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) ≡

∫∞

ln c (v, τ)︸︷︷︸

eM(t−v)−M(τ−v)︸︷︷︸

eθ(t−τ)︸︷︷︸

(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)

∫∞

[w (s+ v)− z (s+ v)] e−[rs+M(s)]ds

∆(u, λ) ≡ eλu+M(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.

IntroductionModel

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) =

−∞

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:

∆(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

BlanchardG−MActual

IntroductionModel

Analysis

Figure 1(B) Mortality Rate of the Population

0 20 40 60 80 100 1200

IntroductionModel

Analysis

Figure 1(c): Expected Remaining Lifetime

0 20 40 60 80 100 1200

IntroductionModel

Analysis

Figure 2(a): Propensity to Consume

∆ (u, θ)=

eθu+M(u)

∫∞

e−[θs+M(s)]ds

0 20 40 60 80 100 1200

BlanchardG−M

IntroductionModel

Analysis

Figure 2(b): Human Wealth

ˆh (v, t) ≡ ∆(u, r) [w − z]

0 20 40 60 80 100 1200

IntroductionModel

Analysis

Figure 2(c): Consumption

ˆc (u) =ˆh (0)

∆ (0, θ)e(r−θ)u

0 20 40 60 80 100 120

IntroductionModel

Analysis

Figure 2(d): Financial Assets

ˆa (u) = ∆ (u, θ) ˆc (u)− ˆh (u)

0 20 40 60 80 100 1200

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

v=−40v=−20v= 0v= 40

−20 0 20 40 60 80 100 120 1400

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

−20 0 20 40 60 80 100 120 1400

time−20 0 20 40 60 80 100 120 1400

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

v=−40v=−20v= 0v= 40

−20 0 20 40 60 80 100 120 1400

time−20 0 20 40 60 80 100 120 1400

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

v=−40v=−20v= 0v= 40

−20 0 20 40 60 80 100 120 1400

time−20 0 20 40 60 80 100 120 1400

IntroductionModel

Analysis

Welfare Effects

change in welfare from shock at time t = 0

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

dΛ(v, 0) = dr

∫∞

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

∫∞

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

Generation−200 −150 −100 −50 0 50−6

Generation

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

−200 −150 −100 −50 0 50−2

Generation−200 −150 −100 −50 0 50−2

Generation

Interest rate, Blanchard Interest rate, Gompertz-Makeham

−200 −150 −100 −50 0 500

Generation−200 −150 −100 −50 0 50

Generation

IntroductionModel

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

time0 50 100 150

−0.1

−0.05

time0 50 100 150

−0.04

−0.02

BlanchardG−M

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

0 50 100 150

−0.2

−0.1

time0 50 100 150

−0.1

time0 50 100 150

−0.1

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

0 50 100 150−0.2

−0.1

time0 50 100 150

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)

∫∞

e (v, τ )1−1/σ

1− 1/σ

e−[θ(τ−t)+M(τ−v)]dτ

e (v, τ ) ≡ c (v, τ ) exp

ζ0 (τ − v)1+ζ

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)

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

IntroductionModel

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

IntroductionModel

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

lecture 3a: the continuous-time overlapping-generations ... · lecture 3a: the continuous-time...

Documents

overlapping generations models - rutgers...

labor market effects of pension reform : an overlapping...

overlapping generations model

asset pricing with idiosyncratic risk and overlapping...

vine mealybug biology & control...

overlapping generations, natural resources and the optimal...

accolley, delali (2014) altruistic overlapping generations...

numerical simulation of the overlapping generations models

on the effect of greenhouse gas abatement in japanese...

organizations and overlapping generations games: memory

overlapping generations models - wouter den haan

overlapping generations model with endogenous ·...

conspicuous consumption and overlapping generations

overlapping generations model roundtable: model ... ·...

chapter 12 overlapping generations in continuous...

market games and the overlapping generations model

monetary equilibrium in an overlapping generations model...

equilibrium for overlapping generations in continuous...

business cycle dynamics of a new keynesian overlapping...

walras' law, pareto efficiency, and intermediation in...