A Framework for Dimensional Analysis of Biophysical Metrics ✩,✩✩

Deepak Malghana

aCentre for Public Policy Indian Institute of Management Bangalore Bannerghatta Road, Bangalore 560076, INDIA

Abstract We develop a generalised stock-fund representation of the economy-ecosystem interaction problem. This represen- tation is used to define dimensionless quantities in the ordinal fund-flux space. We show that our framework for dimensioned quantities in the fund-flux space can help characterise aggregation properties of biophysical metrics. is used as an illustrative example. We also show how a consistent framework for dimensional quantities is central to constructing scale metrics that measure the proportional relationship between the economy and the biophysical system. Key words: Stock-Fund, Biophysical Assessment, Dimensional Consistency, Ecological Footprint.

1. Introduction land area needed to sustain economic activity). Γ is obtained by summing up n different elemental sectors One of the key areas of ecological economics re- of the economy (γ through γ ). For example, the search concerns the development of biophysical metrics 1 n ecological footprint accounting would compute Γ as the that measure the physical size of economic activity. sum of land area demanded by six key sectors of the Some of the more popular metrics include the eco- economy [32, p.9270]. logical footprint [25, 28, 32], human appropriation of the products of photosynthesis [27, 26], and aggregate The ultimate object of any biophysical assessment is material throughput metrics [1, 21, 13, 10, 33, 22]. The to understand the relationship between aggregate eco- central goal of these biophysical assessments of human nomic activity and the ability of the biophysical system activity is to determine if the scale of human activity is to support to this economic activity. Thus, apart from sustainable [29, 16, 15]. Despite the increasing use of aggregating economic activity, biophysical accounting biophysical metrics in policy deliberations, there does methodologies also aggregate the ‘ecological space’ not exist a consistent theoretical framework to study the available to support human activity: aggregation properties of biophysical metrics [19]. n Θ = θ (2) Stripped of all the (obviously important) details, any i Xi=1 biophysical metric, Γ is an aggregation that can be rep- resented as: In equation - 2, Θ is the aggregate ecological space n available and computed by aggregating individual θs. Γ = γi (1) For example, in the ecological footprint accounting, Xi=1 k equation - 2 takes the form of H(t) = i=1 Pi(t) where In the above equation, Γ is the aggregate economic H(t) is the sum-total of bioproductive landP available at activity measured in some biophysical unit (for example any time t, obtained by aggregating k different types of land area [32]. ✩I am grateful to Indian Institute of Management Bangalore for funding my conference participation. ✩✩I would like to thank , Joshua Farley, Jack Santa- Given Γ and Θ, we can construct a scale metric, Ω Barbara, Matthias Ruth, and Robert Ulanowiscz for simulating dis- that characterises the proportional relationship between cussions, and comments on an earlier version. All remaining errors the economy and the ecosystem: are mine. Email address: [email protected] (Deepak Malghan) Ω = S (Γ, Θ) (3) Preprint submitted to Proceedings of the United States Society for Ecological Economics, 2009 May 25, 2009 where S is the ‘scale function’ [19]. The scale-metric 2. Stock-Flow and Fund-Flux Ω represents a proportional relationship between the economy and the ecosystem that contains and sustains We begin by reviewing the stock-fund representa- it. Γ, Θ, and thus Ω are calculated at local, regional, tion of the economy-ecosystem interaction problem national, and finally global levels. Indeed some of the [19, 18]. Consider the elementary ontological picture biophysical assessments have even been used at the of ecological economics – the economy as an open individual or household level (the footprint calculator subsystem of the larger ecosystem that contains and for example). At every level of of aggregation from the sustains the economy. A simplified representation of individual to the planet, Γ, Θ, and Ω are used as indices this ontological vision is presented in figure - 1. This to track progress on achieving biophysical sustainabil- picture represents the economy-ecosystem interac- ity. To serve as effective indicators of , tion problem at every level of economic-geographic any biophysical metric must consistently rank states of aggregation. The economy as an open subsystem is the world that it characterises [19, 17].             The key problem of consistency is being able to rank      Economy the states of the world represented by a biophysical       metric. For example, if we have Γi > Γ j, what can we      say about the corresponding states of the world? Does Ecosystem     moving from Γ to Γ represent an improvement or a  i j  regression on society’s sustainability goals? Formally,  we are looking for a consistent interpretation of Γi ≻ Γ j Figure 1: A Simplified Representation of Ecological Economics’ On- tological Vision – more specifically, given Γi ⋚ Γ j, what can we say about Γi ≻ Γ j? For example, if Γ represents the footprint aggregation, how do we rank two states of the connected to the larger ecosystem through two different kinds of ‘flows’ that are fundamentally different from world with Γi = 5 and Γ j = 3.5? Is j more sustainable than i because Γ > Γ ? Answering these questions each other. The first flow is the familiar material i j 1 requires us to look at the ‘aggregation mechanics’ that throughput. The ecosystem is the ultimate physical aggregated individual γs into Γ. resource base for the economy and is also the ultimate sink for waste products that are an inevitable part of any economic process. We will use the term resource In this paper, we consider a small part of the larger flow to denote this physical throughput. It will be consistency problem – the dimensional consistency of Γ used to denote the material throughput on both the and Θ. In particular we are interested in the aggregation source-side as well as the sink-side. Thus we will use properties of dimensioned biophysical quantities. For the term ‘resource flow’ to refer to both the amount of example, the ecological footprint metric aggregates timber harvested by the paper mill in a given year and a land areas, and some of the material flow metrics particular effluent released by the paper factory.2 aggregate mass of the throughput [1, 21]. We are interested in understanding the dimensional consistency The forest is not just a stock of timber but also pro- of aggregating on physical dimensions like area or mass. vides valuable services like micro-climate stabilisation.

The remainder of this paper is organised as follows. 1In this paper, we illustrate our framework with examples of ma- In the next section, we develop a generalised stock-fund terial throughput. Much of the this discussion is applicable to energy representation of the economy-ecosystem interaction throughput with suitable modifications [19, 18]. problem. In section - 3 we develop definitions for 2The relationship between stock of resources and the corre- sponding resource flow is readily understood on the source-side but dimensioned quantities in the fund-flux space. In needs careful and somewhat more nuanced analysis on the sink-side particular, we develop working definitions for dimen- [18, 19]. On both the source-side and the sink-side, the stock at any sionless quantities that are consistent across both the t˜ instant of time is given by x(t˜) = x(0) + ( fin(t) − fout(t)) dt where stock-flow space as well as the fund-flux space. In R0 section - 4 we review the dimensional consistency fin(t) is the flow into the stock at any time t and fout(t) is the outflow of the ecological footprint metric, and conclude in from the stock. t˜ is the current time period and x(0) is the reference stock at time t = 0. This is of course simply the solution of the differ- section - 5. ential equation that is an accounting identity that holds good on both dx the source-side as well as the sink-side dt = fin(t) − fout(t). 2 Unlike material flows, there is no way to write out an stabilisation service. accounting identity where a ‘flow’ of micro-climate stabilisation service accumulates into any stock. An- The service derived from a fund is not a physical alytically intractable as they may be, these services flow like the resource flow derived from the stock provided by the ecosystem are vastly more crucial function of the ecosystem. However, it is nevertheless than the material flows derived from natural stocks. treated as a flow because it has a ‘per unit time period’ There is of course a definite connection between the kind of dimension to it.4 Services derived from the magnitude of material flows and the more abstract ecosystem in its role as a fund usually have very small service like micro-climate stabilisation. The stock of ‘rates of flow.’ We will use the term service flux to timber that is the source of material flows is after all distinguish ecosytem services (derived from the fund one of the constituents of the forest. Indeed, the aim of function of the ecosystem), from resource flows (that any biophysical assessement is to understand the nature are derived from the stock function). A flux unlike of the relationship between resource flows and valuable a flow is invisible but is nonetheless impressionable. ecosystem services like micro-climate stabilisation [19]. A flux is not amenable to simple additive arithmetic of flows. The different service fluxes derived from A fund is a special configuration of a given stock various ecosystems are critical not just for the survival of material(s). For example a special configuration of the human economy but all biological life. In the of the given stock of steel, aluminium, plastic, etc. ecological economics’ ontological conception, the constitutes the automobile which is a fund of trans- human economy can indeed be conceived as one of the portation services. The operative words here are special groups of stocks that makes up a fund of ecosystem configuration — the same stock of steel and aluminium service fluxes.5 in any other configuration does not constitute a fund of transportation services though they continue to contain The integrity of the fund is related to an harmonious the exact same amount of physical material. The most balance between the different stocks that make up the obvious example will be an automobile which has met fund. We will see how a fund needs to be constantly with an accident, is completely mangled, and is on its nourished by resource flows to help maintain the essen- way to the junk yard. Thus while simple conservation tial “configuration” of the fund. In the main, analytical laws are sufficient to fully characterise the relationship ecological economics concerns itself with developing between stocks and flows, the relationship between a a coherent accounting framework to reconcile how the fund and the service derived from it are more complex. human economy generated material and energy flows The fund-service relationship is more appropriately affect the ecosystem in its role as a fund. characterised by laws that follow the spirit of the entropy law in thermodynamics.3 The ability of the fund to provide a service is contingent on a particular 2.1. Natural Capital and Natural Income configuration of the stocks that constitute a fund. Like an automobile, the ability of the forest to provide One of the highlights of development of ecological valuable services in its role as a fund is contingent on a economics in the last two decades has been the ascen- particular configuration of the stocks that make up the dency of the the concept of “natural capital.” Like the forest. Moreover, the natural regeneration rate of the concept of “human capital,” the concept of natural cap- any constituent stock is dependent on the structure of ital is rather amorphously defined. In the same way as the underlying configuration. Thus a captive plantation human capital yields an income, natural capital yields a with the same standing stock of timber as a forest flow that is the “natural income.” The use of the term with diverse species will regenerate at a different rate ‘capital’ is based on the understanding that broadly and will provide a different level of micro-climate defined, capital is anything that yields income and the traditional use of the term in economics to refer to

