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A hierarchical model is calibrated and used to illustrate labour market flows within a firm. The model establishes a link between the models of the firm from the literature on industrial organisation and the description of labour market dynamics in the flow approach to labour markets. It describes the decision of the personnel management of the firm whether to fire workers, and/or whether to hire workers from the internal or external labour market. The decision is based on firing costs, hiring costs, training costs and the availability of qualified workers within and outside the firm. The firm tries to adjust the size and composition of its personnel as much as possible to the optimal size and hierarchical structure which follows from maximisation of the net revenues in each period. Data in this maximisation process of the firm are productivity, wage differentials, economies of scale from cooperation, foregone production due to supervising and training, and average hiring and firing costs. Yet, usually immediate adjustment to the optimal composition of the personnel is not possible due to unanticipated quits and due to lack of qualified workers. In that case vacancies may remain open for a prolonged time. The parameters of the model are calibrated in order to mimic a representative firm of average size, which has external and internal labour flows in accordance with data found in the (scanty) empirical literature. Heterogeneity amongst firms is introduced by means of changes in the major parameters which dictate the hierarchical structure and optimal size of the firm.
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F.A.G. den Butter
E. van Gameren
A hierarchical model is calibrated and used to illustrate labour
market flows within a firm. The model establishes a link between
the models of the firm from the literature on industrial
organisation and the description of labour market dynamics in the
flow approach to labour markets. It describes the decision of the
personnel management of the firm whether to fire workers,
and/or whether to hire workers from the internal or external
labour market. The decision is based on firing costs, hiring costs,
training costs and the availability of qualified workers within and
outside the firm. The firm tries to adjust the size and composition
of its personnel as much as possible to the optimal size and
hierarchical structure which follows from maximisation of the net
revenues in each period. Data in this maximisation process of the
firm are productivity, wage differentials, economies of scale from
cooperation, foregone production due to supervising and
training, and average hiring and firing costs. Yet, usually
immediate adjustment to the optimal composition of the
personnel is not possible due to unanticipated quits and due to
lack of qualified workers. In that case vacancies may remain
open for a prolonged time. The parameters of the model are
calibrated in order to mimic a representative firm of average size,
which has external and internal labour flows in accordance with
data found in the (scanty) empirical literature. Heterogeneity
amongst firms is introduced by means of changes in the major
Applied Labour Economics Research Team (ALERT), Free University and TI, De Boelelaan 1105,
1081 HV Amsterdam, The Netherlands. The paper has benefited from comments by Peter Doeringer,
Giorgio Brunello and participants of workshops in Amsterdam, Århus and Newcastle-upon-Tyne.
parameters which dictate the hierarchical structure and optimal
size of the firm.
key words:
hierarchical structure of a firm, internal labour market, hiring and
firing costs, job mobility, costs and benefits of training
1. Introduction
Analysis of large longitudinal micro data sets at the level of the firm has recently
provided much insight into labour market dynamics. The seminal work by Leonard
(1994), Dunne
et al
. (1989) and Davis & Haltiwanger (1990, 1992) has shown that gross
job and labour flows are much larger than net changes in employment. Many replication
studies and extensions of the work of Davis & Haltiwanger on job creation and job
destruction indicate a considerable convergence of international evidence with respect to
various characteristics of labour market dynamics (see e.g. Davis, Haltiwanger & Schuh,
1996, and the references therein). One major finding is that there is much heterogeneity
amongst firms and establishments, and that the largest part of job turnover (job creation
+ job destruction) takes place within the same regions and branches of industry. It
implies that job creation and job destruction is much more driven by idiosyncratic, firm
specific shocks, than by demand and supply shocks at the macro level.
In recent years also much effort has been spent on small theoretical models which may
explain these stylised facts of labour market dynamics. A major background is
equilibrium search theory, which explains how dynamic unemployment equilibria are the
result of transaction costs in the search process between employers and employees.
Heterogeneities at the labour market are a main reason for these transaction costs, which
make the matching process between job seekers and employers who have vacant
positions, time consuming. Because of the transaction costs and the possibility of
mismatches, there is a surplus value of a successful match, which is to be shared between
the worker and the employer.
