Alexia Prskawetz

Gustav Feichtinger

Vladimir M. Veliov

English

Research Papers in Economics

Workshop on “Age-structured models in population
dynamics and economics” at the Vienna Institute of
Demography, October 27–28, 2003
Gustav Feichtinger, Alexia Prskawetz, and Vladimir M. Veliov
The workshop brought together specialistsin thetheory and in differentareasof
application of age- and/or duration-structured dynamical systems. Among those ar-
eas are demography, population dynamics, economics, epidemiology, ecology and
others. The main topics were:
● Dynamics and control of human populations
● Descriptive and control models in population economics
● Vintage models of capital accumulation and labour distribution
● Structured epidemiological models and control of infectious diseases
● Structured dynamic models in social sciences
All talks presented at the workshop had in common that age-structured popula-
tions(inaliteralormoregeneralsense)wereinvolved.Theworkshopmanifestedthe
recentsubstantialprogressinboththemathematicaltechniquesemployedinmodel-
ling of age-heterogeneous populations, and the scope of applicability of these tech-
niquesindifferentareas.Theworkshop corroborated thetendencytomergepopula-
tion and economic dynamic considerations on the basis of unified modelling. An-
other point to mention is the considerable presence of control models in a dynamic
optimisationorviabilitycontext.Suchmodelsareemployednotonlyfromanorma-
tivepointofview,butevenmoretounderstandthebehaviourofpresumablyrational
agents, or for the generation of non-trivial objective-oriented scenarios.
The talks presented at the workshop are briefly summarised below. The first
group of talks address issues concerning purely biological aspects of populations,
while the talks in the second group involve economic considerations.
Structured Populations in Demography and Epidemiology
InhistalkEndemicstatesofanimmunisinginfectionwithpositivecasefatalityin
a growing population, Klaus Dietz (Tuebingen/Germany) presented a joint paper
withM.SafanandK.P.Hadeleronanextensionforagrowingpopulationoftheclas-
sicalepidemiologicalmodelbyDanielBernoulliforapotentially fatalinfectionina
constantpopulation.Age-dependentbirthanddeathratesareconsideredwithaposi-
tiveMalthusianparameterintheabsenceoftheinfection.Criticalboundsforage-de-
pendent case-fatality rates are derived which are compatible with an endemic state.
Gustav Feichtinger, Alexia Prskawetz, and Vladimir M. Veliov 333Theforceofinfectionisalsoallowedtodependonage.Generalformulasfortheba-
sicreproductionnumberintermsofthemodelparametersandintermsoftheequilib-
riumprevalenceofthesusceptiblesarederived.Itisassumedthatthedurationofthe
infectiousperiodisnotaffectedbythemortalityinducedbytheinfection.Theanaly-
sis is focused in particular on infections whose infectious period is of negligible
duration compared to the life expectancy of an individual.
RuiDilão(Lisbon/Portugal)discussedinhistalkPopulationcyclesintheagede-
pendentMcKendrickmodeltheEasterlincycles.Theseareoscillationsinthepopula-
tion growth predicted by Easterlin in the 1960s with a very simple two-generation
model. The author reported existence of demographic cycles in the context of a
McKendrick model with age-dependent birth and death rates. For a population with
only one reproductive age class, the temporal evolution of the population size is al-
ways modulated by a time periodic function with a period equal to the age of the re-
productiveageclass.Forapopulationwithacontinuousdistributionofreproductive
ageclasses,theexistenceofdampedcyclesisproved.Theperiodicityofthedamped
cycles is associated with the age of the first reproductive age class. Damping in-
creases as the dispersion of the fertility function around the age class with maximal
fertilityincreases.Ingeneral,theperiodofthedemographiccyclesisassociatedwith
the time that a species takes to reach reproductive maturity.
In his talk Some results on the modelling of tuberculosis, Mimmo Iannelli
(Trento/Italy) presented a general epidemic model that includes some peculiar as-
pects of tuberculosis (TB). Long periods of latency and the emergence of antibiotic
resistance due to incomplete treatment are very important features of TB dynamics.
A two-strain TB model is formulated, with an arbitrarily distributed delay in the la-
tent stage of individuals infected with the sensitive strain, and the effects of variable
periodsoflatencyonthediseasedynamicsareinvestigated.Theauthordiscussedthe
invasionandcoexistenceofthetwostrainsversusasignificantparametersuchasthe
