Fiscal and Monetary Policy Interactions with Low Interest Rates

Fiscal and Monetary Policy Interactions in a Model with Low Interest Rates Proof of Proposition 1. — The entrepreneur’s objective is to solve the following dynamic programming problem: (A1) Vt (Kjt−1 , Bjt−1 , Djt−1 , εjt ) = max {Ijt ,Djt ,Bjt } Cjt + βEt Λt+1 Vt+1 (Kjt , Bjt , Djt , εjt+1 ) , Λt subject to (A2) (A3) (A4) (A5) Kjt = (1 − δ)Kjt−1 + εjt Ijt , Bjt ≥ −µKjt−1 , Rt−1 Rt−1 Cjt + Ijt + Bjt + Djt = Rkt Kjt−1 + Bjt−1 + Djt−1 , Πt Πt Cjt ≥ 0. We conjecture that the value function takes the following form (A6) Vt (Kjt−1 , Bjt−1 , Djt−1 , εjt ) = ϕkt (εjt ) Kjt−1 + ϕbt (εjt ) Bjt−1 + ϕdt (εjt ) Djt−1 , where ϕit (εjt ), i ∈ {k, b, d}, satisfy (A7) (A8) (A9) Z Λt+1 ϕkt+1 (ε)dF (ε), = βEt Λt Z Λt+1 ϕbt+1 (ε)dF (ε), 1 = βEt Λt Z Λt+1 1 = βEt ϕdt+1 (ε)dF (ε). Λt qtk Substituting (A2), (A4), and the above conjecture into the Bellman equation (A1), we have (A10) Vt (Kjt−1 , Bjt−1 , Djt−1 , εjt )   Z Λt+1 k = max Rkt + (1 − δ)βEt ϕt+1 (ε)dF (ε) Kjt−1 Λt {Ijt ,Djt ,Bjt }   Z Λt+1 Rt−1 Rt−1 + Bjt−1 + Djt−1 + βEt ϕkt+1 (ε)dF (ε)εjt − 1 Ijt Πt Πt Λt     Z Z Λt+1 Λt+1 b d + βEt ϕt+1 (ε)dF (ε) − 1 Bjt + βEt ϕt+1 (ε)dF (ε) − 1 Djt . Λt Λt Optimal choices of Bjt and Djt imply that (A8) and (A9) must hold in equilibrium. Otherwise, all entrepreneurs will either save or borrow at the same time, contradicting the market-clearing conditions for bonds. Since Ijt ≥ 0 and Cjt ≥ 0, it follows that Ijt = 0 if εjt < 1/qtk ≡ ε∗t ; but the firm makes as much investment as possible so that Cjt = 0 if εjt > ε∗t . It follows from (A4) that when εjt > ε∗t , we have (A11) (A12) Ijt = −Bjt − Djt + Rkt Kjt−1 + Djt = 0, Rt−1 Rt−1 Bjt−1 + Djt−1 , Πt Πt Bjt = −µKjt−1 . Consider first the case where εjt < ε∗t and Ijt = 0. The entrepreneurs are indifferent between borrowing and saving. Substituting the decision rules into (A10) and reorganizing yield Vt (Kjt−1 , Bjt−1 , Djt−1 , εjt )   Rt−1 Rt−1 Bjt−1 + Djt−1 . = max Rkt + (1 − δ)qtk Kjt−1 + Πt Πt {Ijt ,Djt ,Bjt } Notice that (A8) and (A9) ensure that the indeterminacy of Bjt and Djt does not affect the value function. Matching the coefficients, we have ϕkt (εjt ) = Rkt + (1 − δ)qtk , Rt−1 ϕbt (εjt ) = ϕdt (εjt ) = . Πt Next we consider the case where εjt > ε∗t . Substituting (A11) and (A12) into (A10) and reorganizing yield Vt (Kjt−1 , Bjt−1 , Djt−1 , εjt )      = max Rkt + (1 − δ)qtk + Rkt qtk εjt − 1 − µ 1 − qtk εjt Kjt−1 {Ijt ,Djt ,Bjt } + Rt−1  k  Rt−1  k  qt εjt Bjt−1 + qt εjt Djt−1 . Πt Πt Matching the coefficients yields      εjt εjt k = Rkt 1 + −1 + (1 − δ)qt + µ −1 , ε∗t ε∗t    εjt Rt−1  k  Rt−1 qt εjt = 1+ − 1 . ϕbt (εjt ) = ϕdt (εjt ) = Πt Πt ε∗t ϕkt (εjt ) Combining the two cases above, we have      εjt εjt k = Rkt 1 + max − 1, 0 + (1 − δ)qt + µ max − 1, 0 , ε∗t ε∗t    εjt Rt−1 1 + max − 1, 0 , ϕbt (εjt ) = ϕdt (εjt ) = Πt ε∗t ϕkt (εjt ) for εjt ∈ [εmin , εmax ] . Substituting these two equations into (A7), (A8) and (A9), we