




































Ecology, Economy and Society–the INSEE Journal 1 (1): 17–31, April 2018 

 
THEMATIC ESSAY 

 

On the Normalization of Dimensioned Variables in 
Ecological Economics 
 

Deepak Malghan  
 
1. INTRODUCTION 

The fundamental concern of ecological economics is to accurately model all 
aspects of the economy–ecosystem interaction problem — the myriad ways 
in which the economic and ecological systems are connected to each other. 
Almost all the monetary and physical variables used to describe economy–
ecosystem interactions are dimensional in nature. The exact cardinal value 
taken by dimensioned variables is contingent on the particular measurement 
unit used. While several papers on the subject have pointed to the care 
required in using dimensioned variables in ecological economics, there is 
little consensus on how dimensional variables must be incorporated in 
economy–ecosystem interaction models (Mayumi and Giampietro 2010; 
Malghan 2011; Chilarescu and Viasu 2012; Baiocchi 2012; Mayumi and 
Giampietro 2012). Mayumi and Giampietro (2010) inaugurated the debate 
by making the provocative claim that many models in economics and 
ecological economics that make use of transcendental functions like the 
logarithm are fundamentally flawed when these functions use what are 
apparently dimensioned variables. Malghan (2011) claimed that several 
popular biophysical sustainability indicators are dimensionally inconsistent 
because they neglect the ‗qualitative residual‘ that is the defining 
characteristic of any social–ecological system (Georgescu-Roegen 1971). In 
a brief comment, Chilarescu and Viasu (2012) showed that the critique that 
a neoclassical production function (Arrow et al. 1961) is dimensionally 
inconsistent does not consider that the parameters of a production function 

                                                        
 Indian Institute of Management Bangalore, Bannerghatta Road, Bengaluru 560076, India 
and Ashoka Trust for Research in Ecology and the Environment, Srirampura, Bengaluru 
560064, India; dmalghan@iimb.ac.in, dmalghan@atree.org 

Copyright © Malghan 2018. Released under Creative Commons Attribution-
NonCommercial 4.0 International licence (CC BY-NC 4.0) by the author.  

Published by Indian Society for Ecological Economics (INSEE), c/o Institute of Economic 
Growth, University Enclave, North Campus, Delhi 110007.  

ISSN: 2581-6152 (print); 2581-6101 (web). 

DOI: https://doi.org/10.37773/ees.v1i1.7  

https://doi.org/10.37773/ees.v1i1.7


Ecology, Economy and Society–the INSEE Journal [18] 

are dimensioned variables, too. Thus, in the familiar Cobb-Douglas 
production function, of the form LAKLKFY  ),( , the parameter A 

has appropriate dimensions (contingent on   and  ) such that the 

function itself has the exact same dimension as Y (Chilarescu and Viasu 
2012). In an earlier debate on a similar subject, Folsom and Gonzalez 
(2005) had shown, in response to the dimensional inconsistency claim made 
by Barnett-II (2003), how the parameters of the Cobb-Douglas production 
function are assumed to have implicit dimensions required by dimensional 
consistency. To critique the claim in Mayumi and Giampietro (2010), 
Baiocchi (2012) used examples from a variety of disciplines, including the 
IPAT identity and Environmental Kuznets Curve, and also offered a critical 
historical literature review of dimensional analysis. 

Unfortunately, this debate on dimensional consistency in ecological 
economics has only helped to muddy the waters rather than provide a 
consistent framework for achieving dimensional consistency while studying 
the economy-ecosystem interaction problem. It is trivial to demonstrate 
that a logarithmic function cannot have dimensioned variables as its 
argument. The more pertinent question is whether it might be possible to 
non-dimensionalize basic models of economy-ecosystem interaction that 
are of interest to ecological economists. We illustrate the problem with the 
transcendental logarithm function that has been at the centre of the recent 
debate. In their rejoinder to Chilarescu and Viasu (2012), Mayumi and 
Giampietro use the familiar Maclaurian expansion of )1ln( z and )1ln( z  

to obtain a polynomial expansion for the natural logarithm of any positive 

real number, z  (Mayumi and Giampietro 2012, equation 5):  






































 ....

