Introduction
Model
Analysis
Lecture 3A: The Continuous-Time
Overlapping-Generations Model:
Beyond the Basic Model
Ben J. Heijdra
Department of Economics, Econometrics & Finance
University of Groningen
6 January 2012
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Introduction
Model
Analysis
Outline
1
Introduction
2
Model
Individual choices
Demographic features
Steady-State Profiles
3
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
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Introduction
Model
Analysis
Introductory Remarks
Blanchard-Yaari model is based on rather unrealistic
demographic assumptions: constant mortality rate
Recently a number of authors have started to incorporate a
more realistic demographic structure into the continuous-time
overlapping generations model
Today we discuss two topics:
Topic 1: what does a realistic demography look like and how
can we incorporate it in an analytical overlapping-generations
model of a small open economy? Do the demographic details
matter? (Heijdra & Romp, 2008)
Topic 2: what are the effect of annuity market imperfections
on economic growth and welfare? (Heijdra & Mierau, 2009)
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Introduction
Model
Analysis
Motivation (1)
Gompertz-Makeham Law of Mortality:
“It is possible that death may be the consequence of two
generally co-existing causes; the one, chance, without
previous disposition to death or deterioration; the other, a
deterioration or an increased inability to withstand
destruction.” (Benjamin Gompertz, 1825)
How have macroeconomists incorporated this “fact of life” into
their models so far?
Barro (1974) and many others:
connected finite-lived generations
operative bequests lead to Ricardian equivalence
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Introduction
Model
Analysis
Motivation (2)
Yaari (1965) gives the micro story:
disconnected agents
heavier discounting of future felicity due to uncertainty of
survival
actuarially fair life insurance opportunities
Blanchard (1985)–Buiter (1988)–Weil (1989) add:
general equilibrium representation
constant death rate: all living “dynasties” have same expected
remaining lifetime
aggregation possible
cannot capture life-cycle pattern
Calvo & Obstfeld (1988):
general mortality process
focus on optimal time-consistent policy
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Introduction
Model
Analysis
Motivation (3)
Recent related work in this area:
de la Croix & Licandro (1999); Boucekkine, de la Croix, and
Licandro (2002):
human capital and endogenous growth
infinite intertemporal substitution elasticity
d’Albis (2007)
model similar to Heijdra & Romp (2008)
focusses on different issues [e.g. efficiency property of steady
state]
Rios-Rull (1996)
calibrated stochastic RBC model of the Auerbach-Kotlikoff
OLG type
....OLG feature does not matter to impulse-response functions
with respect to technology shocks
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Introduction
Model
Analysis
Motivation (4)
Hansen & Imrohoroglu (2008)
what if annuities markets do not exist?
absence of annuities markets can account for hump-shaped
consumption pattern
Focus of Heijdra & Romp (2008):
realistic demography in a small open economy
factor prices exogenous (and typically constant)
aggregation not necessary
model can be solved analytically: complementary to large-scale
CGE models
demographic realism matters!
maintained assumption: actuarially fair annuities
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Key Assumptions
small open economy facing constant world interest rate
labour only factor of production (capital could be added easily)
savings instruments:
foreign assets
government debt
perfect substitutes: same rate of return
life-time uncertainty; actuarially fair life insurance
no aggregate uncertainty
rational agents blessed with perfect foresight
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Key Equations (1)
expected remaining lifetime utility at time t of agent born at
time v (t ≥ v)
Z ∞
(τ −v) θ(t−τ )
ln c̄ (v, τ ) eM (t−v)−M
Λ(v, t) ≡
{z
} |e {z }dτ
| {z } |
t
(a)
(b)
(c)
(a) felicity: unitary intertemporal substitution elasticity
(b) lifetime uncertainty: Probability that household of age t − v
reaches age τ − v. Process not memoryless, i.e.
M (t − v) − M (τ − v) 6= M (t − τ ).