obtain (13) and (14). Finally, for the entrepreneur’s objective to be finite, the value function must satisfy the following condition by the Bellman equation (A1): lim Et i→∞ β i Λt+i Vt+i (Kj,t+i−1 , Bj,t+i−1 , Dj,t+i−1 , εj,t+i ) = 0. Λt Using equations (A6)-(A9) we can derive the transversality condition (16). Proof of Lemma 1. — To simplify notations, we define (A13) Mt+1 =  βΛt+1 βΛt+1  Dt−1 Rt−1 l l , Mt+1 = 1 + qt+1 , xt = . Λt Λt Πt Then we can rewrite (34) as   l − Mt+1 xt+1 . xt = St + Et Mt+1 xt+1 + Et Mt+1 Q.E.D. Leading the above equation by one period and multiplying by Mt+1 , we obtain   l Mt+1 xt+1 = Mt+1 St+1 + Et+1 Mt+1 Mt+2 xt+2 + Et+1 Mt+1 Mt+2 − Mt+2 xt+2 . Following similar procedures recursively until period t + T, we have Mt+1 Mt+2 ...Mt+T xt+T = Mt+1 Mt+2 ...Mt+T St+T + Et+T Mt+1 Mt+2 Mt+T +1 xt+T +1   l +Et+T Mt+1 Mt+2 ...Mt+T +1 Mt+T +1 − Mt+T +1 xt+T +1 . Taking conditional expectations Et on the two sides of above system of T + 1 equations and using (A13), we obtain (A14) T T X X Dt+i Rt+i β T +1 Λt+1+T Dt+T Rt+T Dt−1 Rt−1 β i Λt+i β i+1 Λt+i+1 l = Et St+i +Et qt+i+1 +Et . Πt Λt Λt Πt+i+1 Λt ΠT +1 i=0 i=0 Summing over j in (16) and using the market-clearing conditions, we have lim Et i→∞  β i Λt+i  k qt+i Kt+i + Dt+i = 0. Λt k > 0, we have Since Kt+i > 0 and qt+i (A15) lim Et i→∞ β i Λt+i Dt+i = 0. Λt l Since qt+1+T ≥ 0, it follows from (14) that 0 ≤ Et β T Λt+T Λt+1+T Dt+T Rt+T β T +1 Λt+1+T Dt+T Rt+T l ≤ Et β (1 + qt+1+T ) Λt Πt+T +1 Λt Λt+T Πt+T +1 T β Λt+T βΛt+1+T Dt+T Rt+T β T Λt+T l Et+T (1 + qt+1+T ) = Et Dt+T . = Et Λt Λt+T Πt+T +1 Λt Thus, lim Et T →∞ β T +1 Λt+1+T Dt+T Rt+T = 0. Λt Πt+T +1 Taking limit in (A14) as T → ∞ gives (36). Q.E.D. Proof of Lemma 2. — Taking derivative of Rk (ε∗ ) in (39) and reorganizing yields (A16) µ ∂Rk (ε∗ ) = ∗ ∂ε R εmax ε∗ εdF (ε) − (β −1 (1 + g) − 1 + δ)F (ε∗ ) . hR i2 εmax ∗ ) dF (ε) max (ε, ε εmin The numerator in (A16) is strictly decreasing in ε∗ , with the maximum and the minimum being µE [ε] ≥ 0 and −(β −1 (1 + g) − 1 + δ) < 0, respectively. Hence, by the intermediate value theorem, there exists a unique threshold εk ∈ [εmin , εmax ] such that Z εmax µ εdF (ε) − (β −1 (1 + g) − 1 + δ)F (εk ) = 0. εk And it follows that ∂Rk (ε∗ )/∂ε∗ > 0 if ε∗ < εk ; ∂Rk (ε∗ )/∂ε∗ ≤ 0 if ε∗ ≥ εk . Moreover, we have εk = εk (µ) strictly increasing and limµ→0 εk = εmin . Q.E.D. Proof of Lemma 3. — By Lemma 2, on [εk , εmax ], Rk (ε∗ ) is decreasing and thus Φ(ε∗ ) is increasing. By (39), we compute (A17) Rk (εk ) = R εmax εdF (ε) + µεk (1 − F (εk )) R εmax . εk F (εk ) + εk εdF (ε) (1 + g)/β − 1 + δ − µ By Lemma 1, we have εk ∂Rk (ε∗ ) |εk = 0. ∂ε∗ Thus, Z (A18) εmax µ εdF (ε) − (β −1 (1 + g) − 1 + δ)F (εk ) = 0. εk Using this equation, we can eliminate F (εk ) in (A17) to obtain Rε (1 + g)/β − 1 + δ − µ εkmax εdF (ε) R εmax Rk (εk ) = . εdF (ε) εk Substituting this expression into (40) yields (β −1 − 1)(1 + g) Φ(εk ) = − R εmax < 0. εdF (ε) εk Since Φ(εmax ) = +∞ and Φ(εk ) < 0 and Φ is increasing on [εk , εmax ], it follows from the intermediate value theorem that there exists a unique value εl ∈ (εk , εmax ) such that Φ(εl ) = 0. By (39), we have Rk (εmin ) = (1 + g)/β − 1 + δ − µ(E [ε] − εmin ) . E [ε] Substituting this expression into (40) yields Φ(εmin ) = − (β −1 − 1)(1 + g) + µεmin < 0. Eε When µ = 0, we have (β −1 − 1)(1 + g) < 0. Eε By (A18), εk is an implicit continuous function of µ and εk → εmin as µ → 0. By continuity, for sufficiently small µ ≥ 0, we have Φ(ε∗ ) < 0 for ε∗ ∈ [εmin , εk ] . Q.E.D. Φ(εmin ) = − Proof of Proposition 2. — By the assumption and Lemma 3, the investment cutoff ε∗ in any steady state must satisfy ε∗ ≥ εk . Since Φ(εl ) = 0, by (40) and setting ε∗ = εl , we have d = 0. Thus (41) or (43) is satisfied for s = 0. We deduce that ε∗ = εl is the steady-state cutoff for s = 0. This is the only steady state with d = 0 because Φ(ε∗ ) increases with ε∗ ∈ [εk , εmax ] by Lemma 2. Suppose that there is another steady state with d > 0 if Rr (εl ) > 1+g. Then (41) implies that Rr (ε∗ ) = 1 + g for s = 0. Since Rr (ε∗ ) increases with ε∗ and since Rr (εl ) > 1 + g, we must have the steady state cutoff ε∗ < εl . Since Rk (ε∗ ) decreases with ε∗ on (εk , εl ) , it follows (40) that Φ increases with ε∗ on (εk , εl ). Thus we have Φ (ε∗ ) < Φ (εl ) = 0 for ε∗ ∈ (εk , εl ), contradicting equation (40) as d > 0 and Rr > 0. If Rr (εl ) < 1 + g, we show that there is another steady state with d > 0. It follows from (41) we must have Rr = 1 + g. Since Rr (ε∗ ) is a continuous and increasing function and since Rr (εl ) < 1 + g and Rr (εmax ) = (1 + g)/β > 1 + g, by the intermediate value theorem there is a unique solution ε∗ = εh ∈ (εl , εmax ) such that Rr (ε∗ ) = 1 + g. We then have Rr = Rr (εh ) = 1 + g in the steady state. It follows from (40) that Rr d/k = Φ(εh ). Q.E.D. Proof of Proposition 3:. — Recall that εl satisfies Φ(ε∗l ) = 0. Total differentiating this equation and reorganizing yield  ∂Rk (εl ) ∂µ R ε max εdF (ε) εl dεl = − ∂R (ε ) R ε . max k l dµ εdF (ε) − (µ + Rk (εl ))εl F ′ (εl ) ∂εl 1+ εl By (39), we have 1 + ∂Rk (εl )/∂µ > 0 and that ∂Rk (εl )/∂εl < 0. Thus we have dεl /dµ > 0. By (38), Rr (ε∗ ) increases with ε∗ . It follows that both εl and Rr (εl ) increase with µ. Q.E.D. Proof of Proposition 4:. — By Lemma 2, for a sufficiently small µ ≥ 0, we only need to consider steady-state the investment cutoffs in [εk , εmax ] . It follows from Lemma 1 that Rk (ε∗ ) is a decreasing function of ε∗ ∈ [εk , εmax ] . Thus Φ(ε∗ ) increases with ε∗ ∈ [εk , εmax ] by (40). We also know that Rr (ε∗ ) increases with ε∗ ∈ [εmin , εmax ] . By (43) we have (A19)   1+g αpw Ψ (ε∗ ) = 1 − r ∗ Φ(ε∗ ). R (ε ) Rk (ε∗ ) Thus Ψ (ε∗ ) is a product of three increasing functions on [εk , εmax ] . Since Φ(εl ) = 0 and Φ(ε∗ ) < Φ(εl ) = 0 for ε∗ ∈ [εk , εl ] , we will focus on the region [εl , εmax ] as equation (40) must hold with Rr d ≥ 0. On this region Φ(ε∗ ) ≥ 