( % CHANGE FROM NO - UNCERTAINTY ) 0.02 BB Preferences Alternative - PowerPoint PPT Presentation

U NCERTAINTY S HOCKS IN A M ODEL OF E FFECTIVE D EMAND : C OMMENT Oliver de Groot University of St Andrews Alexander W. Richter Federal Reserve Bank of Dallas Nathaniel A. Throckmorton College of William & Mary The views expressed in this presentation are our own and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System.

I NTRODUCTION • Do uncertainty shocks have big effects in macro models? • Basu and Bundick (2017): demand uncertainty shocks generate meaningful declines in output and positive comovement between consumption and investment. • Demand uncertainty is modeled as a stochastic volatility shock to a household’s intertemporal preferences within an Epstein and Zin (1991) recursive preference specification. • If the distributional weights on current and future utility do not sum to 1 , there is an asymptote in the response to the shock with unit intertemporal elasticity of substitution (IES). • In BB the sum of the weights is not 1 and the IES is 0 . 95 , so the asymptote significantly magnifies the responses. DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

P REFERENCE S PECIFICATION • BB preferences: = [ a t (1 − β ) u ( c t , n t ) (1 − σ ) /θ + β ( E t [( U BB U BB t +1 ) 1 − σ ]) 1 /θ ] θ/ (1 − σ ) t • Distributional weights: a t (1 − β ) and β . If a t = 1 for all t , U BB = u ( c t , n t ) 1 − β ( E t [( U BB t +1 ) 1 − σ ]) β/ (1 − σ ) , ψ = 1 t • When a t � = 1 , the weights do not sum to 1 and ψ → 1 − U BB lim = 0 ( ∞ ) for a t > 1 ( < 1) , t ψ → 1 + U BB lim = ∞ (0) for a t > 1 ( < 1) . t • Alternative preferences: � [(1 − a t β ) u ( c t , n t ) (1 − σ ) /θ + a t β ( E t [( U ALT t +1 ) 1 − σ ]) 1 /θ ] θ/ (1 − σ ) 1 � = ψ > 0 U ALT = t u ( c t , n t ) 1 − a t β ( E t [( U ALT t +1 ) 1 − σ ]) a t β/ (1 − σ ) ψ = 1 DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

E NDOWMENT E CONOMY • Model Setup: ◮ c 0 = 1 − w , c 1 = rw , c t = 1 for t ≥ 2 ◮ a t = 1 for t = 0 , 2 , 3 , . . . ◮ a 1 = a H = 1 + ∆ w.p. p and a L = 1 − ∆ w.p. 1 − p • Solve the model with the BB and alternative preferences • Equilibrium ( V j : value function, j ∈ { BB, ALT } ): ◮ BB Preferences:   � 1 − 1 � � 1 /ψ � c BB ( V BB ) 1 − σ θ 0 1 1 = βrE 0  a 1  c BB E 0 [( V BB ) 1 − σ ] 1 1 ◮ Alternative preferences:   � 1 − a 1 β � � � 1 /ψ � � 1 − 1 c ALT ( V ALT ) 1 − σ θ 0 1 1 = βrE 0   c ALT E 0 [( V ALT ) 1 − σ ] 1 − β 1 1 • Use nonlinear solver to back out c j 0 DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

E NDOWMENT E CONOMY A SYMPTOTE ( % CHANGE FROM NO - UNCERTAINTY ) 0.02 BB Preferences Alternative Preferences 0.01 Consumption (%) 0 -0.01 -0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

A UGMENTED D ISCOUNT F ACTOR • Write the equilibrium condition as 1 = ˜ β j r ( c j 0 /c j 1 ) 1 /ψ β is an augmented discount factor. where ˜ • Define W j 1 ≡ ( V j 1 ) 1 /ψ − σ . Then � � E 0 [ W j a j 1 , W j 1 ] 1 + cov 0 (˜ 1 ) β j ≡ β × ˜ × ( E 0 [( W j E 0 [ W j 1 ) θ/ ( θ − 1) ]) ( θ − 1) /θ 1 ] � �� � � �� � Risk Aversion Term Covariance Term a BB a ALT where ˜ = a 1 and ˜ = (1 − a 1 β ) / (1 − β ) . 1 1 β j at c j • Without loss of generality, evaluate ˜ 1 = βr/ (1 + β ) , no-uncertainty level of period- 1 consumption when ψ = 1 . DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

D ECOMPOSITION BB Preferences Alternative Preferences Risk Aversion Term Covariance Term 1.0004 1.0004 1.0002 1.0002 1 1 0.9998 0.9998 0.9996 0.9996 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) a 1 , V j a 1 , W j 2 × 10 -4 2 × 10 -4 cov 0 (˜ 1 ) cov 0 (˜ 1 ) 0 0 -2 -2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

