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The New-Keynesian Liquidity Trap

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The New-Keynesian Liquidity Trap The New-Keynesian Liquidity Trap John H. Cochrane Univeristy of Chicago Booth School of Business New Keynesian models: Diagnosis I Consensus NK diagnosis: “Natural” rate r << 0 (?), i = 0, p = 2.5%, i p too high. I Fix Et ct+¥. “Too high” real rate consumption grows too fast, level too low. ! I Why do we not see more p = r? model. ! 40 Consumption 3% trend 2000•2007 New•K eynesian 35 PIH 30 25 20 Percent 15 10 5 0 2002 2004 2006 2008 2010 2012 2014 New Keynesian models: Policy I Policy: All the laws of economics change sign at the zero bound. 1. Open mouth operations, forward guidance, commitments work. Further in the future has larger effect today. 2. Expected inflation is good, raises output. 3. Totally wasted government spending, even if financed by taxes, can have arbitrarily high multipliers. 4. Capital destruction, technical regress, hurricanes, broken windows are good. 5. All get stronger as price frictions diminish, with ¥ limit. 6. Though stickiness is the central friction causing our troubles, don’t fix! it! Making prices sticker is good. I And if you don’tagree, you’re(unprintable) I NK model is not Paleo-Keynesian. MPC = 0. Intertemporal substitution. ¥ 1 ct = Et lim ct+T s (it+s rt+s pt+s ) ds T ¥ s=0 ! Z This paper I Write down the model. Confirm NK diagnosis & policy predictions. I Show they come from one, particular, arbitrary equilibrium choice. "Most" (plausible?) equilibria do not have these features. I NK/Taylor choice: Fed has two, independent policy tools: interest rate policy and equilibrium-selection policy. 1. All the predictions come from expectations of equilibrium selection policy, not interest rate policy. 2. Look at it. Does anyone have these expectations of equilibrium selection policy? I Play by NK rules of the game. Complain about the model/find correct model some other day. I Simplest possible model to show the basic mechanisms. Not realistic/match to data. I Bottom line: Save the NK model, not criticize it! New Keynesian Model (Werning 2012) dxt 1 = s (it rt pt ) (1) dt dpt = rpt kxt . (2) dt rt = “natural” rate From 1 xt = Et xt+1 + s [it rt Et pt+1] r pt = e Et pt+1 + kxt or ¥ rs pt = k e xt+s ds Zs=0 Scenario Scenario 6 4 2 Interest rate 0 Percent •2 •4 Natural rate •6 0 1 2 3 4 5 6 7 8 9 10 Time it rt = irt > 0; t < T ; it rt = irt = 0; t > T . The frictionless equilibrium Frictionless equilibrium, k = ¥ 6 x p 5 4 3 Percent 2 1 0 •1 0 1 2 3 4 5 6 7 8 9 10 1 dpt xt = rpt .k ¥; xt = 0 k dt ! it rt = pt Solution with price stickiness d xt 0 1 xt irt = + dt pt k r pt 0 I t > T (irt = 0) p m lm (t T ) m kxt l l e zT l < 0 = p t T ; p pt 1 1 el ( )w l > 0 T wT = 0 (no quarrel today). Leaves zT = pT , multiple stable lm (t T ) equilibria e pT , index by pT I t < T : choose zT ,wT to paste at pT . lm (t T ) p kxt r e l lm (t T ) = ir ir ·· p + pT e . pt 1 el (t T ) 1 ·· lm (t T ) lp (t T ) lm (t T ) I Both e and e terms are potentially active. e m explode going back in time. p adds el (t T ) terms. f T g Inflation in all equilibria Inflation across equilibria 10 5 0 •5 Percent •10 •15 •20 0 1 2 3 4 5 6 7 8 9 10 The standard equilibrium choice Standard equilibrium, varying k 10 x p 0 •10 •20 Percent •30 •40 •50 0 1 2 3 4 5 6 7 8 9 10 I pT = 0; backward-explosive eigenvalue. Root of all wild predictions. I Depression (with growth). Deflation. Less friction worse. ! The backwards-stable equilibrium Local to frictionless equilibrium, varying k 6 x p 5 4 3 Percent 2 1 0 •1 0 1 2 3 4 5 6 7 8 9 10 I p : no backward explosions. Nice frictionless limit T ! I Exactly the same interest rate policy. I Et pT = 0 with no glide path assumption has big implications! The no-inflation-jump equilibrium No•jump equilibrium, varying k 6 x p 5 4 3 2 1 Percent 0 •1 •2 •3 •4 •5 0 1 2 3 4 5 6 7 8 9 