However, these equilibrium search models and the related labour demand models, in
which transaction costs play a major role in explaining unemployment equilibria and
sluggish adjustment dynamics, are based on elegant but highly abstract assumptions on
labour market behaviour of firms and workers. For instance, the idea that the employer
and the successful applicant bargain in their wage negotiations over the distribution of
the surplus value of the match, looks somewhat far-off from the way wages of new
workers are set according to fixed salary scales in the real world. Hence, the abstract
models of equilibrium search do not provide insight into the various mechanisms that
govern labour market flows at the level of the firm (and do not purport to do so).
Moreover, these models do not reckon with labour market flows within the firm, i.e. with
internal job mobility. These internal labour flows are, by the way, often neglected in the
empirical analysis of labour market dynamics along the lines of Davis & Haltiwanger (see
et al.
, 1996, for a counter example).
We want to unveil the labour mobility within the black boxof the firm. Obviously,
internal and external labour flows are closely connected. Some of the fundamental
questions about the relation between jobs and employees can be answered only at the
level of the firm. When does a firm open a vacancy, and what determines the design of
the vacancy? Are job requirements only based on the needs of the production or do
personnel managers adjust the job to fit the available employees? When a firm has a
vacancy - either through creation of a new job, through a quit or through an internal
move - the costs of hiring an external candidate are to be balanced against the costs of
moving an internal candidate. Factors like the career prospect within the firm, and the
employability in other firms are important in the study of labour mobility and the process
of job creation and destruction.
In this paper we focus on the construction of a model that formalises the decisions of
personnel managers. The main feature of the model is that explicitly describes the choice
between external mobility and labour mobility within the firm. We design a stylised
hierarchical simulation model of the firm that combines elements of dynamic search
theory with elements of internal labour market theory. In this model we are able to
follow the career of each employee in the firm, including his entry and exit. Hereto we
extend Williamsons (1967) model of the relationship between firm size and hierarchical
control with a number of features which allow us to generate internal and external labour
market flows using the simulation model. These features are the inclusion of
the stochastics of quits,
hiring, firing, training and supervision costs,
the building up of skills of incumbent workers through learning by doing and training,
defining the level of skills required at each hierarchical layer, and
the skill distribution of job seekers available at the external labour market.
Obviously such a simulation model is not a neat and elegant model which can be
analytically solved, like the models of the equilibrium search theory. Our model can only
be solved numerically. It has to be calibrated with respect to numerous functional forms
and parameter values. In this calibration procedure we take account of unexpected
idiosyncratic shocks which hit the firm through the specification of the stochastics of
quits and the skill distribution of job seekers which mimic the type of shocks considered
in the models of the flow approach to the labour market. Heterogeneity amongst firms is
included in the model through changes in the parameter values and functional forms of
e.g. the production function, the skill requirements in the various hierarchical layers, the
span of control of supervisors (defining the flatness of the organisation), the hiring, firing
and training costs etc.. As far as we know equilibrium search models of the labour
market do not take account of these kinds of heterogeneity between firms.
The model is calibrated in a simulation exercise with 100 replications so that its dynamic
base line simulation reproduces some stylised facts from empirical studies of the
industrial organisation on the size distribution of firms. Moreover, in the calibration the
average sizes of the internal and external labour market flows should be in line with what
seems plausible and with the scant empirical evidence in this respect.
The next section reviews some of the stylised facts and discusses some relevant
assumptions and implications of equilibrium search theory and internal labour market
theory. Section 3 presents Williamsons original hierarchical model of the firm and
discusses the necessary extensions to the model inspired by search theory so that the
model generates labour flows at the level of the firm. Section 4 gives the main features
and characteristics of the simulation model, whereas section 5 discusses the central
projection of the calibrated model. Section 6 illustrates the sensitivity of the working of
the model in case of specification changes and changes in parameter values, in order to
gain insight in the sources of observed heterogeneity amongst firms. Section 7 concludes
and gives some suggestions for further research in which the model of this paper can be a
2. Stylised facts and implications from internal labour market theory and the
flow approach to labour markets
The conventional neo-classical labour market theory, with homogeneous labour supply,
elastic labour conditions, perfect information, perfect competition and rational behaviour
by representative agents does not seem to be much in line with the observed behaviour of
employers and employees. There is much less mobility between jobs then implied by the
neo-classical theory. Tenure is often long, and mobility low. To capture this
phenomenon, the concept of dual or segmented labour markets, and more specific the
concept of the internal labour market is developed.