rate at which infected people develop drug-resistant TB, due to incomplete treat-
ment. Control policies through screening programs are proposed for the case of a
single (drug-sensitive) strain.
Hisashi Inaba (Tokyo/Japan) presented a talk on Endemic threshold results in an
age-structured population model for HIV/AIDS epidemic. Within an age-dura-
tion-structured model of the HIV infection in a homogeneous community, the inva-
sion problem of the HIV/AIDS epidemic is investigated using positiveoperator the-
ory. Conditions for the existence and uniqueness of an endemic steady state are
given. In particular, it is shown that a backward bifurcation can occur at the thresh-
old,i.e.,multipleendemicsteadystatescanexistforthestructuredmodelofHIVin-
fection. Stability results for the endemic steady states are discussed.
NanLi(Berkeley/USA) presented thepaperPopulation momentum forrealpop-
ulationandlinearfertilitytransition.Populationmomentumistheratioofapopula-
tion’s ultimate size after a demographic transition to its initial size before the transi-
tion. For stable population and instantaneous drop to replacement fertility, Keyfitz
found a simple formula for the momentum. Keyfitz’s formula has been extended to
334describe gradual demographic transitions. The stable initial population, however,
still obstructs approaching reality. Extending the solution of a Lotka equation with
time-varying vital rates to negative values of time, this paper provides an approxi-
mateformulaforthemomentumofanyinitialpopulationthatundergoeslinearfertil-
ity transition. Examples using data from the United Nations indicate that this
formula works well both for age structures far away from or close to stable.
Population Economics
Christian Almeder and Gernot Tragler (Vienna/Austria) presented a joint talk on
Optimalpreventionprogramsinanage-structureddruginitiation model.Abriefre-
viewofdynamicmodelsofillicitdruguseconsistingofODE’s(bothdescriptiveand
normative) is given. Even though these models do a good job in approximating em-
piricaldatasuchasthenumbersof‘light’and‘heavy’cocaineusersintheU.S.,they
neglectoneimportant fact,whichisthestrong dependenceofdrug initiation onage.
Therefore a model for drug initiation that extends traditional dynamic models by
consideringexplicitlytheagedistributionoftheusersisintroduced.Onthebasisofa
2-groups model in which the population is split into a user and a non-user group the
advantage of a continuous age distribution is shown by considering more details,
yielding new results. Neglecting death rates reduces the model to a single state
(1-group)descriptivemodelwhichcanstillsimulatesomeofthecomplexbehaviour
of drug epidemics such as repeated cycles. Furthermore, prevention programs—es-
pecially school-based programs—can be targeted to certain age classes. Optimal
controlmodelsareusedforbestallocationofresourcestopreventionprogramsover
different age classes.
NoëlBonneuil(Paris/France)presentedthepaperPreservingTransferBenefitfor
Present and Future Generations (jointly with R. Boarini). The main issue is: how
shouldstateinterventionsbecalibratedintimeandmagnitudetoguaranteeapositive
accountforeachgenerationaswellasaminimalwelfareandsomeequityacrossage
classes?Thesocialcontractturnsintothedynamicmaintenanceofapositiveaccount
for every present and future generation. It requires either accelerating or slowing
downbothpensionandeducationspending.Theensuingdynamictrade-offbetween
generational benefit and equity delineates an optimal interest rate in human capital.
Moreover,thefavourableeffectofpopulation growthonthecapacitytopayforpen-
sionsisthwartedwhenfairnessbetweenageclassestogetherwiththepreservationof
apositivenetbenefitforfuturegenerationsisalsoconsidered.Thespeakerpresented
an informal introduction to the viability theory, which is the main tool the authors
employ in the analysis.
Raouf Boucekkine (Louvain/Belgium) presented the overview Modelling vin-
tagestructureswithdifferential-differenceequations:specification,stabilityassess-
mentandoptimalcontrol.FirstavintagecapitalmodeloftheSolowtypeisdiscussed
in order to simply illustrate the main specifications within this framework, and the
Gustav Feichtinger, Alexia Prskawetz, and Vladimir M. Veliov 335typical stability and numerical solution tools. Then Ramsey vintage capital models
with and without a stationary environment are considered. The dynamic properties
ofsuchmodels,togetherwiththeassociatedanalyticaltools,arebrieflysummarised.
