Risk and Ambiguity in Models of Business Cycles Dave Backus, Axelle - PowerPoint PPT Presentation

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Risk and Ambiguity in Models of Business Cycles Dave Backus, Axelle Ferriere, and Stan Zin Carnegie-Rochester-NYU Conference April 18, 2014 This version: April 19, 2014 Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex What happened? What we see ◮ Much deeper recession than usual ◮ Slower recovery [??] Like the Kydland-Prescott model with productivity shocks? ◮ Relative magnitudes look right ◮ Comovements look right ◮ But... measured productivity didn’t fall very much More What’s missing? Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex What happened? Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex What happened? Marco Buti, Director General of the European Commission Economic theory suggests that uncertainty has a detrimental effect on economic activity by giving agents the incentive to postpone investment, consumption and employment decisions until uncertainty is resolved, and by pushing up the cost of capital through increased risk premia. Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex What happened? Nick Bloom The onset of the Great Recession was accompanied by a massive surge in uncertainty . The size of this uncertainty shock was so large it potentially accounted for around one third of the 9% drop in GDP versus trend during 2008-2009. Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex What happened? Gary Becker, Steve Davis, and Kevin Murphy Faced with a highly uncertain policy environment , the prudent course is to set aside or delay costly commitments that are hard to reverse. The result is reluctance by banks to increase lending, reluctance by businesses to undertake new capital expenditures or expand work forces, and decisions by households to postpone major purchases. Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex What we do Take a streamlined business cycle model Ask: How does uncertainty affect the dynamics of output, consumption, and investment? ◮ How are business cycle properties affected by fluctuations in uncertainty? ◮ Can “uncertainty shocks” magnify downturns or produce slow recoveries? Potential issues: Barro-King, Tallarini Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Modeling ingredients Streamlined business cycle model - Recursive preferences - Unit root in productivity - Fixed labor supply With fluctuations in uncertainty - Risk (stochastic volatility) - Ambiguity (unobservable long-term growth) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Preview of results Solution method Transparent loglinear approximation Checked numerically Business cycle properties Separation property Intertemporal elasticity of substitution vs. risk aversion Risk vs. ambiguity Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Risk and uncertainty Recursive references U t = V [ c t , µ t ( U t +1 )] t + βµ t ( U t +1 ) ρ ] 1 /ρ [(1 − β ) c ρ = [ E t ( U α t +1 )] 1 /α µ t ( U t +1 ) = V , µ t homogeneous of degree one, α, ρ < 1, σ = 1 / (1 − ρ ) Stochastic structure of productivity a t log ( a t / a t − 1 ) = log g + e ⊤ x t (“productivity growth”) log g t = Ax t + v 1 / 2 x t +1 = Bw 1 t +1 (“news”) t v t +1 = (1 − ϕ v ) v + ϕ v v t + τ w 2 t +1 (“risk”) ( w 1 t , w 2 t ) = iid standard normals Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Stationarity Bellman equation � � J ( k t , x t , v t , a t ) = max V c t , µ t [ J ( k t +1 , x t +1 , v t +1 , a t +1 )] c t s.t. k t +1 = f ( k t , a t n ) − c t plus shocks & initial conditions Assume f hd 1: f ( k , an ) = k ω ( an ) 1 − ω + (1 − δ ) k Rescaled Bellman equation [˜ k t = k t / a t , ˜ c t = c t / a t ] J (˜ c t , µ t [ g t +1 J (˜ � � k t , x t , v t , 1) = max V ˜ k t +1 , x t +1 , v t +1 , 1)] ˜ c t g t +1 ˜ k t +1 = f (˜ s.t. k t , n ) − ˜ c t plus shocks & initial conditions More Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Recursive business cycle model Bellman equation t + βµ t ( g t +1 J t +1 ) ρ ] 1 /ρ c ρ J t = max [(1 − β )˜ ˜ c t k t +1 = [ f (˜ ˜ s.t. k t , n ) − ˜ c t ] / g t +1 plus shocks & initial conditions First-order and envelope conditions c ρ − 1 βµ t ( g t +1 J t +1 ) ρ − α E t [( g t +1 J t +1 ) α − 1 J kt +1 ] (1 − β )˜ = t J 1 − ρ βµ t ( g t +1 J t +1 ) ρ − α E t [( g t +1 J t +1 ) α − 1 J kt +1 ] f kt J kt = t c ρ − 1 J ρ − 1 ⇒ (1 − β )˜ = J kt / f kt t t Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Loglinearization strategy Goal: loglinear decision rule and (controlled) law of motion h ck log ˜ k t + h ⊤ log ˜ c t = cx x t + h cv v t log ˜ h kk log ˜ k t + h ⊤ k t +1 = kx x t + h kv v t − log g t +1 Dynamic programming version of Campbell (JME, 1994) Loglinearization around the stochastic steady-state Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Loglinear approximation Guess loglinear value function p k log ˜ k t + p ⊤ log J t = x x t + p v v t + p 0 Loglinearize capital’s marginal product and law of motion λ r log ˜ log f kt = k t + λ 0 log ˜ λ k log ˜ k t +1 = k t − λ c log ˜ c t + λ 1 − log g t +1 More Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Loglinearized policies 1 The law of motion for capital is log ˜ h k 0 + h kk log ˜ k t + h ⊤ k t +1 = kx x t + h kv v t − log g t +1 h kk = λ k − λ c [ σ + (1 − σ ) p k + σλ r ] h ⊤ kx = − λ c (1 − σ ) p ⊤ x h kv = − λ c (1 − σ ) p v Tallarini ( σ = 1) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Loglinearized policies 2 The law of motion for capital is log ˜ h k 0 + h kk log ˜ k t + h ⊤ k t +1 = kx x t + h kv v t − log g t +1 h kk = λ k − λ c [ σ + (1 − σ ) p k + σλ r ] h ⊤ kx = − λ c (1 − σ ) p ⊤ x h kv = − λ c (1 − σ ) p v Separation property ◮ h kk is independent of risk and risk aversion. � �� � ρ p k − 1 − λ r = ρ p k − 1 λ k − λ c [(1 − σ ) p k + σ (1 + λ r )] . Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Loglinearized policies 3 The law of motion for capital is log ˜ h k 0 + h kk log ˜ k t + h ⊤ k t +1 = kx x t + h kv v t − log g t +1 h kk = λ k − λ c [ σ + (1 − σ ) p k + σλ r ] h ⊤ kx = − λ c (1 − σ ) p ⊤ x h kv = − λ c (1 − σ ) p v Exposure to shocks ( x t , v t ) ◮ If σ < 1, an increase in v t lowers consumption ◮ Magnitude depends on α ( p v ∝ α ) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk in a business cycle model Ambiguity in a business cycle model Conclusion Annex Loglinear approximation 3.8 0.15 numerical solution 3.7 0.12 3.6 3.5 density measure 0.09 log k(t+1) 3.4 0.06 3.3 3.2 0.03 3.1 3 0 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 log k(t) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity