 ≈ vn y yˆt (i) + (1 + η) yˆt2 (i) − (1 + η) yˆt (i) Aˆt − η yˆt (i) υˆt 2 +t.i.p.,
(24)
where I used the steady state expression of the production function with A = 1. In order to relate the marginal utility of consumption to the marginal disutility of labor in the steady state, note that under monopolistic competition, we have vn θ−1 1 = , Uc θ (1 − τ )
(25)
where τ denotes a constant wage subsidy which is financed by a lump-sum tax. In the competitive equilibrium, it holds
vy Uc
= 1, hence, the steady state in the model is distorted, unless τ = 1θ . I will
adapt this assumption for the remainder of the analysis.3 Thus, integrating over i, (24) becomes Z
1 0
n h i 1 v (yt (i) /At , ξt ) di ≈ UC Y Ei yˆt (i) + (1 + η) (Ei yˆt (i))2 + vari yˆt (i) − 2Aˆt Ei yˆt (i) 2 o −η υˆt Ei yˆt (i) + t.i.p. 1 1 o n 1 ≈ UC Y Yˆt + (1 + η) Yˆt2 − 2Aˆt Yˆt + + η vari yˆt (i) − η υˆt Yˆt 2 2 θ +t.i.p.,
3
(26)
Since I consider optimal fiscal policy, it appears consistent that the policymaker chooses subsidy τ optimally.
8
where I have used Yˆt ≈ Ei yˆt (i) +
1 2
1−
1 θ
vari yˆt (i).
Finally, the last term in (21) is approximated by n o 2 ˆ t + 1 1 − ω −1 G ˆ t + ω −1 ψˆt G ˆ t + t.i.p. g (Gt , ξt ) ≈ gG Y G 2
(27)
Considering again the non-distorted steady state, optimal government spending must satisfy
UC = gG .
I then combine (23), (26) and (27) to 2 θ−1 + η o 2 n gap 1 σ gap −1 ˆ ˆ ˆ Vt ≈ − U C Y σ + η Y t − Γ Gt vari yˆt (i) . + Γ 1 + − Γ Gt + −1 2 ω σ +η Employing the demand function for individual good i, the variance of the log of the individual production level can be related to the variance of the log of price i vari log yt (i) = θ2 vari log Pt (i) .
Woodford (2003) shows that price dispersion vari log Pt (i) ≡ ∆t approximately evolves according to4 ∆t = α∆t−1 +
α π ˆ2, 1−α t
We can then express the quadratic approximation of discounted lifetime utility as follows 2 o 2 X tn 2 1 ˆ gap ˆ gap + λG G β π , ˆt + λ Yˆtgap − ΓG β Vt = − UC Y κ−1 θ σ −1 + η E E0 t t 2 t=0 t=0 (28) ∞ X
4
∞
t
See Woodford (2003), Proposition 6.3.
9
where (1 − α) (1 − αβ) −1 σ +η α (1 + ηθ) κ λ = θ σ λG = λΓ 1 + − Γ . ω κ =
C
Numerical algorithm
C.1
Policy function approximation under discretion
h Let Z = π ˆ
Yˆ
h i′ ˆ , and Ze = Z ′ G
i′ ˆ .5 I approximate Z by a linear combination of n basis R
functions ψi , i = 1, ..., n. In matrix notation
Z (u, d) ≈ CΨ (u, d) ,
where
cπ1
··· Y C= c1 · · · cG ··· 1
cπn cYn cG n
(29)
ψ1 (u, d) .. Ψ (u, d) = . ψn (u, d)
,
.
