The Zero Bound Based on work by: Eggertsson and Woodford, 2003, The Zero Interest Rate Bound and Optimal Monetary Policy, Brookings Panel on Economic Activity. Christiano, Eichenbaum, Rebelo, When is the Government Spending Multiplier Big? (JPE, 20)
Nonlinearity The New Keynesian model suggests that an economy may be vulnerable to welfarereducing volatility when the zero lower bound on the nominal interest rate is binding. Exploring this idea and what to do about it requires solving a non linear model. Will apply the Eggertsson Woodford approach Deterministic Shooting Algorithm. 2
The ZLB Analysis (Over) Simplified Identity: expenditures GDP If one group reduces spending, then GDP must fall unless another group increases. Another group increases if real rate drops: R e If R is at lower bound and e cannot rise, have a problem.
The ZLB Analysis, cnt d Several reasons e may not rise.all presume a lack of commitment in monetary policy Ex post, monetary authority would not deliver high inflation (Eggertsson). Real world monetary authorities spent years persuading people they would not use inflation to stabilize economy. Fears consequences of loss of credibility in case they now raise e for stabilization purposes. In the presence of commitment, ZLB not a big problem.
The ZLB Analysis (Over) Simplified Recession likely to follow, as real rate fails to drop. The recession could be very severe if a deflation spiral occurs. R e The decrease in spending leads to a fall in marginal cost, which makes firms cut prices. When there are price frictions, downward pressure on prices is manifest as a reduction in inflation. 5
Deflation Cycle in Zero Bound Low spending
Deflation Cycle in Zero Bound Low spending Low marginal cost
Deflation Cycle in Zero Bound Low spending Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
Deflation Cycle in Zero Bound Low spending High real interest rate Low marginal cost Low expected inflation
The Whole Analysis, cnt d The preceding indicates that the drop in output might be substantial. Options for solving zlb problem Direct: by interrupting destructive deflation spiral, increase government spending may have a very large effect on output. Tax credits Investment tax credit cash for clunkers Increase anticipated inflation Convert to a VAT tax in the future (Feldstein, Correia Fahri Nicolini Teles). Don t: cut labor tax rate or subsidize employment (Eggertsson)
Outline Analysis in normal times when zlb constraint on interest rate can be ignored. Show that the government spending multiplier is fairly small. Analysis when zlb is binding. Government spending can have a big, welfareimproving impact on output. 20
Derivation of Model Equilibrium Conditions Households First order conditions Firms: final goods and intermediate goods marginal cost of intermediate good firms Aggregate resources Monetary policy Three linearized equilibrium conditions: Intertemporal, Pricing, Monetary policy Results 2
