Lecture 3A: The Continuous-Time Overlapping-Generations ... · Lecture 3A: The Continuous-Time...

IntroductionModel

Analysis

Lecture 3A: The Continuous-TimeOverlapping-Generations Model:

Beyond the Basic Model

Ben J. Heijdra

Department of Economics, Econometrics & Finance

University of Groningen

6 January 2012

NAKE Dynamic Macroeconomic Theory Lecture 3A (January 6, 2012) 1 / 41

IntroductionModel

Analysis

Outline

1 Introduction

2 ModelIndividual choicesDemographic featuresSteady-State Profiles

3 AnalysisShocks and transitionWelfare and aggregate effectsExtensions and concluding remarks


IntroductionModel

Analysis

Introductory Remarks

Blanchard-Yaari model is based on rather unrealisticdemographic assumptions: constant mortality rate

Recently a number of authors have started to incorporate amore realistic demographic structure into the continuous-timeoverlapping generations model

Today we discuss two topics:

Topic 1: what does a realistic demography look like and howcan we incorporate it in an analytical overlapping-generationsmodel of a small open economy? Do the demographic detailsmatter? (Heijdra & Romp, 2008)Topic 2: what are the effect of annuity market imperfectionson economic growth and welfare? (Heijdra & Mierau, 2009)


IntroductionModel

Analysis

Motivation (1)

Gompertz-Makeham Law of Mortality:

“It is possible that death may be the consequence of twogenerally co-existing causes; the one, chance, withoutprevious disposition to death or deterioration; the other, adeterioration or an increased inability to withstanddestruction.” (Benjamin Gompertz, 1825)

How have macroeconomists incorporated this “fact of life” intotheir models so far?

Barro (1974) and many others:

connected finite-lived generationsoperative bequests lead to Ricardian equivalence


IntroductionModel

Analysis

Motivation (2)

Yaari (1965) gives the micro story:

disconnected agentsheavier discounting of future felicity due to uncertainty ofsurvivalactuarially fair life insurance opportunities

Blanchard (1985)–Buiter (1988)–Weil (1989) add:

general equilibrium representationconstant death rate: all living “dynasties” have same expectedremaining lifetimeaggregation possiblecannot capture life-cycle pattern

Calvo & Obstfeld (1988):

general mortality processfocus on optimal time-consistent policy


IntroductionModel

Analysis

Motivation (3)

Recent related work in this area:

de la Croix & Licandro (1999); Boucekkine, de la Croix, andLicandro (2002):

human capital and endogenous growthinfinite intertemporal substitution elasticity

d’Albis (2007)

model similar to Heijdra & Romp (2008)focusses on different issues [e.g. efficiency property of steadystate]

Rios-Rull (1996)

calibrated stochastic RBC model of the Auerbach-KotlikoffOLG type....OLG feature does not matter to impulse-response functionswith respect to technology shocks


IntroductionModel

Analysis

Motivation (4)

Hansen & Imrohoroglu (2008)

what if annuities markets do not exist?absence of annuities markets can account for hump-shapedconsumption pattern

Focus of Heijdra & Romp (2008):

realistic demography in a small open economyfactor prices exogenous (and typically constant)aggregation not necessarymodel can be solved analytically: complementary to large-scaleCGE modelsdemographic realism matters!maintained assumption: actuarially fair annuities


IntroductionModel

Analysis

Individual choicesDemographic featuresSteady-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 assetsgovernment debtperfect substitutes: same rate of return

life-time uncertainty; actuarially fair life insurance

no aggregate uncertainty

rational agents blessed with perfect foresight


IntroductionModel

Analysis


Key Equations (1)

expected remaining lifetime utility at time t of agent born attime v (t ≥ v)

Λ(v, t) ≡

∫∞

t

ln c (v, τ)︸︷︷︸

(a)

eM(t−v)−M(τ−v)︸︷︷︸

(b)

eθ(t−τ)︸︷︷︸

(c)

dτ

(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


IntroductionModel

Analysis


Key Equations (2)

mortality factor and mortality rate:

M (τ − v) ≡

∫ τ−v

0m (s) ds

m (s) is instantaneous mortality rate, i.e. hazard rate ofhazard rate of the stochastic distribution of the date of death:

m (s) ≡φ (s)

1− Φ (s)

φ (s) = density functionΦ (s) = distribution (or cumulative density) functionin this paper: m (s) depends only on household age [stationarydemography]


IntroductionModel

Analysis


Key Equations (3)

budget identity:

˙a (v, τ ) = [r +m (τ − v)] a (v, τ ) + w (τ)− z (τ)− c (v, τ )

a (v, τ ) = financial assetsr = world interest rate [patient country, r > θ]r +m (τ − v) = annuity rate of interestw (τ) = wage ratez (τ ) = lump-sum taxc (v, τ ) = consumption