The coefficients cji , i = 1, 2, ..., n; j ∈ {π, Y, G}, are set such that (29) holds exactly at n selected collocation nodes Z X(k,:) = CΨ X(k,:) , for k = 1, ..., n, where
′
′
(ιd ⊗ u) X= ′ d ⊗ ιu
is a n × 2 matrix, and X(k,:) refers to the elements in row k of matrix X. The column vectors u and d contain the grid points of the cost-push shock and the efficient real interest rate shock, respectively. The vectors have length nh , h ∈ {u, d}. It holds n = nu · nd . ιh is a column vector of ones ˆ refer to the output gap and the government spending gap, respectively. The superscript gap has been Here, Yˆ and G eliminated for notational convenience. 5
10
with length nh . Here, I use linear spline basis functions, where the breakpoints coincide with the collocation nodes. The iterative solution algorithm to obtain the policy function approximations then works as follows. I start with an initial guess on the coefficient matrix C (0) . For fixed C (s) in iteration s, first update the expectations functions Eπ ˆ (s) X(k,:)
E Yˆ (s) X(k,:)
ˆ (s) X(k,:) EG
=
m X
̟l C(1,:) Ψ ρu X(k,1) + ǫ(l,1) , ρd X(k,2) + ǫ(l,2)
=
m X
̟l C(2,:) Ψ ρu X(k,1) + ǫ(l,1) , ρd X(k,2) + ǫ(l,2)
=
m X
(s) ̟l C(3,:) Ψ ρu X(k,1) + ǫ(l,1) , ρd X(k,2) + ǫ(l,2) ,
(s)
l=1
(s)
l=1
l=1
for k = 1, ..., n. A Gaussian quadrature scheme is used to discretize the normally distributed random variables, where ǫ is a m × 2 matrix of quadrature nodes and ̟ is vector of length m containing the weights. Assuming first, that the zero bound is not binding at any collocation node, the optimality conditions for the discretionary policy regime imply ′ Z˜ (s) X(k,:) = A−1 · B + A−1 · F · EZ (s) X(k,:) + A−1 · D · X(k,:) , for k = 1, ..., n, where
1 −κ κΓ 0 0 1 −1 σ A= κ λ −λΓ 0 λG 0 0 0 1−Γ
,
0 0 B = , 0 0
β 0 0 σ 1 −1 F = , 0 0 0 0 0 0
1 0 0 1 D= . 0 0 0 0
ˆ (s) X(k,:) < −r∗ , the update is replaced For those k for which the zero lower bound is violated, i.e. R
by
ˆ + Aˆ−1 · F · EZ (s) X(k,:) + Aˆ−1 · D · X ′ , Z˜ (s) X(k,:) = Aˆ−1 · B (k,:)
11
where
1
−κ
κΓ
0
0 1 −1 σ Aˆ = , (1 − Γ) κ (1 − Γ) λ λG − (1 − Γ) λΓ 0 0 0 0 1 I then update C (s+1) = Z (s) X(1,:)
until vec C (s+1) − C (s) ∞ < δ.
···
Z (s) X(n,:)
ˆ= B
0 0 0 −r∗
.
and continue the iteration procedure
For the baseline calibration, I set n = 4800, nu = 40, nd = 120, and m = 25. The collocation nodes are equally distributed with a support covering ± 4 unconditional standard deviations of the two exogenous state variables. I use MATLAB routines from the CompEcon toolbox of Miranda and Fackler (2002) to obtain the Gaussian quadrature approximation of the innovations to the cost-push shock and the efficient real rate of interest, and to evaluate the spline functions.
C.2
Policy function approximation under commitment
h Let Z = π ˆ
Yˆ
h i′ ˆ , and Ze = Z ′ G
i′ ˆ .6 I approximate Z by a linear combination of n basis R
functions ψi , i = 1, ..., n. In matrix notation
Z (u, d, φ−1 , µ−1 ) ≈ CΨ (u, d, φ−1 , µ−1 ) ,
(30)
where
cπ1
··· Y C= c1 · · · cG ··· 1
cπn cYn cG n
,
ψ1 (u, d, φ−1 , µ−1 ) .. Ψ (u, d, φ−1 , µ−1 ) = . ψn (u, d, φ−1 , µ−1 )
.
ˆ refer to the output gap and the government spending gap, respectively. The superscript gap has been Here, Yˆ and G eliminated for notational convenience. 6
12
The coefficients cji , i = 1, 2, ..., n; j ∈ {π, Y, G}, are set such that (30) holds exactly at n selected collocation nodes Z X(k,:) = CΨ X(k,:) , for k = 1, ..., n, where
(ιdφµ ⊗ u)′
ιφµ ⊗ d ⊗ ιu ′ X= ′ ιµ ⊗ φ−1 ⊗ ιud ′ µ−1 ⊗ ιudφ
′
and X(k,:) refers to the elements in row k of matrix X. The column vectors u, d, φ−1 and µ−1 contain the grid points of the cost-push shock, the efficient real interest rate shock and the two Lagrange multipliers, respectively. The vectors have length nh , h ∈ {u, d, φ, µ}. It holds n = nu · nd · nφ · nµ . ιh is a column vector of ones with length nh , and ιhq is a vector of ones with length nh · nq , where q ∈ {u, d, φ, µ}. Here, I use linear spline basis functions, where the breakpoints coincide with the collocation nodes. The iterative solution algorithm to obtain the policy function approximations then works as follows. I start with an initial guess on the coefficient matrix C (0) . For fixed C (s) in iteration s, I first update the guesses for the policy functions of the two Lagrange multipliers based on the optimality conditions derived in section 4 of the main paper φ(s) X(k,:) µ(s) X(k,:)