Model Household preferences and constraints: E 0 t 0 t C t Nt v G t P t C t B t W t N t R t B t T t, T t ~lump sum taxes and profits Optimality conditions marginal benefit tomorrow from saving more today marginal cost of giving up one unit of consumption to save uc,t E t u c,t extra goods tomorrow from saving more today R t t, marginal cost (in units of goods) of labor effort u N,t u c,t marginal benefit of labor effort Wt P t 22
Model Household preferences and constraints: E 0 t 0 t C t Nt King Plosser Rebelo (KPR) preferences. v G t P t C t B t W t N t R t B t T t, T t ~lump sum taxes and profits Optimality conditions marginal benefit tomorrow from saving more today marginal cost of giving up one unit of consumption to save uc,t E t u c,t extra goods tomorrow from saving more today R t t, marginal cost (in units of goods) of labor effort u N,t u c,t marginal benefit of labor effort Wt P t 23
Firms Final, homogeneous good Y t Yt i 0 di, Efficiency condition: P t i P t i th intermediate good Y t Y t i Y t i N t i Optimize price with probability θ, otherwise P t i P t i 24
Monetary Policy Monetary policy rule (after linearization) dr t R dr t R t k 2 Ŷ t l dr t R t R, R Ŷ t Y t Y Y k,l 0,. 25
Pulling All the Equations Together IS equation: Ŷ t gĝ t Ŷ t gĝ t g dr t d t Phillips curve: t t g N N Ŷ t g g Ĝ t Monetary policy rule: dr t R dr t R t k 2 Ŷ t l 26
The Equations in Matrix Form g 0 0 0 l R 2 k R 0 Ŷ t t dr t 2 0 g Ŷ t 0 0 0 Ŷ t g N N 0 t 0 0 0 t l R 2 k R dr t 0 0 R dr t g g 0 0 Ĝ t g g g g 0 Ĝ t, or, 0 z t z t 2 z t 0 s t s t 0. s t Ps t t, s t Ĝ t, P 27
Solution: Undetermined coefficients, A and B: z t Az t Bs t A and B must satisfy: 0 A 2 A 2 0 0 AB BP B 0 P 0. When R 0, 2 0 A 0 works. 28
Results Fiscal spending multiplier small, but can easily be bigger than unity (i.e., C rises in response to G shock) Contrasts with standard results in which multiplier is less than unity Typical preferences in estimated models: E 0 t 0 t C t N t v G t,,, 0. Marginal utility of C independent of N for CGG Marginal utility of C increases in N for KPR. 29
APR % of gdp %deviation from ss APR Simulation Results Benchmark parameter values: 0. 035, 0. 99,. 5, 2 0, N 0.23, g 0. 2, 2, 0. 8, R 0 output inflation 4 3.5 3.2 2.5 0.8 2.5 0.5 0.6 0.4 0.2 5 0 5 20 5 0 5 20 2 interest rate 4 3.5 Ghat G t G Y G t G G G Y Ĝ tg.5 3 2.5 2.5 0.5 0.5 5 0 5 20 5 0 5 20 30
APR % of gdp %deviation from ss APR Simulation Results Benchmark parameter values: 0. 035, 0. 99,. 5, 2 0, N 0.23, g 0. 2, 2, 0. 8, R 0 output inflation 4 3.5 3.2 2.5 0.8 2.5 0.5 0.6 0.4 0.2 Multiplier =.05, constant. 2 5 0 5 20 interest rate 4 3.5 5 0 5 20 Ghat G t G Y G t G G G Y Ĝ tg.5 3 2.5 2.5 0.5 0.5 5 0 5 20 5 0 5 20 3
dy/dg dy/dg dy/dg dy/dg dy/dg dy/dg Multiplier for Alternative Parameter Values =.5, = 0, = 0, = 0.8, = 0.03, 2 R = 0.99, = 0.2857, N = 0.33333, g = 0.2, = 2.2.5.2.. 0.8.05 0.95 0.9 2 3 0.9 0.8 0 2 0.6 0.4 0.2 0 0.8.2.4.6.8.2.2 0.9. 0.8.5 0.7 0.6 0.5.05 0 0.2 0.4 0 0.2 0.4 0.6 0.8 2 R. 0.9 0.8 0 0.2 0.4 0.6 0.8 Results: multiplier bigger the less monetary policy allows R to rise. the more complementary are consumption and labor (i.e., the bigger is ). the smaller the negative income effect on consumption (i.e., the smaller is ). smaller values of κ (i.e., more sticky prices) 32
