Eco 554, part 1, Spring 2005 Lars Svensson 3/6/05

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1 Eco 554, part, Spring o7_05.tex Lars Svensson 3/6/05 Monetary policy in a liquidity trap for an open economy The zero (lower) bound for nominal interest rate, ĩ t 0. Japan (Bernanke, Krugman, Posen, Meltzer, McCallum) A lost decade Continued stagnation, deflation Zero interest rate, deflation, positive real interest rate, recession, output < potential output, high unemployment, expansionary fiscal policy, big budget deficit, high public debt Objective for macro policy: Escape from liquidity trap and recession, jump-start economy, get to small positive inflation rate, eliminate output gap and unemployment gap Objective for micro policy: Reforms, financial-sector clean-up (easier when out of recession). US (Bernanke 02) c 2005 Lars E.O. Svensson. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personal use or distributed free. Liquidity trap, real balances in excess of satiation level, ĩ =0(if ĩ m 0 interest rate on money, ĩ =ĩ m ) Representative consumer, m τ M τ P τ real balances, U(C, m) concave Satiation level m s (C) X E t τ=t ( δ)δ τ t U(C τ, m τ ) U m (C, m) > 0 for m < m s (C) U m (C, m) = 0 for m m s (C) m s (C) min{ m 0 U m (C, m) =0} m U C m (C, m) = 0 for m > m s (C) Money demand ĩ>0: ĩ t =0: U m (C, m) U C (C, m) = ĩ +ĩ m = g(c, ĩ) lim g(c, ĩ) = ĩ 0+ ms (C) m m s (C) 2

2 Period home-country private-sector budget constraint: (Bt h and B f t domestic- and foreign-currency one-period zero-coupon bonds, respectively, paying unit of domestic or foreign currency in period t +; Bt h issued by home government; ĩ t and ĩ t domestic- and foreign-currency interest rates, respectively; S t exchange rate (domestic currency/foreign currency)) P t C t + M t + B h t + S t +ĩ t +ĩ B f t = P t Y t P t T t + M t + Bt h + S t B f t t P t C t = Pt h Ct h + P f t C f t P t = P (Pt h,p f t ) C t = C(Ct h,c f t ) ĩ t =0: P t C t +(M t + Bt h )+S t +ĩ B f t = P t Y t P t T t +(M t + Bt )+S h t B f t t ĩ t =0 Nominal bonds and money perfect substitutes in period t (as long as M t P t m s (C t ) 3 Intertemporal private-sector budget constraint (perfect foresight), W t M t + Bt h + S t B f t (nominal financial wealth in the beginning of period t) X µ X D t,τ P τ C τ + ĩτ M τ = D t,τ P τ (Y τ T τ )+W t. +ĩ τ τ=t Uncovered interest-rate parity Fisher equation Nominal (market) discount factor τ=t = +ĩ t +ĩ t = +ĩ t + r t S t S t+ P t P t+ Real (market) discount factor D t,t = D t,τ Yj j=t +ĩ j (τ>t) d t,t = d t,τ = Yj j=t + r j = D t,τ P τ P t 4

3 Assumptions (in nominal form): No-Ponzi condition W t + X D t,τ P τ (Y τ T τ ) 0 Finite present value of after-tax income X D t,τ P τ (Y τ T τ ) < Transversality condition τ=t τ=t lim τ D t,τw τ =0 5 Assume ĩ τ =0for τ t Fisher equation and UIP P t+ = P t, S t+ = + r t +ĩ S t t D t,τ = Transversality condition lim W τ =0 τ Assume () B f τ bounded: Bf τ < b < (2) r τ > 0 lim τ P τ =0 (3) ĩ τ > 0 lim τ S τ =0 (4) Home government nominal liabilities bounded below, M τ + Bτ h > B >0 lim τ W τ =lim τ M τ + Bτ h + S τ B f τ 6=0 Not ĩ τ =0for all τ 0 Hence: Only temporary liquidity trap. Else, government insolvent (Woodford 03 Chap. 2; Benhabib, Schmitt-Grohé, Uribe 02) 6

4 TheMagicoftheExchangeRate: Optimal Escape from a Liquidity Trap in Small and Large Open Economies Plan. Intro 2. A world of two large countries 3. A liquidity trap in a simple case of a small open economy 4.Themagicoftheexchangerate 5. The international impact in a world of two large economies 6. Conclusions 7 6. Conclusions Optimal escape from a liquidity trap involves private-sector expectations of a higher future price level (Krugman, Eggertsson-Woodford) Credibility problem, difficult to make higher future price level credible (Krugman) Themagicoftheexchangerate(Svensson0,03): Current exchange rate indicates private-sector expectations of the future price level Intentional depreciation and crawling peg can induce correct private-sector expectations and implement optimal escape; solves credibility problem (Jeanne and Svensson 03) Foolproof Way: () price-level target path, (2) depreciation and peg, (3) exit strategy [Optimal Foolproof Way (i t =0), slightly different from original FPW (i t 0)] 8

