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Demographics, Redistribution, and Optimal
In‡ation
James Bullard, FRB of St. Louis
Carlos Garriga, FRB of St. Louis
Christopher J. Waller, FRB of St. Louis
2012 BOJ-IMES Conference
Demographic Changes and Macroeconomic Performance
The views expressed herein do not necessarily re‡ect those of
the FOMC or the Federal Reserve System.
In‡ation and Demography
I
Can observed low in‡ation outcomes be related to
demographic factors such as an aging population?
I
A basic “back-of-the-envelop” suggests NO
I
Consider an economy were capital and money are perfect
substitutes
1
r = δ+n =
1+π
I
The e¤ect of a permanent increase in n0 > n increases the
return of capital r 0 > r and in‡ation decreases to π 0 .
I
Countries with relatively young (old) populations would have
relatively low (high) in‡ation rates, all else equal.
In‡ation and Demography: Japan
In‡ation and Demography: USA
Objective
I
Understand the determination of central bank objectives when
population aging shifts the social preferences for redistribution
and its implications for in‡ation.
I
The intergenerational redistribution tension is intrinsic in
life-cycle models.
I
I
I
Young cohorts do not have any assets and wages are the main
source of income.
Old generations cannot work and prefer a high rate of return
from their savings.
When the old have more (less) in‡uence over redistributive
policy, the rate of return of money is high (low).
Objective
I
Approach: Based on Bullard and Waller (2004) but with
dynamics
I
Use a direct mechanism to decide the allocations. A baby
boom corresponds to putting more weight on the young of a
particular generation relative to past and future generations.
I
This mechanism can replicate any steady state allocation
arising from a political economy model with population
growth or decline.
Outline Presentation
I
E¢ cient economy and intergenerational redistribution
I
Constrained e¢ ciency and redistribution
I
Optimal Wedges: Capital taxes=In‡ation
I
Numerical examples
I
I
Transitory demographics
Persistent demographic changes
Model
Environment
I
Two-period OLG model with capital.
I
Discrete time t = ...,
I
Population growth Nt = (1 + n )Nt
I
Preferences: U (c1,t , c2,t +1 ) = u (c1,t ) + βu (c2,t +1 )
I
Neoclassical production F (Kt , Nt ) and constant depreciation δ
I
Per capital resource constraint
c1,t +
1
c1,t
1+n
1
2,
1, 0, 1, 2, ...
1
where N0 = 1
+ (1 + n) kt +1 = f (kt ) + (1
δ) kt .
Social Preferences and Optimal Allocations
I
The objective function weights current and future generations
according to
∞
V (k0 ) = max f βλ
1 u (c2,0 ) +
∑ λt [u (c1,t ) + βu (c2,t +1 )]g.
t =0
subject to the resource constraint.
I
Optimality conditions imply
u 0 (c1,t )
λt 1
=
β (1 + n )
0
u (c2,t )
λt
and
(1 + n )
u 0 (c1,t )
λ t +1
=
1
0
u (c1,t +1 )
λt
δ + f 0 (kt +1 ) .
Steady State
I
E¢ cient production: the steady state stock of capital k s is
determined by
f 0 (k s ) = (1 + n ) λ
1
+δ
1,
For λ < 1, the economy is dynamically e¢ cient. When λ = 1,
the economy satis…es the golden rule f 0 (k ) = n + δ.
I
E¢ cient consumption c1s and c2s solve
u 0 (c1s ) = β(1 + n )u 0 (c2s )
c1s +
c2s
+ ( δ + n )k s = f (k s ).
1+n
Market Implementation: Intergenerational Redistribution
I
Consumers: Representative newborn solves
max u (c1,t ) + βu (c2,t +1 )
s.t. c1,t + st = wt lt + T1,t ,
c2,t +1 = (1
δ + rt +1 )st + T2,t +1 .
The optimality condition
u 0 (wt lt
I
st + T1t ) = βu 0 [(1
δ + rt +1 )st + T2,t +1 ] (1 + rt +1 ).
Intergenerational redistribution:
T1,t +
T2,t
= 0.