dy/dg dy/dg dy/dg dy/dg dy/dg dy/dg Multiplier for Alternative Parameter Values =.5, = 0, = 0, = 0.8, = 0.03, 2 R = 0.99, = 0.2857, N = 0.33333, g = 0.2, = 2.2.5..05 dr t R dr t R 0.95 0.9 2 3.2. t 2 0.9 Ŷ t 0.8 0 2 0.8 0.6 0.4 0.2 0 0.8.2.4.6.8.2.2 0.9. 0.8.5 0.7 0.6 0.5.05 0 0.2 0.4 0 0.2 0.4 0.6 0.8 2 R. 0.9 0.8 0 0.2 0.4 0.6 0.8 Results: multiplier bigger the less monetary policy allows R to rise. the more complementary are consumption and labor (i.e., the bigger is ). the smaller the negative income effect on consumption (i.e., the smaller is ). smaller values of κ (i.e., more sticky prices) 33
dy/dg dy/dg dy/dg dy/dg dy/dg dy/dg Multiplier for Alternative Parameter Values =.5, = 0, = 0, = 0.8, = 0.03, 2 R = 0.99, = 0.2857, N = 0.33333, g = 0.2, = 2.2.5..05 0.95 0.9 0.9 0.8 2 3.2. 0.9 0.8.2 u C t,n t.5 0 2 0.8 0.6 0.4 0.2.2. 0 0.8.2.4.6.8 C t N t 0.7. 0.9 0.6 0.8 0.5.05 0 0.2 0.4 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 2 u c,t C t R N t Results: multiplier bigger the less monetary policy allows R to rise. the more complementary are consumption and labor (i.e., the bigger is ). the smaller the negative income effect on consumption (i.e., the smaller is ). smaller values of κ (i.e., more sticky prices) 34
dy/dg dy/dg dy/dg dy/dg dy/dg dy/dg Multiplier for Alternative Parameter Values =.5, = 0, = 0, = 0.8, = 0.03, 2 R = 0.99, = 0.2857, N = 0.33333, g = 0.2, = 2.2.5.2.. 0.8.05 0.95 0.9 0.9 0.8 Ĝ t Ĝ 2 3 t 0 t 2 0.6 0.4 0.2 0 0.8.2.4.6.8.2.2 0.9. 0.8.5 0.7 0.6 0.5.05 0 0.2 0.4 0 0.2 0.4 0.6 0.8 2 R. 0.9 0.8 0 0.2 0.4 0.6 0.8 Results: multiplier bigger the less monetary policy allows R to rise. the more complementary are consumption and labor (i.e., the bigger is ). the smaller the negative income effect on consumption (i.e., the smaller is ). smaller values of κ (i.e., more sticky prices) 35
dy/dg dy/dg dy/dg dy/dg dy/dg dy/dg Multiplier for Alternative Parameter Values =.5, = 0, = 0, = 0.8, = 0.03, 2 R = 0.99, = 0.2857, N = 0.33333, g = 0.2, = 2.2.5.2.. 0.8.05 0.95 0.9 2 3 0.9 0.8 0 2 0.6 0.4 0.2 0 0.8.2.4.6.8.2.2 0.9. 0.8.5 0.7 0.6 0.5.05 0 0.2 0.4 0 0.2 0.4 0.6 0.8 2 R. 0.9 0.8 0 0.2 0.4 0.6 0.8 Results: multiplier bigger the less monetary policy allows R to rise. the more complementary are consumption and labor (i.e., the bigger is ). the smaller the negative income effect on consumption (i.e., the smaller is ). smaller values of κ (i.e., more sticky prices) 36
Analysis of Case when the Nonnegativity Constraint on the Nominal Interest Rate is Binding Need a shock that puts us into the lower bound. One possibility: increased desire to save. Seems particularly relevant at the current time. Other shocks will do it too... Discount rate shock. 37
Monetary Policy Monetary policy rule (after linearization) Z t R R R t R R t 2 Ŷ t Ŷ t Y t Y Y, R R t Z t if Z t 0 0 ifz t 0. nonlinearity 38
Eggertsson Woodford Saving Shock Preferences: u C 0,N 0, G 0 r E 0 u C, N,G r 2 u C 2, N 2,G r 2 r 3 u C 3, N 3,G 3... Before t=0 System was in non stochastic, zero inflation steady state, r t R R t R Ĝ t 0, for all t 39