Discussions about the theory of the internal labour market often use the definition of
Doeringer & Piore (1971) as the starting point. They define the internal labour market as
an administrative unit, such as a manufacturing plant, within which the pricing and
allocation of labor is governed by a set of administrative rules and procedures
(p. 1,
2). It is clear that every internal labour market is linked to an external labour market,
where the conventional (neo-classical) theory of pricing, allocating and training decisions
governed directly by economic variables is assumed to be applicable. Optimising
behaviour, which is a basic part of the neo-classical theory, plays a minor role in internal
labour markets, but it is too simple to conclude that the ILM takes an anti efficiency
approach. The occurrence of ILMs can be efficient when the assumptions underlying the
neo-classical theories are relieved. For example, the existence of turnover costs can make
it profitable to have more or less fixed job ladders instead of competition between all
workers. Adjustment costs, i.e. costs involved with hiring, firing and training of
employees, will play an important role when workers are heterogeneous and when firm-
specific skills are necessary. Both the employee and the employer can benefit from the
existence of an internal labour market: the employee has more job security while the
employer is guaranteed of good workers at relatively low costs.
Of course, as the internal and external markets are interconnected, there must be some
job classifications at which entry to and exit from internal markets is possible. In general,
the ILM-concept gives certain rights and privileges to employees already incorporated in
the firm, e.g. career paths, and to a certain extent they are protected from competition
with persons who try to enter the firm. Doeringer & Piore make a difference between
closed and open ILMs. The closed ILM has only one entry level, where the open ILM is
characterised by the existence of many entrance ports. Governance by economic
variables like the unemployment rate, wages paid by competitors and so on, will be
important only at these entry levels. Wages in ILMs are related to jobs and not to the
productivity of individual workers, as in neo-classical theory. Only the wages for jobs at
the entry level must have a competitive level. Promotions are not solely based on ability,
other factors like seniority often play a major role.
It is questionable whether the definition of Doeringer & Piore, although it is generally
used and accepted, is useful in practical research. An option for practical research is to
follow the flows of employees through the firm, as it is difficult to discover the
underlying rules, procedures, habits etc., and their effects on the final decision on internal
and external hiring. However, due to lack of data, research on internal flows is rare and
often derived from information on external flows. See, e.g., the study on the
characteristics of ILMs by Van Bergeijk & De Grip (1986), which uses data from the
Annual Social Reports of 19 large Dutch firms. They observe that on average about one-
third of the employees have job tenure of over ten years, while about two-third of the
personnel has more than five years of tenure. Between 1974 and 1983, tenure in these
firms has increased from 8 to just over 10 years on average. In the early 80s, almost all
firms have (external) mobility rates of less than 10%. Information on the occurrence of
flows through the organisation is not available. An exceptional case where information
on internal flows is available is Hamermesh
et al.
, (1996). This study estimates the sizes
of various internal and external worker flows in a large sample of Dutch firms. Flows of
workers within firms appear to be small compared to flows into and out of firms
An elaborated case study is done by Baker
et al.
(1993,1994), who identified the
hierarchical structure of a medium-sized U.S. firm in a services industry, based on the
personnel files over the years 1969-1988, and by observing the career paths of the
(management) employees. A large number of employees have lengthy careers in the firm,
and these careers are characterised by movements through numerous jobs, which is an
indication for the existence of an ILM in the firm. However, there is no evidence on the
existence of specific ports of entry and exit. Both occur in all jobs and at all levels. There
is little difference in quality between personnel hired externally and employees who are
hired internally; firm-specific human capital does not seem to be important. Career
performance does not differ between internal and external hires, tenure does not result in
better career attainment, and fast tracksdo exist. These are all indications that
promotions are based on performance and qualities of employees, and not so much on
seniority. There is strong evidence for the importance of job levels as determinants of the
wages, but wage jumps at promotions are much smaller than differences across levels
and within levels. In general, administrative rules in wage determination seem to be
important so that the neo-classical assumption that wages reflect marginal productivity
seems to be falsified.