10 I pT : p0 = 0. I Not deflation. I All equilibria with limited p0 jump have normal limits. Inflation in all equilibria Inflation across equilibria 10 5 0 •5 Percent •10 •15 •20 0 1 2 3 4 5 6 7 8 9 10 Output gaps in all equilibria Output gap across equilibria 10 5 0 •5 Percent •10 •15 •20 0 1 2 3 4 5 6 7 8 9 10 Magical multipliers and paradoxical policies dpt = rpt k (xt + gt ) . dt I g = totally useless spending, technical regress, high wage mandates, capital destruction, etc. I p channel, not consumption function. All else constant, more g means more p, less i r. pt constant means ¶x/¶g = 1 f g I Consider gt = g > 0 for t < T , then gt = 0 for t > T . The solution is unaffected t > T . For t < T : lm (t T ) kxt kg e = (irt ) + ·· p kg pt 0 el (t T ) ·· p l m + p (g)el (t T ) 1 T I Calculate ¶xt /¶g, private-output multiplier. Steady state, ¶x/¶g = 1. With p.... Magical multipliers Multiplier 20 Standard equilibrium Local•to•frictionless 18 16 14 12 10 8 6 4 2 0 •2 0 1 2 3 4 5 6 Years I Backward-explosive solutions+big p dynamics, ¶xt /¶gt is big I Without big inflation dynamics ...(xt + gt )... ¶xt /¶gt = 1 Low rate promise —standard equilibrium I Interest rate stays at zero for time t after the trap ends t = 0 .0 t = 0 .5 10 10 0 0 •10 •10 Percent Percent •20 •20 x p •30 •30 0 2 4 6 8 0 2 4 6 8 t = 1 .0 t = 2 .0 30 30 20 20 10 10 Percent Percent 0 0 •10 •10 0 2 4 6 8 0 2 4 6 8 I Far away promises have huge effects — backward explosive solution Low rate promise —backwards-stable t = 0.0 t = 0.5 6 6 4 4 2 2 P ercent P ercent 0 0 x •2 p •2 0 2 4 6 8 0 2 4 6 8 t = 1.0 t = 2.0 6 6 4 4 2 2 P ercent P ercent 0 0 •2 •2 0 2 4 6 8 0 2 4 6 8 I Faraway promises have little current effects. Werning’sequilibrium selection Why pT = 0? Answer: expectations of an equilibrium-selection policy, apart from interest rate policy Taylor rule I Why pT = 0? I NK: Two Fed policies, “interest rate policy” it and “equilibrium selection policy” to pick one p from pt consistent with i . t f g t I Taylor form of selection. Fed also specifies p from pt , then adds t f g it = i + f (pt p). t p t f > 1+ “no explosive solutions” = select p k pk t I The same as “Wicksellian” (optimal) “stochastic intercept” "temporary deviations” policy it i = f (pt p) t p t it = (i f p) + f pt t p t p it = ı¯t + fppt . I Expectations about equilibrium-selection policy/stochastic intercept/Wicksellian! response, not about expected interest rates, drive the whole result. Taylor rule I Equilibrium selection policy it = i + f (pt p). t p t I But do people really have these “expectations about equilibrium selection policy?” I Let’slook! Also “what if people expect a pure Taylor rule with no accommodation ı¯t for end-of-trap glidepath?” t > T : it = r + f pt . i = r, p = 0, t > T p $ t T I Continuous time: needs partial adjustment rule dit q [f pt (it r)] if it > 0 = p . dt max q [f pt (it r)] , 0 if it = 0 f p g I No change for t < T . Different expectations for t > T select t < T equilibrium. Taylor rule and the standard solution Inflation across equilibria with a Taylor rule 10 5 0 •5 Percent •10 •15 •20 0 1 2 3 4 5 6 7 8 9 10 Taylor rule and the standard solution Output gap across equilibria with a Taylor rule 10 5 0 •5 Percent •10 •15 •20 0 1 2 3 4 5 6 7 8 9 10 Interest rate across equilibria with a Taylor rule 10 5 Percent 0 •5 0 1 2 3 4 5 6 7 8 9 10 I Not: “if pT > 0, Fed tightens too much, lowers inflation too fast” I Yes: “if pT > 0 people expect the Fed to explode the economy.” Or those equilibria are not ruled out. A Taylor rule could select the glidepath too Inflation across equilibria with a Taylor rule 10 5 0 •5 Percent •10 •15 •20 0 1 2 3 4 5 6 7 8 9 10 A Taylor rule could select the glidepath too Output gap across equilibria with a Taylor rule 10 5 0 •5 Percent •10 •15 •20 0 1 2 3 4 5 6 7 8 9 10 I The issue is expectations of a “equilibrium selection” policy.
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