R ISK A VERSION AND U NCERTAINTY Risk Aversion ( σ ) Uncertainty ( ∆ ) 0.12 0.12 BB ( σ = 2) BB ( ∆ = 0 . 02) 0.08 0.08 BB ( σ = 5) BB ( ∆ = 0 . 05) ALT ( σ = 2) ALT ( ∆ = 0 . 02) ALT ( σ = 5) ALT ( ∆ = 0 . 05) Consumption (%) Consumption (%) 0.04 0.04 0 0 -0.04 -0.04 -0.08 -0.08 -0.12 -0.12 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

F ULL BB M ODEL Textbook New Keynesian Model: • Endogenous labor supply • Endogenous investment with capital adjustment costs ( φ K ) • Variable capital utilization • Sticky prices from Rotemberg price adjustment costs ( φ P ) • Central bank follows a Taylor rule • Intertemporal preference ( a ) and technology shocks ( z ) • Solved with third-order perturbation methods DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

BB M ODEL A SYMPTOTE BB Preferences Alternative Preferences Level Shock Level Shock Level Shock -0.02 0.3 -0.2 Consumption (%) Investment (%) Output (%) -0.04 -0.4 0.2 -0.06 -0.6 0.1 -0.08 -0.8 -0.1 0 -1 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) IES ( ψ ) Volatility Shock Volatility Shock Volatility Shock 0.15 0.15 0.15 Consumption (%) Investment (%) 0.1 0.1 0.1 Output (%) 0.05 0.05 0.05 0 0 0 -0.05 -0.05 -0.05 -0.1 -0.1 -0.1 -0.15 -0.15 -0.15 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

A LTERNATIVE P REFERENCES AND C APITAL A DJUSTMENT C OSTS (D ASHED L INE : B ASELINE V ALUE ) φ K = 0 φ K = 2 . 09 φ K = 4 φ K = 16 × 10 -4 × 10 -4 × 10 -3 8 0 Consumption (%) 5 Investment (%) 6 Output (%) -5 4 0 2 -10 0 -5 -15 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

A LTERNATIVE P REFERENCES AND R ISK A VERSION (D ASHED L INE : B ASELINE V ALUE ) σ = 2 σ = 80 σ = 160 σ = 1000 × 10 -3 × 10 -3 × 10 -3 1 6 0 Consumption (%) Investment (%) Output (%) 0 4 -1 -1 2 -2 -2 -3 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

A LTERNATIVE P REFERENCES AND P RICE A DJUSTMENT C OSTS (D ASHED L INE : B ASELINE V ALUE ) φ P = 0 φ P = 100 φ P = 200 φ P = 1000 × 10 -4 × 10 -4 × 10 -3 10 15 0 Consumption (%) Investment (%) Output (%) 10 -5 5 5 -10 0 0 -15 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

BB P REFERENCES L ARGER S HOCKS (S OLID L INE : B ASELINE V ALUE ) σ a = 0 . 0026 σ a = 0 . 0053 σ a = 0 . 0263 0.2 0.2 0.2 Consumption (%) Investment (%) 0.1 0.1 0.1 Output (%) 0 0 0 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

ALT P REFERENCES L ARGER S HOCKS (S OLID L INE : B ASELINE V ALUE ) σ a = 0 . 0026 σ a = 0 . 0053 σ a = 0 . 0263 × 10 -3 × 10 -3 0.06 0 6 Consumption (%) Investment (%) Output (%) 0.04 -5 4 0.02 -10 2 0 -15 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 IES ( ψ ) IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

D ISASTER R ISK S HOCKS • Preferences: t u ( c t , n t )) (1 − σ ) /θ + β ( E t [( U BB U BB = [(1 − β )( a d t +1 ) 1 − σ ]) 1 /θ ] θ/ (1 − σ ) t • Asymptote no longer appears with IES = 1 • a d t = ( a BB ) 1 − 1 /ψ , so the volatility of a d t rises as IES → 0 t Output (%) Consumption (%) Investment (%) 0 0 0 -0.05 -0.05 -0.05 -0.1 -0.1 -0.1 -0.15 -0.15 -0.15 -0.2 -0.2 -0.2 0 1 2 0 1 2 0 1 2 IES ( ψ ) IES ( ψ ) IES ( ψ ) DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT

C ONCLUSION 1. BB results rest on an—until now—undetected asymptote 2. Without the influence of the asymptote, demand uncertainty shocks have very little effect on real activity 3. Future work: resolve the uncertainty puzzle—why models struggle to generate sizeable movements in economic activity in response to changes in uncertainty DE G ROOT , R ICHTER , AND T HROCKMORTON : C OMMENT