The Lower Bound in DSGE Models by Lawrence J. Christiano 1 - PowerPoint PPT Presentation

... The Lower Bound in DSGE Models by Lawrence J. Christiano 1

Background • Countries Have Fought and Won a Tough Battle Against Inflation. – Problem Now is to Figure Out How to Keep Inflation Low. – One Possibility is to Target a Low Inflation Rate! – Recent Literature (Krugman, Eggertsson-Woodford) Suggests This Exposes An Economy to Risk of Economic Collapse When the Lower Bound on the Nominal Interest Rate Binds – Some Argue that Japan’s ‘Lost Decade’ is a Consequence of Hitting the Lower Bound, and that Japan Therefore Illustrates the Real Danger Associated with Low Inflation. 2

Background ... • Eggertson-Woodford Construct a Simple Model Environment Which Poten- tially Rationalizes the Concerns. – Example is Dramatic: Things Can Go Badly Wrong. – Simple: You Can Work it Out on a Napkin Over Beer. • Model Suggests a Solution to the Problem: Price Level Targetting – Interestingly, Does not Require High Inflation. – Need to Inject a Small Amount of Inflation After Certain Shocks. 3

Questions • Is the Lower Bound Still a Matter of Concern In Models that Incorporate Investment And Open Economy Considerations? – Answer: Lower Bound is Much Less Likely With Investment and Open Economy • What Does Lower Bound Imply for Effects of Fiscal Shocks? – Answer: Predicts that Government Spending Multiplier Huge 4

Outline • Simple Intuition of E-W Example • Introducing Capital into E-W Model, Rexamining the Likelihood of Hitting the Lower Bound. • The Output Effects of Government Spending in the Lower Bound 5

Model • Household Preferences: X ∞ β t [ u ( C t , M t /P t ) − v ( H t )] , E 0 t =0 • Discount Rate: 1 ¡ ¢ , β t = (1 + r n 0 ) (1 + r n 1 + r n 1 ) · · · t − 1 1 β t +1 = . 1 + r n β t t 6

Model ... • Experiment: r n 0 low, and remains low with probability p with probability 1 − p, it jumps back up to its steady state and remains there • Monetary Policy: Set Nominal Interest Rate, i t , so that π t = P t /P t − 1 = 1 , if possible otherwise, set i t = 0 and let market forces determine π t 7

Figure 1: Consequence of Increase in Saving When there is Lower Bound on Real Interest Rate, For Two Investment Elasticities Real Inelastic Rate Investment Elastic Investment Saving A Lower bound C B Saving, Investment

Simple Algebra of Eggertsson-Woodford • Linearized Intertemporal Euler Equation (‘IS Curve’) r n x t = E t x t +1 − σ (ˆ ı t − E t ˆ π t +1 − ˆ t ) – Here: x t = c t − c c ı t = i t − i ( 1 ˆ 1 + i = β ) 1 + i π t = π t − π ˆ ( π = 1 ) π t = r n t − r n 1 r n ˆ 1 + r n = β ). ( 1 + r n • Linearized Calvo Equation: π t = βE t ˆ ˆ π t +1 + κx t . 8

Simple Algebra of Eggertsson-Woodford ... • Implications of Zero Bound For ˆ ı t : ı t ≡ i t − i ˆ 1 + i = β ( i t + 1) − 1 so, ˆ ı t ≥ β − 1 • Monetary Policy: Set ˆ π t = 0 , unless this Implies ˆ ı t < β − 1 If ˆ π t = 0 Implies ˆ ı t < β − 1 , Set ˆ ı t = β − 1 and Let ˆ π t Be Determined Endogenously 10

Simple Algebra of Eggertsson-Woodford ... • Equations of Model: r n x t = E t x t +1 − σ (ˆ ı t − E t ˆ π t +1 − ˆ t ) π t = βE t ˆ ˆ π t +1 + κx t . • In Steady State: r n x t = ˆ π t = ˆ ı t = ˆ t = 0 r n • Suppose ˆ t < 0 r n r n – If ˆ t ≥ β − 1 , Set ˆ ı t = ˆ t , And x t = ˆ π t = 0 Is Still Equilibrium r n – If ˆ t < β − 1 , ˆ ı t = β − 1 , ˆ π t is Free 11

Simple Algebra of Eggertsson-Woodford ... r n • What Happens if ˆ t < β − 1? • Depends on Expectations About the Future! • Here is the E-W Setup: – In Period 0 and 1 : r n r n ˆ 0 < β − 1 = ˆ l ½ ˆ r n probability p r n l ˆ 1 = 0 probability 1 − p – In Period t : r n r n if ˆ t − 1 = 0 , ˆ t = 0 ½ ˆ r n probability p r n l or , ˆ t = 0 probability 1 − p 12

Simple Algebra of Eggertsson-Woodford ... • Equilibrium is Simple to Compute! • In Low State, π t = ˆ ˆ π l , x t = x l • Find These Variables by Solving: r n x l = px l − σ (( β − 1) − pπ l − ˆ l ) π l = βpπ l + κx l • Parameterization: p = 0 . 9 , σ = 0 . 5 , κ = 0 . 02 , β = 0 . 99 , r n l = − . 02 / 4 . x l = − 0 . 14 , π l = − 0 . 0263 . 13