IntroductionModel

Analysis


Key Equations (4)

optimal choices of household with age u ≡ t− v:

˙c (v, τ )

c (v, τ )= r − θ > 0

c (v, t) =1

∆ (u, θ)

[a (v, t) + h (v, t)

]

h (v, t) ≡ eru+M(u)

∫∞

u

[w (s+ v)− z (s+ v)] e−[rs+M(s)]ds

∆(u, λ) ≡ eλu+M(u)

∫∞

u

e−[λs+M(s)]ds, (u ≥ 0, λ > 0)

h (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)


IntroductionModel

Analysis


Lemma 1

Let ∆(u, λ) be defined as on the previous slide and assume thatthe 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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Analysis


Demographic Theory (1)

birth process:L(v, v) = bL(v)

L (v, v) = newborn cohort at time vb = 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, m:

mL (t) =

∫ t

−∞

m (t− v)L (v, t) dv


IntroductionModel

Analysis


Demographic Theory (2)

relative cohort weights [needed for aggregation]:

l (v, t) ≡L (v, t)

L (t)= be−[n(t−v)+M(t−v)]

n ≡ b− m = aggregate population growth rate

for a given birth rate and mortality process, there is an implicitsolution for n such that:

b =1

∆(0, n)


IntroductionModel

Analysis


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 leastsquares:

constant mortality rate [Blanchard]Boucekkine et al. mortality rate [not shown here – see Lecture2]Gompertz-Makeham mortality rate [G-M]

Estimation results in Table 1.

Visualisation of fit in Figure 1.


IntroductionModel

Analysis


Table 1: Estimated Survival Functions

1. Blanchard demography: M (u) ≡ µ0u

σ = 0.2213 µ0

m = 1.15% 0.1147 × 10−1

1− Φ (100) = 31.8% (14.3)

2. G-M demography: M (u) ≡ µ0u+ (µ1/µ2) [eµ2u − 1]

σ = 0.4852 × 10−2 µ0 µ1 µ2

m = 1.02% 0.2437 × 10−2 5.52 × 10−5 0.09641− Φ (100) = 0.1% (65.8) (20.5) (138.2)


IntroductionModel

Analysis


Figure 1(a) Surviving Fraction of the Population

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

age

BlanchardG−MActual


IntroductionModel

Analysis


Figure 1(B) Mortality Rate of the Population

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

age


IntroductionModel

Analysis


Figure 1(c): Expected Remaining Lifetime

0 20 40 60 80 100 1200

50

100

150

age


IntroductionModel

Analysis


Figure 2(a): Propensity to Consume

1

∆ (u, θ)=

[

eθu+M(u)

∫∞

u

e−[θs+M(s)]ds

]−1

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

age

BlanchardG−M


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Analysis


Figure 2(b): Human Wealth

ˆh (v, t) ≡ ∆(u, r) [w − z]

0 20 40 60 80 100 1200

50

100

150

age


IntroductionModel

Analysis


Figure 2(c): Consumption

ˆc (u) =ˆh (0)

∆ (0, θ)e(r−θ)u

0 20 40 60 80 100 120

5

5.2

5.4

5.6

age


IntroductionModel

Analysis


Figure 2(d): Financial Assets

ˆa (u) = ∆ (u, θ) ˆc (u)− ˆh (u)

0 20 40 60 80 100 1200

5

10

15

age


IntroductionModel

Analysis

Shocks and transitionWelfare and aggregate effectsExtensions and concluding remarks

Macroeconomic Shocks

balanced-budget fiscal policy

once-off increase in government consumption and lump-sumtaxes

temporary tax cut

short-run tax cut financed with debtgradual increase lump-sum taxlong-run debt positive

interest rate shock

once-off increase in world interest rate


IntroductionModel

Analysis


Figure 3: Balanced-budget Fiscal PolicyHuman wealth (h), Blanchard Human wealth (h), Gompertz-Makeham

−20 0 20 40 60 80 100 120 1400

20

40

60

80

100

120

time

v=−40v=−20v= 0v= 40

−20 0 20 40 60 80 100 120 1400

20

40

60

80

100

120

time

Financial assets (a), Blanchard Financial assets (a), Gompertz-Makeham

−20 0 20 40 60 80 100 120 1400

5

10

15

20

time−20 0 20 40 60 80 100 120 1400

0.5

1

1.5

2

2.5

3

time


IntroductionModel

Analysis


Figure 4: RET experiment: Temporary tax cutHuman wealth (h), Blanchard Human wealth (h), Gompertz-Makeham

