This paper develops a novel, geometric approach to modelling a firm's hierarchical structure. We model the firm's hierarchy as the sector of a circle, in which the radius represents the height of the hierarchy and the angle of the sector represents the width of the hierarchy. The firm then chooses the height and angle in order to maximise profit. We analyse the impacts of changes in economies of scale, input substitutability and labour productivity on the firm's hierarchical structure. We find that the firm will unambiguously become more hierarchical as specialisation of its workers increases or as its output price increases. The effect of changes in scale economies is contingent on the level of task specialisation and output price

Beard, Rodney

Ng, Evelyn

Tang, Kam Ki

Yanes, Leopoldo

University of Queensland eSpace

The hierarchical structure of a firm: a geometric approach 
Leopoldo Yanes Kam Ki Tang
School of Economics, University of Queensland School of Economics, University of Queensland
Evelyn Ng Rodney Beard
School of Economics, University of Queensland ngevie@yahoo.com
Abstract
This paper develops a novel, geometric approach to modelling a firm’s hierarchical structure.
We model the firm’s hierarchy as the sector of a circle, in which the radius represents the
height of the hierarchy and the angle of the sector represents the width of the hierarchy. The
firm then chooses the height and angle in order to maximise profit. We analyse the impacts of
changes in economies of scale, input substitutability and labour productivity on the firm’s
hierarchical structure. We find that the firm will unambiguously become more hierarchical as
specialisation of its workers increases or as its output price increases. The effect of changes
in scale economies is contingent on the level of task specialisation and output price.
Citation: Yanes, Leopoldo, Kam Ki Tang, Evelyn Ng, and Rodney Beard, (2005) "The hierarchical structure of a firm: a
geometric approach." Economics Bulletin, Vol. 12, No. 13 pp. 1−7
Submitted: June 2, 2005.  Accepted: September 4, 2005.
URL: http://www.economicsbulletin.com/2005/volume12/EB−05L00004A.pdf
 1
The Hierarchical Structure of a Firm: A Geometric Approach 
 
1. Introduction 
 
There is wide variation in the hierarchical structure of firms. Typically, firms in heavy 
manufacturing industries have tall hierarchies (e.g., General Electric). Conversely, firms in 
the services industry, such as consulting firms, are relatively flat and less hierarchical (e.g., 
Ernst & Young). Moreover, the hierarchical structure of a firm rarely stays constant over 
time. Consider Google Inc. as an example. When the company was founded in 1998 it had 
four employees other than its two founders. By 2001, it had grown to 200 employees. 
Currently, it has almost ten organisational departments. 
 
These observations lead to the following research questions: (1) What are the economic 
factors that influence the hierarchical structure of firms? (2) How does the hierarchical 
structure respond to a change in these factors? 
 
The literature on hierarchy can be broadly divided into two branches. One studies the “shape” 
of the firm, while the other focuses on the “distribution of authority”. The distribution of 
authority attempts to explain the internal structure of a hierarchy, that is, how workers are 
organised according to the influence one has over others. A set theoretic framework based on 
simple games, namely, command and control games, has been used to model the authority 
structure. The Shapley-Shubik power index is widely used in this branch of the literature as a 
quantification of the authority distribution and as a measure of each worker’s task 
responsibility. Research in this branch includes the works of Shapley and Palamara (2000), 
and Hu and Shapley (2003a; 2003b). 
 
In contrast, understanding the shape of a hierarchy gives an idea of its external structural 
design. The shape is determined by the number of levels in the hierarchy and by the span of 
control of a superior. The approaches in this branch are diverse, ranging from theories of 
knowledge, promotion tournaments, contracting, costs and optimisation problems to stage 
games. Works in this branch include the papers by Garicano (2000), Mookherjee and 
Reichelstein (2001), Qian (1994), Yang (2002), and Dubey and Haimanko (2003). 
 
We aim to contribute to the second branch of the literature in two ways. Firstly, the paper 
adapts the neoclassical methodologies of profit maximisation to explain the hierarchical 
structure of the firm. However, instead of assuming the span of control and number of levels, 
we specify production in terms of the hierarchy itself. In doing so, our model combines 
certain elements from neoclassical theory as an attempt to shed light on the age-old question 
of “how does a firm organize its workers for production?”. 
 
Secondly, we use a geometric approach to pose the optimisation problem. More specifically, 
we model the shape of the hierarchy using the sector of a circle. This allows us to employ the 
geometric formula of the sector in the construction of the cost function, which keeps the 
analysis simple and tractable. The simplicity of the modelling method provides room for 
further extensions and captures the essential features of the firm’s decision process, albeit in 
an abstract form. 
  