1

1

3

1

1

1
2)ln(

3

z

z

z

z
z  (1) 

It is straightforward to see that equation 1 cannot take a dimensioned z . 
Thus, Mayumi and Giampietro (2012) argue that a regression model that 
includes a term like )ln( L

V  used by Arrow et al. (1961) in their labour–capital 

substitution model is problematic because the logarithm takes on a 
dimensioned quantity (measured in US dollars per person–year of labour 
unit, for example). Even in the 1960s, the classic paper by Arrow et al. 
(1961) had been critiqued for not considering the dimensional consistency 
of production function specifications (De Jong 1967; De Jong and Kumar 
1972; Cantore and Levine 2012). However, Mayumi and Giampietro (2010, 
2012) ignore the fact that it is possible in theory to obtain non-dimensional 

versions of V and L through the well-established process of normalization. 
In principle, there should be no objection to using an expression like )ln( L

V  



[19] Deepak Malghan 

if value added ( V ) and quantity of labour ( L ) are expressed as non-
dimensional variables. 

While Mayumi and Giampietro (2012) cite several examples from 
prominent economists committing the apparent error of using dimensioned 
quantities in the logarithmic functions, we demonstrate in section 2 that 
normalization or non-dimensionalization can in principle address this 
problem. We argue that this is a relatively minor technical point, and that 
the more fundamental problem is that of representing the economy–
ecosystem interaction problem in a dimensionally consistent fashion.  

The remainder of this paper is organized as follows: the next section will 
review normalization and non-dimensionalization using canonical examples 
from economics and ecology. It is not merely sufficient for an ecological 
economics model to be dimensionally consistent. The key question is 
‘whether or not the selected dimensional choice for a given expression has an operational 
meaning or relevance for the purpose [of] analysis’ (Mayumi and Giampietro 2012; 
emphasis in original). While this was the true import of Mayumi and 
Giampietro (2010), the subsequent papers in the debate have missed the 
forest for the trees by focussing exclusively on narrow technical 
dimensional consistency. In section 3, we discuss the limitations of 
normalization and non-dimensionalization procedures. In particular, we 
show that it is nontrivial to normalize dimensioned variables in analytically 
accurate models of economy-ecosystem interaction. 

 

2. NORMALIZATION AND NON-DIMENSIONALIZATION 

Using several canonical (and elementary) examples from ecology and 
economics, we demonstrate in this section that normalization and non-
dimensionalization can address dimensional consistency issues in ecological 
economics. We examine the production function and the consumer‘s utility 
maximization problem from elementary microeconomics; the logistic 
population growth model from ecology; and the normalisation of the 
Gaussian distribution in statistics. 

2.1. Normalization of the Canonical Cobb-Douglas Production Function 

The standard Cobb-Douglas production function for two inputs K and L  
can be represented as: 

 LAKY   (2) 

Now consider a simple constant-returns version of equation 2 such that 
(  1 ) and ( 0,;10  LK ): 

LAKY  1  (3) 



Ecology, Economy and Society–the INSEE Journal [20] 

The central dimensional concern with the Cobb-Douglas function in 

equation 3 is that capital ( K ), and labor ( L ) are measured in units that are 

different from each other, and from the output (Y ). The constant A has a 
dimension that is contingent on the factor-share parameter  ,  such that 

equation 3 is dimensionally consistent. To make this point of A being 
dimensional even more explicit, equation-3 can be rewritten as: 

 LAKAY LK

 1     (4a) 

LK AAA       (4b) 

where the dimensional constants 
KA  and 

LA are the so-called efficiency 

parameters. The presence of these two dimensioned quantities makes 
analytical work and interpretation difficult. However, as shown by De Jong 
(1967) and Cantore and Levine (2012), equation 4 is most easily normalised 
and rendered into a non-dimensional form. Consider a normalization-point 