5 Magnitude and direction of int l impact of optimal escape from liquidity trap depends Case of negative int l output externalities (Complete int l risksharing, σ < η)(worst-case scenario) Noncooperation Lower foreign natural interest rate Foreign recession, if foreign liquidity trap Cooperation Foreign recession optimal (output-gap smoothing across countries) Case of positive int l output externalities (Incomplete int l risksharing) (Good-case scenario) Reduces foreign recession and/or eliminates foreign liquidity trap, if initial foreign liquidity trap Optimal or original FPW is good policy (Also: Simulations by Coenen-Wieland 03, Meredith 03, IMF Multimod, Fed,...) 9 2. A model of a two-country world Variant of open-economy macro model (Benigno-Benigno, Clarida-Galí-Gertler, Corsetti-Pesenti, Obstfeld-Rogoff, IMF GEM) Home country, foreign country, size: α, α. Quantities per household Home representative household X E t j=0 C σ t + V ( M t P c ) N +ϕ t. () σ t +ϕ 0 <δ< subjective discount factor, ρ ln δ rate of time preference; δ j 0

6 C t (aggregate) consumption; σ intertemporal elasticity of substitution; M t home nominal (base) money; P c t CPI; N t labor supply; Satiation level e µ (µ>0) forrealmoney V M t ( c Pt ) 0 µ e M P t c t C ht, C ft consumption of home and foreign final goods; η intratemporal elasticity of substitution; C t [( α) ηc η ht P c t = + α ηc η h ( α)p η t + αp f ηi η t P t and P f t home-currency prices of home and foreign final goods; T t P t f P t terms of trade. ft ] /η, (2) P t ( α)+αtt η η (4) (3) 2

7 Log-linear approximation around a steady state (to be determined) p c t =( α)p t + αp f t = p t + ατ t, (5) τ t p t p f t (6) Producer-currency pricing, perfect exchange-rate pass-through, Law of One Price, p f t = p t + s t, (7) p t (log) foreign-currency price of foreign final goods; s t (log) exchange rate 3 Production of home and foreign final goods, Y t and Yt, perfect competition, inputs of nontraded differentiated home and foreign goods, Y t (ι) and Yt (ι ), 0 ι, 0 ι Y t [ Y t [ Z 0 Z Y 0 Y t (ι) ξ dι] t (ι ) ξ dι ] ξ>elasticity of substitution between differentiated goods Price indices P t = [ P f t = [ Z 0 Z 0 P t (ι) ξ dι] ξ, P f t (ι ) ξ dι ] ξ, /ξ, (8) /ξ, (9) 4

8 Demand for home and foreign differentiated goods Y t (ι) = Y t µ Pt (ι) Yt (ι ) = Yt P t à P f ξ t (ι ) P f t Production, exogenous stochastic productivity A t and A t, monopolistic competition, gross markup ξ/(ξ )! ξ Y t (ι) = A t N t (ι), Yt (ι ) = A t Nt (ι ), N t (ι) and Nt (ι ) home and foreign households input of labor N t = N t = Z 0 Z 0 N t (ι)dι N t (ι )dι 5 Foreign representative household: Same σ, ϕ, η and ρ; Ct, Mt /Pt c, Nt PPP holds p c t = αp t +( α)(p t s t )=p t ( α)τ t (0) p c t = p c t + s t. () 6

9 Complete international risk-sharing, suitable initial conditions Zero steady-state trade balance Market equilibrium implies MUC t =MUC t, c t = c t; (2) p c + c = p + y, (3) p c + c = p + y, (4) c t = y t αητ t, (5) c t = yt +( α)ητ t ; (6) Terms of trade τ t = η (y t y t ). (7) Combination of (2) and (5) (7) gives c t = c t =( α)y t + αy t. (8) Normalize steady state c = c = y = y = τ = Price setting Differentiated goods, monopolistic competition, prices set in advance P t+ (ι) = ξ ξ E tmc t+ (ι) = p t+ (ι) =ln p t (ι) =p t,y t (ι) =y t,n t (ι) =n t,y t = ȧ t + n t ξ ξ E W t+ t. A t+ ξ ξ + w t+ t ȧ t+ t, (9) w t = p c t +(w t p c t)=p c t + ϕn t σ c t, (20) ξ p t+ =ln ξ + p t+ t + +σϕ y t+ t + α( η σ σ )τ τ+ t ( + ϕ)ȧ t+ t. ȧ +ϕ ln ξ ξ, a t ȧ t ȧ p t+ p t+ = p t+ t = p t+ t 8

10 2.2 Potential output Home flexprice equilibrium, given foreign output level = ξ ξ A t W t P t = ξ ξ ξ 0 = ln ξ ȧ t +(p c t p t )+(w t p c t) ξ = ln ξ ȧ t + α η (ȳ t yt )+ϕ(ȳ t ȧ t )+ σ c t Pt c W t A t P t P c, (2) = α η (ȳ t y t )+ϕȳ t + σ [( α)ȳ t + αy t ] ( + ϕ)a t, t 9 ȳ t b a t b 2 y t, (22) σ( + ϕ) b > 0, (23) + σϕ σ b 2 + σϕ α( σ ) > 0 (σ<η) (24) η σ ( α) σ + α η, (25) Terms of trade effect (cf. (4)): yt τ t (p c t p t ) P t c W t P t Pt c ȳ t α η Consumption effect (perfect international risksharing): y t c t MUC t W t P c t P t c W t P t Pt c ȳ t α σ Negative international output externality (σ <η): b 2 > 0) 20

11 Foreign potential output, given home output ȳ t = b a t b 2y t, (26) b σ ( + ϕ) + σ ϕ > 0, b 2 σ + σ ϕ ( α)( σ η ) > 0 σ α σ +( α) η, (σ<η), Negative international output externality (σ <η): b 2 > Real interest rates, natural interest rates, output gaps and the trade balance First-order conditions for intertemporal consumption (z t+j t E t z t+j ) c t = c t+ t σ(r c t ρ), (27) CPI real interest rate i t home nominal interest rate Home (own-good) real interest rate r c t i t (p c t+ t pc t) Relation CPI and own-good real interest rate r t i t (p t+ t p t ) r t = r c t + α(τ t+ t τ t ) 22