0. Suppose that Rr (εl ) > 1 + g. Then we have 1− 1+g 1+g >1− r >0 r ∗ R (ε ) R (εl ) for ε∗ > εl > εk . Since Φ(εl ) = 0, we have Φ(ε∗ ) > 0 for ε∗ > εl . Thus, as a product of three positive increasing functions on [εl , εmax ], Ψ (ε∗ ) increases with ε∗ ∈ [εl , εmax ] . Since Ψ (εl ) = 0 and Ψ (εmax ) = +∞, it follows from the intermediate value theorem that there exists a unique solution εp ∈ (εl , εmax ) to equation (43). Then Rr (εp ) > Rr (εl ) > 1 + g. Suppose that Rr (εl ) < 1 + g. Then Proposition 2 shows that there exists εh ∈ (εl , εmax ) such that Rr (εh ) = 1 + g and Ψ (εh ) = 0. Thus Rr (ε∗ ) > 1 + g for ε∗ ∈ [εh , εmax ] by the monotonicity of Rr (ε∗ ) . It follows that Ψ (ε∗ ) increases with ε∗ ∈ [εh , εmax ] because Ψ (ε∗ ) is a product of three positive increasing functions on [εh , εmax ] . The intermediate value theorem implies that there exists a unique cutoff εp ∈ (εh , εmax ) such that Ψ (εp ) = s/y > 0. Then we have Rr (εp ) > Rr (εh ) = 1 + g. For ε∗ ∈ (εl , εh ) , we have Rr (ε∗ ) < Rr (εh ) = 1 + g and thus Ψ (ε∗ ) < 0. There cannot exist a steady state with s/y > 0 by (43). Q.E.D. Proof of Proposition 5:. — As in the proof of Proposition 4, for a sufficiently small µ ≥ 0, we only need to consider the region [εl , εmax ] for the steady state investment cutoff. By assumption, Rr (εl ) < 1 + g. By the proof of Proposition 4, Ψ (ε∗ ) is positive and increases with ε∗ ∈ (εh , εmax ]. But Ψ (ε∗ ) is negative for ε∗ ∈ (εl , εh ) . Moreover, Ψ (εh ) = Ψ (εl ) = 0. Let s be defined as in the proposition. By the intermediate value theorem, for any s/y ∈ (−s, 0) , there exist at least two steady-state cutoffs ε∗l and ε∗h with εl < ε∗l < ε∗h < εh such that (43) holds. Q.E.D. Appendix B. Detrended Equilibrium System The model exhibits long-run growth. To find a steady state and to study the dynamics around a steady state, we need to detrend the model around a long-run growth path. We consider transformations of xt = Xt /At for any variable Xt ∈ {Kt , Dt , St , Yt , Wt , Ct , It }. For the marginal utility, we denote λt = At Λt . Then the detrended system can be summarized by the following 20 equations in 20 variables {Rkt , kt , Rt , qtk , qtl , ε∗t , dt , τt , Πt , p∗t , Γat , Γbt , ∆t , wt , λt , pwt , Nt , yt , ct , it }, where {R−1 , ∆−1 , d−1 , k−1 } and {zmt , zτ,t , Gat } are given exogenously: 1) The capital return, α−1 1−α Rkt = α (1 + g)1−α pwt kt−1 Nt . (B1) 2) Evolution of capital, (B2)   Z εmax Rt−1 (1 + g)kt = (1 − δ)kt−1 + (µ + Rkt ) kt−1 + dt−1 εdF (ε). Πt ϵ∗t 3) The nominal interest rate, (B3) 1=  λt+1 Rt  β l Et 1 + qt+1 . 1+g λt Πt+1 4) Tobin’s Q, (B4)   β λt+1 λt+1 k βµ λt+1 l β l + Et Rkt+1 1 + qt+1 Et qt+1 (1 − δ) + Et q . qtk = 1+g λt 1+g λt 1+g λt t+1 5) Liquidity premium, (B5) qtl Z εmax = ε∗t   qtk ε − 1 dF (ε). 6) Investment cutoff, ε∗t = 1/qtk . (B6) 7) Government budget constraint, (B7) Rt−1 dt−1 = τt − Gat + dt . Πt 1 + g 8) Fiscal policy rule, (B8) (τt − τ ) /y = ϕd (dt−1 − d)/y + zτ,t . 