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

UNCERTAINTY SHOCKS IN A MODEL

OF EFFECTIVE DEMAND: COMMENT

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.

INTRODUCTION

• 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 GROOT, RICHTER, AND THROCKMORTON: COMMENT

PREFERENCE SPECIFICATION

• BB preferences:

UBB

t

= [at(1 − β)u(ct, nt)(1−σ)/θ + β(Et[(UBB

t+1)1−σ])1/θ]θ/(1−σ)

• Distributional weights: at(1 − β) and β. If at = 1 for all t,

UBB

t

= u(ct, nt)1−β(Et[(UBB

t+1)1−σ])β/(1−σ), ψ = 1

• When at = 1, the weights do not sum to 1 and

lim

ψ→1− UBB t

= 0 (∞) for at > 1 (< 1), lim

ψ→1+ UBB t

= ∞ (0) for at > 1 (< 1).

• Alternative preferences:

UALT

t

=

• [(1 − atβ)u(ct, nt)(1−σ)/θ + atβ(Et[(UALT

t+1 )1−σ])1/θ]θ/(1−σ)

1 = ψ > 0 u(ct, nt)1−atβ(Et[(UALT

t+1 )1−σ])atβ/(1−σ)

ψ = 1

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

ENDOWMENT ECONOMY

• Model Setup:

◮ c0 = 1 − w, c1 = rw, ct = 1 for t ≥ 2 ◮ at = 1 for t = 0, 2, 3, . . . ◮ a1 = aH = 1 + ∆ w.p. p and aL = 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 = βrE0  a1

• cBB

cBB

1

1/ψ (V BB

1

)1−σ E0[(V BB

1

)1−σ] 1− 1

θ

 

◮ Alternative preferences:

1 = βrE0   1 − a1β 1 − β cALT cALT

1

1/ψ (V ALT

1

)1−σ E0[(V ALT

1

)1−σ] 1− 1

θ

 

• Use nonlinear solver to back out cj

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

ENDOWMENT ECONOMY ASYMPTOTE (% CHANGE FROM NO-UNCERTAINTY)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 IES (ψ)

• 0.02
• 0.01

0.01 0.02 Consumption (%)

BB Preferences Alternative Preferences

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

AUGMENTED DISCOUNT FACTOR

• Write the equilibrium condition as

1 = ˜ βjr(cj

0/cj 1)1/ψ

where ˜ β is an augmented discount factor.

• Define W j

1 ≡ (V j 1 )1/ψ−σ. Then ˜ βj ≡ β × E0[W j

1 ]

(E0[(W j

1 )θ/(θ−1)])(θ−1)/θ

• Risk Aversion Term

×

• 1 + cov0(˜

aj

1, W j 1 )

E0[W j

1 ]

• Covariance Term

where ˜ aBB

1

= a1 and ˜ aALT

1

= (1 − a1β)/(1 − β).

• Without loss of generality, evaluate ˜

βj at cj

1 = βr/(1 + β),

no-uncertainty level of period-1 consumption when ψ = 1.

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

DECOMPOSITION

0.5 1 1.5 2

IES (ψ)

0.9996 0.9998 1 1.0002 1.0004

Risk Aversion Term

0.5 1 1.5 2

IES (ψ)

0.9996 0.9998 1 1.0002 1.0004

Covariance Term

0.5 1 1.5 2

IES (ψ)

• 2

2 ×10-4

cov0(˜ a1, V j

1 )

0.5 1 1.5 2

IES (ψ)

• 2

2 ×10-4

cov0(˜ a1, W j

1 )

BB Preferences Alternative Preferences

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

RISK AVERSION AND UNCERTAINTY

0.5 1 1.5 2 IES (ψ)

• 0.12
• 0.08
• 0.04

0.04 0.08 0.12 Consumption (%) Risk Aversion (σ)

BB (σ = 2) BB (σ = 5) ALT (σ = 2) ALT (σ = 5)

0.5 1 1.5 2 IES (ψ)

• 0.12
• 0.08
• 0.04

0.04 0.08 0.12 Consumption (%) Uncertainty (∆)

BB (∆ = 0.02) BB (∆ = 0.05) ALT (∆ = 0.02) ALT (∆ = 0.05)

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

FULL BB MODEL

Textbook New Keynesian Model:

• Endogenous labor supply
• Endogenous investment with capital adjustment costs (φK)
• Variable capital utilization
• Sticky prices from Rotemberg price adjustment costs (φP)
• Central bank follows a Taylor rule
• Intertemporal preference (a) and technology shocks (z)
• Solved with third-order perturbation methods

