Lecture 3C: The Continuous-Time Overlapping-Generations … · 2011. 12. 28. · Lecture 3C: The...

Motivation & ModelAnalysis

Conclusions & Further Research

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

Beyond the Basic Model

Ben J. Heijdra

Department of Economics, Econometrics & FinanceUniversity of Groningen

13 January 2012

NAKE Dynamic Macroeconomic Theory Lecture 3C (January 13, 2012) 1 / 41



Outline

1 Motivation & ModelMotivation & OverviewModel: householdsModel: firms

2 AnalysisInformational assumptionsSeparating equilibriumPooling equilibrium

3 Conclusions & Further Research




Motivation & OverviewModel: householdsModel: firms

Motivation & Overview (1)

In the absence of a bequest motive, annuities are veryattractive insurance instruments in the presence of longevityrisk (Yaari, 1965; Davidoff, Brown, Diamond, 2005)

The macroeconomic literature has mostly restricted attentionto the case of perfect annuities, one in which annuity rates areactuarially fair.

In reality, “money’s worth” calculations reveal that annuityrates are typically not actuarially fair. Possible reasons:

Administrative costs incurred by the annuity companiesnecessitates a loading factor [ignored].Imperfect competition in the annuity sector [ignored].Adverse selection effects: people who believe themselves to behealthier than average are more likely to buy annuities[stressed in this paper].






Based on: Heijdra, B.J. & Reijnders, L.S.M. (2009),Economic growth and longevity risk with adverse selection.CESifo Working Paper, Nr. 2898, December 2009 (Rev.November 2010).

Objective of the paper: to study the growth and welfareimplications of adverse selection effects in the annuity market.

Key model features:

Endogenous growth model of a closed economy.Overlapping generations of heterogeneous finitely-lived agents.Privately known mortality process.Competitive annuity market, offering linear annuity contracts(cf. Pauly, 1974; Abel, 1986).






Main findings of the paper:

If health status were observable, there would be a separatingequilibrium (SE). Each health type would get actuarially fairperfect insurance.With unobservable health, there is a pooling equilibrium (PE).The healthy have an incentive to misrepresent their healthstatus. The healthy get a better than actuarially fair rate,whilst the unhealthy get a less than actuarially fair rate.In the PE the unhealthy encounter a self-imposed borrowingconstraint if they live long enough.For a plausible calibration, the SE is slightly better in welfareterms than the PE.





Individual agents (1)

Finitely-lived agents of different health types

Lifetime utility function of type-j agent:

Λj (v, v) =

∫ v+Dj

v

cj(v, τ )1−1/σ − 1

1− 1/σe−ρ(τ−v)−Mj(τ−v)dτ,

(1)

Dj is the maximum attainable agecj (v, τ ) is consumption of vintage-v agent at time τρ is the pure rate of time preferenceσ is the intertemporal substitution elasticityMj (τ − v) ≡

∫ τ−v

0µj(s)ds is the cumulative mortality rate

µj (s) is the instantaneous mortality rate at s (0 ≤ s ≤ Dj ,µ′

j (s) > 0, µ′′

j (s) > 0, and lims→Djµj (s) = +∞).






Household budget identity:

˙aj (v, τ ) = [r + pj (τ − v)] · aj (v, τ ) + w (τ)− cj (v, τ ) (2)

aj (τ ) is financial assetsr is the interest rate; r + pj (s) is the annuity rate of interestw (τ) is the wage rate

Labour supply is exogenous and equal to unity, so w (τ) alsostands for the household’s wage income

At time v, the agent chooses time paths for cj (v, τ ) andaj (v, τ ) (τ ≥ t) in order to maximize (1) subject to (2),taking into account the initial and terminal conditions onassets, aj (v, v) = aj

(

v, v + Dj

)

= 0, and the TVC.