0Y  such that:  

))(( 0

1

00

 LAKAY LK

     (5) 

Now dividing equation 4 by equation 5 we readily obtain the non-
dimensional version of the constant-returns Cobb-Douglas function:  

  lky  1      (6a) 

000

;;
L

L
l

K

K
k

Y

Y
y   (6b) 

Any econometric model involving logarithms of the non-dimensional 
variables ( lky ,, ) will pose no dimensional issues — for example, a log-log 

model to estimate factor share,  . While equation 6 eliminates the 
dimensional constants, it offers no clarity on how to pick the normalization 
point (

0Y ). In the context of a neoclassical economic growth model, it 

would be most intuitive to use the steady state value as the normalization 
point. While the choice of normalization point is easily determined for the 
present problem, we show below how this can be non-trivial when studying 
the economy-ecosystem interaction problem. Indeed, as we discuss below 
in section 3 below, in ecological economics mass balance problems (say 
when studying stock of timber in a forest), the choice of normalization 
point is critical for model specification. Further, we show why the selection 
of the appropriate normalization point is fundamentally a non-technical 
choice that is value laden.  

Before we take up another canonical example — the logistic growth 
equation from ecology to illustrate the process of non-dimensionalization (a 
homologue of the normalization process discussed here) — it is important 
to note that more general production functions (CES, for example) can be 



[21] Deepak Malghan 

normalized in the same manner as the pedagogically simple case of Cobb-
Douglas discussed here (Klump and La Grandville 2000; Klump and Saam 
2008; Cantore and Levine 2012; Temple 2012). 

2.2. The Logistic Equation 

Consider the logistic equation that has been the pedagogical model of 
choice for students of ecology from the time Alfred Lotka formalised the 
original Verhulst formulation in the context of population growth of 
parasite colonies (Lotka 1925). The simple population growth logistic 
equation with a fixed carrying capacity, K and population growth rate, r can 
be written as: 

0)0(;1 PP
K

P
rP

dt

dP









   (7) 

where P  is the population at any time t ; and the initial population is 

known such that 
0)0( PP  . In the above equation, all the four variables are 

dimensional — P and K have the dimension of [N] (number of individual 
plasmodium parasites in a colony for example); t  has the [T] dimension 
(time, measured in hours or minutes); and r  has the dimension of [T−1] 
(inverse time dimension, measured in per-hour or per-minute, persevering 
with the plasmodium colony growth example). The units in which 
population and time are measured are arbitrary and the parameter values in 
equation 7 will change if we went from measuring time in hours to say, in 
minutes or years. 

It is straightforward to non-dimensionalize equation 7 so that it is invariant 
to particular choices of units for population and time. This is achieved by 
scaling or normalizing the time and population variables as: 











r

t

1
  (8a) 

K

P
x   (8b) 

 d

dx
rK

rd

Kxd

dt

dP


)/(

)(
  (8c) 

K

P
x 0

0    (8d) 

The new variables X and  defined in equation 8 are non-dimensional. 
Substituting equation 8 in equation 7 we obtain the non-dimensionalized 
form of the logistic equation: 



Ecology, Economy and Society–the INSEE Journal [22] 

0)0();1( xxxx
d

dx



 (9) 

Unlike the original dimensioned variables, P , K , t and r , the scaled non-
dimensional variables X and  can be used in any transcendental functions 
like the natural logarithm or the exponential function. Like any non-
dimensionalization process, the scaled variables X and  are related to the 
intrinsic property of the physical phenomenon being studied. The scaled 
population, X represents the population relative to the carrying capacity, 

K and is the intrinsic unit for measuring population in a simple logistic 
model. 1  By measuring population using non-dimensional X , we have 
scaled the problem so that equation 9 applies to a wide variety of 

phenomena following the logistic growth pattern. Similarly r
1 that we used 

to scale time, t to obtain the non-dimensional   is the intrinsic unit for 
measuring time in the context of population growth models. In an 
exponential growth model (the initial part of the logistic growth curve when 

KP  , the population grows by a factor of e in the time interval r
1 – an 

intrinsic unit for measuring time in any exponential growth problems. 
Besides being an intrinsic representation, the scaled non-dimensional form 
of the logistic equation is also the most parsimonious representation of the 
problem of carrying capacity constrained growth. 