(c) pure discounting (θ > 0): impatience
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Key Equations (2)
mortality factor and mortality rate:
Z τ −v
m (s) ds
M (τ − v) ≡
0
m (s) is instantaneous mortality rate, i.e. hazard rate of
hazard rate of the stochastic distribution of the date of death:
m (s) ≡
φ (s)
1 − Φ (s)
φ (s) = density function
Φ (s) = distribution (or cumulative density) function
in this paper: m (s) depends only on household age [stationary
demography]
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Key Equations (3)
budget identity:
ā˙ (v, τ ) = [r + m (τ − v)] ā (v, τ ) + w̄ (τ ) − z̄ (τ ) − c̄ (v, τ )
ā (v, τ ) = financial assets
r = world interest rate [patient country, r > θ]
r + m (τ − v) = annuity rate of interest
w̄ (τ ) = wage rate
z̄ (τ ) = lump-sum tax
c̄ (v, τ ) = consumption
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Key Equations (4)
optimal choices of household with age u ≡ t − v:
c̄˙ (v, τ )
c̄ (v, τ )
= r−θ >0
1
ā (v, t) + h̄ (v, t)
∆ (u, θ)
Z ∞
ru+M (u)
h̄ (v, t) ≡ e
[w̄ (s + v) − z̄ (s + v)] e−[rs+M (s)] ds
u
Z ∞
λu+M (u)
∆ (u, λ) ≡ e
e−[λs+M (s)] ds, (u ≥ 0, λ > 0)
c̄ (v, t) =
u
h̄ (v, t) = human wealth (market value of time endowment,
using annuity rate of interest for discounting)
∆ (u, λ) = demographic factor (plays central role, e.g.
1/∆ (u, θ) is propensity to consume out of total wealth)
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Lemma 1
Let ∆ (u, λ) be defined as on the previous slide and assume that
the mortality rate is non-decreasing, i.e. m′ (s) ≥ 0 for all s ≥ 0.
Then the following properties can be established for ∆ (u, λ):
(i) decreasing in λ, ∂∆ (u, λ) /∂λ < 0;
(ii) non-increasing in household age, ∂∆ (u, λ) /∂u ≤ 0;
(iii) upper bound, ∆ (u, λ) ≤ 1/ [λ + m (u)];
(iv) ∆ (u, λ) > 0 for u < ∞;
(v) for λ → ∞, ∆ (u, λ) → 0.
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Demographic Theory (1)
birth process:
L(v, v) = bL(v)
L (v, v) = newborn cohort at time v
b = birth rate [constant]
L (v) = total population at time v
size of cohort over time:
L (v, τ ) = L (v, v) e−M (τ −v)
aggregate mortality rate, m̄:
Z t
m (t − v) L (v, t) dv
m̄L (t) =
−∞
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Demographic Theory (2)
relative cohort weights [needed for aggregation]:
l (v, t) ≡
L (v, t)
= be−[n(t−v)+M (t−v)]
L (t)
n ≡ b − m̄ = aggregate population growth rate
for a given birth rate and mortality process, there is an implicit
solution for n such that:
b=
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Demographic Estimates
use actual demographic data for the Netherlands
projections on expected survival rates for people born in 1920
three parametric models are estimated with nonlinear least
squares:
constant mortality rate [Blanchard]
Boucekkine et al. mortality rate [not shown here – see Lecture
2]
Gompertz-Makeham mortality rate [G-M]
Estimation results in Table 1.
Visualisation of fit in Figure 1.