In our simulation model, we will concentrate on the mobility of employees in the firm.
We will try to implement a number of empirical facts discovered by Baker
et al.
therefore, keep some distance from the ILM theory of Doeringer & Piore. Decisions
about hiring are taken to find an efficient mode of production. In this decision we assume
that the firm weights the benefits and costs of internal mobility against external hiring; we
abstract from promotions-guided-by-procedures. A case study of a large Dutch
manufacturer by Van Veen (1997) supports our efficiency-approach. Van Veen shows
that there exists a system of formal rules in the firm, but that it is adjusted regularly to
match changes in the environment of the firm and rationalise the mobility of employees.
In the calibration procedure we try to select the specifications and parameter values of
the model in such a way that it reproduces empirical evidence on internal and external
labour market flows found by Hamermesh
et al.
and evidence on personnel management
and hierarchical structure by Baker
et al.
Therefore we are to endogenise the hierarchical
structure, tenure and consequently internal and external labour flows in our simulation
model of the firm and derive these characteristics from optimising behaviour.
3. Hierarchical models of the firm
As a starting point for our simulation model, we use the hierarchical model of Williamson
(1967), a simple model that gives a clear insight in the relation between hierarchical
control and the optimal size of a firm. In this section we will give a description of
Williamsons model, a model that is used as reference model by many authors, and
discuss where we have to extend it in order to describe labour flows and personnel
Williamsons model describes a profit-maximising, price-taking firm with
trative layers. Each employee can supervise
subordinates; the span of control is
assumed to be the same at every layer. Production takes place only at the lowest layer,
employees at higher levels spend their time with supervising the employees at the next
lower level. Each production worker has the same labour productivity
; however, due
to loss of control, productivity decreases by a constant fraction (1-
) for each layer.
These four parameters define the total output
. The costs that are involved
with the production of
consist of wages and non-labour costs. Each production
worker receives a wage
; Williamson assumes that the wage at each level is a multiple
of the wage at the next lower level. Non-labour costs are assumed to be a fixed fraction
of the production. Given this specification of output and costs, the profit maximisation
procedure results in an optimal size of the firm by the determination of the optimal
number of layers
Under reasonable conditions for the parameters, Williamson obtains several conclusions
from his model. The optimal number of layers (
) increases when the degree of
compliance (
) increases and when the span of control (
) increases. The optimal number
of layers decreases when the wage difference between levels increases, and also when the
wages increase relatively to the output price and the non-labour costs. If there is no loss
of control (
=1), then the optimal size of the firm consists of infinitely many layers; the
existence of a loss of control results in the existence of a limit on the size of the firm.
In a more general framework, Calvo & Wellisz (1978) show that loss of control in itself
is not sufficient for the existence of a limit on the size of the firm. When employees
cannot identify the moments at which they are monitored, there is no limit on the firm
size. An optimal size will exist when it is impossible (e.g. due to technological reasons)
to set up a supervision system that forces employees to work 100% of the time they are
being supervised. In the model of Williamson, this is guaranteed by the fixed, strictly
positive control loss fraction. Furthermore, Calvo & Wellisz show that in Williamsons
set-up it is not optimal to set the same wage for two successive layers. This supports
Williamsons assumption of higher wages at each higher level. The literature on
hierarchical models illustrates that the assumptions which ensure a limit to the optimal
size of the firm under the condition of price taking and hence of perfect competition, are
essential for the working of the model. We will pay special attention to this issue in the
construction of our simulation model and show how the optimal size of the firm may
differ when changing the parameter values with respect to these assumptions. An
alternative would be to release the assumption of perfect competition and to specify a
downward sloping demand curve for the product of the firm.