3[9] in addition to providing the original exposition of the concept of fund also speculated on a entropy law modelled on the Second Law 4The difference between resource flows and services on the time- of thermodynamics for matter. This so-called ‘fourth law’ has been dimension is crucial to understanding the difference between ecosys- hotly contested. For our purposes here, it is sufficient to note that con- tems as stocks and ecosystems as funds. See [9, p.227], [19]. servations laws (like the First Law of thermodynamics) alone cannot 5In an analytical study of the economy in the biophysical dimen- completely describe a fund and we need to invoke some mechanism sion the system boundary between economy and ecosystem can be a like the Second Law that allows for qualitative degradation of energy contentious issue. However, in this paper, we limit ourselves to nar- and matter. row anthropogenic concerns. 3 manufactured equipment is only a specific example [4]. the human economy.8 In figure - 2, utility, U consists The concept of natural capital is amorphous because of three parts – Y˘so, from the ecosystem fund in its ecosystem is simultaneously a stock of material flows role as a source of resource flow (x ˙i); Y˘si, from the and a fund of service fluxes. Both the material flows fund function of the ecosystem as the sink for waste as well as the service fluxes are constituent elements flow (x ˙o); and finally Y˘e from the fund function of the of natural income. Natural capital is an aggregate material stock in the economy. The representation of term used to simultaneously denote both the fund and the throughput is straightforward. The parameters ti and stock functions of the ecosystem. The concept of to, a function of economic decision making influence natural capital was developed mainly in the context the source-side and sink-side of the throughput,x ˙i and of the sustainability discourse. The goal of ‘strong x˙o respectively. sustainability’ is to maintain non-diminishing natural capital while that of ‘weak sustainability’ is to maintain a non-diminishing sum of natural and manufactured 2.2.1. Source-Side and Sink-Side: Modelling Regener- capital [23]. ation We now turn to the regeneration portion of figure - 2. We demonstrate here that a clear distinction between On the source-side, Yˆso is the inflow associated with stock and fund aspects of ecosystem directly leads to a the stock xso in every time period. This regeneration formal definition of natural capital and natural income flow is a function of the standing stock (xso) and the 9 that is both theoretically appealing and empirically regeneration parameter, (rso). We will treat Yˆso as tractable for policy. Further, a formal definition of nat- one component of the natural income derived from the ural capital and natural income leads to operational and ecosystem on the source-side with the other component policy relevant definitions of concepts of biocapacity, being the service flux Y˘so. Unlike the service flux, Y˘so regenerative capacity, and carrying capacity. that is not readily quantified, the natural income on the resource flow dimension, Yˆso has the same physical dimensions asx ˙i. For instance, consider the ‘timber 2.2. Stock-Fund Representation from a forest’ example – the annual income represented by Yˆso is simply the amount of new wood that the In figure - 2, below, we present the basic stock-fund forest adds each year and has the same dimensions as representation of the economy-ecosystem interaction x˙i, the throughput of timber into the economy. For problem. In figure - 2, the human economy derives non-renewable resources like coal or oil, there is no benefit from all three stocks – the source, the sink, and income, Yˆso as new coal or oil is not produced in the economy. The benefit derived is shown as a utility time scales that are relevant to the human economic 6 flux, U. This flux is not physically quantifiable and is predicament. Thus on the source-side, we have two made up of three components – the two ecosystem ser- different kinds of income – one from the ecosystem in vice fluxes from the source and sink, and the traditional its role as a stock of raw materials (Yˆso) and the other 7 utility flux from artifacts in the human economy. The from the ecosystem in its role as a fund of service flux utility flux is of course related to the fund aspect of the (Y˘so). three stocks. Utility is derived from an automobile as a fund of transportation service and not as a stock of We turn next to modelling the aspects of natural steel. Similarly, the contribution of the ecosystem to income on the sink-side where things are somewhat the utility flux is from the fund-role of the ecosystem. less analytically tractable. We begin by looking at The timber that is derived form the forest is subsumed physical and tangible waste that is absorbed by the ˘ under Ye, the component of utility flux derived from ecosystem in its sink function and not the various indirect service benefits like climate regulation that are

