...
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 Potentially 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:
E0
∞
X
t=0
β t [u(Ct, Mt/Pt) − v(Ht)] ,
• Discount Rate:
1
¢,
¡
n
n
n
(1 + r0 ) (1 + r1 ) · · · 1 + rt−1
1
=
.
1 + rtn
βt =
β t+1
βt
6
Model ...
• Experiment:
r0n 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 π t = Pt/Pt−1 = 1, if possible
otherwise, set it = 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
Rate
Lower
bound
Inelastic
Investment
Elastic
Investment
Saving
A
C
B
Saving, Investment
Simple Algebra of Eggertsson-Woodford
• Linearized Intertemporal Euler Equation (‘IS Curve’)
xt = Etxt+1 − σ (ı̂t − Etπ̂ t+1 − r̂tn)
– Here:
xt =
ı̂t =
π̂ t =
r̂tn =
ct − c
c
it − i
1+i
πt − π
π n
n
rt − r
1 + rn
(
1
= β)
1+i
(π = 1)
(
1
= β ).
n
1+r
• Linearized Calvo Equation:
π̂ t = βEtπ̂ t+1 + κxt.
8
Simple Algebra of Eggertsson-Woodford ...
• Implications of Zero Bound For ı̂t :
ı̂t ≡
it − i
= β (it + 1) − 1
1+i
so, ı̂t ≥ β − 1
• Monetary Policy:
Set π̂ t = 0, unless this Implies ı̂t < β − 1
If π̂ t = 0 Implies ı̂t < β − 1, Set ı̂t = β − 1 and Let π̂ t Be Determined Endogenously
10
Simple Algebra of Eggertsson-Woodford ...
• Equations of Model:
xt = Etxt+1 − σ (ı̂t − Etπ̂ t+1 − r̂tn)
π̂t = βEtπ̂ t+1 + κxt.
• In Steady State:
xt = π̂ t = ı̂t = r̂tn = 0
• Suppose r̂tn < 0
– If r̂tn ≥ β − 1, Set ı̂t = r̂tn, And xt = π̂ t = 0 Is Still Equilibrium
– If r̂tn < β − 1, ı̂t = β − 1, π̂ t is Free
11
Simple Algebra of Eggertsson-Woodford ...
• What Happens if r̂tn < β − 1?
• Depends on Expectations About the Future!
• Here is the E-W Setup:
– In Period 0 and 1 :
r̂0n < β − 1 = r̂ln
r̂1n
=
– In Period t :
½
r̂ln probability p
0 probability 1 − p
n
if r̂t−1
= 0, r̂tn = 0
or,
r̂tn
=
½
r̂ln probability p
0 probability 1 − p
12
Simple Algebra of Eggertsson-Woodford ...
• Equilibrium is Simple to Compute!
• In Low State,
π̂ t = π̂ l , xt = xl
• Find These Variables by Solving:
xl = pxl − σ((β − 1) − pπ l − r̂ln)
π l = βpπ l + κxl
• Parameterization:
p = 0.9, σ = 0.5, κ = 0.02, β = 0.99, rln = −.02/4.
xl = −0.14, π l = −0.0263.
13
Rate of Discount
Nominal interest rate
Real Rate of Interest
4.5
0
-2
-4
3.5
3
2.5
2
1.5
1
0.5
0
-6
10
Inflation
20
10
0
-10
-20
-30
small shock
medium shock
large shock
larger shock
-40
-50
0
10
Quarters
-0.5
20
20
Percent Deviation from Steady State Output
0
Annual Percentage Return
Annual Percentage Return
2
-60
50
4
Annual Percentage Return
Annual Percentage Return
4
30
20
10
0
0
10
20
Consumption
10
0
-10
-20
-30
-40
-50
40
0
10
Quarters
20
0
10
20
Percent Deviation from Steady State Output
Figure 3: Discount Rate Shock in Model without Investment, Three Discount Rate Shocks
Output
20
10
0
-10
-20
-30
-40
-50
0
10
Quarters
20
Model With Investment
• Household Preferences:
E0
∞
X
t=0
• Discount Rate:
β t [u(Ct, Mt/Pt) − v(Ht)] ,
1
¢,
¡
βt =
n
n
n
(1 + r0 ) (1 + r1 ) · · · 1 + rt−1
1
β t+1
=
.
n
βt
1 + rt
• Household Budget Constraint:
PtCt + Mt + Bt+1 ≤ Mt−1 + Bt(1 + it+1) +
Z
1
Ptwt(j)Ht(j)dj + Tt
0
8
Model With Investment ...