2.3. The Standard Normal, and the Box-Cox Transform 

Consider a random variable X  that is distributed with mean  and variance 

),(~( 22  NX . The Gaussian distribution for )(, XfX is given by: 
2

2

1

2

1
)(








 

 





X

exf  (10) 

Any normally distributed variable can be expresses in terms of the standard 
normal, Z (where )1,0(~ NZ ). 

2

2

1

2

1
)(

Z

eZf





 (11) 

As every beginning student of statistics is taught, for any random variable 









X
ZNX ),,(~ 2 is a standard normal, or )1,0(~ N

X




. Besides 

helping with statistical inference, this normalization process is of 

                                                        
1 One could have also carried out the non-dimensionalization of equation-7 by setting x = 
P/P0. A non-dimensional x that is a scaled by the initial population is however not intrinsic 
to the system as the carrying capacity (for a system with time invariant K). 



[23] Deepak Malghan 

significance for our discussion about dimensioned variables. X is a 
dimensioned variable (has the dimensions of [T] for example if X  was 
measuring some temporal phenomenon). However, the normalized variable 
Z is dimensionless as   and  have the same dimensions as X – [T] in the 

present example. Thus, while X cannot be used as an argument in 
transcendental functions, an expression of the form )ln(ZY  can be 

evaluated using equation 1. This normalization process is even more 
significant if one considers the fact that the sum of a sufficiently large set of 
independent random variables (with finite variance) will converge to a 
normal distribution (the central limit theorem). 

Statisticians have long recognized the centrality of transformations in 
studying dimensioned variables. While the pedagogical example presented 
here is familiar even to a beginning student, there is a well-developed 
literature on generalizing transformations starting with the seminal 
contribution of John Tukey (1957), and the celebrated paper of Box and 

Cox (1964). The Box-Cox transformation (
X ) of variable X is simply: 







1


X
X  (12) 

It is straightforward to show that the CES production function (of which 
the Cobb Douglas corresponds to a specific parametric value) is a special 
case of the Box-Cox transformation. 2  Further, as 0 , the Box-Cox 

transform is the log transform ( )log( XX 
). It is for this reason that a 

Box-Cox transform with 0 finds numerous applications in applied 
economics. 

2.4. The Numéraire Good and Consumer’s Utility Maximization 
Problem 

The most widely used example of normalization in economics – by a wide 
margin – is the numéraire good. All prices in the pure theory of exchange are 
relative prices — prices that have been normalised by an appropriate 
numéraire. Money (dollars for example) is simply the most common choice 
for the numéraire. In principle, any other commodity can be used as a 
numéraire. 

Consider an individual‘s utility function defined by a Cobb-Douglas 
function (in a simple two-good case) as follows: 

YXU  1  (13) 

                                                        
2 I thank an anonymous referee for this pedagogical suggestion. 



Ecology, Economy and Society–the INSEE Journal [24] 

Following our discussion in equation 5, we can write out a corresponding 
utility normalization point as: 

0,; 000

1

00   YXYXU   (14) 

Dividing equation 13 by equation 14 we obtain a non-dimensional analogue 
of equation-6: 

 yxu  1  (15a) 

000

;;
Y

Y
y

X

X
x

U

U
u   (15b) 

All three variables (utility, and the quantity of two goods that are 
consumed) in equation 15 are non-dimensional. While )ln(U is not defined, 

equation 1 can be used to evaluate )ln(u . Before we consider the 

consumer‘s utility maximization problem, we write out the budget 
constraint faced by the consumer: 