6Here we do not discuss how U is aggregated across different members of the society. We have reported particular difficulties else- 8See [19] for a detailed discussion on the merits of modelling util- where ([19, p.98-104] and [17]). Here we assume that the social ity as as a function of stocks rather than flows. 9 choice problem of specifying U(Y˘so, Y˘e, Y˘si) has been solved. Figure - 2 shows the regeneration rate represented by the param- 7 We recollect that this paper even while recognising the intrinsic eter rso to be independent of xso. While we have not shown how rso worth of ecosystem and the resource flows and service fluxes derived is related to the health of the ecosystem fund (xso) the discussion here from it, abstracts from all non-anthropogenic concerns as our primary and elsewhere in this paper is based on the recognition that regenera- focus here is the relationship between the human economy and the tion rate is endogenously determined by the underlying fund structure supporting ecosystem. of xso. 4 ti to

Yˆ Yˆ so x˙i x˙o si x˜ x xe si so xsi

rso Y˘ so Y˘ e Y˘ si rsi

U˘

Figure 2: Source, Sink, Fund, Flux, and Throughput

indeed derived from the ability of ecosystems to absorb However, the regenerative flow on the sink-side, Yˆsi rep- wastes generated by economic processes. The service resents a ‘flow of new regenerative capacity.’ To use benefits are represented by Y˘si in figure - 2 and are an analogy from electrical engineering and physics, re- similar to Y˘so on the source-side. Here we model the source regeneration on the source-side and sink capacity sink-analogue of the regenerative resource flow, Yˆso on regeneration are like electrons and holes. Holes are ab- the source-side of the stock-flow space. In figure - 2, stract theoretical constructs while electrons are a phys- the waste stream from the economy,x ˙o flows into the ical reality. Nevertheless it is useful to study regenera- sink. The waste absorption process involves a change tion of ecosystem’s ability to absorb waste in the same in the chemical and/or physical structure of the waste. way as it is useful to study positive charge in semicon- When acid rain falls on a lake, the lake water has the ductor physics as ‘holes’. In figure - 2 we have the fol- ability to neutralise some of the acid. River water, lowing simple accounting identity relating waste stock, similarly, has the ability to get rid of certain amount xso and holes,x ˜so. of sewage waste every year. The amount of waste that is physically or chemically absorbed and is no longer dxsi d˜xsi = − = x˙o − Yˆsi (4) distinguishable as waste is the natural income from dt dt ! the ecosystem in its role as a sink in the stock-flow space. If new wood is what is regenerated as natural As a consequence of this simplified representation on income on the source side in the timber example, the the flow-dimension, the stock representing the ecosys- corresponding natural income on the sink side of our tem in its sink function is more complex that the ecosys- river example is the regeneration of waste absorption tem stock on the source-side. On the sink side the capacity. Thus as an analogue of resource regeneration ecosystem stock is represented by a combination of Yˆso on the source-side, what is being regenerated on waste stock (xsi) and holes (x ˜si). The sink is in con- the sink-side is absorption capacity for the waste stream. stant flux with new holes being created by the regen- eration process, and holes ‘consumed’ by the waste- In figure - 2 this regeneration is represented by Yˆ . stream. If the waste flow (x ˙o) exceeds the rate at which si ˆ Unlike the regeneration flow, Yˆ on the source-side, holes are created (Ysi), there will be an accumulation of so waste in the ecosystem and if the regeneration exceeds Yˆsi is an abstraction from the actual physical process. The actual physical process would be represented by the waste flow into the sink, there will be excess holes in the sink.10 waste stream,x ˙o flowing into the sink and