In a follow up paper, Calvo & Wellisz (1979) derive several propositions that are useful
in building a simulation model of internal labour flows. First, they show that the choice of
an optimal policy at each level in the hierarchy is independent of the optimisation at other
levels. Second, it is optimal to assign employees with better qualities to jobs that are
higher on the hierarchical ladder, even when jobs at every level are equally difficult.
Third, it is shown that the optimal span of control is higher, the higher the rank of a job.
Finally, Calvo & Wellisz show that it is optimal to pay higher wages to higher-ranked
employees. The optimal wage differentials between layers is greater than the differentials
in effective labour per worker, i.e. the number of hours worked corrected for shirking
and loss of control.
An alternative explanation for the wage differentials between hierarchical levels is
based on the marginal product of employees. In the traditional view, agents with better
abilities are placed in higher-ranked jobs and generate a downward externalityfor the
productivity of workers lower ranks (Bhattacharya & Guasch 1988). Also tournament
models give a similar wage/productivity distribution. See e.g. Lazear & Rosen (1981),
who show that wages based upon the rank in the hierarchy can induce the same efficient
allocation as wages based on individual output levels.
Qian (1994) shows, using a model that is essentially the same as the model of
Calvo & Wellisz (1979), that their conclusion of decreasing wages and span of control
throughout the hierarchy is caused by the fact that they fix the number of layers and
allow the number of production workers to vary. When these two are interchanged
(hence a variable number of layers but a fixed number of production workers), the result
is a wage and a span of control that is constant throughout the hierarchy. Profit is
unlimited when both the number of layers and the number of production workers are
flexible. This statement is important for our model in which we want to keep both factors
free. Underlying Qians results is the choice of the effort levels: employees are working
fully efficient, or they shirk. In case of a continuous choice of the optimal effort level, he
shows decreasing wage and effort levels at lower hierarchical levels. The result on the
span of control however is ambiguous.
The model of Williamson is unfit for the description of worker flows through the firm,
because most determinants which may explain the existence of internal worker flows are
fixed in the model. First, the span of control is the same on each level, for each job and
for each worker. Second, for workers on the production level, there exists equality in
labour productivity. Hence, all workers are equal and there is only a difference between
production workers and supervisors. Qualities of individuals do not play any role in the
model. Furthermore, other parameters like the control loss parameter, the wage for
production workers, and the factor that describes the difference in wages between
different levels, are equal for each job. In short, there is no distinction between individ-
uals in this model. Some of the restrictions are loosened in Calvo & Wellisz (1979), who
However, Keren & Levhari (1979) develop a model and conclude the opposite. They find that the
optimal span of control is higher in lower-ranked jobs.
bring in heterogeneous workers. However, a major drawback for making labour flows
endogenous remains the fact that it is a static model. Before we can study the flows of
workers through a hierarchically structured firm, a time factor has to be added and we
have to introduce adjustment costs in the model. If we simply create a series of
Williamson-type models over time, then there do not exist factors that prevents the firm
from instantaneous changes in the number of hierarchical levels, employees and so on,
because costs that are involved in the adjustment do not occur. However, in practice, it
will be costly to change the hierarchical system dramatically, and even small changes will
not be free of costs. We note that these adjustment and search costs are major elements
in theoretical models of the flow approach to labour markets.
A very neat but difficult way to introduce the time factor and adjustment costs in the
model is the development of a dynamic optimisation model. In that case, expectations of
all parties should be modelled, and the optimal distribution of employees (and their
qualities) over the jobs must be determined. Generally, the optimal solution of a dynamic
optimisation problem is based on an initial state condition, for example the current
distribution of workers over jobs, and eventually on a final state. It is hard to allow for
unexpected quits of employees in such a framework and solve a stochastic dynamic
optimisation problem. It seems possible to introduce some conditions on intermediate
states, in order to keep the optimisation model manageable, but the main consequence of
this kind of requirements is that the dynamic optimisation problem is broken down in a
series of short termproblems, where each problem has to satisfy its own final
condition. The result of each problem is the initial condition for the optimisation problem
in the next period. However, in doing so we have a dynamic optimisation problem that is
similar to the dynamic model that we referred to as the simplest solution to introduce the
time dimension in the model: a series of static models.