σ (s) X(k,4) + C(1,:) Ψ X(k,:) β 1 λG (s) X(k,4) − λC(2,:) Ψ X(k,:) + κφ(s) X(k,:) , β λG − λΓ (1 − Γ)
= X(k,3) − =
13
for k = 1, ..., n. Next, I update the expectation functions
Eπ ˆ
(s)
X(k,:)
E Yˆ (s) X(k,:) ˆ (s) X(k,:) EG
=
m X
=
m X
=
l=1
l=1 m X l=1
(s) ̟l C(1,:) Ψ ρu X(k,1) + ǫ(l,1) , ρd X(k,2) + ǫ(l,2) , φ(s) X(k,:) , µ(s) X(k,:)
(s) ̟l C(2,:) Ψ ρu X(k,1) + ǫ(l,1) , ρd X(k,2) + ǫ(l,2) , φ(s) X(k,:) , µ(s) X(k,:)
(s) ̟l C(3,:) Ψ ρu X(k,1) + ǫ(l,1) , ρd X(k,2) + ǫ(l,2) , φ(s) X(k,:) , µ(s) X(k,:) ,
for k = 1, ..., n. A Gaussian quadrature scheme is used to discretize the normally distributed random variables, where ǫ is a m × 2 matrix of quadrature nodes and ̟ is vector of length m containing the weights. We can now solve for updated policy function approximations Z (s) . First, assume that the zero bound is not binding at any collocation node. The updating then follows π ˆ (s) X(k,:) ˆ (s) X(k,:) G Yˆ (s) X(k,:) ˆ (s) X(k,:) R
= = = =
−1 2 κ κ + σ βλ κ λ Eπ ˆ (s) X(k,:) + X − X + X λ + κ2 λ + κ2 (k,1) λ + κ2 (k,3) βσ −1 (λ + κ2 ) (k,4) 1−Γ − X βλG (k,4) κ κ + σ −1 ˆ (s) X(k,:) − κ π ˆ (s) X(k,:) − X(k,3) + X ΓG λ λ βλσ −1 (k,4) ˆ (s) X(k,:) − Yˆ (s) X(k,:) − G ˆ (s) X(k,:) + X(k,2) σ −1 E Yˆ (s) X(k,:) − E G +E π ˆ (s) X(k,:) ,
ˆ (s) X(k,:) < −r∗ , the for k = 1, ..., n. For those k for which the zero lower bound is violated, i.e. R
update is replaced by ˆ (s) X(k,:) R ˆ (s) X(k,:) G Yˆ (s) X(k,:)
π ˆ (s)
= −r∗
ˆ (s) = G
=
1 (s) µ X(k,:) − X(k,4) β (s) ˆ (s) X(k,:) − E π ˆ (s) X(k,:) − σ R ˆ (s) X(k,:) X(k,:) − E G X(k,:) + E Yˆ
1−Γ λG
+X(k,2) ˆ (s) X(k,:) + βE π ˆ (s) X(k,:) + X(k,1) . X(k,:) = κ Yˆ (s) X(k,:) − ΓG
14
Let C
(s+1)
h = Z (s) X(1,:)
···
Z (s) X(n,:)
i
.
I then update C (s+1) = ζC
(s+1)
+ (1 − ζ) C (s) ,
(s+1)
− C (s) where ζ ∈ (0, 1], and continue the iteration procedure until vec C
< δ. ∞
For the baseline calibration, I set n = 10829, nu = 7, nd = 17, nφ = 7, nµ = 13, and m = 25.
The collocation nodes are equally distributed with a support covering ± 4 unconditional standard deviations of the two exogenous state variables. The support of endogenous state variables is chosen such that it covers all realizations of the two Lagrange Multipliers when simulating the model for the welfare analysis. I use MATLAB routines from the CompEcon toolbox of Miranda and Fackler (2002) to obtain the Gaussian quadrature approximation of the innovations to the cost-push shock and the efficient real rate of interest, and to evaluate the spline functions.
D
Equivalent consumption loss
A permanent reduction in private consumption C by the share χ reduces the utility of the representative agent by an amount equivalent to
1 1−β UC Cχ.
Using (28), we have
1 1 UC Cχ = UC Y κ−1 θ σ −1 + η L, 1−β 2 where L refers to the policy objective function introduced in the main paper. Solving for χ gives 1 χ = (1 − β) 2
C Y
−1
κ−1 θ σ −1 + η L.
Note, that the standard deviation of the efficient rate shock has been expressed in percentage points, hence losses obtained from the simulations have to be divided by 1002 , or, if we want to express the consumption loss in percentage terms, by 100.
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References Calvo, Guillermo. (1983). “Staggered contracts in a utility-maximizing framework.” Journal of Monetary Economics 12, 383-398. Miranda, Mario J., and Paul L. Fackler. (2002). Applied Computational Economics and Finance. The MIT Press. Woodford, Michael. (2003). Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton: Princeton University Press.
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