9) Monetary policy rule,  (B9) Rt = R Πt Π  ϕπ exp(zmt ). 10) Pricing rule, p∗t = (B10) σ Γat . σ − 1 Γbt 11) Numerator in the pricing rule, (B11) Γat  = λt pwt yt + βξEt Πt+1 Π σ Γat+1 . 12) Denominator in the pricing rule, (B12) Γbt = λt yt + βξEt  Πt+1 Π σ−1 Γbt+1 . 13) Evolution of inflation, (B13) # 1 "   1−σ Π 1−σ ∗1−σ . 1= ξ + (1 − ξ) pt Πt 14) Price dispersion, (B14) ∆t = (1 − ξ)p∗−σ t  +ξ Π Πt −σ ∆t−1 . 15) Labor demand, (B15) α wt = (1 − α) (1 + g)−α pwt kt−1 (Nt )−α . 16) Labor supply, (B16) wt = ψ . λt 17) Marginal utility, (B17) λt = 1/ct . 18) Aggregate output, −α α yt = ∆−1 kt−1 (Nt )1−α . t (1 + g) (B18) 19) Aggregate investment, (B19)   Rt−1 (1 + g)it = (µ + Rkt ) kt−1 + dt−1 (1 − F (ε∗t )) . Πt 20) Resource constraint, (B20) ct + it + Gat = yt . For the real version of our model, we set pwt = 1 − 1/σ, Πt = ∆t = 1, and Rt = Rtr in the above system and the detrended equilibrium system consists of 13 equations (B1)-(B7), and (B15)-(B20) in 13 variables {Rt , Rkt , λt , ε∗t , qtk , qtl , wt , dt , kt , Nt , yt , ct , it }. Appendix C. Steady-State System We study the nonstochastic steady state of the detrended system with s/y and Π being exogenously given. Define real interest rate as Rr = R/Π. Let variables without time subscripts denote their steady state values. By the steady-state version of (B13), we obtain p∗ = 1. It then follows from (B14) that ∆ = 1. Combining (B10), (B11), and (B12), we have pw = 1 − 1/σ, Γa = pw Γb = pw λy/(1 − βξ). With w and λ being eliminated by using (B16) and (B17), and noting that zτ = zm = 0, we obtain a steady-state system of 11 equations in 11 variables {Rr , Rk , ε∗ , q k , d, k, N , y, c, i, q l } : 1) The capital return, (C1) Rk = (1 − 1/σ) α (1 + g)1−α k α−1 N 1−α . 2) Evolution of capital, r (C2) (g + δ)k = ((µ + Rk ) k + R d) Z εmax εdF (ε). ϵ∗ 3) Nominal interest rate, (C3) 1=   β Rr 1 + q l . 1+g 4) Tobin’s Q, (C4)   β β β k l q = Rk 1 + q + q (1 − δ) + µq l . 1+g 1+g 1+g k 5) Liquidity premium, Z l (C5) εmax q =   q k ε − 1 dF (ε). ε∗ 6) Investment cutoff, ε∗ = 1/q k . (C6) 7) Government budget constraint,  (C7) Rr −1 1+g  d s = . y y 8) Labor demand, (C8) ψc = (1 − 1/σ) (1 − α) (1 + g)−α k α N −α . 9) Aggregate output, y = (1 + g)−α k α N 1−α . (C9) 10) Aggregate investment, (C10) (1 + g)i = [(µ + Rk ) k + Rr d] (1 − F (ε∗ )) . 11) Resource constraint, (C11) c + i + Ga = y. As discussed in Section II, the investment cutoff ε∗ can be solved for by combining (C3), (C4), (C5), (C6), and (C7). Given the inflation target Π, we obtain the nominal interest rate R = Rr (ε∗ )Π. By (C6), q k = 1/ε∗ . By (C5), we derive q l . With Rr = Rr (ε∗ ), Rk = Rk (ε∗ ) and Rr d/k = Φ(ε∗ ), we can determine y/k from Rk = (1 − 1/σ)(1 + g)αy/k and d/k = Φ(ε∗ )/Rr (ε∗ ). Noticing that equation (C10) pins down the value of i/k, we can derive i/y = (i/k)/(y/k). Using the resource constraint and the exogenously given Ga /y by calibration, we obtain c/y = 1 − Ga /y − i/y. Dividing (C8) over (C9) and reorganizing yield the steady-state value of labor: N = (1 − 1/σ) 1−α c /( ). ψ y Then by noting that Rk = Rk (ε∗ ) = (1 − 1/σ)α(1 + g)1−α k α−1 N 1−α , we can solve for k. Combining with the ratios given above, we can then determine y, d, i, c, w, and s. Finally, we have Γa = (1 − 1/σ)Γb = (1 − 1/σ)(y/c)/(1 − βξ). Appendix D. Linearized System Let x bt = (xt − x)/x denote the log deviation from steady state for any variable xt except for the surplus st and public debt dt . For these two variables we consider level deviation relative to output, det = (dt − d) /y and τet = (τt − τ ) /y, instead of log deviation, because d may be zero and τ may be negative. By standard linearization of the DNK model, we know the deviation of price dispersion b t is of second-order. Thus we ignore the law of motion for the price dispersion. Moreover, ∆ the supply block can be summarized by the New-Keynesian Phillips curve. Hence, we can b at , and Γ b bt . Then the linearized model can be summarized by a system further eliminate pb∗t , Γ bt , N bkt , b bt , qbk , qbl , εb∗ , det , τet , Π b t , pbwt , w bt , ybt , b of 16 equations in 16 variables, R kt , R bt , λ ct , t t t b−1 , de−1 , and b b at are exogenous and bit , where R k−1 are predetermined, and zτ,t , zmt , and G AR(1) processes: 1) The capital return, (D1) bkt = pbwt + (α − 1)b bt . R kt−1 + (1 − α)N 2) Evolution of capital, (D2)   Rr d ∗ 2 ∗ ∗ b b (1 + g)kt =(1 − δ)kt−1 − µ + Rk + ε f (ε )b εt k   Z εmax  Rr y rd  R bkt + bt−1 − Π bt + εdF (ε) (µ + Rk )b kt−1 + Rk R + R det−1 . k k ε∗ 3) Nominal interest rate,   bt − λ bt+1 − bt − Et Π b t+1 = Et λ R (D3) ql l Et qbt+1 . 1 + ql 4) Tobin’s Q, (D4) β Rk (1 + q l ) b Et Rkt+1 1+g qk β (µ + Rk ) q l β l k + Et qbt+1 + (1 − δ)Et qbt+1 . k 1+g 1+g q   bt+1 − λ bt + qbtk =Et λ 5) Liquidity premium, R εmax qbtl (D5) =− ε∗ εdF q l ε∗ εb∗t . 6) Investment cutoff, εb∗t = −b qtk . (D6) 7) Government budget constraint, (D7) τet + det =  Ga b Rr e Rr d  b bt . Gat + dt−1 + Rt−1 − Π y 1+g 1+gy 8) Fiscal policy rule, (D8) τet = ϕd det−1 + zτ,t . 9) Monetary policy rule, b t + zmt . b t = ϕπ Π R (D9) 10) New-Keynesian Phillips curve, (D10) b t+1 + κb b t = βEt Π pwt , Π where κ = (1 − ξ)(1 − βξ)/ξ. 11) Labor demand, (D11) bt . w bt = pbwt + αb kt−1 − αN 12) Labor supply, bt . w bt = −λ (D12) 13) Marginal utility, bt = −b λ ct . (D13) 14) Aggregate output, (D14) bt . ybt = αb kt−1 + (1 − α)N 15) Aggregate investment, (D15)   rd rd ry ib R R R ∗ bkt + bt−1 − bt + (1 + g) it = [1 − F (ε )] (µ + Rk )b kt−1 + Rk R R Π det−1 k k k k   Rr d − µ + Rk + f (ε∗ )ε∗ εb∗t . k 16) Resource constraint, (D16) c b ct + y ib Ga b it + Gat = ybt . y y Appendix E. Additional Results In this appendix we present some additional results not reported in the main text. First, Figure E1 shows the determinacy region for the steady state in which the interest rate is higher than the economic growth rate. We set the long-run s/y = 4.45% and