a flow out of the sink. However the stock that represents the sink in figure - 2 has both the flows flowing into the box. The 10Equation - 4, is not a good description of a pristine sink (x ˙ = 0). waste stream from the economy,x ˙o, has the same phys- If we assume Yˆsi > 0 (absorption capacity being continually regen- ical dimensions as the source-side of the throughput,x ˙i. erated), then equation - 4 implies that the ‘stock’ of holes or x˜si is 5 2.3. Representation A fund that supports this n-component throughput is To begin formalising our discussion about stocks, represented as: flows, funds, and fluxes we start with a simple case with one-component throughput. In figure- -2, we fo- xso1 xe1 x˜si1 xsi1 . . . . cus on the single physical stock in the economy, xe. The  . . . .    throughput consists of two parts. The flow from source Fn =  x x x˜ x  (10)  so j e j si j si j  into the economy,x ˙i, and the waste flow from the econ-  . . . .   . . . .  omy back to the ecosystem,x ˙o. For the single stock, xe,  . . . .    we can represent the associated throughput as:  xson xen x˜sin xsin      T = x˙i x˙o (5) The fund in the n-component case corresponding to N   can be written as: The throughput T connects three different stocks (four if we accounted for the fact that the sink-side contains xso1 x˜si1 xsi1 holes as well as ecosystem generated waste). We define . . .  . . .  X as the ecosystem stock:   Nn =  x x˜ x   so j si j si j  (11) X = x x˜ x (6)  . . .  so si si  . . .     . . .  The single stock, x can be part of a larger fund   soi  xso x˜si xsi   n n n  with several other stocks. Even with a one-component   throughput, the source-side stock, xso j can be part of the same fund as the sink-side stock, x . si j In equations 10 and 11, the constituent elements of both the funds Fn and Nn are cardinal stocks. Funds F = xso xe x˜si xsi (7) j j j j Fn Nn   and do not merely represent a collection of car- As a matter of notational convention, we will use dinal stocks but a particular configuration among those boldface fonts to represent cardinal variables in the constituent stocks. The defining characteristic of a fund stock-flow space (for example, X or T), and blackboard is its underlying configuration. A fund is defined not fonts to represent ordinal variables in the fund-flux only by the magnitudes of underlying stocks but also F space (for example, ). the qualitative interrelationship between the constituent Fn Fn stocks. In particular, two funds say 1 and 2 can be In equation - 7 we included the economy-stock, xe as (and indeed are in many cases) different even when they F part of the fund, . This is consistent with the ecologi- support the same throughput, Tn. Our canonical forestry cal economics ontology that represents the economy as example is a good illustration of why this might be true. an open subsystem of the larger ecosystem. However, Consider a simple n = 2 case where wood from the for- for analytical purposes it is useful to define a fund that est is used for making paper, and as building material. consists of only the ecosystem stock, X. Fn Let 1 that represents a forestry industry in the tropics N and Fn represent one in a temperate region. Even when = xso1 x˜si1 xsi1 (8) 2   the wood contained in the forest, both sectors of the re- 2.3.1. The n-Component Case spective economies, and the waste stocks are the same in both places, Fn cannot possibly be the same as Fn. In the general case, a given stock xso j or xsi j is part 1 2 of a larger fund that supports a throughput of n compo- This point will become even more clearer once we have nents. This n-component throughput can be represented defined the n-component analogue of X in equation - 6. as: xso x˜si xsi x˙i1 x˙o1 1 1 1 . . . . .  . .   . . .   . .    n   n =  x x˜ x  T =  x˙i x˙o  (9) X  so j si j si j  (12)  j j   . . .   . .   . . .   . .   . . .       