 2
2. The Model 
 
Consider the sector of a circle as an abstract representation of the hierarchical structure of a 
firm, as depicted in Figure 1. The height of the firm is represented by H, and can be 
interpreted as the total number of levels in the hierarchy. The angle, T, can be interpreted as 
span of control from any supervisor’s perspective. In the figure, h represents the distance 
from the highest ranking worker. Thus h is inversely related to authority, with h=0 
representing the highest ranking worker (namely the CEO, who is located at the tip of the 
sector), and h=H representing the lowest echelon of the hierarchy. b represents the number of 
workers in a particular hierarchical level (h). b can be interpreted as the width of the firm at 
level h. Using the arc length formula, we have that b=hT. 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 1. A Sector of a Circle Representation of a Firm’s Hierarchy 
 
With this abstract representation, we reduce the problem of designing the hierarchical 
structure of the firm to one of choosing the height ( H ) and the angle (T ) of the sector. In 
treating the firm’s design problem as one of choosing the dimensions of a geometric sector, 
we make the assumption that the hierarchy features a continuum of workers, both 
horizontally and vertically. Thus, the area of the sector represents the total number of workers 
in the firm. This specification helps keep the model tractable, whilst allowing us to capture 
some essential features of the design of a firm’s hierarchy. 
 
The profit function of the firm is specified as follows: 
 
 S =
0
( , ) ( )
H
PY b h w h dhT ³  (1) 
 
where P is output price, Y is output, ( , )b h hT T  is the width of the hierarchy (i.e., arc 
length) at any hierarchical level h, ( )w h  is the wage rate (which is a convex and decreasing 
function of the hierarchical level, h). 
 
The hierarchy is incorporated into the firm’s production function rather than being treated as 
an ‘outside’ factor, thereby capturing the forces shaping the hierarchical structure.  
 
The major departure of our model from the standard neoclassical model of firm is that inputs 
are measured in terms of H  and T  in the form of a constant elasticity of substitution (CES) 
production function: 
 
b
T 
  H h
 3
    ;    0, 0Y H HEU U UT T  ! ! , 1, 0 and 0U U E z !  (2)  
 
The capital-labour ratio is assumed to be exogenously determined, thus we need only 
consider the distribution and quantity of labour within the hierarchy (which would then be 
associated to a certain level of capital)1. The structure of production implies that, for a given 
number of workers, it is the organization of the hierarchy which determines output. 
 
As argued by Erich Gutenberg (1951), a firm can be seen as a ‘productive relationship’ 
whereby productivity depends more on the organisation and complementarity of factor inputs 
rather than just the potential productivity of each input (Albach et al., 1999). This concept of 
productivity is atypical in the sense that the firm’s productivity is not only determined by 
neoclassical input-output relations but is also influenced by the organisation of workers. 
Several other authors such as Greenan (2003), Carlson (2001) and Foray (2004) have also 
pointed out that the organisation of workers (combined with other factors) influences how 
successful a firm is in achieving its objective. In other words, the firm can be seen as a 
system where the way in which its elements (in our case, workers) are organised will 
determine the extent to which it can achieve its objective. These authors argue that 
characteristics such as teamwork and the monitoring of workers are therefore crucial to the 
workings of the firm. In short, this strand of the literature affirms that a firm’s production 
depends on its hierarchical structure. 
 
In (2),E  measures economies of scale where  
 
 
3 : decreasing returns to scale
= 3: constant returns to scale
3: increasing returns to scale
E
­°®°!¯
 (3) 
 
To see this, note that the total number of workers is equal to the area of a sector, 2 / 2HT . So, 
if both T and H increase by factor g, the total input will be equal to 3 2 / 2g HT , while output 
will be equal to ( , ) ( , )Y g gH g Y HET T . Therefore, returns to scale depend on whether E  is 
bigger than, equal to, or less than 3. 
 
The elasticity of substitution between H and T is measured by U1
1 . If workers at different 
levels perform different tasks, then an increase in U  indicates a decrease in the task 
specialisation of workers within the firm. In other words, there is a decrease in intra-firm 
specialisation. An indicator of intra-firm specialisation is whether subordinates are able to fill 
in their superior’s role when she/he is absent, and vice versa. For example, intra-firm 
specialisation is expected to be high in the IT sector, and low in fast food catering: In the IT 
sector, qualifications of subordinates are not too different from supervisors’, while the 
opposite may hold in the fast food industry. In particular, as H  and T  become more 
substitutable, for example, the firm would be more inclined to give up one superior for 
another lower-level worker since that worker is cheaper to hire. 
                                                 
1 Qian (1994) makes use of a similar assumption to dispatch the problem of substitution between capital and 
workers in order to make his model tractable. The extension of the model to consider capital we will be the topic 
of future research. 
 4
 
The wage rate schedule is exogenously determined (presumably by the CEO), and is uniform 
across each level but increasing convexly as we move up the hierarchy. The total wage bill of 
workers at level h for a given T  is given by ( , ) ( )b h w hT . Therefore, the total wage bill for the 
firm is the integral of ( , ) ( )b h w hT  from 0h   (the CEO) to h H (the lowest level worker). 
The wage schedule is specified as: 
 
 w h I  (4) 
 
whereI  is (the absolute value of) the wage elasticity with respect to h. 
 