• Final Goods Production Function:
Yt =
∙Z
0
1
θ
¸ θ−1
θ−1
yt(j) θ di
, 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
µ
¶
kt+1(i)
kt(i)
kt(i)
10
Model With Investment ...
• Objective of Firms:
Et
∞
X
j=0
β t+j Λt+j {(1 + τ )Pt+j (i)yt+j (i) − Pt+j wt+j (i)ht+j (i) − Pt+j It+j (i)} .
• Subsidy Eliminates Monopoly Power Distortions
θ
1+τ =
θ−1
• Resource Constraint and Production Technology:
Ct + It + Gt = Yt
It =
Z
1
It(i)di.
0
11
Rate of Discount
Nominal interest rate
Real Rate of Interest
4.5
0
-2
-4
3.5
3
2.5
2
1.5
1
0.5
0
10
Inflation
20
10
0
-10
-20
small shock
medium shock
large shock
larger shock
-30
-40
-50
0
10
Quarters
-0.5
20
20
Percent Deviation from Steady State Output
0
0
10
30
20
10
20
Consumption
10
0
-10
-20
-30
-40
-50
40
0
0
10
Quarters
20
0
Percent Deviation from Steady State Output
-6
Annual Percentage Return
Annual Percentage Return
2
-60
50
4
Annual Percentage Return
Annual Percentage Return
4
10
20
Investment
80
60
40
20
0
-20
0
10
Quarters
20
Percent Deviation from Steady State Output
Figure 4: Discount Rate Shock in Model with Investment, Three Discount Rate Shocks
Output
20
10
0
-10
-20
-30
-40
-50
0
10
Quarters
20
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
Rate of Discount
Nominal interest rate
0
-2
-4
8
3.5
7
3
2.5
2
1.5
1
0.5
0
10
Inflation
Annual Percentage Return
0
-2
-4
-6
-8
-10
0
10
Quarters
-0.5
20
20
0
10
4
3
2
1
Consumption
4
2
0
-2
-4
-6
-8
-10
-12
0
10
Quarters
-1
20
6
-14
5
0
20
Percent Deviation from Steady State Output
0
Percent Deviation from Steady State Output
-6
6
0
10
20
Investment
Output
25
20
15
10
5
0
-5
0
10
20
Government Spending Multiplier (dY/dG)
20
35
30
25
15
20
15
ratio
2
Real Rate of Interest
4
Annual Percentage Return
4
Annual Percentage Return
Annual Percentage Return
6
Percent Deviation from Steady State Output
Figure 7: Dynamic Response to Small Shock, With (*) and Without (-) Increase in Gov't Spending
10
10
5
5
0
-5
-10
0
10
Quarters
20
0
0
10
Quarters
20
Rate of Discount
Nominal interest rate
0
-2
-4
8
3.5
7
3
2.5
2
1.5
1
0.5
0
10
Inflation
Annual Percentage Return
0
-2
-4
-6
-8
-10
0
10
Quarters
-0.5
20
20
0
10
4
3
2
1
Consumption
4
2
0
-2
-4
-6
-8
-10
-12
0
10
Quarters
-1
20
6
-14
5
0
20
Percent Deviation from Steady State Output
0
Percent Deviation from Steady State Output
-6
6
0
10
20
Investment
Output
25
20
15
10
5
0
-5
0
10
20
Government Spending Multiplier (dY/dG)
20
35
30
25
15
20
15
ratio
2
Real Rate of Interest
4
Annual Percentage Return
4
Annual Percentage Return
Annual Percentage Return
6
Percent Deviation from Steady State Output
Figure 8: Dynamic Response to Larger Shock, With (*) and Without (-) Increase in Gov't Spending
10
10
5
5
0
-5
-10
0
10
Quarters
20
0
0
10
Quarters
20
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.
12