MYPXP Yx   (16) 

In equation 16, M is the disposable income available to the consumer; and 

XP and 
YP are respectively prices (say in dollars per unit) of goods X and 

Y respectively. The budget constraint when expressed using dimensionless  

x and y (instead of dimensioned quantities X and Y ) can be written out as: 

MyPxP YX 
~~

   (17a) 

XX PXP 0

~

     (17b) 

YY PYP 0

~

     (17c) 

In equation 17 XP
~

and 0

~

Y are simply prices corresponding to normalised 

(and dimensionless) quantities of X and Y . Money, measured in dollars ($) 

is the numéraire in both equations (16) and (17). While
XP and 

YP have the 

dimensions of XP
quantity

dollar ~

,  and 
~

YP have dimensions of dollars. One of the 

fundamental insights from consumer‘s problem is that the neither the 
budget set nor the budget constraint is affected by our choice of numéraire. 
Now, if we normalize equation 17 using x as the numéraire good, we can 
rewrite the budget constraint as: 

__

MyPx Y   (18a) 

~

~
_

X

Y
Y

P

P
P   (18b) 



[25] Deepak Malghan 

~

_

XP

M
M   (18c) 

Every term in the budget constraint represented by equation 18 is 

dimensionless. As the relative price 
_

yP  and 
_

M  are dimensionless they can 

be used as arguments in a transcendental function. Thus a regression 

equation that uses )ln(
_

iM poses no dimensional problems (where 
_

iM  the 

disposable income of household i). 

We can now write out the consumer‘s utility maximization problem using 
equations (15) and (18). The dimensionless Lagrangian is simply: 

__
1 )()( MyPxyxL Y     (19) 

By setting 0,0 









y

L

x

L  and eliminating  we obtain the dimensionless 

first order condition for the consumer‘s utility maximization problem: 
























 
_

11

YP
x

y



  (20) 

Every single variable in equation 20 is dimensionless. 

 

3. OBJECT LESSONS FOR BIOPHYSICAL AND ECOLOGICAL 
ECONOMICS 

We have demonstrated using canonical examples from economics, ecology, 
and statistics that normalization and non-dimensionalization can transform 
dimensional forms into their dimensionless counterparts. The examples 
presented in the previous section show that in theory, normalization can 
circumvent the objections raised by Mayumi and Giampietro (2010) in the 
recent debate over dimensions. However, as pointed out by Mayumi and 
Giampietro (2012) in their rejoinder, the more relevant question is one of 
delineating the physical basis for normalization. In the examples that we 
have considered, the non-dimensionalization procedure for the logistic 
equation or the construction of the standard normal statistic is well-
grounded. From the two economics‘ examples we have considered, 
normalization using an arbitrary choice of the numéraire good in the 
consumer problem is well-established. A production function on the other 
hand must not only be dimensionally consistent but also reflect the physical 
basis for production. Normalization only solves the technical problem of 



Ecology, Economy and Society–the INSEE Journal [26] 

dimensional consistency but the normalized representation of the 
production process is only as good as the original dimensioned 
representation. An accurate physical representation of the production 
process has been one of the founding tenets of ecological economics 
(Georgescu-Roegen 1971; Kraev 2002; Røpke 2004). 

As an illustration of the difficulties involved in selecting a normalization 
point in realistic models of economy-ecosystem interaction, consider any 

model that includes a throughput variable (
.

X ), say measured in kilograms 

per year so that 
.

X  has the dimensions of [MT−1]. The throughput 
.

X cannot 
be an argument in any transcendental function. It is trivial to normalize the 

throughput with some reference throughput, 
.

