The IRPUD Model: Ageing Submodel In the Ageing Submodel all changes of zonal stock
variables are modelled which are assumed to result from biological,
technological or long-germ socio-economic trends or originating outside of
the model, i.e. which are not treated as
Change of zonal employment occurs in the model in four different submodels:
In the t) of industry s
situated on land use category l in zone j at time t.
There are three different ways for E to change in this
submodel:_{slj}a) Sectoral Decline Declining industries make workers redundant. This occurs not necessarily at the same rate all over the region, but is more likely where locational conditions are less favourable: (1) is the number of workers in industry t)
indicates total employment of industry s in the region and
E(_{s}t+1) is the exogenous
projection of total regional employment for time t+1. The utility
u(_{slj}t) expresses the
attractiveness of land-use category l in zone j for industry
s (see below). R is set to
zero for growing industries._{slj}b) Relocation Some industries are very stationary, while others easily
move from one location to another. If (2) is the number of workplaces relocated from land-use
category c) Lack of Building Space In most industries, mechanisation and automation tend to
increase the building floorspace per workplace. Accordingly, in each
period, a number of jobs (3) where t+1) is the projected
floorspace per workplace in industry s in zone j at time
t+1, which will be always greater than or equal to its value at time
t. Where redundancies exceed relocations due to lack of space,
S is set to zero._{slj}For the workers made redundant, later new buildings will
have to be provided in the
The population projection model predicts zonal
population by age, sex and nationality exclusive of migration. Changes of
zonal population by migration into, out of or within the region are
modelled in the
Changes of population due to fertility and mortality are modelled by a cohort-survival model subject to exogenous forecasts of fertility and mortality rates. To reduce data requirements, a simplified version of the cohort-survival population projection model with five-year age groups is applied. The method starts by calculating survivors for each age group and sex: (4) where t+1) are surviving
persons of age group a and sex s in zone i in year
t+1, P(_{asi}t) is population of age group
a and sex s in year t and
d(_{asi'}t) is the average annual death rate of age
group a and sex s between years t and t+1 in
the group of zones i' to which zone i belongs.Next it is calculated how many persons change from one age group to the next through ageing employing a smoothing algorithm: (5) where t,t+1) is the
number of persons of sex s changing from age group a to age
group a+1 in region r. Surviving persons in year
t+1 are then (6) with special cases (7) and (8) where t,t+1) are
births of sex s in zone i between years t and
t+1:(9) where t,t+1) are average number
of births of sex s by women of child-bearing five-year age groups a,
a = 4,10 (15 to 49 years of age) in the group of zones i' to
which zone i belongs between years t and t+1,
and d(_{0si}t,t+1) is the death rate
during the first year of life of infants of sex s in that group of
zones.If the duration of the simulation period is more than one year, the population projection model is executed once in each year of the simulation period.
Households are represented in the model as a
four-dimensional distribution classified by:
Similarly, housing of each zone is represented as a
four-dimensional distribution of dwellings classified by
All changes of households and housing during the
simulation are computed for these 120 household types and 120 housing
types. However, where households and housing are cross-classified together,
these households and housing types are aggregated to
The cross-classification of households and housing is
performed in the h, h = 1,
..., H, occupying a dwelling of type k, k = 1, ...,
K, the total matrix contains all households occupying a dwelling or
all dwellings occupied by a household.
In addition, there are for each zone three vectors
representing households without a dwelling or dwellings without a
household.
By incorporating the zonal dimension Changes occurring to households and housing affect either households only, dwellings only, or both households and dwellings. Households come into existence, grow, get older, separate or merge, get more or less income, finally shrink and disappear. Dwellings are built, maintained or upgraded, or deteriorate and eventually are torn down. The association of households with dwellings changes through occupations and vacations, and this leads to changes of the composition and price of housing supply. Two principal kinds of changes can be distinguished: changes that are decision-based and changes that are not. For instance, migration and housing investments are normally based on rational decisions and are therefore modelled in decision models. The ageing of households and dwellings, however, depends only on the course of time and can therefore be modelled by probabilistic transition rates. Other changes are in reality decision-based, such as changes of household status through births, marriage or divorce, but are modelled probabilistically as the motivations behind these changes are of no interest for the purpose of the model. Still other consequences are merely consequences of events occurring in other sectors of the model, e.g. changes of household income due to employment changes. A last category of changes are exogenous, i.e. directly specified by the user such as public housing programmes. Following this classification, changes of households and housing are modelled in different submodels:
In the Because of the association of households and housing in
the occupancy matrix, households and housing are aged simultaneously in a
common semi-Markov model with dynamic transition rates. A transition rate
is defined as the probability that a household or a dwelling of a certain
type changes to another type during the simulation period from time
and two for the four housing quality groups:
Not all household events occur to every household. Some
are applicable only to singles, some only to families, some only to adults,
some only to children. Some households events are followed by housing
events, and vice versa: where a household dissolves, a dwelling is vacated
and where an occupied dwelling is demolished, a household is left without a
dwelling. The housing events contain only those changes of the housing
stock which can be expected to occur under normal conditions in any housing
area, i.e. a normal rate of deterioration and demolition. More demolition
may occur in the
The basic event probabilities are aggregated to
zone-specific transition rates between household or dwelling types in two
matrices, d for dwellings using the
disaggregate (120-type) household and housing distributions of each zone as
weights. The matrices _{i}h and
_{i}d are of dimensions _{i}H x H and
K x K, respectively, where the rows indicate the source state
and the columns the target state. Most events are independent of each other
and can be aggregated multiplicatively; but some exclude others, i.e. are
the complement to each other. Multiplication of h
and _{i}d with the occupancy matrix
_{i}R yields the occupancy matrix aged by one
simulation period:_{i}(10) where h. This
procedure assumes that all households of type _{i}h in zone i
have the same transition rates no matter in which dwelling they live, and
that all dwellings of type k in zone i have the same
transition rates irrespective of their occupancySpecial provisions are necessary for events which create new households without a dwelling or vacant dwellings. New households without a dwelling are created by the events marriage of child, new household of child or divorce or by demolition of dwelling: (11) where t,t+1)
is an H x H matrix containing current household formation
probabilities calculated from the above events and
q(t,t+1) is a _{i}K x 1 vector of demolition
rates of housing types. An element
n(_{hh'i}t,t+1) is defined as the
probability that a new household of type h is produced by a
household of type h' in zone i during the simulation
period.Similarly, vacant dwellings may be generated by dissolution of households: (12) where t,t+1)
is an 1 x H vector of household dissolution rates aggregated from
events such as marriage, relative joins household and death. Of course,
vacant dwellings may also result from housing construction, but this is
modelled in the
Public Programmes Submodel,
and
Private Construction Submodel
and becomes effective only in the following
simulation period.In addition is it necessary to age households and
dwellings outside the occupancy matrix (13) and also vacant dwellings deteriorate or may be torn down: (14) where t-
1,t) is the 1 x K vector of new dwellings constructed in
the previous simulation period.
Wegener, M. (1985): The Dortmund housing market model: a
Monte Carlo simulation of a regional housing market. In: Stahl, K. (ed.):
© 1998 Michael Wegener, IRPUD |