Solutions:

˙cj(v, t)

cj(v, t)= σ

[

r + pj (t− v)− µj (t− v)− ρ]

(3)

cj(v, v) =

∫ v+Dj

v w (τ) e−r(τ−v)−Pj(τ−v)dτ∫ v+Dj

v e−(1−σ)[r(τ−v)+Pj(τ−v)]−σ[ρ(τ−v)+Mj(τ−v)]dτ(4)

aj (v, t) e−r(t−v)−Pj(t−v) =

∫ t

vw (τ) e−r(τ−v)−Pj(τ−v)dτ

−cj(v, v)

∫ t

ve−(1−σ)[r(τ−v)+Pj(τ−v)]−σ[ρ(τ−v)+Mj(τ−v)]dτ (5)





Demography

Non-zero population growth

Population proportion of type-j people, πj, is constantDemographic steady state:

βj ·

∫ Dj

0e−ns−Mj(s)ds = 1 (6)

βj is the crude birth rate of type j cohortsn is the growth rate of the population (for a given value of nand a given mortality process Mj (s), (6) defines the coherentsolution for βj)

Relative cohort size of type j agents of age t− v evolvesaccording to:

lj (v, t) ≡Lj (v, t)

L (t)=

βjπje−n(t−v)−Mj(t−v) for 0 ≤ t− v ≤ Dj

0 for t− v > Dj

(7)





Figure 1(a): Demographics, surviving fraction

e−Mj(u)

20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

biological age (u+18)

HealthyUnhealthy





Figure 1(b): Demographics, instantaneous mortality rate

µj(u)

20 30 40 50 60 70 80 900

0.01

0.02

0.03

0.04

0.05

0.06


HealthyUnhealthy





Aggregate household sector (1)

Focus on steady-state growth path:

w (t) = w (v) · eg(t−v) (8)

Per capita consumption:

cj (t) ≡

∫ t

t−Dj

lj (v, t) cj (v, t) dv

Can be written as:

cj (t)

w (t)= βjπj

[

cj(v, v)

w (v)

∫ Sj

0e−(n+g)s−(σ+1)Mj(s)+σ(r−ρ)s+σPj(s)ds

+

∫ Dj

Sj

e−ns−Mj(s)ds

]

(9)





Aggregate household sector (2)

Per capita asset holdings of type j agents,aj (t) ≡

∫ tt−Dj

lj (v, t) aj (v, t) dv, evolves over time accordingto:

aj (t) = (r − n) aj (t) + πjw (t)− cj (t)

+

∫ t

t−Dj

[

pj (t− v)− µj (t− v)]

lj (v, t) aj (v, t) dv (10)

Per capita total assets, a (t) ≡∑

j aj (t), satisfy thedifferential equation:

a (t) = (r − n) a (t) +w (t)− c (t) (11)





Firms (1)

Firm-level relationships:

Yi (t) = Ω (t)Ki (t)ε Li (t)

1−ε (13)

w (t) = (1− ε)Ω (t) ki (t)ε (14)

r (t) + δ = εΩ (t) ki (t)ε−1 (15)

External effect as in Heijdra & Mierau:

Ω (t) = Ω0k (t)1−ε (16)

Aggregate relationships:

Y (t) = Ω0K (t) (17)

w (t)L (t) = (1− ε)Y (t) (18)

r (t) = r = εΩ0 − δ (19)




Informational assumptionsSeparating equilibriumPooling equilibrium

Assumption about the annuity market

Assumption (A1) The annuity market is perfectly competitive. Alarge number of firms offer annuity contracts toindividuals. Firms entry and exit is unrestricted.

Assumption (A2) Annuity firms do not use up any real resources.

Assumption (A3) The annuitant’s health status is privateinformation and cannot be observed by the annuitycompanies. Annuity firms know all the features of themortality process of each health group.

Assumption (A4) The annuitant’s age is public information andcan thus be observed by the annuity companies.

Assumption (A5) Annuitants can buy multiple annuities fordifferent amounts and from different annuity firms.Individual annuity firms cannot observe an annuitant’sholdings with their competitors.





Rothschild-Stiglitz-Wilson versus Pauly-Abel (1)

Most papers on adverse selection use Rotschild-Stiglitz (1976QJE ) or Wilson (1977 JET ) equilibrium concept (R-S-Whereafter).

non-linear pricing (price and quantity controlled)requires annuity companies (a) to perfectly monitor annuitypurchases of all annuitants and (b) to prevent customers frombuying more than one annuity contract.relevant for some insurance markets but not for annuities.

Reasons for rejection of R-S-W:

Monitoring is very difficult. Receipts of annuity payments fromother annuity firms are not observable.Death of the annuitant ends the liability of the annuity firm:withholding payments and investigating compliance notpossible (as they are with fire- or car insurance)Menus with co-payments (as with medical insurance) areinfeasible.





