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

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The Lower Bound in DSGE Models

by Lawrence J. Christiano

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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.

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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.

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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

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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

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Model

• Household Preferences:

E0

∞

X

t=0

βt [u(Ct, Mt/Pt) − v(Ht)] ,

• Discount Rate:

βt = 1 (1 + rn

0) (1 + rn 1) · · ·

¡ 1 + rn

t−1

¢, βt+1 βt = 1 1 + rn

t

.

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Model ...

• Experiment:

rn

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, it, so that πt = Pt/Pt−1 = 1, if possible

• therwise, set it = 0 and let market forces determine πt

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

Simple Algebra of Eggertsson-Woodford

• Linearized Intertemporal Euler Equation (‘IS Curve’)

xt = Etxt+1 − σ (ˆ ıt − Etˆ πt+1 − ˆ rn

t )

– Here:

xt = ct − c c ˆ ıt = it − i 1 + i ( 1 1 + i = β) ˆ πt = πt − π π

(π = 1)

ˆ rn

t = rn t − rn

1 + rn

(

1 1 + rn = β).

• Linearized Calvo Equation:

ˆ πt = βEtˆ πt+1 + κxt.

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Simple Algebra of Eggertsson-Woodford ...

• Implications of Zero Bound For ˆ

ıt : ˆ ıt ≡ it − i 1 + i = β (it + 1) − 1

so, ˆ

ıt ≥ β − 1

• Monetary Policy:

Set ˆ

πt = 0, unless this Implies ˆ ıt < β − 1

If ˆ

πt = 0 Implies ˆ ıt < β − 1, Set ˆ ıt = β − 1 and Let ˆ πt Be Determined Endogenously

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Simple Algebra of Eggertsson-Woodford ...

• Equations of Model:

xt = Etxt+1 − σ (ˆ ıt − Etˆ πt+1 − ˆ rn

t )

ˆ πt = βEtˆ πt+1 + κxt.

• In Steady State:

xt = ˆ πt = ˆ ıt = ˆ rn

t = 0

• Suppose ˆ

rn

t < 0

– If ˆ

rn

t ≥ β − 1, Set ˆ

ıt = ˆ rn

t , And xt = ˆ

πt = 0 Is Still Equilibrium

– If ˆ

rn

t < β − 1, ˆ

ıt = β − 1, ˆ πt is Free

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Simple Algebra of Eggertsson-Woodford ...

• What Happens if ˆ

rn

t < β − 1?

• Depends on Expectations About the Future!
• Here is the E-W Setup:

– In Period 0 and 1 :

ˆ rn

0 < β − 1 = ˆ

rn

l

ˆ rn

1 =

½ ˆ rn

l

probability p

0 probability 1 − p

– In Period t : if ˆ

rn

t−1 = 0, ˆ

rn

t = 0

• r, ˆ

rn

t =

½ ˆ rn

l

probability p

0 probability 1 − p

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Simple Algebra of Eggertsson-Woodford ...

• Equilibrium is Simple to Compute!
• In Low State,

ˆ πt = ˆ πl, xt = xl

• Find These Variables by Solving:

xl = pxl − σ((β − 1) − pπl − ˆ rn

l )

πl = βpπl + κxl

• Parameterization:

p = 0.9, σ = 0.5, κ = 0.02, β = 0.99, rn

l = −.02/4.

xl = −0.14, πl = −0.0263.

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10 20

• 60
• 50
• 40
• 30
• 20
• 10

10 20 Inflation Annual Percentage Return Quarters small shock medium shock large shock larger shock 10 20

• 50
• 40
• 30
• 20
• 10

10 20 Output Percent Deviation from Steady State Output Quarters 10 20

• 50
• 40
• 30
• 20
• 10

10 Consumption Quarters Percent Deviation from Steady State Output 10 20

• 0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 Nominal interest rate Annual Percentage Return 10 20

• 6
• 4
• 2

2 4 Rate of Discount Annual Percentage Return

Figure 3: Discount Rate Shock in Model without Investment, Three Discount Rate Shocks

10 20 10 20 30 40 50 Real Rate of Interest Annual Percentage Return

Model With Investment

• Household Preferences:

E0

∞

X

t=0

βt [u(Ct, Mt/Pt) − v(Ht)] ,

• Discount Rate:

βt = 1 (1 + rn

0) (1 + rn 1) · · ·

¡ 1 + rn

t−1

¢, βt+1 βt = 1 1 + rn

t

.

• Household Budget Constraint:

PtCt + Mt + Bt+1 ≤ Mt−1 + Bt(1 + it+1) + Z 1 Ptwt(j)Ht(j)dj + Tt

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Model With Investment ...