At time t=0, Saving Shock, cnt d r r l 0 Prob r t r r t r l p Prob r t r l r t r l p Prob r t r l r t r 0 Discount rate drops in t=0 and is expected to return permanently to its normal level with constant probability, p. 40
Zero Bound Equilibrium simple characterization: l,ŷ l,r 0, Z l 0 while discount rate is low t Ŷ t 0, R r as soon as discount rate snaps back up 4
Fiscal Policy Government spending is set to a constant deviation from steady state, during the zero bound. That is, Ĝ t may be nonzero while r t r l, Ĝ t 0 when r t r 42
Equations With Discount Shock IS equation: Ŷ t g Ĝ t g R t r t E t t E t Ŷ t g E t Ĝ t Ŷ l g Ĝ l g 0 r l p l pŷ l g pĝ Phillips curve: t E t t g Monetary Policy: R t 0 N Ŷ N t g Ĝ g t l p l g N N Ŷ l g g Ĝl Z t R R R t R R t 2 Ŷ t 0 43
Solving for the Zero Bound Allocations Is equation: Ŷ l g Ĝ l g 0 r l p l pŷ l g pĝ Phillips curve: l p l g N Ŷ l g N g Ĝl Two equations in two unknowns! Solve for and verify that Ŷ l, l Z l 0 44
Solution Inflation: l g N N g Ĝ l g p rl p g N N p g p g g Ĝl Output: Ŷ l g Ĝ l g p rl p l 45
Zl dy/dg Yhatl Numerical Simulations 40 20 00 80 60 40 20 0.028 0.03 0.032 0.034 0.036 kappa -2-4 -6-8 -0 0.028 0.03 0.032 0.034 0.036 kappa - -2 =.5, = 0, = 0, = 0.8, 2 R = 0.03, = 0.99, = 0.2857, N = 0.33333, g = 0.2, Ghat = 0, s -3-4 Problem is worse with more flexible prices! 0.028 0.03 0.032 0.034 0.036 kappa Results: multiplier 3.7 at benchmark parameter values and may be gigantic. 46
Zl inflation dy/dg Yhatl benchmark parameter values: =.5, = 0, = 0, = 0.8, = 0.03, 2 R = 0.99, = 0.2857, N = 0.33333, g = 0.2, k = 0, l = 0, Ghat = 0, sig = 2, p = 0.8, r l = -0.0 70 60 50 40 30 20 0 dy/dg at baseline: 3.6857 - -2-3 -4-5 0.72 0.74 0.76 0.78 0.8 p 0.72 0.74 0.76 0.78 0.8 p -0.5 - -.5-0.5 - -2 0.72 0.74 0.76 0.78 0.8 p -.5 0.72 0.74 0.76 0.78 0.8 p As p increases, zero bound becomes more severe this is because with higher p, fall in output is more persistent and resulting negative wealth effect further depresses consumption. 47
Fiscal Expansion in Zero Bound Highly Effective, But is it Desirable? Intuition: Yes. the vicious cycle produces a huge, inefficient fall in output in the first best equilibrium, output, consumption and employment are invariant to discount rate shocks If G helps to partially undo this inefficiency, then surely it s a good thing 48
Fiscal Expansion in Zero Bound Highly Effective, But is it Desirable? Preferences t 0 p r l t C l N l v G l p r l C l N l v G l p r l Compute optimal (i) v G l = 0, (ii) N Ŷ l Ng Ĝ l N Ŷl Ĝ l v Ng Ĝ l v G g G, g chosen to rationalize g 0.2 as optimal in steady state 49
C over C steady state nominal rate of interest inflation Utility Y over Y steady state Zl Case Where G is not Valued phi =.5, phi2 = 0, rhor = 0, rho = 0.8, kap = 0.03, bet = 0.99, gam = 0.2857, N = 0.33333, g = 0.2, k = 0, l = 0, Ghat = 0, sig = 2, p -0.96-0.97-0.98-0.99 - -.0 -.02 0 0.2 0.4 Ghat.05 0.95 0.9 0.85 0.8 0 0.5 Ghat 0.04 0.02 0-0.02-0.04-0.06 0 0.5 Ghat 0.8 0.75 x 0-3 5 4 3-0.0-0.02 0.7 2-0.03 0.65 0.6 0 0.2 0.4 Ghat 0-0 0.2 0.4 Ghat -0.04-0.05 0 0.5 Ghat 50