−20 0 20 40 60 80 100 120 1400

20

40

60

80

100

120

time

v=−40v=−20v= 0v= 40

−20 0 20 40 60 80 100 120 1400

20

40

60

80

100

120

time


−20 0 20 40 60 80 100 120 1400

5

10

15

20

25

30

time−20 0 20 40 60 80 100 120 1400

2

4

6

8

10

time


IntroductionModel

Analysis


Figure 5: Increase in the World Interest RateHuman wealth (h), Blanchard Human wealth (h), Gompertz-Makeham

−20 0 20 40 60 80 100 120 1400

20

40

60

80

100

120

time

v=−40v=−20v= 0v= 40

−20 0 20 40 60 80 100 120 1400

20

40

60

80

100

120

time


−20 0 20 40 60 80 100 120 1400

10

20

30

40

50

60

time−20 0 20 40 60 80 100 120 1400

2

4

6

8

10

time


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Analysis


Welfare Effects

change in welfare from shock at time t = 0

existing agents (v ≤ 0): evaluate dΛ (v, 0):

dΛ(v, 0) = dr

∫∞

0

τe−θτ−M(τ−v)+M(−v)dτ +∆(−v, θ) ln ΓE(v)

ΓE(v) ≡ˆa(−v) + h(v, 0)

ˆa(−v) + ˆh(−v)

future agents (v > 0): evaluate dΛ (v, v):

dΛ(v, v) = dr

∫∞

0

se−[θs+M(s)]ds+∆(0, θ) ln ΓF (v)

ΓF (v) ≡h(v, v)

ˆh(0)


IntroductionModel

Analysis


Figure 6: Welfare EffectsBalanced budget, Blanchard Balanced budget, Gompertz-Makeham

−200 −150 −100 −50 0 50−6

−4

−2

0

Generation−200 −150 −100 −50 0 50−6

−4

−2

0

Generation

Temporary tax cut, Blanchard Temporary tax cut, Gompertz-Makeham

−200 −150 −100 −50 0 50−2

−1

0

1

Generation−200 −150 −100 −50 0 50−2

−1

0

1

Generation

Interest rate, Blanchard Interest rate, Gompertz-Makeham

−200 −150 −100 −50 0 500

0.1

0.2

Generation−200 −150 −100 −50 0 50

0

0.02

0.04

Generation


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Analysis


Figure 7: Aggregate effects of the shocks

Human wealth, BB HW, Temporary tax cut HW, Interest rate

0 50 100 150

−0.2

−0.1

0

time0 50 100 150

−0.1

−0.05

0

0.05

time0 50 100 150

−0.04

−0.02

0

time

BlanchardG−M

Consumption, BB C, Temporary tax cut C, Interest rate

0 50 100 150

−0.2

−0.1

0

time0 50 100 150

−0.1

0

0.1

time0 50 100 150

−0.1

0

0.1

time

Financial assets, BB FA, Temporary tax cut FA, Interest rate

0 50 100 150−0.2

−0.1

0

time0 50 100 150

0

2

4

time0 50 100 150

0

2

4

time


IntroductionModel

Analysis


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:

˙c (v, τ )

c (v, τ )= r − (θ +m(τ − v))

˙c (v, τ ) > 0 for young and ˙c (v, τ ) < 0 for old agents


IntroductionModel

Analysis


Extensions (2)

hump-shaped consumption due to diminishing needs as onegets older:

Λ(v, t) ≡ eM(t−v)

∫∞

t

[

e (v, τ )1−1/σ

− 1

1− 1/σ

]

e−[θ(τ−t)+M(τ−v)]dτ

e (v, τ ) ≡ c (v, τ ) exp

{

ζ0 (τ − v)1+ζ

1

1 + ζ1

}

, ζ0 > 0, ζ1 > 0

σ = intertemporal substitution elasticitye (v, τ) = effective consumptionEuler equation becomes:

˙c (v, τ )

c (v, τ )= σ (r − θ)− (1− σ) ζ0 (τ − v)

ζ1

for 0 < σ < 1, ˙c (v, τ ) > 0 for young and ˙c (v, τ ) < 0 for oldagents


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Analysis


Extensions (3)

endogenous labour supply, schooling, and retirement

in progressapplications: ageing, educational subsidies, inequality, pensionreform, and optimal retirementeducation at start of life [cf. de la Croix, Licandro, andBoucekkine papers mentioned above]application: growth effects of ageing

realistic demography in a closed economy

steady state easydifficult to get analytical results for transitional dynamicsapproximate solutions may be attainable


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Analysis


Closing Remarks

in the context of a small open economy [or with constantmarginal product of capital] there is no need to use modelsbased on an unrealistic description of the demographic process

using a realistic demographic process matters because.....

individual behaviour is differentimpulse-response functions are differenttransition speed is affectedwelfare effects may be non-monotonic


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