The firm’s problem is to choose H  and T  to maximise profits, just like a neoclassical firm 
chooses capital and labour to maximise profits. Solving the firm’s problem yields the 
following solutions 
 
 
 
 2
1
3
**
3
2
P
E U
U
I U
U
I EIT E
I

 
 ª º« » « »« »¬ ¼
 (5) 
                                        
   
1
3
**
1
3
2
H P
E U I E
U
E U
U
IE
I
  
 
ª º« » « »« »¬ ¼
 (6) 
 
In order to obtain tractable comparative statics results, we restrict the wage elasticity, I, to 
one. This simpler model allows us to pinpoint precisely how the firm’s hierarchical structure 
responds to changes in the economic parameters, and also enables a more stylized analysis. In 
this simpler setting, the solutions in (5) and (6) become: 
 
 
1
2
* * 2H P
E U E
UT E
 ª º  « »¬ ¼
 (7)   
 
Lemma 1. If 1I  , the second-order conditions for profit-maximisation require 0 2E   
and  0U  . 
Proof: The Lagrangian for the profit maximization problem is  P H HEU U UT T  L  when 
1I  . It can be shown that * * ( 2) / 2
HH TT E U   L L  and 
* * ( 2 ) / 2
H HT T E U  L = L . 
The second-order conditions for *H  and *T  to be profit maximising values are 
* *  0, 0HH TT L L and * * * 2)( ) ( )HH HTT T!(L L L , which simplify to 0 2E   and  0U  . 
Q.E.D. 
 
The upper bound of E  set by the second order condition (i.e. 2) is in fact equal to the 
constant returns to scale value of E  (i.e. 3) minus the height elasticity of the wage schedule 
 5
(i.e. 1I  ). From (3), it can be seen that the second order conditions imply the firm operates 
under decreasing returns to scale: Given that the wage is a decreasing function of h , the 
workers at the bottom will be paid progressively less as the firm's hierarchy extends. If 
workers' productivity does not decrease fast enough down the hierarchy, then a profit-
maximising firm will expand vertically indefinitely. The intuition for restricting the 
substitutability between H  and T  (i.e. U  < 0) can be explained by similar reasoning. If H  
and T  are almost perfectly substitutable (i.e. U o  1), then the firm can save labour costs by 
continuously expanding vertically and shrinking horizontally. In summary, 0 2E   and 
0U   are conditions that prevent the firm from expanding infinitely in the vertical direction. 
 
The following propositions summarise the comparative statics properties of the model: 
 
PROPOSITION 1. The firm will expand vertically and horizontally when output price 
rises. 
Proof: Let * * *Z H T  . Then 
* *
0
(2 )
Z Z
P P E
w  !w  , by the second-order condition: 2E  . 
Q.E.D. 
 
As output price rises, marginal revenue increases and becomes greater than marginal cost. 
The firm takes advantage of this by expanding its hierarchy. In doing so, its marginal benefit 
decreases while its marginal cost increases until they equate again. The result suggests that, in 
industries in which output prices are protected (e.g. by regulation), firms would be ‘bigger’ in 
terms of the number of workers they hire. 
 
PROPOSITION 2. The firm becomes less hierarchical as intra-firm specialisation 
decreases. 
 
Proof. A decrease in intra-firm specialisation is represented by an increase in substitutability 
between * *  and H T , i.e., an increase in U . 
* *
2 ln 2 02
Z Z E
U EU
w § ·  ¨ ¸w © ¹ , by the second-order 
conditions: E < 2 and U < 0. Q.E.D. 
 
Given that U  must be less than zero, an increase in U  causes marginal revenue to decrease. 
Therefore, the optimal response for the firm is to structure its hierarchy to cut costs until 
marginal cost equates marginal revenue again. The result implies that firms would be smaller 
(in terms of the size of their labour force) in industries in which division of labour happens to 
a lesser extent. 
 
PROPOSITION 3. An increase in scale economies leads to a hierarchical expansion if either 
output price is sufficiently large for given input substitutability, or intra-firm specialisation is 
sufficiently high for a given output price. 
 
Proof: 
* *
*1 1 ln 2 ln
(2 )
Z Z ZE E E U
ª ºw   « »w  ¬ ¼ . For this to be positive, we require either 
2 2
2eP
E
E UU
E



!  or , equivalently, 2 ln 2[1 ln 2 ln( )] 2P
EU E E    . Q.E.D. 
 6
 
3. Concluding Remarks 
 
In this paper, we have introduced a geometric approach to model the hierarchy of firms. A 
simple and stylised model has enabled us to examine how the firm’s hierarchical structure 
changes with certain economic factors, including output price, economies of scale in 
production and intra-firm specialisation. By introducing a small modification in the standard 
neoclassical structure, the model sheds light on possible causes of variation in hierarchical 
structure across firms and possibly even within a given firm through time. We have taken a 
completely different perspective by directly incorporating the hierarchical structure into the 
production process. The model described in the paper is (purposefully) a basic test case of 
this geometric approach. The specification of the model can be relaxed in many directions to 
explain more complex issues. Possible extensions include relaxing the assumption that the 
elasticity of the wage schedule is equal to one, relaxing the assumption of a fixed capital to 
labour ratio, while letting the firm choose capital for each hierarchical level, as well as 
generalizing the functional forms for the production function and wage schedule. 
 
 
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