C over C steady state nominal rate of interest inflation Utility Y over Y steady state Zl Case Where G is not Valued phi =.5, phi2 = 0, rhor = 0, rho = 0.8, kap = 0.03, bet = 0.99, gam = 0.2857, N = 0.33333, g = 0.2, k = 0, l = 0, Ghat = 0, sig = 2, p Optimal G is substantial, around 5%. -0.96-0.97-0.98-0.99 - -.0 -.02 0 0.2 0.4 Ghat.05 0.95 0.9 0.85 0.8 0 0.5 Ghat 0.04 0.02 0-0.02-0.04-0.06 0 0.5 Ghat 0.8 0.75 x 0-3 5 4 3-0.0-0.02 0.7 2-0.03 0.65 0.6 0 0.2 0.4 Ghat 0-0 0.2 0.4 Ghat -0.04-0.05 0 0.5 Ghat 5
C over C steady state nominal rate of interest inflation Utility Y over Y steady state Zl Case Where Gov t Spending is Desirable phi =.5, phi2 = 0, rhor = 0, rho = 0.8, kap = 0.03, bet = 0.99 gam = 0.2857, N = 0.33333, g = 0.2, k = 0, l = 0, Ghat = 0, sig = 2, psig -.6.05 0.04 psig=0.05226 -.7 0.02 -.8 0.95 0 -.9 -.2 0 0.2 0.4 Ghat 0.9 0.85 0.8 0 0.5 Ghat -0.02-0.04-0.06 0 0.5 Ghat Ĝ t Ĝ t Ĝ t Ĝ t 0.8 0.75 x 0-3 5 4 3-0.0-0.02 0.7 2-0.03 0.65 0-0.04 0.6 0 0.2 0.4 Ghat - 0 0.2 0.4 Ghat -0.05 0 0.5 Ghat 52
C over C steady state nominal rate of interest inflation Utility Y over Y steady state Zl Case Where Gov t Spending is Desirable Optimal Y higher than before crisis phi =.5, phi2 = 0, rhor = 0, rho = 0.8, kap = 0.03, bet = 0.99 gam = 0.2857, N = 0.33333, g = 0.2, k = 0, l = 0, Ghat = 0, sig = 2, psig -.6 -.7 -.8 -.9 -.2 0 0.2 0.4 Ghat.05 0.95 0.9 0.85 0.8 0 0.5 Ghat 0.04 0.02 0-0.02-0.04-0.06 0 0.5 Ghat Ĝ t Ĝ t Ĝ t Ĝ t psig=0.05226 The high level of output is necessary to get partial recovery in consumption 0.8 0.75 0.7 0.65 0.6 0 0.2 0.4 Ghat x 0-3 5 4 3 2 0-0 0.2 0.4 Ghat -0.0-0.02-0.03-0.04-0.05 0 0.5 Ghat 53
Introducing Investment Inclusion of investment does not have a large, qualitative effect. Financial frictions could make things much worse. Deflation hurts net worth of investors with nominal debt, and this forces those agents to cut spending by more. 54
Conclusion of G Multiplier Analysis Government spending multiplier in a neighborhood of unity in normal times. Multiplier can be large when the zero bound is binding (because R constant then). Increase in G is welfare improving during lower bound crisis. Caveat: focused exclusively on multiplier Increasing G may be bad idea because hard to reverse. May be other ways of accomplishing similar thing (e.g., transition to VAT tax over time). 55
Can Zero Bound, in Conjunction with Other Shocks Account for Recent Data? Suppose that something (thing, x) happened in 2008Q3. Identify impulse response of economy to x by comparing what actually happened with forecast as of 2008Q2. Assume x is a shock to households desire to save (the saving rate did go up), and wedge in the rate of return on capital (spreads did go up). 56
First, what happened?... 57
Actual and 2008Q2 Forecasts -2.85 per capita GDP -3.35 Consumption 2.8 investment -2.9-3.4 2.7-2.95-3.45 2.6-3 -3.5 2.5 998 2000 2002 2004 2006 2008 200 998 2000 2002 2004 2006 2008 200 998 2000 2002 2004 2006 2008 200-4.98-5 St&loc purchases -4.55 Gov C and I 2.5 cpi -5.02-5.04-4.6 2-5.06.5-5.08 998 2000 2002 2004 2006 2008 200-4.65 998 2000 2002 2004 2006 2008 200 998 2000 2002 2004 2006 2008 200 6 5 4 Funds rate 7 6 5 priv saving rate actual univariate forecast 3 4 2 3 2 998 2000 2002 2004 2006 2008 200 998 2000 2002 2004 2006 2008 200 58