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Table 1: Estimated Survival Functions
1. Blanchard demography:
M (u) ≡ µ0 u
σ̂ = 0.2213
m̄ = 1.15%
1 −\
Φ (100) = 31.8%
µ̂0
0.1147 × 10−1
(14.3)
2. G-M demography:
M (u) ≡ µ0 u + (µ1 /µ2 ) [eµ2 u − 1]
σ̂ = 0.4852 × 10−2
m̄ = 1.02%
1 −\
Φ (100) = 0.1%
µ̂0
0.2437 × 10−2
(65.8)
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µ̂1
5.52 × 10−5
(20.5)
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µ̂2
0.0964
(138.2)
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Figure 1(a) Surviving Fraction of the Population
1
0.8
0.6
0.4
Blanchard
G−M
Actual
0.2
0
0
20
40
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Figure 1(B) Mortality Rate of the Population
0.5
0.4
0.3
0.2
0.1
0
0
20
40
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Figure 1(c): Expected Remaining Lifetime
150
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50
0
0
20
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Figure 2(a): Propensity to Consume
−1
Z ∞
1
θu+M(u)
−[θs+M(s)]
= e
e
ds
∆ (u, θ)
u
0.5
Blanchard
G−M
0.4
0.3
0.2
0.1
0
0
20
40
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Figure 2(b): Human Wealth
ˆ (v, t) ≡ ∆ (u, r) [ŵ − ẑ]
h̄
150
100
50
0
0
20
40
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Figure 2(c): Consumption
c̄ˆ (u) =
ˆ (0)
h̄
e(r−θ)u
∆ (0, θ)
5.6
5.4
5.2
5
0
20
40
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Introduction
Model
Analysis
Individual choices
Demographic features
Steady-State Profiles
Figure 2(d): Financial Assets
ˆ (u)
ˆ (u) = ∆ (u, θ) c̄ˆ (u) − h̄
ā
15
10
5
0
0
20
40
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Macroeconomic Shocks
balanced-budget fiscal policy
once-off increase in government consumption and lump-sum
taxes
temporary tax cut
short-run tax cut financed with debt
gradual increase lump-sum tax
long-run debt positive
interest rate shock
once-off increase in world interest rate
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Figure 3: Balanced-budget Fiscal Policy
Human wealth (h̄), Blanchard
Human wealth (h̄), Gompertz-Makeham
120
120
100
100
80
80
60
60
v=−40
v=−20
v= 0
v= 40
40
20
0
−20
0
20
40
20
40
60
time
80
100
120
140
Financial assets (ā), Blanchard
20
0
−20
0
20
40
60
time
80
100
120
140
Financial assets (ā), Gompertz-Makeham
3
2.5
15
2
10
1.5
1
5
0.5
0
−20
0
20
40
60
time
80
100
120
140
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0
−20
0
20
40
60
time
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Figure 4: RET experiment: Temporary tax cut
Human wealth (h̄), Blanchard
Human wealth (h̄), Gompertz-Makeham
120
120
100
100
80
80
60
60
v=−40
v=−20
v= 0
v= 40
40
20
0
−20
0
20
40
20
40
60
time
80
100
120
140
Financial assets (ā), Blanchard
0
−20
0
20
40
60
time
80
100
120
140
Financial assets (ā), Gompertz-Makeham
30
10
25
8
20
6
15
4
10
2
5
0
−20
0
20
40
60
time
80
100
120
140
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0
20
40
60
time
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100
120
140
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Figure 5: Increase in the World Interest Rate
Human wealth (h̄), Blanchard
Human wealth (h̄), Gompertz-Makeham
120
120
100
100
80
80
60
60
v=−40
v=−20
v= 0
v= 40
40
20
0
−20
0
20
40
20
40
60
time
80
100
120
140
Financial assets (ā), Blanchard
0
−20
0
20
40
60
time
80
100
120
140
Financial assets (ā), Gompertz-Makeham
60
10
50
8
40
6
30
4
20
2
10
0
−20
0
20
40
60
time
80
100
120
140
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0
20
40
60
time
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140
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Welfare Effects