12 Home natural interest rates (home flexprice equilibrium, given foreign output) r t r c t + α( τ t+τ t τ t ) r c t + α η [(ȳ t+ t ȳ t ) (y t+ t y t )] r c t ρ + σ ( c t+ t c t ) ρ + σ [( α)(ȳ t+ ȳ t )+α(y t+ t y t )] r t = ρ + d (a t+ t a t )+d 2 (y t+ t y t ) (28) d b σ +ϕ + σϕ > 0, d 2 b 2 ϕ σϕ + σϕ α( σ η ) > 0 (σ<η), 23 d 2 :Effect of unit increase in y t+ t y t Terms-of-trade change: α( τ t+τ t τ t ) : α η Consumption growth (perfect int l risksharing): r t c : α σ Net effect, for given ȳ t+ t ȳ t : α( σ η ) Effect on ȳ t+ t ȳ t : b 2 Total effect: d 2,fractionofα( σ η ) 24

13 Aggregate-demand relation y t = y t+ t σ[r t ρ α( σ η )(y t+ t y t )], (29) ȳ t ȳ t+ t σ[ r t ρ α( σ η )(y t+ t y t )], (30) Output gap x t y t ȳ t, (3) x t = x t+ t σ(r t r t ) (32) 25 Real interest-rate parity Nominal interest-rate parity, CPI real interest rates are equalized τ t = τ t+ t (r t r t ), (33) s t = s t+ t (i t i t ). (34) r c t = r c t. (35) 26

14 Home trade balance (net export share in GDP) Linear approximation is nx t P ty t P c t C t P t Y nx t = y t c t (p c t p t ) = α(y t y t ) ατ t = α(η )τ t = α( η )(y t y t ). Marshall-Lerner condition: η> Money and the zero lower bound The zero lower bound, i t 0. (36) Log of first-order condition for optimal money holdings m t p c t = lnf (e c t,i t ) < µ (i t > 0), (37) m t p c t ln F (e c t, 0) = µ (i t =0), (38) m t (log) nominal money supply, F(C t,i t ) C t > 0, F(C t,i t ) i t < 0 (i t > 0) Satiation level of money demand, µ Use (5), (7) and (8): m t p t = l(y t,yt,i t ) (i t > 0), (39) m t p t l(y t,yt, 0) (i t =0), (40) l(y t,y t,i t ) α η (y t y t )+lnf (e ( α)y t+αy t,i t ) l(y t,y t,i t ) l(y t,y t, 0) α η (y t y t )+µ Home central bank controls i t by setting m t 28

15 p c t = p t + ατ t, τ t p t p f t = η (y t y t ), p f t Summary of model = p t + s t c t = c t = ( α)y t + αyt, c = c = y = y = τ = 0, p t+ = p t+ t, ȳ t = b a t b 2 yt (b 2 σ η > 0 σ<η) r t i t (p t+ t p t ) (p t+ t p t ) s t = s t+ t (i t i t ) r t = ρ + d (a t+ t a t )+d 2 (yt+ t y t ) (d 2 σ η > 0 σ<η) x t y t ȳ t = x t+ t σ(r t r t ) m t p t = l(y t,yt,i t ) (i t > 0) m t p t l(y t,yt, 0) (i t =0) 29 Flexible own-inflation targeting Home inflation target π 0: X E t j=0 Foreign inflation target π 0: X E t j=0 2.5 Monetary-policy objectives ( δ)δ j 2 [(p t+j p t+j π) 2 + λx 2 t+j] ( δ)δ j 2 [(p t+j p t+j π ) 2 + λ x 2 t+j] 30

16 3. A liquidity trap and optimal escape in a simple case of a small open economy Assume foreign country exogenous (home country small open economy) Assume foreign country in steady state (π 0) yt = y = 0, rt = r t = ρ > 0, p t = π + p t ˆp t, i t = i ρ + π > 0, for all t. Assume a t iid ȳ t and r t also iid: a t+ t =0 ȳ t r t = b a t = ρ d a t a t a t+ t a t, r t Alternative (Jeanne-Svensson): Assume a t random walk. In period t, getinformationabout a t+ and thereby r t. 3 Prob{ r t + π<0} small; Prob{ r t + π 0} large, normal case Ideal equilibrium p t+ = p t+ t = p t + π, (4) x t = 0, (42) i t = r t + π 0, (43) r t = r t, (44) m t = p t + l(ȳ t, 0, r t + π). (45) 32

17 Consider home country in period ( present ). Assume expected ideal equilibrium from period 2 on From (39): p 2 and m 2 linked by p 3 = p 2 + π, (46) x 2 = 0, y 2 = ȳ 2 = y = 0, (47) r 2 = r 2 = ρ > 0, i 2 = ρ + π > 0. p 2 = m 2 + l(0, 0,ρ+ π). (48) Period-2 price level Period- price level predetermined, p 2 = p 2 p = p 0 33 Aggregate-demand relation (since x 2 =0) x = σ[i (p 2 p ) r ] σ( r + p 2 p ), (49) i 0, (50) By above assumptions r = ρ d a. Relevant loss function L = 2 [λx2 + δ(p 2 ˆp 2 ) 2 ], (5) ˆp 2 p + π (52) More formal version of Krugman 98. Simplified version of Eggertsson-Woodford