Deterministic Simulation of Medium Size Model Features: Habit persistence in preferences Adjustment costs in change of investment Shock to discount rate and to wedge in rate of return on capital wedge k R t k t k r t k P t k P t Wedge designed to capture increased financial frictions. Model taken from Altig Christiano Eichenbaum Linde (200). 59
Simulating a Lower Bound Episode In periods t<0, the system is in deterministic steady state. In periods t=,2,,t (=0) the shocks that make the lower bound bind occur. In periods t>t all shocks go back to their steady state. Events in periods t 0 completely deterministic. 60
Monetary Policy Z t RdR t R t 2 Ŷ t R t Z t Z t 0 zero bound not binding 0 Z t 0 binding zero bound 6
Equilibrium Conditions Period t endogenous variables (simplified): z t dn t d t dk t dr t di t dz t endogenous variables s t k dr t d t Ĝ t exogenous variables System for t>t : 0 z t z t 2 z t 0, solution: z t Az t Bottom equation corresponds to R t Z t : bottom row of 0, 2, 0, composed of zeros. bottom row of : 0 0 0 0 62
Equilibrium Conditions Period t endogenous variables (simplified): z t dn t d t dk t dr t di t dz t endogenous variables s t k dr t d t Ĝ t exogenous variables System for t>t : 0 z t z t 2 z t 0, solution: z t Az t Bottom equation corresponds to R t Z t : bottom row of 0, 2, 0, composed of zeros. bottom row of : 0 0 0 0 63
Equilibrium Conditions Period t endogenous variables (simplified): z t dn t d t dk t dr t di t dz t endogenous variables s t k dr t d t Ĝ t exogenous variables System for t>t : 0 z t z t 2 z t 0, solution: z t Az t Bottom equation corresponds to R t Z t : bottom row of 0, 2, 0, composed of zeros. bottom row of : 0 0 0 0 64
Equilibrium Conditions Period t endogenous variables (simplified): z t dn t d t dk t dr t di t dz t endogenous variables s t k dr t d t Ĝ t exogenous variables System for t>t : 0 z t z t 2 z t 0, solution: z t Az t Bottom equation corresponds to R t Z t : bottom row of 0, 2, 0, composed of zeros. bottom row of : 0 0 0 0 65
Equilibrium Conditions Period t endogenous variables (simplified): z t dn t d t dk t dr t di t dz t endogenous variables s t k dr t d t Ĝ t exogenous variables System for t>t : 0 z t z t 2 z t 0, solution: z t Az t Bottom equation corresponds to R t Z t : bottom row of 0, 2, 0, composed of zeros. bottom row of : 0 0 0 0 66
Equilibrium Conditions In binding zlb, dr t The difference equation has to be modified to accommodate the change for t,2,...,t : 0 z t z t 2 z t d 0 s t s t 0 for t T : z t Az t Here, s T+=0 bottom row of : 0 0 0 0 0 d 0 0 0 0 0 67
Equilibrium Conditions In binding zlb, dr t The difference equation has to be modified to accommodate the change for t,2,...,t : 0 z t z t 2 z t d 0 s t s t 0 for t T : z t Az t Here, s T+=0 bottom row of : 0 0 0 0 0 d 0 0 0 0 0 68
Equilibrium Conditions In binding zlb, dr t The difference equation has to be modified to accommodate the change for t,2,...,t : 0 z t z t 2 z t d 0 s t s t 0 for t T : z t Az t Here, s T+ =0 bottom row of : 0 0 0 0 0 d 0 0 0 0 0 69