1+n
No Intergenerational Redistribution (Ramsey)
I
In the absence of intergenerational redistribution
∞
V (k0 ) = max f βλ
1 u (c2,0 ) +
∑ λt [u (c1,t ) + βu (c2,t +1 )]g
t =0
s.t. c1,t = fl (kt ) l
c2,t = [1
I
(1 + n)kt +1 ,
δ + fk (kt )] kt ,
Optimality conditions (endogenous multipliers γ1,t , γ2,t )
γ
λt 1
u 0 (c1,t )
=
β(1 + n ) 1,t
0
u (c2,t )
λt
γ2,t
Markets Redistribute: Capital taxes/in‡ation
I
The intergenerational decision of savings (capital) is more
complicated:
(1 + n) u 0 (c1,t ) =
| {z }
"savings
l
λ t +1 0
u (c1,t +1 )fl ,k
λt
| 1{z+ n}
"wage rate
0
+ βu (c2,t +1 )[1
δ + fk +
st
]
fk ,k
1
+
n
| {z }
#return all savings
I
E¢ cient wedges
1
u 0 (c1,t )
=
0
βu (c2,t +1 )
st 1
k
1 +n (1 + φ t +1 )
(1 + φtλ+1 )
δ + fk
.
where
φk =
kfk ,k
< 1;
fk
φtλ+1 = λ
u 0 (c1,t +1 )
fk ,k
u 0 (c1,t )
st 1
1+n
l
<1
1+n
Money and Capital
I
The optimal intergenerational redistribution determines the
equilibrium interest rate.
I
These parameters determine in‡ation when capital and money
are perfect substitutes.
I
Per capita money growth rate evolves
Mt +1 (1 + n ) = (1 + zt )Mt
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Arbritrage then implies that
fk (kt ) =
I
1+n
1
=
.
1 + πt
1 + zt
Money is priced as an asset that is held in zero net supply
(Woodford’s (2003) “cashless” economy).
Quantitative Illustration
Functional Forms
I
Preferences:
1 σ
1 σ
c1,t
c2,t
U (c1,t , c2,t +1 ) =
+ β +1 ,
1 σ
1 σ
I
Technology:
f (k ) = Ak α
Parameterization
Parameter
α
A
l=δ
σ
n
β
Value
0.35
10
1
2
0.99630
0.97930
Steady State
Capital Stock
2.4
E fficient
Constrained
2.2
2
Capital Stock
1.8
1.6
1.4
1.2
1
0.8
0.6
0.6
0.7
0.8
0.9
1
1.1
1.2
Intergenerational Discounting(λ)
1.3
1.4
1.5
Consumption Young
0.65
Efficient
Constrained
Consumption Young(%)
0.6
0.55
0.5
0.45
0.6
0.7
0.8
0.9
1
1.1
1.2
Intergenerational Discounting ( λ)
1.3
1.4
1.5
In‡ation (n<0)
-3
x 10
Efficient
Constrained
20
Inflation
15
10
5
0
-5
0.6
0.7
0.8
0.9
1
1.1
1.2
Intergenerational Discounting(λ)
1.3
1.4
1.5
In‡ation (n=0)
Efficient
Constrained
0.01
Inflation
0.005
0
-0.005
-0.01
-0.015
0.6
0.7
0.8
0.9
1
1.1
1.2
Intergenerational Discounting( λ)
1.3
1.4
1.5
Transitional Dynamics:
Demographics and In‡ation
Intergenerational Redistribution: Transitory
1.6
Intergenerational Discounting (λ)
1.4
1.2
1
0.8
0.6
0.4
0
10
20
30
40
50
60
70
80
90
In‡ation and Demographics: Transitory
0.015
Inflation (Annualized)
0.01
Efficient
Constrained
0.005
0
-0.005
-0.01
10
20
30
40
50
60
70
80
90
Interest Rates and Demographics: Transitory
0.2
Interest Rates (% Change)
0.1
0
-0.1
Efficient
Constrained
-0.2
-0.3
-0.4
-0.5
0
10
20
30
40
50
60
70
80
90
100
Intergenerational Redistribution: Permanent
1.6
Intergenerational Discounting (λ)
1.4
1.2
1
0.8
0.6
0.4
0
10
20
30
40
50
60
70
80
90
In‡ation and Demographics: Permanent
-3
x 10
Efficient
Constrained
14
12
Inflation (Annualized)
10
8
6
4
2
0
-2
-4
-6
10
20
30
40
50
60
70
80
90
Interest Rates and Demographics: Permanent
0.2
Efficient
Constrained
Interest Rates (% Change)
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
10
20
30
40
50
60
70
80
90
100
Conclusions
I
Study the interaction between population demographics, the
desire for redistribution in the economy, and the optimal
in‡ation rate.
I
The intergenerational redistribution tension is intrinsic in
life-cycle models.
I
I
I
Young cohorts do not have any assets and wages are the main
source of income.
Old generations cannot work and prefer a high rate of return
from their savings.
When the old have more (less) in‡uence over redistributive
policy, the rate of return of money is high (high).