18 3. Optimal escape from a liquidity trap Optimal escape from a liquidity trap: Optimal policy under commitment with a binding ZLB Combine constraints (49) and (50): x σ( r + p 2 p )= σ( r + π + p 2 ˆp 2 ), (53) Infer i from (49). Lagrangian L = 2 [λx2 + δ(p 2 ˆp 2 ) 2 ] φ [ σ( r + π + p 2 ˆp 2 ) x ], Lagrange multiplier φ 0, complementarity slackness conditions φ [ σ( r + p 2 p ) x ]=0. First-order condition with respect to p 2 δ(p 2 ˆp 2 ) φ σ =0. First-order condition with respect to x λx + φ =0. 35 Optimal targeting rule (Svensson 03 JEL) (N) No liquidity trap: If possible, set p 2 =ˆp 2 and choose i 0 so as to fulfill the target criterion x =0. (Choose i = r + π 0; thenr = r and x =0.) (L) Liquidity trap: If this is not possible, set i =0and choose p 2 > ˆp 2 so as to fulfill the target criterion p 2 ˆp 2 = λ σ δ x > 0. (54) 36

19 Case N: No liquidity trap if and only if Then φ =0, ideal equilibrium r + π 0. p 2 = ˆp 2, x = 0, r = r, (55) i = r + π 0, m = p + l(ȳ, 0, r + π) L = 0. (56) Case L: Liquidity trap if and only if r + π<0, (57) Then φ > 0 and i = 0, (58) m p + l(ȳ + x, 0, 0), (59) p 2 = ˆp 2 λ σ2 δ + λ σ 2( r + π) p 2 > ˆp 2, (60) r = (π + p 2 ˆp 2 ) = (π + p 2 ˆp 2 ) r > r (6) x = σ( r + π + p 2 ˆp 2 ) = δ σ δ + λ σ 2( r + π) x < 0 L > 0. (62) 37 ^ p 2 p 2 x ~ = σ ( r - + π + p ^ 2 ) ( 2 p2 E C (r- + π) A p ~ λσ 2 p ^ 2 = x δ Q ~ ^ p 2 p 2 B D ^ x ~ x O x 38

20 The price level in period 2 overshoots the price-level target. Inflation expectations exceed the inflation target, p 2 p = p 2 p = π λ σ2 δ + λ σ 2( r + π) >π. (63) The optimal policy under commitment hence trades off the right amount of overshooting the future price-level/inflation target for the appropriate reduction in the magnitude of the output gap. Main insight: Krugman 98. Precise derivation: Jung-Teranishi-Watanabe 0, Eggertsson-Woodford 03 Good equilibrium, point Q The credibility problem Krugman 98: Why optimal policy credible? Why not p 2 =ˆp 2, π 2 =ˆπ? Commitment to being irresponsible in the future : m 2 = m 2 p 2 + l(ȳ 2, 0, r 2 + π) No commitment mechanism Auerbach-Obstfeld 03: Permanent expansion of money supply But current m p + l(ỹ, 0, 0) mayverywellbelargerthan m 2 Japan: Quantitative easing, mon. base up 60+% Mar 0-Summer 04. Assume optimal policy not credible: Bad equilibrium, higher loss i = 0, m p + l(ȳ +ˆx, 0, 0), p 2 = ˆp 2 < p 2, r = π ˆr > r, x = σ( r + π) ˆx < x ˆL > L. 40

21 4. Themagicoftheexchangerate How to get from the bad equilibrium to the good equilibrium? No zero bound for the exchange rate Currency depreciation can stimulate the economy (net export) (Bernanke, McCallum, Meltzer, Orphanides-Wieland) Currency depreciation and peg serves as a conspicuous commitment to a higher future price level, induces higher private-sector inflation expectations, reduces the real interest rate (Svensson 0, FPW) The right peg can implement the optimal escape from a liquidity trap (Svensson 03 JEP) 4 Find exchange-rate path for bad and good equilibrium, respectively By (6) and (7), private-sector expectations fulfill By (7) and the above assumptions, we have It follows that Bad equilibrium: From (64) and (65) Optimal escape: p 2 p 2 + s 2 + τ 2. τ 2 = τ =0. s 2 = p 2 p 2 (64) s = s 2 (i i ). (65) s 2 = ˆp 2 p 2 ŝ 2, s = ŝ 2 + i ŝ. s 2 = p 2 p 2 s 2 > ŝ 2, s = s 2 + i s > ŝ. (66) 42

22 s t, p t ~ ~ s = s 2 + i* ^ ^ s = s 2 + i* ~ s 2 = p 2 p^ 2 * ~ p 2 ~ p s 2 = p 2 p 2 * p 2 = p + π t 2 ^ ^ ^ ^ 43 Result : s varies one-to-one with p 2. Exchange rate is indicator of expectations of future price level Japan: Quantitative easing, no depreciation, policy failed Result 2: Depreciate currency to s = s and implement crawling peg with rate of crawl = i. Induces s 2 = s 2 and p 2 = p 2! Intentional currency depreciation and crawling peg can implement optimal escape from liquidity trap Pegging strong currency always feasible; pegging weak currency difficult. Does not require portfoliobalance effects. Just commit to s = s and rate of crawl. During the initial defense of the peg, the CB may end up accumulating substantial foreign-exchange reserves. Balance-sheet incentive to maintain peg! (Jeanne-Svensson, flexible CPI targeting) Once the peg has become credible and private-sector expectations have adjusted, the peg is no longer necessary and binding. 44