The hierarchical models of Williamson and Calvo & Wellisz can be solved analytically
without much problems, and it may be feasible to develop a dynamic model in this vein
that can be solved analytically as well. This is true for models which do not describe the
individual characteristics and careers of employees. However, in our case, where we
have to keep track of each individual, it seems utterly impossible to construct a model
which can be solved analytically, even when we consider a series of static models only.
For that reason we will restrict ourselves to building a model which can only be solved
numerically by means of simulations. In this numerical process, we adopt the kind of
myopic behaviour of constructing a series of static models in order to keep our
optimisation problem manageable.
4. The simulation model
As we discussed in the previous section, we will not make an attempt to develop a model
that can be solved analytically. We want to construct a model in which the career paths
of heterogeneous individual employees can be followed. This section discusses the set-up
of the model. The calibration procedure and simulation results for the basic version of
the model will be presented in the next section.
The central assumption in our hierarchical model is that each employee takes the decision
on whether he or she spends his/her time on the production of output or on supervising
subordinates (or a combination of both) independently from the others in the firm.
may consider this decision for each individual to be taken by the management of the firm
instead of by the employees themselves. Starting with the highest-ranked person in the
firm, she takes a decision on the number of subordinates that is optimal in her
circumstances, by maximisation of the function
, with respect to
, the (potential)
number of subordinates.
In this function, the index
indicates the hierarchical level of
the supervisor (
=1 for the top level), while the subordinates (who are identified by index
) are employed at level
+1. The index
indicates time; in our simulations we will assume
that time is measured in years. Two parts can be distinguished in the profit function. The
first part, formed by the terms under the summation sign, describes the production
) and its direct costs: the wage (
) and the supervision costs (
) which
measure the foregone production. All three variable are measured per subordinate. The
production of an individual subordinate is given by (
), but the actual production in
case of co-operation with other workers can be higher or lower. We introduce a function
that describes the benefits of co-operation,
). It is specified as a concave parabola,
implying the existence of an optimal scale of production. This specification of the
production function, with increasing returns to co-operation for the first few
subordinates and decreasing returns to co-operation for the latter subordinates, is the
main determinant for the span of control of the supervisor. Co-operation increases
production, but this effect is offset by its direct costs that are described by the function
labelled Supervision Costs. It is an indicator for the value of the time that the
entrepreneur has to spend on supervision of each of her subordinates; while supervising,
she cannot contribute to the production. Each subordinate requires a certain amount of
supervision, dependent on his qualities. These two functions together endogenously
For the sake of convenience we will assume in the remainder of the paper that the highest ranked
person in the firm is female and that all others are male.
The production of the supervisor and her wage are not represented in the profit function because they
do not affect the optimal number of subordinates.
determine the optimal span of control. Span of control is not fixed, as it was in many
earlier models.
The individual measures of productivity and costs are summed over all
subordinates to
determine the total (potential) direct profit.
The second part of the profit function,
(.), describes the costs that are involved in the adjustment process. Hence it is a
function of both
, the optimal number of subordinates, and
, the actual number of
subordinates that remains from last period. Generally, a change in the number of
subordinates is not free of costs since both hiring and firing are costly. Without
adjustment costs it would be optimal to choose the number of subordinates that gives
maximum direct profit, but the existence of costly adjustment causes a slower adjustment
towards the optimal level. This is an essential feature which enables the model to
describe internal and external flows of labour, and to link the model with search theory
on the labour market. Moreover, the introduction of adjustment costs in the model and
the fact that we want to allow for heterogeneity amongst workers makes it necessary to
keep a track record of each individual worker in the simulation program. The set up of
the model enables us to monitor the quality of each individual worker and to register
changes over time of this quality. Information on individual qualities is used in the
selection procedure when filling vacancies.