X  to obtain a non-

dimensional version 

X

Xx

..
 such that the normalized throughput, 

.

x  has 

no physical dimensions and can be used as arguments in transcendental 
functions. Indeed, such a measure is homologous to the rapidity measure 
used in physics to characterize speed relative to the speed of light.3 Unlike 

relativity-physics however, the choice of reference throughput, 


X  is not 
universal but highly context dependent. A possible candidate for reference 
throughput is the maximum sustainable throughput — the throughput 
above which the integrity of the underlying biophysical system in jeopardy. 
Consider an illustrative example — throughput of timber from a forest. 
The maximum sustainable throughput is a function of the health of the 
underlying forest ecosystem and will vary across both space and time. A 
tropical forest will necessarily have a different maximum sustainable 
throughput from a temperate forest. Even in a single location, maximum 
sustainable throughput will vary with time. The determination of maximum 
sustainable throughput is a function of ecosystems as funds rather than 
stocks (Malghan 2011). 

The fundamental economy-ecosystem interaction problem, or the 
biophysical connections between the economy and the ecosystem is not 
reducible to cardinal arithmetic that governs stocks and flows.4 Instead, any 
reasonably complete account of the economy–ecosystem interaction 
problem must also account for funds and fluxes that cannot be described in 
a cardinal space (Georgescu-Roegen 1971; Malghan 2011). A fund is a 
'special configuration of a given stock of materials(s)' (Malghan 2011). 
Consider, for example, an automobile. It is a stock of various material 

                                                        
3 In physics, rapidity, φ is defined as )(tanh 1

c

v where c is the speed of light. 

4 This part of the essay is abstracted from the presentation in Malghan (2011). 



[27] Deepak Malghan 

stocks – steel, aluminium, plastic, etc. However, these stocks have to come 
together in a specific configuration to constitute a fund of useful 
transportation services. An automobile that has been ‗totalled‘ in an 
accident still retains all the original stock but is no longer a fund of 
transportation services. Thus, while simple mass conservation laws can 
account for stocks, funds are best described by laws that 'follow the spirit of 
entropy law in thermodynamics' (Malghan 2011). More formally, equation-
21 below fully describes the evolution of stock over time, funds are not 
reducible to the cardinal space. 

dtxxxtx out

t

in )()0()(
.

0

.

   (21) 

where )(tx is the quantity of stock at any time t ; )0(x is the initial stock at 

0t ; and inx
.

and outx
.

are inflows and outflows respectively. The stock x  is 

in steady state if ttxtx outin  )()(
..

. No such simple steady state condition 

can written out for a fund, as illustrated in the totalled automobile example 
above where all the stocks in the car in an approximate steady state. In 
addition to providing the original exposition of the concept of ‗fund‘, 
Georgescu-Roegen (1971) also speculated on an entropy law modelled on 
the Second Law of thermodynamics for matter. This so-called ‗fourth law‘ 
has been hotly contested (Cleveland and Ruth 1997; Ayres 1998, 1999; 
Hammond and Winnett 2009). For our purposes here, it is sufficient to 
note that conservations laws (like the first law of thermodynamics) alone 
cannot completely describe a fund, and we need to invoke some mechanism 
like the Second Law that allows for qualitative degradation of energy and 
matter (Malghan 2011). 

The distinction between stocks and funds introduced here, using the 
automobile as a pedagogical example, has  direct implications for 
dimensional consistency in even the most elementary models of the 
economy–ecosystem interaction in biophysical economics or ecological 
economics. Continuing with the sustainable throughput example discussed 
above, consider a very simple measure of the scale of a forestry industry: 




Y

x
S

.