System Shooting Algorithm for t,2,...,t : 0 z t z t 2 z t d 0 s t s t 0 for t T : z t Az t Initial conditions: z t =0. Fix z arbitrarily. Compute z 2,z 3,...,z T recursively, to solve: 0 z 2 z 2 z 0 d 0 s 2 s 0 0 z 3 z 2 2 z d 0 s 3 s 2 0 0 z T z T 2 z T d 0 s T s T 0 Adjust z until z T Az T 0 This is zero by assumption. 70
System Shooting Algorithm for t,2,...,t : 0 z t z t 2 z t d 0 s t s t 0 for t T : z t Az t Initial conditions: z 0 =0. Fix z arbitrarily. Compute z 2,z 3,...,z T recursively, to solve: 0 z 2 z 2 z 0 d 0 s 2 s 0 0 z 3 z 2 2 z d 0 s 3 s 2 0 0 z T z T 2 z T d 0 s T s T 0 Adjust z until z T Az T 0 This is zero by assumption. 7
System Shooting Algorithm for t,2,...,t : 0 z t z t 2 z t d 0 s t s t 0 for t T : z t Az t Initial conditions: z 0 =0. Fix z arbitrarily. Compute z 2,z 3,...,z T recursively, to solve: 0 z 2 z 2 z 0 d 0 s 2 s 0 0 z 3 z 2 2 z d 0 s 3 s 2 0 0 z T z T 2 z T d 0 s T s T 0 Adjust z until z T Az T 0 This is zero by assumption. 72
System Shooting Algorithm for t,2,...,t : 0 z t z t 2 z t d 0 s t s t 0 for t T : z t Az t Initial conditions: z 0 =0. Fix z arbitrarily. Compute z 2,z 3,...,z T recursively, to solve: 0 z 2 z 2 z 0 d 0 s 2 s 0 0 z 3 z 2 2 z d 0 s 3 s 2 0 0 z T z T 2 z T d 0 s T s T 0 Adjust z until z T Az T 0 This is zero by assumption. 73
System Shooting Algorithm for t,2,...,t : 0 z t z t 2 z t d 0 s t s t 0 for t T : z t Az t Initial conditions: z 0 =0. Fix z arbitrarily. Compute z 2,z 3,...,z T recursively, to solve: 0 z 2 z 2 z 0 d 0 s 2 s 0 0 z 3 z 2 2 z d 0 s 3 s 2 0 0 z T z T 2 z T d 0 s T s T 0 Adjust z until z T Az T 0 This is zero by assumption. 74
Complications 0 In practice, is singular. Apply QZ decomposition. In practice, zero bound does not necessarily bind in first and last few periods. Have to adjust equations in an obvious way. 75
Results: Model Replicates Impulse Responses Reasonably Well model, 2008Q3-205Q2 data, 2008Q3-200Q 0 net inflation 0 net nominal rate of interest 0 - output -0.2-0.5-2 APR -0.4-0.6-0.8 - -.2 2009 200 20 202 203 204 205 APR - -.5 2009 200 20 202 203 204 205 % dev from ss -3-4 -5-6 -7 2009 200 20 202 203 204 205 2 Z consumption investment.5-0 APR 0.5 0-0.5 - -.5 % dev from ss -.5-2 -2.5 % dev from ss -5-0 -5-20 2009 200 20 202 203 204 205 2009 200 20 202 203 204 205 2009 200 20 202 203 204 205 76
Government consumption multiplier: dy t dg 0.49,2.0,2.2,2.3,2.3,2.3,2.2,2.0,.8,.7,.5,.3,0.02, t,2,...,3. Denominator: change in G operative while G is up (i.e., periods 2 to 2). 77
Conclusion of Simulation Can account for dynamics of recent data as reflecting the operation of the zero bound and two particular shocks. Many people would expect this not to be possible. Mindful of the deflation spiral, they would anticipate that the drop in inflation would be too great. In fact, incorporate a very small slope to Phillips curve. However, prices sticky on average only a year. This reflects that capital is firm specific. A fully stochastic simulation, with non linear equilibrium conditions much more interesting. Method for dealing with occasionally binding constraints (Christiano Fisher, JEDC2000). See Adams Billie, Anton Braun, Villaverde Juan Rubio. 78