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 decision-based. These changes are effected by probabilistic ageing or updating, or semi-Markov, models with dynamic transition rates. There are three such models for employment, population and households and housing.

Change of Employment

Change of zonal employment occurs in the model in four different submodels:


Decline of zonal employment due to sectoral decline, lack of building space and intraregional relocation of firms is modelled in this submodel.


Changes of zonal employment due to the location or removal of large plants exogenously specified by the user are executed in the Public Programmes Submodel.


Changes of zonal employment due to new jobs in vacant industrial or commercial buildings, in newly built industrial or commercial buildings or in converted residential buildings are modelled in the Private Construction Submodel.

In the Ageing Submodel decline of zonal employment due to sectoral decline, lack of building space and intraregional relocation of firms is modelled. Each of the forty industries of the model is considered a separate submarket. The model starts from existing employment Eslj(t) of industry s situated on land use category l in zone j at time t. There are three different ways for Eslj to change in this submodel:

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:


is the number of workers in industry s made redundant on land-use category l in zone j between t and t+1. Es(t) indicates total employment of industry s in the region and Es(t+1) is the exogenous projection of total regional employment for time t+1. The utility uslj(t) expresses the attractiveness of land-use category l in zone j for industry s (see below). Rslj is set to zero for growing industries.

b) Relocation

Some industries are very stationary, while others easily move from one location to another. If rs is a sectoral mobility rate, then


is the number of workplaces relocated from land-use category l in zone j during the period. The mobility rate rs is exogenous.

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 Sslj have to be relocated because of lack of space:


where bsj(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, Sslj is set to zero.

For the workers made redundant, later new buildings will have to be provided in the Public Programmes Submodel or in the Private Construction Submodel. Where decline of employment is large, buildings remain vacant, but these may be reused by other industries later.

Population Change

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 Housing Market Submodel.

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:


where P'asi(t+1) are surviving persons of age group a and sex s in zone i in year t+1, Pasi(t) is population of age group a and sex s in year t and dasi'(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:


where gasi(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


with special cases




where Bsi(t,t+1) are births of sex s in zone i between years t and t+1:


where basi'(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 d0si(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.

Household Change

Households are represented in the model as a four-dimensional distribution classified by:
- nationality (native, foreign)
- age of head (16-29, 30-59, 60+)
- income/skill (low, medium, high, very high)
- size (1, 2, 3, 4, 5+ persons)

Similarly, housing of each zone is represented as a four-dimensional distribution of dwellings classified by
- type of building (single-family, multi-family)
- tenure (owner-occupied, rented, public)
- quality (very low, low, medium, high)
- size (1, 2, 3, 4, 5+ rooms)

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 H household and K housing types, with H and K not exceeding 30.

The cross-classification of households and housing is performed in the occupancy matrix. The occupancy R of a zone is an H x K matrix representing the association of households with dwellings in the zone. Each element Rhk of the matrix contains the number of households of type 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. S is an H x 1 vector of subtenant households, V is an 1 x K vector of vacant dwellings and N is an 1 x K vector of dwellings newly constructed in the previous period and released to the market now.

By incorporating the zonal dimension i, i = 1, ..., I, the matrix R becomes three-dimensional, and the vectors S, V and N become two-dimensional matrices. R, S, V and N are a complete representation of the household/housing system at the outset of the simulation period. All changes occurring to households and housing during the period can be represented by transitions into, within, or out of these four matrices.

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:


Ageing of households and housing and other demographic changes of household status are modelled in this submodel (see Figure 1).


Public housing programmes specified by the model user are executed in the Public Programmes Submodel.


Private housing maintenance/upgrading and new construction investments and the resulting changes in housing and land prices are modelled in the Private Construction Submodel.


Changes of household income induced by changes in employment status are modelled in the Labour Market Submodel.


Changes of the association of households with housing and the resulting changes in housing prices are modelled in the Housing Market Submodel.

Figure 1. Household and housing change in the Ageing Submodel.

In the Ageing Submodel all changes of households and housing are modelled that in the model are treated as merely time-dependent. For households such changes include demographic changes of households status in the household's life cycle such as ageing and death as well as birth, marriage and divorce and all new or dissolved households resulting from these changes, plus changes of nationality. On the housing side they include deterioration by ageing (filtering down the quality scale) and eventually demolition.

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 t to time t+1. The transition rates are computed as follows. The time-dependent changes to be simulated are interpreted as events occurring to a household or dwelling with a certain probability in time interval t to t+1. The basic event probabilities and their expected future development are determined exogenously or are taken from the demographic submodel (see Population Change above). Eleven basic event probabilities were identified for each of the three household age groups:


change of nationality
birth, native
birth, foreign
relative joins household
death of child
marriage of child
new household of child

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 Private Construction Submodel, where housing may have to make way for industrial or commercial land uses. Maintenance/upgrading and new housing construction are assumed to be demand-generated, i.e. decision-based, and are therefore treated in the Private Construction Submodel.

The basic event probabilities are aggregated to zone-specific transition rates between household or dwelling types in two matrices, hi for households and di for dwellings using the disaggregate (120-type) household and housing distributions of each zone as weights. The matrices hi and di are of dimensions 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 hi and di with the occupancy matrix Ri yields the occupancy matrix aged by one simulation period:


where h'i is the transpose of hi. This procedure assumes that all households of type 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 occupancy

Special 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:


where Ni(t,t+1) is an H x H matrix containing current household formation probabilities calculated from the above events and qi(t,t+1) is a K x 1 vector of demolition rates of housing types. An element nhh'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:


where ri(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 R, as also households without dwelling get older:


and also vacant dwellings deteriorate or may be torn down:


where Ni(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.): Microeconomic Models of Housing Markets. Lecture Notes in Economics and Mathematical Systems 239. Berlin/Heidelberg/New York: Springer Verlag, 144-191.

© 1998 Michael Wegener, IRPUD