• Final Goods Production Function:

Yt = ∙Z 1 yt(j)

θ−1 θ di

¸ θ

θ−1

, θ > 1.

• Intermediate Goods Production (Capital is firm-specific)

yt(i) = Kt(i)f µ ht(i) Kt(i) ¶ .

• Intermediate Goods Investment Technology:

It(i) = I µkt+1(i) kt(i) ¶ kt(i)

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Model With Investment ...

• Objective of Firms:

Et

∞

X

j=0

βt+jΛt+j {(1 + τ)Pt+j(i)yt+j(i) − Pt+jwt+j(i)ht+j(i) − Pt+jIt+j(i)} .

• Subsidy Eliminates Monopoly Power Distortions

1 + τ = θ θ − 1

• Resource Constraint and Production Technology:

Ct + It + Gt = Yt It = Z 1 It(i)di.

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10 20

• 60
• 50
• 40
• 30
• 20
• 10

10 20 Inflation Annual Percentage Return Quarters small shock medium shock large shock larger shock 10 20

• 50
• 40
• 30
• 20
• 10

10 20 Output Percent Deviation from Steady State Output Quarters 10 20

• 20

20 40 60 80 Investment Quarters Percent Deviation from Steady State Output 10 20

• 50
• 40
• 30
• 20
• 10

10 Consumption Quarters Percent Deviation from Steady State Output 10 20

• 0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 Nominal interest rate Annual Percentage Return 10 20

• 6
• 4
• 2

2 4 Rate of Discount Annual Percentage Return

Figure 4: Discount Rate Shock in Model with Investment, Three Discount Rate Shocks

10 20 10 20 30 40 50 Real Rate of Interest Annual Percentage Return

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.

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10 20

• 10
• 8
• 6
• 4
• 2

Inflation Annual Percentage Return Quarters 10 20

• 5

5 10 15 20 25 Output Percent Deviation from Steady State Output 10 20

• 10
• 5

5 10 15 20 25 30 35 Investment Quarters Percent Deviation from Steady State Output 10 20

• 14
• 12
• 10
• 8
• 6
• 4
• 2

2 4 6 Consumption Quarters Percent Deviation from Steady State Output 10 20

• 0.5

0.5 1 1.5 2 2.5 3 3.5 4 Nominal interest rate Annual Percentage Return 10 20

• 6
• 4
• 2

2 4 6 Rate of Discount Annual Percentage Return 10 20

• 1

1 2 3 4 5 6 7 8 Real Rate of Interest Annual Percentage Return

Figure 7: Dynamic Response to Small Shock, With (*) and Without (-) Increase in Gov't Spending

10 20 5 10 15 20 Government Spending Multiplier (dY/dG) ratio Quarters

10 20

• 10
• 8
• 6
• 4
• 2

Inflation Annual Percentage Return Quarters 10 20

• 5

5 10 15 20 25 Output Percent Deviation from Steady State Output 10 20

• 10
• 5

5 10 15 20 25 30 35 Investment Quarters Percent Deviation from Steady State Output 10 20

• 14
• 12
• 10
• 8
• 6
• 4
• 2

2 4 6 Consumption Quarters Percent Deviation from Steady State Output 10 20

• 0.5

0.5 1 1.5 2 2.5 3 3.5 4 Nominal interest rate Annual Percentage Return 10 20

• 6
• 4
• 2

2 4 6 Rate of Discount Annual Percentage Return 10 20

• 1

1 2 3 4 5 6 7 8 Real Rate of Interest Annual Percentage Return

Figure 8: Dynamic Response to Larger Shock, With (*) and Without (-) Increase in Gov't Spending

10 20 5 10 15 20 Government Spending Multiplier (dY/dG) ratio Quarters

Conclusion

• E-W Have Produced a Very Sharp Example of the Sort of Things People Might

Have in Mind When they Worry About Low Inflation.

• It is Interesting to Investigate Robustness to:

– Presence of Investment – Other types of Shocks, other Frictions

• Analysis Suggests that DSGE Models Do Form a Case that Inflation Targetting

in a Low Inflation Environment Exposes an Economy To Risks Due to Lower Bound Considerations. – In Worst Case Scenario, Can Expand Fiscal Policy

• Is Japan in a Low rn E-W Trap?

– Conjecture: Model Predicts Large Y Effect From Positive G. – Japan did Increase G, So What Is Happening in Japan Must Not Reflect the Lower Bound Considerations Raised by Eggertsson and Woodford.

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