fix other parameter values as in Table 1. The implied debt to GDP ratio is 120%. Next we study welfare for different policy parameter mixes ϕd ∈ [−0.2, 0.2] and ϕπ ∈ [0, 3] given adverse financial shocks as in Section IV. We consider parameter values in the set such that the model admits a unique equilibrium. Figures E2, E3, and E4 present the welfare losses in terms of the consumption equivalent relative to the steady state without the financial shock for the equilibria around the three steady states, respectively. We find that the welfare loss is the smallest when ϕd = −0.2 and ϕπ = 0 in regime F. Figure E1. : Determinacy region for the steady state with Rr > 1 + g. Figure E2. : Welfare loss in response to financial shocks under different policy mixes around steady state L. Figure E3. : Welfare loss in response to financial shocks under different policy mixes around steady state H. Figure E4. : Welfare loss in response to financial shocks under different policy mixes around the steady state with Rr > 1 + g. Appendix F. The Standard New Keynesian Block Retailers are monopolistically competitive. Their role is to introduce nominal price rigidities. In each period t they buy intermediate goods from entrepreneurs at the real price pwt and sell good j at the nominal price Pjt . Intermediate goods are transformed into final goods according to the CES aggregator Z Yt = 1 Yjt σ−1 σ σ  σ−1 dj , σ > 1. 0 Thus retailers face demand given by  (F1) Yjt = Pjt Pt −σ Yt , where the price index is given by Z Pt ≡ (F2) 1  1−σ 1 Pjt 1−σ dj . 0 Aggregating equation (F1) yields aggregate output equation (31). To introduce price stickiness, we assume that each retailer is free to change its price in any period only with probability 1−ξ, following Calvo (1983). To introduce trend inflation, we follow Erceg, Henderson and Levin (2000) and assume that whenever the retailer is not allowed to reset its price, its price is automatically increased at the steady-state inflation rate. The retailer selling good j chooses the nominal price Pjt∗ in period t to maximize the discounted present value of real profits max ∗ Pt ∞ X " k ξ Et k=0 β k Λt+k Λt Πk Pjt∗ − pw,t+k Pt+k # ! ∗ Yjt+k , subject to the demand curve ∗ Yjt+k = Πk Pjt∗ Pt+k !−σ Yt+k , k ≥ 0, where Π denotes the steady-state inflation target. We use the household intertemporal marginal rate of substitution as the stochastic discount factor because retailers must hand in all profits to households who are the shareholders. The first-order condition gives the pricing rule Pjt∗ = Pt∗ P∞ k σ Y k −σ σ Et k=0 (βξ) Λt+k pw,t+k Pt+k t+k (Π ) ≡ P∞ σ−1 σ − 1 Et k=0 (βξ)k Λt+k Pt+k (Πk )1−σ Yt+k for all j. Let p∗t = Pt∗ /Pt . We can then write the pricing rule in a recursive form as follows p∗t = σ Γat , σ − 1 Γbt where Γat Γbt  Πt+1 σ a Γt+1 , = Λt pwt Yt + βξEt Π   Πt+1 σ−1 b = Λt Yt + βξEt Γt+1 . Π  It follows from (F2) and Calvo price setting that "   # 1 1−σ Π 1−σ 1= ξ + (1 − ξ) p∗1−σ . t Πt