Table 1. Set-up of the simulation model
Maximisation of the profit function by the entrepreneur is the first step in the
construction of a hierarchically structured firm. The whole simulation process can be
decomposed in five steps, which are shown in table 1. After the determination of the
optimal number of subordinates (step 1), the second step in the building of the firm is to
be taken: if necessary, i.e. if the optimal number of subordinates
differs from the
currently available number
, then the optimisation process will be followed by a
The summation starts from
=0, what reflects the possibility that a supervisor has no subordinates.
STEP 1 determination of optimal number of subordinates (per supervisor, per time period)
STEP 2 in case of vacancies: search for employees
- promotion of insiders (causes vacancy chains)
- hire of outsiders (training might be necessary)
in case of superfluous workers: fire the least qualified subordinates
(the result of this step might be that there remain unfilled vacancies)
repeat steps 1 and 2 for each subordinate until you reach the layer where the (optimal)
number of subordinates equals zero.
STEP 4 determine the number of employees, production (optimal, actual), hiring, firing (flows,
costs), organizational structure of the firm
STEP 5 - random quits will take place
- increase in the experience of the employees who stay (learning by doing)
- repeat steps 1 to 4 for the next period
search process, or by the firing of employees. In the latter case, where we have
superfluous employees (
), the subordinates with the lowest qualities are fired. If
such superfluous employee is located in the higher ranks of the firm, this firing process
implies that the whole branch (or department) of the firm, which apparently is not
profitable, is closed down. In the former case, with
, there exist one or more
vacant positions, for which we have to find employees who can fill them. The question is,
who are available and what are the consequences when someone is hired. Here, we
assume that there is a preference for internal candidates, where we select among all
employees in the firm who are working on the next lower level (not only direct
subordinates, but also employees on this level in other branchesof the hierarchical
tree). If there are not enough internal candidates, or if the internal candidates fail to meet
the minimum requirements, then search will be extended to external applicants. It is
assumed that the qualities of external candidates follow a uniform random distribution.
Two remarks have to be made. First, we assume higher minimum requirements for
internal candidates because if an internal candidate is hired, the consequence is that his
old position becomes vacant and a search process has to be started here as well. Second,
if the qualities of an external candidate are almost sufficient, he will receive a training
procedure to boost his qualities. We assume that the benefits of the training can be used
in the current production, without delay. We do not consider training for incumbents.
Qualities are equally important in each job, thus training would not give them any
additional qualities that would make it optimal to leave their current job and move to
another job inside the firm. However, if neither the internal candidate nor the external
candidate has a sufficient quality level, the job will remain vacant, and as a consequence
the actual number of subordinates is smaller than the optimal number of subordinates. If
this is the case, the supervisor will spend less time on supervising and has a higher
residual production.
For each filled job, i.e. for each subordinate, we will repeat the optimisation and search
procedure (step 1 and 2), taking into account the central assumption that each employee
takes independent decisions on whether he spends his time on the production of output,
on supervising subordinates, or a combination of both. The determination of the optimal
number of subordinates does not depend on the decisions that are taken at other levels
and at other branchesin the hierarchy: it is a purely individual decision. The structure
of the profit function has to guarantee that there is a limit on the size of firm, i.e. that
there exists a layer for which it is not optimal to have subordinates or for which it is
impossible to find applicants that meet the minimum requirements. If this is certified, the
number of repetitions of step 1 and 2 is finite and it will be possible to describe the firm,
by means of (e.g.) the number of employees, the organisational structure, the output that
is generated, and by the number of unfilled vacancies (step 4 in table 1). Note that both
the number of hierarchical layers as well as the span of control is determined within the
model; a setting that caused troubles in Qians model.
After these four steps, we have established the hierarchical set up of the firm at the start
of period
. All workers in the hierarchy stay at their job for (at least) one time period,
and produce output during this period. At the end of period
(or at the beginning of
+1), a random number of employees decides to quit the firm. Here we can think
of workers finding a job elsewhere, of retirements or of workers who have other reasons
for leaving the labour force. The result is the opening of vacancies at their old positions.
Furthermore, the passing of time generates an increase in the experience of those
employees who are in the firm, which is depicted by an increase in their quality measure.