23 4. The Foolproof Way Original FPW (Svensson 0) () Upward-sloping price-level target path, price gap to be undone (Bernanke 00, 03) (2) Depreciation and (approximately) fixed peg, s t = s (t ), i = i > 0. Unnecessarily high p 2, not quite optimal (Figure) (3) Exit strategy: Floating and flexible inflation/price level targeting once price-level target met Optimal FPW () Price-level target path ˆp o t = p 2 +(t 2)π (t ), Price gap ˆp o p = p 2 ˆp 2 = λ σ2 δ + λ σ 2( r + π) > 0. (2) Depreciation and crawling peg, (3) Exit strategy as above s t = s (t )i (t ), Japan: π =%/yr. US:π =2%/yr,i =%. Original FPW: i = i + π π =0. Optimal FPW: i =0. In this case, no difference The international impact International impact: Foreign country no longer exogenous Simple case: Foreign ideal equilibrium expected from period 2 Period- foreign price level predetermined: p = p 0. Foreign output gap, potential output and the natural interest rate p 3 = p 2 + π, x 2 = 0, y2 = ȳ 2 = y = 0, (67) r2 = r 2 = ρ > 0, i 2 = ρ + π > 0. x = σ [i (p 2 p ) r ] σ ( r + p 2 p ), (68) i 0 (69) ȳ and r both depend on y ; y and r both depend on y Foreign flexible own-inflation targeting L ˆp 2 = 2 [λ x 2 + δ(p 2 ˆp 2) 2 ] p + π 46

24 Discuss international impact in (x,x )-space Potential outputs ȳ = b a b 2 y Output gaps ȳ = b a b 2y x = y ȳ Output in terms of output gaps x = y ȳ y = y + f x f 2 x y = y + f x f2 x World (flexprice) home and foreign potential outputs (0 b 2 b 2 < ) y b a b 2 b a b 2 b 2 y b a b 2b a b 2 b 2 f b 2 b 2 > 0 f 2 b 2 b 2 b 2 > 0 (σ<η) f 2 b 2 b 2 b 2 > 0 (σ<η). 47 Home output decreasing in foreign output gap (σ <η) f 2 > 0: x y y 48

25 Natural interest rates in terms of output gaps World home and foreign natural interest rates r increasing in x,decreasinginx (σ <η) g 2 > 0: x y r r = ρ d a d 2 y r + g x g 2 x r (x,x ) r = ρ d a d 2y r + gx g2x r (x,x ) r ρ d a d 2 y r ρ d a d 2y g d 2 f2 > 0, g 2 d 2 f > 0 (σ<η), g d 2f 2 > 0, g2 d 2f > 0 (σ<η), 49 Rewrite constraints 5. Noncooperative commitment equilibrium x σ[ r (x,x )+π + p 2 ˆp 2 ], (70) x σ [ r (x,x )+π + p 2 ˆp 2] (7) Home chooses p 2 and x so as to minimize L under commitment, subject to (70), taking y and r as given Foreign chooses p 2 and x so as to minimize L under commitment, subject to (7), taking y and r as given 50

26 Home targeting rule (N) No liquidity trap: If possible, set p 2 =ˆp 2 and choose i 0 so as to fulfill the target criterion x =0. (L) Liquidity trap: If this is not possible, set i =0and choose p 2 > ˆp 2 so as to fulfill the target criterion p 2 ˆp 2 = λ σ δ x > 0. (72) Foreign targeting rule (N*) No liquidity trap: If possible, set p 2 =ˆp 2 and choose i 0 so as to fulfill the target criterion x =0. (L*) Liquidity trap: If this is not possible, set i =0and choose p 2 > ˆp 2 so as to fulfill the target criterion p 2 ˆp 2 = λ σ δ x > 0. (73) Combining this with the constraints x = min{ σ[r + g x g 2 x + π λ σ δ x ], 0}, (74) x = min{ σ [r + gx g2x + π λ σ δ x ], 0}. (75) 5 x * δσ = δ ( ~ * = * * ( r + π ) ~ * * * ~ *2 σ g ) + λ σ C* x x D ~ = r + ~ * 2x ~ 2 δσ ( π g ) = δ ( σg ) + λσ x * B = x = r = + π g 2 = * r +π g * 2 Q* O P Q * x B* * x C δσ = δ ( ~ = ( r + π ) ~ ~ 2 σg) + λσ x * δσ = δ ~ * = * * * ( r + π g2x ) ~ * * * ~ *2 ( σ g ) + λ σ D* 52

27 Point B to the left of O, point B* above O 5.. The special case (L, N*) r + π < 0 r + π > 0 (N* ) Foreign not in liquidity trap, ideal equilibrium p 2 = ˆp 2 x = 0 i = r (0,x )+π r g2x + π 0 53 (L) Home in liquidity trap, bad equilibrium V, good equilibrium Q Home moves from V to Q No impact on x,lower r, i C* x D ~ = r σ ( x = + π g ~ σg B * 2x ) x ~ x^ B* O Q V * x F C D* 54

28 Suppose B* between V and Q (r + π < 0) C* x D B O * x Q B* V F C D* Home moves from bad to good equilibrium, move from V to Q, foreign liquidity trap, x < 0. Problem? Cf. optimal int l cooperation! 55 Constraints World loss Lagrangian 5.2 Cooperative commitment equilibrium x σ(r + π + g x g 2 x + p 2 ˆp 2 ), x σ (r + π + gx g2x + p 2 ˆp 2) ( α)l + αl =( α) 2 [λx2 + δ(p 2 ˆp 2 ) 2 ]+α 2 [λ x 2 + δ(p 2 ˆp 2) 2 ] L = ( α)l + αl ( α)φ { σ[r + π + g x g 2 x + p 2 ˆp 2 ] x } αφ { σ [r + π + g x g 2x + p 2 ˆp 2] x } 56