Figure 3: Discount Rate Shock in Model without Investment, Three Discount Rate Shocks Rate of Discount Nominal interest rate Real Rate of Interest Output Percent Deviation from Steady State Output 4.5 20 4 50 4 Annual Percentage Return Annual Percentage Return 3.5 Annual Percentage Return 10 2 40 3 0 2.5 0 30 -10 2 -2 20 1.5 -20 1 -30 -4 10 0.5 -40 0 -6 0 -0.5 -50 0 10 20 0 10 20 0 10 20 0 10 20 Quarters Inflation Consumption Percent Deviation from Steady State Output 10 20 0 10 Annual Percentage Return 0 -10 -10 -20 -20 -30 -30 small shock -40 medium shock -40 large shock -50 larger shock -60 -50 0 10 20 0 10 20 Quarters Quarters

Model With Investment • Household Preferences: X ∞ β t [ u ( C t , M t /P t ) − v ( H t )] , E 0 t =0 • Discount Rate: 1 ¡ ¢ , β t = (1 + r n 0 ) (1 + r n 1 + r n 1 ) · · · t − 1 β t +1 1 = . 1 + r n β t t • Household Budget Constraint: Z 1 P t C t + M t + B t +1 ≤ M t − 1 + B t (1 + i t +1 ) + P t w t ( j ) H t ( j ) dj + T t 0 8

Model With Investment ... • Final Goods Production Function: ∙ Z 1 ¸ θ θ − 1 θ − 1 θ di Y t = y t ( j ) , θ > 1 . 0 • Intermediate Goods Production (Capital is firm-specific) µ h t ( i ) ¶ y t ( i ) = K t ( i ) f . K t ( i ) • Intermediate Goods Investment Technology: µ k t +1 ( i ) ¶ I t ( i ) = I k t ( i ) k t ( i ) 10

Model With Investment ... • Objective of Firms: ∞ X E t β t + j Λ t + j { (1 + τ ) P t + j ( i ) y t + j ( i ) − P t + j w t + j ( i ) h t + j ( i ) − P t + j I t + j ( i ) } . j =0 • Subsidy Eliminates Monopoly Power Distortions θ 1 + τ = θ − 1 • Resource Constraint and Production Technology: C t + I t + G t = Y t Z 1 I t = I t ( i ) di. 0 11

Figure 4: Discount Rate Shock in Model with Investment, Three Discount Rate Shocks Rate of Discount Nominal interest rate Real Rate of Interest Output Percent Deviation from Steady State Output 4.5 20 4 50 4 Annual Percentage Return Annual Percentage Return 3.5 Annual Percentage Return 10 2 40 3 0 2.5 0 30 -10 2 -2 20 1.5 -20 1 -30 -4 10 0.5 -40 0 -6 0 -0.5 -50 0 10 20 0 10 20 0 10 20 0 10 20 Quarters Inflation Consumption Investment Percent Deviation from Steady State Output Percent Deviation from Steady State Output 10 80 20 0 60 10 Annual Percentage Return 0 -10 40 -10 -20 20 -20 small shock -30 -30 medium shock 0 large shock -40 larger shock -40 -20 -50 -60 -50 0 10 20 0 10 20 0 10 20 Quarters Quarters Quarters

Increasing Government Spending When the Lower Bound Binds • In Steady State, G = 0 . 18 × Y steady state . • I Set G = . 1925 × Y steady state , for t = 1 , 2 , ..., 14 • With Small Preference Shock: – Lower Bound Not Binding and Multiplier Small (0.76 initially, and 0.41 eventually) – This is the Normal Government Spending Multiplier in DSGE Models. • With Largest Preference Shock, Government Spending Has Huge Impact. – This is What Happens in Textbook ‘Paradox of Thrift’ Analysis. 12

Figure 7: Dynamic Response to Small Shock, With (*) and Without (-) Increase in Gov't Spending Rate of Discount Nominal interest rate Real Rate of Interest Output Percent Deviation from Steady State Output 6 4 8 25 3.5 7 4 20 Annual Percentage Return Annual Percentage Return Annual Percentage Return 3 6 2 2.5 5 15 2 4 0 10 1.5 3 -2 1 2 5 0.5 1 -4 0 0 0 -6 -0.5 -1 -5 0 10 20 0 10 20 0 10 20 0 10 20 Inflation Consumption Investment Government Spending Multiplier (dY/dG) Percent Deviation from Steady State Output Percent Deviation from Steady State Output 6 35 20 0 4 30 Annual Percentage Return 2 25 15 -2 0 20 -2 15 -4 ratio -4 10 10 -6 -6 5 -8 5 0 -10 -8 -5 -12 -10 -14 -10 0 0 10 20 0 10 20 0 10 20 0 10 20 Quarters Quarters Quarters Quarters

Figure 8: Dynamic Response to Larger Shock, With (*) and Without (-) Increase in Gov't Spending Rate of Discount Nominal interest rate Real Rate of Interest Output Percent Deviation from Steady State Output 6 4 8 25 3.5 7 4 20 Annual Percentage Return Annual Percentage Return Annual Percentage Return 3 6 2 2.5 5 15 2 4 0 10 1.5 3 -2 1 2 5 0.5 1 -4 0 0 0 -6 -0.5 -1 -5 0 10 20 0 10 20 0 10 20 0 10 20 Inflation Consumption Investment Government Spending Multiplier (dY/dG) Percent Deviation from Steady State Output Percent Deviation from Steady State Output 6 35 20 0 4 30 Annual Percentage Return 2 25 15 -2 0 20 -2 15 -4 ratio -4 10 10 -6 -6 5 -8 5 0 -10 -8 -5 -12 -10 -14 -10 0 0 10 20 0 10 20 0 10 20 0 10 20 Quarters Quarters Quarters Quarters