If the growth in the qualities of an employee is insufficient to keep track of the growth of
the requirements, then the employee will be fired and his job becomes vacant. Instead of
immediately starting the search for candidates who can fill these vacancies (and the
vacancies that remained unfilled after the last search process), we return to step 1, the
optimisation process. There, it is determined whether it is optimal to try to find an
employee for the vacancy, or whether it is better to close the vacancy without starting
the search process for a new employee. It might turn out to be optimal that even more
vacancies will be opened. The next steps in the table are carried out successively, as
described above. In principle, it is possible to continue the process indefinitely, but we
will restrict our simulations to a predetermined number of years.
Note that we do not care about the actual production and the actual costs of the firm.
The optimisation process is directed at the expected number of employees and expected
qualities of applicants, and tries to find out the optimal number of subordinates. The
results of the search process (is it possible to find an employee for a vacancy, or will the
position remain vacant) does not affect the optimisation process for the current
employee. Of course, it affects the optimisation procedure for subsequent layers and for
the next period, because it determines the starting point for these processes.
We refer to the appendix for an extensive discussion of the model specification. Here, we
will only give a description of the basic structure of the profit function and the
adjustment costs. Basically, the functions consist of a shift-parameter, a slope-parameter
that defines the differences between hierarchical levels and a part that takes account of
the qualities of the employee occupying the job.
The slope-part is structured in such a
way that workers on lower levels produce and earn less than higher-ranked employees.
The difference in the slope-parameters has to set a limit to the size of the firm, to
A time-varying part (trend, business cycle) is not included in the simulations, but the model is capable
to handle variance over time; it is included in the model in the appendix. Moreover, some functions do
not contain the slope-parameter or the quality-dependent part but only the shift-parameter.
guarantee that the difference between production and costs becomes so large that extra
subordinates are not beneficial any more. This is a crucial feature of the model because
we assume perfect competition, what implies that the price of output is not a function of
the total production. Hence, we do not have a downward sloping demand curve for the
product of the firm which would set a limit to the size of the firm. For simplicity we
normalised the output price to one. The individual-specific part of the functions is
specified to ensure that better-qualified employees produce and earn more, and need less
supervision. Following our central assumption, there are no direct links between the
production at different levels. The only requirement is that the position of the (direct)
supervisor does not remain vacant. Production, or the quality of an employee, at level
does not affect the production at other levels, neither does it affect the wages at other
The adjustment costs
), depend on the number of vacancies
. In the
literature, quadratic adjustment cost functions -or more general strictly convex
adjustment cost functions- are most widespread, because it is possible to derive explicit
labour demand functions for them. See Nickell (1986) for a discussion on shapes of cost
functions. For the sake of simplicity we assume linear cost functions, which seems not to
be at variance with empirical evidence as well (see Pfann and Verspagen, 1989). We
assume not only asymmetry between fires and hires, but we take into account the kind of
applicants that can be expected: we make a distinction between costs of search among
internal candidates and search costs for external applicants. If there are vacancies, i.e.
>0, we assume that the supervisor has insight in the number of subordinates that
are available on the next lower level in the hierarchy, and that these employees are ready
to fill the vacancy. Hence, in determining the optimal number of vacancies, it is
that the employees on the next lower level are the first group that can fill the vacancy. If
the number of vacancies is higher than the number of internal candidates, the remaining
vacancies are
to be filled by external applicants.
5. Characteristics of the basis version of the calibrated model
Now we are to find plausible functional forms and realistic parameter values for the
processes of production, coordination and search distinguished by the model. As
mentioned before, there are only very few studies from which we can borrow empirical
results on the labour market dynamics and on the hierarchical organisation of firms. For
our calibration of quits, fires, internal and external worker flows, we base the
characteristics of the benchmark representative firm of our basic simulation on a study by
et al.
(1996). They use data from a survey amongst firms with 10 or more
employees, a stratified sample of about 2000 firms. About half of these could be used to
make estimates of worker flows in 1990. These estimates are presented in table 2. It
shows clearly that the majority of flows is to and from existing jobs, for inflows and
outflows, as well as for internal mobility. We will consider this table as a representative
firm and we calibrate upon these flows.
Table 3 presents the parameters of