Finally, two vintage capital models displaying endogenous growth are presented.
Thefirstonedescribesthedynamicsofhumancapitalaccumulationunderarealistic
survival law and a vintage human capital structure. The second is the one-hoss shay
modelwithconstantreturntocapital.Handlingthismodeltheauthorprovidesasur-
vey on some of the optimal control material required to treat the presence of lagged
controls in this model.
Alexia Fuernkranz-Prskawetz (Vienna/Austria) presented the paper Optimal
age-specific labour force: effects of education, productivity and wages (jointly with
P.KortandV.Veliov)whichanalysestheoptimalhiring andfiring policyofasingle
firm that is subject to hiring and firing costs. In contrast to previous contributions,
this paper allows for age-dependent labour force, adjustment costs, wages, and pro-
ductivity.Inparticular,thepaperinvestigatestheeffectoftheage-specificwage-pro-
ductivity gap on the optimal age schedule of the work force.
Peter Kort (Tilburg/Netherlands) presented the paper Environmental policy, the
Porterhypothesisandthecompositionofcapital:Effectsoflearningandtechnologi-
calprogress(jointly withG.Feichtinger,R.HartlandV.Veliov).Thispaperinvesti-
gates the effect of environmental policy on the composition of capital. By allowing
for non-linearities it generalises an earlier paper by Xepapadeas and De Zeeuw
(1999) and determines scenarios in which their results do not carry over. In particu-
lar, it is shown that the way acquisition cost of investment decreases with the age of
thecapitalstock isof crucialimportance. Alsoitisobtained thatenvironmental pol-
icyhasoppositeeffectsontheaverageageofthecapitalstockinthecaseofeitherde-
teriorationordepreciation. Furthermore,itturnsoutthatinthepresenceoflearning,
implementingastricterenvironmentalpolicywiththeaimtoreachacertaintargetof
emissionsreductionhasastrongernegativeeffectonindustry profits,whichimplies
quite the opposite from what the Porter hypothesis describes.
The paper Optimal harvesting of age class structured forest resources by Olli
Tahvonen (Helsinki/Finland) studies optimal harvesting of multiple age class forest
resourcesunder discounting and strictly concaveoptimisation criteria.The problem
is analysed as a discrete time, infinite horizon and any number of state and control
variables optimisation problem. The necessary and sufficient optimality conditions
are obtained using the Karush-Kuhn-Tucker theorem in non-linear programming. It
is shown that at the optimal stationary state, timber harvest may evolve cyclically
over time. The cycles occur because in the forest age class model space is measured
continuously and timeindiscreteunits. Cyclesdisappear asperiod length movesto-
wardzero,whentheoptimalrotationperiodisnotuniqueorwhenthediscountfactor
approaches unity.
In his talk Optimal taxation and governmental investment: an age-structured
model, T. Tsachev (Sofia/Bulgaria) presented a macroeconomic model in which the
population and the capital are age-structured. The government’s revenue consists of
336asingletax,modelledasapercentageoftheGNP.Themodelisintheformofanopti-
mal control problem. The control variables are the collected tax and the percentage
of it which the government spends for investments.The discounted social welfareis
to be maximised over a prescribed time interval. Identification and solution
approaches are discussed.
InthetalkentitledHeterogeneouscontrolsystems:ageneralperspectiveandap-
plications in population dynamics and economics, Vladimir Veliov (Vienna/Aus-
tria) presented the concept of a heterogeneous control system which extends that of
age-structuredsystemsbyallowingforparametersofheterogeneityotherthanageor
duration. Thebasictheory isbriefly outlined, including necessaryoptimality condi-
tions,secondordersufficientconditions,andnumericalapproaches.Thensomepar-
ticularheterogeneous control modelsarisingindemography,economics,andepide-
miology are discussed, with more details devoted to a model of the spread of an in-
fectious disease in a population which is heterogeneous with respect to the level of
risky behaviour.
Gustav Feichtinger, Alexia Prskawetz, and Vladimir M. Veliov 337

Workshop on "Age-structured models in population dynamics and economics at the Vienna Institute of Demography, October 27-28, 2003

http://epub.oeaw.ac.at/0xc1aa500d_0x0006201b

Workshop on "Age-structured models in population dynamics and economics at the Vienna Institute of Demography, October 27-28, 2003

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Research Papers in Economics