Rothschild-Stiglitz-Wilson versus Pauly-Abel (2)

In favour of Pauly (1974 QJE ) – Abel (1986 Ectrica) linearpricing:

Quantity of annuity purchases cannot be controlled, only theprice of annuities.Alternative interpetation: Nash equilibrium. Firms deviatingfrom the zero-profit price incur losses.





Separating equilibrium (1)

If health type is observable to annuity firms:

pj (u) = µj (u)

Full longevity risk insurance

Analytical model in Table 1

Visualized in Figures 2(a)–(d)

Quantified in Table 2





Table 1: Balanced growth in the separating equilibrium

(a) Microeconomic relationships:

cj (v, v)

w (v)=

∫ Dj

0 e−(r−g)s−Mj(s)ds∫ Dj

0 e−(1−σ)rs−σρs−Mj(s)ds(T1.1)

(b) Macroeconomic relationships:

c (t)

w (t)=

∑

j

βjπjcj (v, v)

w (v)

∫ Dj

0e−(n+g)s−Mj(s)+σ(r−ρ)sds (T1.2)

g ≡k (t)

k (t)= r − n+

[

1−c (t)

w (t)

]

·w (t)

k (t)(T1.3)

w (t)

k (t)= (1− ε)Ω0 (T1.4)





Table 2: Growth and retirement: quantitative effects(a) SE (b) PE (c) SE (d) PENo retirementb With retirementc

cH (v, v)

w (v)0.7668 0.7396 0.8267 0.8147

cU (v, v)

w (v)0.7858 0.7775 0.8492 0.8476

SH (years) DH DH DH DH

SU (years) DU 61.63 DU 63.93

c (t)

w (t)1.0714 1.0753 0.8966 0.8979

g (%year) 2.00 1.89 2.00 1.96

w (t)

k (t)0.2800 0.2800 0.3346 0.3346

ΛH (v0, v0) 27.4619 26.4639 11.9477 11.8281

ΛU (v0, v0) 21.8047 20.6189 10.3314 9.7873





Figure 2(a): Separating equilibrium, scaled individual

consumption

cj(v,v+u)w(v)

20 30 40 50 60 70 80 900

1

2

3

4

5

6

7

8

9


HealthyUnhealthy





Figure 2(b): Separating equilibrium, scaled cohort assets

aj(v,v+u)w(v)

20 30 40 50 60 70 80 900

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


HealthyUnhealthy





Separating equilibrium (2)

Proposition 1. Consider the separating equilibrium (SE) inwhich annuity firms can observe the health type of annuitants.Provided σr − g > ρ, agents of all health types are net saversthroughout life, i.e. aj(v, v) = aj(v, v + Dj) = 0 andaj(v, v + u) > 0 for 0 < u < Dj .





Pooling equilibrium (1)

If health type is unobservable to annuity firms, the competitivepooling rate is:

p (u) =

µH (u) · aH (v, v + u) + µU (u) · aU (v, v + u)

aH (v, v + u) + aU (v, v + u)for 0 < u ≤ DU

µH (u) for DU < u < DH

(25)

For 0 < u ≤ DU , µH (u) ≤ p (u) ≤ µU (u). No actuarialfairness.

Analytical model in Table 3

Visualized in Figures 3(a)–(d)





Pooling equilibrium (2)

Proposition 2. Consider the case in which annuity firms areunable to observe the health type of annuitants. Assume thata pooling equilibrium (PE) exists and that σr − g > ρ. Then:

(i) healthy agents are net savers throughout life, i.e.aH(v, v) = aH(v, v + DH) = 0 and aH(v, v + u) > 0 for0 < u < DH ;

(ii) unhealthy agents are net savers until age SU < DU afterwhich they adopt a self-imposed borrowing constraint, i.e.aU (v, v) = 0, aU (v, v + u) > 0 for 0 < u < SU , andaU (v, v + u) = 0 and cU (v, v + u) = w (v + u) forSU ≤ u ≤ DU .