29 First-order conditions (plus complementary slackness) δ(p 2 ˆp 2 ) φ σ = 0 δ(p 2 ˆp 2) φ σ = 0 λx φ σg + φ + α α φ σ g2 = 0 λ x φ σ g + φ + α α φ σg 2 = 0 Optimal targeting rule: Home (N) No liquidity trap: If possible, set p 2 ˆp 2 =0and choose i 0 so as to fulfill the target criterion x = α δg2 α λ (p 2 ˆp 2) 0. (L) Liquidity trap: If this is not possible, set i =0and choose p 2 > ˆp 2 so as to fulfill the target criterion λ σ p 2 ˆp 2 = δ( σg ) x α σg2 (p 2 α σg ˆp 2) > 0. Foreign: (N*) No liquidity trap: If possible, set p 2 =ˆp 2 and choose i 0 so as to fulfill the target criterion x = α δg 2 α λ (p 2 ˆp 2 ) 0. (L*) Liquidity trap: If this is not possible, set i =0and choose p 2 > ˆp 2 so as to fulfill the target criterion p 2 λ σ ˆp 2 = δ( σ g )x α σ g 2 α σ g (p 2 ˆp 2 ) Consider case (L, N*) Targeting rule home: Targeting rule foreign Result: x = α α Smooth output gaps across countries p 2 =ˆp 2 λ σ p 2 ˆp 2 = δ( σg ) x > 0 δg 2 λ (p 2 ˆp 2 )= α α x < ex < 0, ex < x = 0. g 2 λ λ σ x < 0 σg 58

30 Output-gap smoothing across countries (ray OG) C* x D B W B* O Q Q V * x C C G F D* Similar to the case when optimal escape under noncooperation leads to a foreign liquidity trap Positive international output externalities (b 2 < 0,d 2 < 0, also w/ incomplete risksharing?) x D* B* F C F* R T C* S P Q* Q V O B D * x 60

31 The case (L, N*) x D* O Q W B D * x C C Q V F B* G C* 6 B*betweenVandQ x D* Q O W B D * x C Q C B* V F C* G 62

32 6. Conclusions Optimal escape from a liquidity trap involves private-sector expectations of a higher future price level (Krugman, Eggertsson-Woodford) Credibility problem, difficult to make higher future price level credible (Krugman) Themagicoftheexchangerate(Svensson0,03): Current exchange rate indicates private-sector expectations of the future price level Intentional depreciation and crawling peg can induce correct private-sector expectations and implement optimal escape; solves credibility problem (Jeanne and Svensson 03) Foolproof Way: () price-level target path, (2) depreciation and peg, (3) exit strategy [Optimal Foolproof Way (i =0), slightly different from original FPW (i 0)] 63 Magnitude and direction of int l impact of optimal escape from liquidity trap depends Case of negative int l output externalities (Complete int l risksharing, σ < η)(worst-case scenario) Noncooperation Lower foreign natural interest rate Foreign recession, if foreign liquidity trap Cooperation Foreign recession optimal (output-gap smoothing across countries) Case of positive int l output externalities (Incomplete int l risksharing) (Good-case scenario) Reduces foreign recession and/or eliminates foreign liquidity trap, if initial foreign liquidity trap Optimal or original FPW is good policy (Also: Simulations by Coenen-Wieland 03, Meredith 03, IMF Multimod, Fed,...) 64

33 Political problems (Japan) Reactions from BOJ (summer 2000) (Ueda 00) (Ito-Mishkin 04) Since one cannot be absolutely sure, that any given policy action or change in the monetary policy regime will succeed in getting the economy out of the liquidity trap, it is safer not to try. Prudent policy: Try a number of suggested remedies (as long as not inconsistent), some may work Foolproof way might be useful if needed, but not needed now. Problems are over! Direct proposal to MOF and US Treasury. Myopic bureaucratic interests and technical details before national welfare Lack of Rooseveltian resolve (Bernanke 00) 65 Coenen and Wieland 03: Comparison of Orphanides-Wieland, McCallum and Svensson methods for Japan Three-region model of Japan, euro area and US Three ways to stimulate Japan out of liquidity trap and recession. Oprhanides-Wieland: Expand monetary base, rely on portfolio-balance effect to depreciate currency 2. McCallum: When i t =0, switch to reaction function for exchange rate 3. Svensson: FPW Simulations Series of bad shocks during 20 quarters drives Japan into recession and deflation Methods start in quarter All three methods work Moderate repercussions on the US (smallest for the FPW) 66