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

BB MODEL ASYMPTOTE

0.5 1 1.5 2

IES (ψ)

• 0.15
• 0.1
• 0.05

0.05 0.1 0.15

Output (%) Volatility Shock

0.5 1 1.5 2

IES (ψ)

• 0.15
• 0.1
• 0.05

0.05 0.1 0.15

Consumption (%) Volatility Shock

0.5 1 1.5 2

IES (ψ)

• 0.15
• 0.1
• 0.05

0.05 0.1 0.15

Investment (%) Volatility Shock

0.5 1 1.5 2

IES (ψ)

• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

Output (%) Level Shock

0.5 1 1.5 2

IES (ψ)

0.1 0.2 0.3

Consumption (%) Level Shock

0.5 1 1.5 2

IES (ψ)

• 1
• 0.8
• 0.6
• 0.4
• 0.2

Investment (%) Level Shock

BB Preferences Alternative Preferences

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

ALTERNATIVE PREFERENCES

AND CAPITAL ADJUSTMENT COSTS

(DASHED LINE: BASELINE VALUE)

0.5 1 1.5 2

IES (ψ)

• 5

5

Output (%)

×10-4 0.5 1 1.5 2

IES (ψ)

• 15
• 10
• 5

Consumption (%)

×10-4 0.5 1 1.5 2

IES (ψ)

2 4 6 8

Investment (%)

×10-3 φK = 0 φK = 2.09 φK = 4 φK = 16

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

ALTERNATIVE PREFERENCES

AND RISK AVERSION

(DASHED LINE: BASELINE VALUE)

0.5 1 1.5 2

IES (ψ)

• 2
• 1

1

Output (%)

×10-3 0.5 1 1.5 2

IES (ψ)

• 3
• 2
• 1

Consumption (%)

×10-3 0.5 1 1.5 2

IES (ψ)

2 4 6

Investment (%)

×10-3 σ = 2 σ = 80 σ = 160 σ = 1000

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

ALTERNATIVE PREFERENCES

AND PRICE ADJUSTMENT COSTS

(DASHED LINE: BASELINE VALUE)

0.5 1 1.5 2

IES (ψ)

5 10 15

Output (%)

×10-4 0.5 1 1.5 2

IES (ψ)

• 15
• 10
• 5

Consumption (%)

×10-4 0.5 1 1.5 2

IES (ψ)

5 10

Investment (%)

×10-3 φP = 0 φP = 100 φP = 200 φP = 1000

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

BB PREFERENCES LARGER SHOCKS (SOLID LINE: BASELINE VALUE)

0.5 1 1.5 2

IES (ψ)

• 0.2
• 0.1

0.1 0.2

Output (%)

0.5 1 1.5 2

IES (ψ)

• 0.2
• 0.1

0.1 0.2

Consumption (%)

0.5 1 1.5 2

IES (ψ)

• 0.2
• 0.1

0.1 0.2

Investment (%)

σa = 0.0026 σa = 0.0053 σa = 0.0263

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

ALT PREFERENCES LARGER SHOCKS (SOLID LINE: BASELINE VALUE)

0.5 1 1.5 2

IES (ψ)

2 4 6

Output (%)

×10-3 0.5 1 1.5 2

IES (ψ)

• 15
• 10
• 5

Consumption (%)

×10-3 0.5 1 1.5 2

IES (ψ)

0.02 0.04 0.06

Investment (%)

σa = 0.0026 σa = 0.0053 σa = 0.0263

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

DISASTER RISK SHOCKS

• Preferences:

UBB

t

= [(1 − β)(ad

t u(ct, nt))(1−σ)/θ + β(Et[(UBB t+1)1−σ])1/θ]θ/(1−σ)

• Asymptote no longer appears with IES = 1
• ad

t = (aBB t

)1−1/ψ, so the volatility of ad

t rises as IES → 0 1 2

IES (ψ)

• 0.2
• 0.15
• 0.1
• 0.05

Output (%)

1 2

IES (ψ)

• 0.2
• 0.15
• 0.1
• 0.05

Consumption (%)

1 2

IES (ψ)

• 0.2
• 0.15
• 0.1
• 0.05

Investment (%)

DE GROOT, RICHTER, AND THROCKMORTON: COMMENT

CONCLUSION

• 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 GROOT, RICHTER, AND THROCKMORTON: COMMENT