change in welfare from shock at time t = 0
existing agents (v ≤ 0): evaluate dΛ (v, 0):
Z ∞
τ e−θτ −M(τ −v)+M(−v) dτ + ∆(−v, θ) ln ΓE (v)
dΛ(v, 0) = dr
0
ΓE (v)
≡
ˆ(−v) + h̄(v, 0)
ā
ˆ (−v)
ˆ(−v) + h̄
ā
future agents (v > 0): evaluate dΛ (v, v):
Z ∞
se−[θs+M(s)] ds + ∆(0, θ) ln ΓF (v)
dΛ(v, v) = dr
0
ΓF (v) ≡
h̄(v, v)
ˆ (0)
h̄
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Figure 6: Welfare Effects
Balanced budget, Blanchard
Balanced budget, Gompertz-Makeham
0
0
−2
−2
−4
−4
−6
−200
−150
−100
−50
Generation
0
50
Temporary tax cut, Blanchard
−6
−200
−100
−50
Generation
0
50
Temporary tax cut, Gompertz-Makeham
1
1
0
0
−1
−1
−2
−200
−150
−150
−100
−50
Generation
0
50
Interest rate, Blanchard
−2
−200
−150
−100
−50
Generation
0
50
Interest rate, Gompertz-Makeham
0.2
0.04
0.1
0
−200
0.02
−150
−100
−50
Generation
0
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−200
−150
−100
−50
Generation
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50
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Figure 7: Aggregate effects of the shocks
Human wealth, BB
HW, Temporary tax cut
HW, Interest rate
0.05
0
0
Blanchard
G−M
0
−0.1
−0.02
−0.05
−0.2
0
50
100
150
−0.1
0
50
time
100
−0.04
150
Consumption, BB
C, Temporary tax cut
0.1
−0.1
0
0
−0.2
100
150
−0.1
0
50
time
100
150
−0.1
FA, Temporary tax cut
4
−0.1
2
2
0
50
50
100
150
100
150
FA, Interest rate
4
0
150
time
0
−0.2
0
time
Financial assets, BB
100
C, Interest rate
0.1
50
50
time
0
0
0
time
0
0
time
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0
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Extensions (1)
effects of demographic change:
embodied: mortality rate depends on date of birth m (v, s)
disembodied: mortality rate depends on calender date m (t, s)
hump-shaped consumption due to absent annuity markets [cf.
Hansen & Imrohoroglu (2008)]
Euler equation becomes:
c̄˙ (v, τ )
= r − (θ + m(τ − v))
c̄ (v, τ )
c̄˙ (v, τ ) > 0 for young and c̄˙ (v, τ ) < 0 for old agents
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Extensions (2)
hump-shaped consumption due to diminishing needs as one
gets older:
Λ(v, t)
ē (v, τ )
"
#
1−1/σ
ē (v, τ )
− 1 −[θ(τ −t)+M(τ −v)]
≡ e
e
dτ
1 − 1/σ
t
)
(
1+ζ
ζ 0 (τ − v) 1
, ζ 0 > 0, ζ 1 > 0
≡ c̄ (v, τ ) exp
1 + ζ1
M(t−v)
Z
∞
σ = intertemporal substitution elasticity
ē (v, τ ) = effective consumption
Euler equation becomes:
c̄˙ (v, τ )
ζ
= σ (r − θ) − (1 − σ) ζ 0 (τ − v) 1
c̄ (v, τ )
for 0 < σ < 1, c̄˙ (v, τ ) > 0 for young and c̄˙ (v, τ ) < 0 for old
agents
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Extensions (3)
endogenous labour supply, schooling, and retirement
in progress
applications: ageing, educational subsidies, inequality, pension
reform, and optimal retirement
education at start of life [cf. de la Croix, Licandro, and
Boucekkine papers mentioned above]
application: growth effects of ageing
realistic demography in a closed economy
steady state easy
difficult to get analytical results for transitional dynamics
approximate solutions may be attainable
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Introduction
Model
Analysis
Shocks and transition
Welfare and aggregate effects
Extensions and concluding remarks
Closing Remarks
in the context of a small open economy [or with constant
marginal product of capital] there is no need to use models
based on an unrealistic description of the demographic process
using a realistic demographic process matters because.....
individual behaviour is different
impulse-response functions are different
transition speed is affected
welfare effects may be non-monotonic
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