34 Optimal escape from a liquidity trap Credible Commitment to Optimal Escape from a Liquidity Trap: TheRoleoftheBalanceSheet of an Independent Central Bank Olivier Jeanne Lars E.O. Svensson. Introduction Create expectations of higher future inflation (Krugman, Eggertsson-Woodford) Not credible without commitment mechanism (Krugman) Open economy: the Foolproof Way (Svensson) () Price-level target path, (2) currency depreciation and crawling peg, (3) exit strategy Renege by future appreciation? Peg credible? Commitment mechanism: Central-bank capital Future appreciation implies future capital loss on FX reserves Independent CB prevents capital to fall below minimum level Manage balance sheet: Keep capital at minimum level A simple model One traded good (the foreign good), foreign-currency price unity, Low of One Price, home-currency price/exchange rate S t One traded asset (foreign-currency bond), constant continuously compounded interest rate r > 0 Home country: Private sector (household, firms), public sector (central bank, government) Household consumes foreign good, nontraded home good, and supplies labor X E t τ=0 δ τ [( α)lnc h,t+τ + α ln C f,t+τ + U( M t+τ P h,t+τ ) N t+τ ], δ e r discount factor, M t home currency, P ht home-currency price of home goods U(M t /P ht ) liquidity services of real money (time saved in the transactions of the home good) 68

35 U(M t /P ht ) liquidity services of real money (time saved in the transactions of the home good) M U ( P t ht ) U 0 0 µ e M P t ht 69 CPI P t = Pht α St α. (2.) Firms produce nontraded goods in two stages Final stage, perfect competition Y t [ Z 0 Y t (ι) ξ dι] /ξ, (2.2) ξ>elasticity of substitution between intermediate inputs Price index Z P ht =[ P ht (ι) ξ dι] ξ, (2.3) 0 Demand for intermediate good ι [0, ] µ ξ Pht (ι) Y t (ι) =Y t. (2.4) P ht Initial stage, intermediate good ι produced by firm ι with linear technology, country-wide exogenous stochastic productivity, A t, Y t (ι) =A t N t (ι), Monopolistic competition, gross markup ξ/(ξ ) over marginal cost 70

36 Budget constraint for the home household in period t P ht C ht + S t C ft + M t + B t + S t B t = P ht Y t + M t + e i t B t + S t e r Bt + Z t, (2.5) B t home-currency bonds, i t continuously compounded interest rate, Bt 0 foreign-currency bonds, Z t 0 home-currency value of net transfers from the government Budget constraint for the central bank Z t + S t R t = M t M t + S t e r R t, (2.6) Z t home-currency value of CB s dividend paid to the government, R t 0 foreign-currency bonds held as foreign-exchange reserves (only asset held by CB) The government collects the dividend from the CB and passes it on as a lumpsum transfer to the household No home-currency bonds held in the foreign country, net supply zero, B t =0. (2.7) Equilibrium relationships Equilibrium Add (2.5), (2.6); use (2.7), (2.8) Home country s net foreign assets C ht = Y t. (2.8) C ft + F t = e r F t, (2.9) F t B t + R t 0 Optimal intertemporal consumption of the foreign good e r = δe t/c f,t+ (2.0) /C ft δ = e r implies that C ft and F t will be constant C ft = (e r )F t C f, (2.) F t = F t. Current account is constant and unaffected by monetary policy (separable utility) 72

37 The firm ι [0, ] producing intermediate good ι sets its price for period t, P ht (ι), oneperiodin advance P h,t (ι) = ξ ξ E t, (2.2) A t W t the nominal wage in period t Gross markup times the expected home-currency marginal cost Labor supply: MRS of home goods for labor W t = = Y t P ht ( α)/y t α, (2.3) W t Potential output P h,t (ι) =P ht E t Y t Ȳ t, (2.4) Ȳ t = ξ ( α)a t ξ (2.5) Y t E t = Ȳ t (2.6) 73 Real exchange rate equals MRS of the home good for the foreign good Q t S t α/c ft = = α Y t, (2.7, 2.8) P ht ( α)/c ht α C f The real exchange rate is proportional to home output Natural real exchange rate. Q t = α Ȳ t (2.9) α C f 74

38 Optimal intertemporal consumption of the home good will be e i t P ht Y t = δe t, P h,t+ Y t+ (2.20) The real (CPI) interest rate, r t,willfulfill µ e r t P ht Y t /P t Yt = δe t = δe t P h,t+ Y t+ /P t+ α, (2.2) The natural (CPI) interest rate, r t,theflexprice real (CPI) interest rate µ α e r Ȳt t = δe t. (2.22) Ȳ t+ The home-currency interest rate fulfills the zero lower bound Y t+ i t 0. (2.23) 75 Given the assumptions about the liquidity-services function, money demand can be written M t P ht = G(Y t,i t ) (i t > 0) M t P ht e µ (i t =0) G(Y t,i t ) <e µ G, > 0, G < 0 for i t > 0, Y t i t G(Y t, 0) = e µ. (2.24) 76

39 2.3 Productivity Stochastic process (logs lowercase) a t = a (t ), a t = a b (t 2). b stochastic, announced in period Potential output, natural real exchange rate ȳ t =ȳ and q t = q (t ), ȳ t =ȳ b and q t = q b (t 2), µ ξ ȳ ln ξ q ln ( α) + a, α α +ȳ c f. (2.25) 77 Natural interest rate Rate of time preference, ln δ = r Expected productivity growth, r t = r +( α)(ȳ t+ ȳ t ) = r +( α)(a t+ a t ). (2.26) a t+ a t =0 (t 6= ), a t+ a t = b (t =). r t = r (t 6= ) r t = r ( α)b r (t =) (2.27) b>0 one-time fall fall in expected productivity growth Announcement of b>0 surprise (appendix: anticipated) 78