Table 3: Balanced growth in the pooling equilibrium

(a) Microeconomic relationships:

cH (v, v)

w (v)=

∫ DH0 e−(r−g)s−P(s)ds

∫ DH0 e−ρ∗s−(1−σ)P (s)−σMH (s)ds

(T3.1)

cU (v, v)

w (v)=

∫ SU0 e−(r−g)s−P (s)ds

∫ SU0 e−ρ∗s−(1−σ)P (s)−σMU (s)ds

(T3.2)

cU (v, v)

w (v)= e

−(σ(r−ρ)−g)SU+σ[MU (SU )−P (SU )] (T3.3)

aH (v, v + u)

w (v)= βHπHe

(r−n)u−MH (u)+P(u)[ ∫

u

0e−(r−g)s−P(s)

ds

−cH (v, v)

w (v)

∫

u

0e−ρ∗s−(1−σ)P(s)−σMH (s)

ds

]

, (0 ≤ u ≤ DH ) (T3.4a)

aU (v, v + u)

w (v)= βUπU e

(r−n)u−MU (u)+P(u)[ ∫

u

0e−(r−g)s−P (s)

ds

−cU (v, v)

w (v)

∫

u

0e−ρ∗s−(1−σ)P (s)−σMU (s)

ds

]

, (0 ≤ u < SU ) (T3.4b)

aU (v, v + u)

w (v)= 0, (SU ≤ u ≤ DU ) (T3.4c)

p (u) =µH (u)aH (v, v + u) + µU (u)aU (v, v + u)

aH (v, v + u) + aU (v, v + u)(T3.5)





Table 3: Balanced growth in the pooling equilibrium

(b) Macroeconomic relationships:

c (t)

w (t)= βHπH

cH (v, v)

w (v)

∫

DH

0e(r−n−g−ρ∗)s−(1+σ)MH (s)+σP (s)

ds

+βUπU

[

cU (v, v)

w (v)

∫

SU

0e(r−n−g−ρ∗)s−(1+σ)MU (s)+σP (s)

ds

+

∫

DU

SU

e−ns−MU (s)

ds

]

(T3.6)

g ≡k (t)

k (t)= r − n +

[

1 −c (t)

w (t)

]

w (t)

k (t)(T3.7)

w (t)

k (t)= (1 − ε)Ω0 (T3.8)





Figure 3(a): Pooling equilibrium, scaled individual

consumption

cj(v,v+u)w(v)

20 30 40 50 60 70 80 900

1

2

3

4

5

6

7

8

9


HealthyUnhealthyWage income





Figure 3(b): Pooling equilibrium, scaled cohort assets

aj(v,v+u)w(v)

20 30 40 50 60 70 80 900

0.02

0.04

0.06

0.08

0.1

0.12


HealthyUnhealthy





Figure 3(c): Pooling equilibrium, relative pooling rate (U)

p(u)− µU (u)

20 30 40 50 60 70 80 90−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0


annu

al p

erce

ntag

e po

ints





Figure 3(d): Pooling equilibrium, relative pooling rate (H)

p(u)− µH(u)

20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7


annu

al p

erce

ntag

e po

ints





Welfare analysis

Clearly ΛSEj (v0, v0) > ΛPE

j (v0, v0) for j ∈ H,U

Lost growth years (were you born too early?):

LGYij =

1

gi·ΛSEj (v0, v0)− Λi

j (v0, v0)∫ Dj

0 e−ρs−Mj(s)ds. (28)

From Table 2:

Difference between SE and PE is smallFirst version of the paper: difference between PE and NAE islarge. Depends on recycling of accidental bequests. Tragedy ofannuitization.Adverse selection problems are small!





Robustness check

Two conclusions emerge from the analysis conducted thus far:

Conclusion (C1): Unhealthy individuals adopt a self-imposedborrowing constraint fairly early on during old age.Conclusion (C2): The welfare effects of the annuity marketimperfection due to asymmetric information are rather modest.

Robustness checks with respect to:

Mandatory retirement under a PAYG systemEndogenous versus exogenous growth model




Main findings

SE is slightly better in welfare terms than PE.

But an H-type individual has an incentive to misrepresenthealth status. SE is not credible.

In PE, H-types benefit for part of life from presence ofU -types.

In PE, U -types run out of assets if they live long enough.

Conclusions are robust. They hold under mandatory retirementand also in an exogenous growth model.




Extensions

Endogenous labour supply, optimal retirement, and growth.

Transitional dynamics.

Health shocks

Differential productivity.

Long-term annuities and lock-in effects


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