 (22) 

where 
^

Y  is the rate at which timber regenerates in the forest, and 
.

x  is the 

throughput. Now suppose we have S measured for two different places: 

1S = 0.8 in a tropical forest; and 
2S = 0.7 in a temperate forest. Can we 

automatically conclude that the forestry industry in the temperate forest is 



Ecology, Economy and Society–the INSEE Journal [28] 

more sustainable than the one in the tropics because 
2S < 

1S ? We cannot 

because equation 22 does not contain any information about the underlying 
fund. S is dimensionless only in the stock-flow space in the sense that both 

.

x  and 
^

Y  have the same dimensions – say, tons/year. In order to be able to 
make comparisons across space and time, we will need metrics that are 
appropriately normalized to render them dimensionless in both the stock-
flow space and the fund–flux space. In particular, it is important to note that 

S ≤ 1 does not automatically imply sustainable throughput. During 
regeneration of a degraded forest, a sustainable throughput might as well be 

S = 0 (Malghan 2011). 

Several extant aggregate biophysical metrics are not dimensionless in the 
fund–flux space (Malghan 2011). Examples of such metrics include the 
ecological footprint (Rees 1992; Wackernagel and Rees 1996; Wackernagel 
et al. 2004); human appropriation of the products of photosynthesis 
(Vitousek et al. 1986; Rojstaczer et al. 2001); and aggregate material 
throughput metrics (National-Research-Council, 2004; Adriaanse et al. 1997; 
Matthews et al. 2000; Klee and Graedel 2004; Gordon et al. 2006; Wernick 
and Ausubel 1995). These aggregate metrics are all appropriately normalized 
in the stock-flow space but not in the fund-flux space. For example 
aggregating multiple flows (as is done in the material throughput) is simple 
enough in the stock-flow space but fraught with problems in the fund-flux 
space. Consider a simple material flow metric: 







ni

i

ixX
1

..

 (23) 

The sum in equation 23 is dimensionally valid only in the stock flow space, 
but not in the fund–flux space. If the n elementary flows are summed into 

the aggregate throughput, 
.

X , the sum is defined as long as all the n flows 
are measured in a common unit – say tons/year. However, in the fund flux 
space, it makes little sense to sum up tons/year of some metal and tons/year 
of water. For the sum to be defined in the fund-flux space, the individual 
flows have to be appropriately normalized so that they are 'strictly 
dimensionless' in the fund-flux space (Malghan 2011). 

The difficulty with determining an appropriate normalization point in the 
throughput example above is related to a more general problem of mapping 
ordinal and cardinal variables in a dimensionally consistent fashion. An 
accurate representation of the economy-ecosystem interaction problem 
requires accounting for ecosystem as a fund in addition to ecosystem as 
simply a collection of stocks. Unlike stocks and flows, funds and fluxes are 
ordinal and subject to additional dimensional consistency constraints. The 



[29] Deepak Malghan 

mapping between an ordinal fund-flux space and the cardinal stock-flow 
space is at the heart of economics of ecosystem services (Farley 2012; 
Malghan 2011). In the current debate of dimensioned variables in ecological 
economics, the true import of the first salvo fired by Mayumi and 
Giampietro (2010) was the fact that several empirical models are not careful 
about making the distinction between fund and stock functions of 
ecosystem. We would be missing the forest for the trees if we focused the 
dimensions debate exclusively on technical aspects of normalization. 
Normalization procedure, as demonstrated using elementary examples is 
well-established for cardinal variables but the cardinal stock-flow space 
alone is inadequate for accurately modelling the economy-ecosystem 
interaction problem. There is a need for ecological economics to develop 
models of economy-ecosystem interaction that are at once realistic 
representation of the problem and are dimensionally consistent. 

 

ACKNOWLEDGEMENTS 

I thank Kanchan Chopra, Herman Daly, Federico Demaria, Joshua Farley, 
Philip Lawn, Joan Martinez Alier, Matthias Ruth, Anish Sugathan, Hema 
Swaminathan, Sneha Thapliyal, and Robert Ulanowicz for critical 
comments on earlier versions. Suggestions and comments from the two 
anonymous referees were pivotal in further improving the exposition. I am 
most grateful to Nandan Nawn and the editorial team at EES for help with 
typesetting the paper that was originally written using a typesetting program 
that is not compatible with the normal journal production process at EES. 
Usual disclaimers apply. 

 

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