40 Money m t p ht = g(y t,i t ) (i t > 0), m t p ht µ (i t =0), (2.28) Monetary policy Flexible (CPI) inflation targeting, inflation target π 0 X E t τ=0 ( δ)δ τ L t+τ (2.29) L t = 2 [(π t π) 2 + λ(y t ȳ t ) 2 ], (2.30) Discretion: Minimize (2.30) each period, for given p ht, π t+ t, and constraints π t p t p t = p ht + αq t p t = α q t + p ht p t + α(y t ȳ t ) (2.3, 2.33) q t q t = y t ȳ t (2.32) π t π t t = α(y t y t t )=α(y t ȳ t ) 80

41 Proposition 2.. Under discretion, for t 0 and t 2, y t =ȳ t, q t = q t,andp ht p h,t = π t = s t s t = π A liquidity trap in a period y ȳ = α (i π r ) α ( r + π). (2.35) π π = α(y ȳ) (2.36) Proposition 2.2. The economy falls in a liquidity trap in period if and only if the natural interest rate is sufficiently negative, r < π 0, that is, if and only if the fall in expected productivity growth is sufficiently large, b> r + π α > 0. ŷ ȳ = α ( r + π) < 0 (2.38) α ˆπ π = α ( r + π) < 0, (2.39) ˆr r = ( r + π) > The optimal policy under commitment Commitment equilibrium: Lower loss, π 2 >π,lowerr,highery ȳ Relevant loss function in period L + 2 δ(π 2 π) 2 = 2 [(π π) 2 + λ(y ȳ) 2 + δ(π 2 π) 2 ]. (2.40) Minimize (2.40) subject to (2.36) and y ȳ α ( r + π 2 ). (2.35) (π 2 replaces π) Targeting rule under commitment in a liquidity trap: Set m p h + µ and thereby i =0,andchooseπ 2 >π, and thereby y ȳ = α ( r + π 2 ), (2.4) so as to fulfill the target criterion α π 2 π = δ( α) (π λ π) δ( α) (y ȳ) > 0. (2.42) Combine (2.36), (2.4), and (2.42) δ( α) ỹ ȳ = δ( α) 2 + α 2 + λ ( r + π) < 0, (2.43) α 2 + λ π 2 π = δ( α) 2 + α 2 + λ ( r + π) > 0, (2.44) δ( α) 2 r r = δ( α) 2 + α 2 + λ ( r + π) > 0, 82

42 Compare commitment and discretion equilibrium ỹ ȳ>ŷ ȳ. Optimal policy trades off period-2 overshoot of the inflation target, π 2 π>0, fortheright amount of increase in the period- output and partial closing of the output gap Implementing the commitment equilibrium Implement commitment equilibrium by committing to period-2 money supply m 2 = p h2 + g(ȳ b, r + π) = p 2 α( q b)+g(ȳ b, r + π), But, suppose no commitment to future money supply (Krugman) (Auerbach-Obstfeld, Japan) 84

43 Positive CB capital not needed 3. The balance-sheet concerns of central banks CBs are concerned about their capital, for political-economy reasons (independence) How an independent central bank can commit to a future price level Commit to π 2 by committing to future exchange rate s 2 Commit to s 2 by balance-sheet management. Price levels and exchange rates Discretion p 2 = ˆp + π ˆp 2, s 2 = p 2 +( α)q 2 = ˆp 2 +( α)( q b) ŝ 2, Commitment s = s 2 + r =ŝ 2 + r ŝ, p 2 = p + π 2 p 2, s 2 = p 2 +( α)( q b) s 2, s = s 2 + r s. (4.) 86

44 Compare s ŝ = s 2 ŝ 2 = p ˆp + π 2 π = α(ỹ ŷ)+ π 2 π = α ( π 2 π) α 2 + λ = ( α)[δ( α) 2 + α 2 + λ] ( r + π) > 0, Parallel shift up of the (log) exchange-rate path Nonlog exchange rate levels: S > Ŝ and S 2 > Ŝ Commit to S 2 by appropriate period- balance-sheet management Z t + S t R t = M t M t + S t e r R t, (2.6) CB s capital at the end of period t V t S t R t M t. (4.) By (2.6) and (4.), V t = S t e r R t M t Z t, (4.2) V t depends on S t and Z t Independent CB: Nonnegative lower bound, V, V t V 0. (4.3) CB controls Z t, no capital injection Z t 0. (4.4) 88

45 Proposition 4.. The CB can implement the commitment solution S, S 2 by pegging the exchange rate at S and by setting its capital equal to the minimum level in period, V = V. In period 2, for given R and M, (4.2) (4.4) imply a lower bound for S 2 S 2 M + V e r R. (4.7) Chose Z = Z, R = R, M = M >P h e µ so M + V = S 2. e r R Then, for Z 2 =0and S 2 = S 2, V 2 S 2 e r R M = V. (4.8) Furthermore, for S = S, since S = S 2 e r (i =0) Private sector verifying commitment CB publishes balance sheet Check V = V for S = S, M >P h e µ V = S R M S 2 e r R M = V 2 = V. (4.9) S 2 = S 2 ;butthens = S = S 2 e r only equilibrium 89 CB implement S = S > Ŝ? Announce S Commit to buy and sell unlimited amounts of FX at S If private-sector doubts, speculative attack to appreciate, excess demand for home currency Excess demand can always be fulfilled Big difference defining peg for strong/weak currency 90

46 5. Conclusions Liquidity trap: i =0, r too high, y too low Optimal policy: π 2 >π, r lower, y higher Credibility problem Foolproof Way: () Price-level target path, (2) currency depreciation and peg, (3) exit strategy Peg credible? Future appreciation? Independent CB: Commit to peg by balance-sheet management, minimum CB capital 9