x˙ x˙   xson x˜sin xsin   in on          The right hand sides of equation - 11 and equation - 12 monotonically increasing. This apparent anomaly is explained by the n fact that our representation of the sink in figure - 2 does not account are identical. However, X only represents a collection for ‘natural death’ of holes. of stocks without regard to the qualitative relationship 6 between the constituent stocks that is captured by Nn.11 In equation - 16, U˘ n includes fluxes from both the n n In particular, we want to emphasize that X1 = X2 does ecosystem stocks (X) and the economy-stock, For the Nn ∼ Nn not in any way also imply 1 2. sake of completeness, we will extend our simple 1- component definitions to the n-component case. The stock-income, Yˆ n and the fund income, Y˘ n can be com- 2.3.2. Regeneration in the n-component case bined to obtain the aggregate natural income, Yn. The regeneration associated with the n-component ˆ ˘ ˆ ˘ throughput is represented as: Yso1 Yso1 Ysi1 Ysi1  . . . .  Yˆ Yˆ . . . . so1 si1   . . Yn =  Yˆ Y˘ Yˆ Y˘  (17) . .  so j so j si j si j   . .       . . . .  Yˆ n =  Yˆ Yˆ  (13)  . . . .   so j si j     . .   ˆ Y˘ ˆ Y˘   . .   Yson son Ysin sin   . .         Yˆso Yˆsi  It will prove useful to also define the source-side and  n n    sink-side components of the aggregate natural income,  Yˆ n  The regeneration matrix, represents the stock aspect Yn on the source-side and Yn on the sink-side. of the natural income derived from the n-component so si system. Note that Yˆ n is represented as a fund-variable Yˆso Y˘ so Yˆsi Y˘ si despite the fact that every element of the matrix in equa- 1 1 1 1  . .   . .  tion - 13 is a cardinal quantity. While Yˆ j is itself a  . .   . .  n   n   cardinal variable, its value is determined by the ecosys- Y =  Yˆso Y˘ so  and Y =  Yˆsi Y˘ si  (18) so  j j  si  j j  tem fund, N and not merely the extent of the ecosystem  . .   . .   . .   . .  stock, X. We will represent the cardinal n-component      ˆ Y˘   ˆ Y˘   Yson son   Ysin sin  regeneration as:     ˆ n Yˆ n     Y = | | (14) Following equation - 14, we can write out the cardinal The fund aspect of the natural income from this system regeneration as: can be represented by: Yˆ n = |Yn| (19) ˘ ˘ Yso1 Ysi1 . .  . .  Yˆ n = |Yn | (20)   so so Y˘ n =  Y˘ Y˘   so j si j  (15)    . .  ˆ n Yn  . .  Ysi = | si| (21)    Y˘ Y˘   son sin    3. Dimensional Analysis in the Fund-Flux Space In addition to the natural stocks, xso on the source-side and xsi,x ˜si on the sink-side, there is another stock in our The central goal of any biophysical metric is to de- representation – the economy-stock, xe. The total flux, scribe the throughput (Tn) in both the stock-flow space Un Fn that is derived from the fund in equation - 10 is (xn) as well as the fund-flux space (Nn). The aggregate represented as: economic activity, Γ, in Equation - 1 is simply an alternate representation of the n-component throughput, Y˘ so Y˘ si Y˘ e 1 1 1 Tn defined by equation - 9.12 While the throughput,  . . .   . . .  Tn is supported by the underlying fund, Nn, Tn is n   U˘ =  Y˘ Y˘ Y˘  (16) typically measured by biophysical metrics as a cardinal  so j si j e j   . . .  variable (Γ). In this section, we develop a framework  . . .    for describing the dimensional properties of Tn (or Γ)  Y˘ Y˘ Y˘   son sin en  in the stock-flow space as well as the fund-flux space.   11Indeed, a more accurate way of representing Xn would be to write Xn = |Nn|. Developing the ‘stock-magnitude’ of any arbitrary fund is 12Indeed, the aggregate measure of economic activity, Γ can be involved and beyond the scope of this paper. We address this issue in written as Γ = k|Tn|k. k|x|k is not be confused with the regular ma- an ongoing research effort [20]. trix norm, kxk (c.f. [11, 20]). 7 A consistent representation of dimensioned quantities However, in the fund-flux dimension, tons(wood)/year is critical to studying the cross-section and time-series and tons(discarded - paper)/year have a different properties of a biophysical metric. qualitative dimension. Thus what is dimensionless is determined by the context. M is dimensionless if Consider a very simple one-dimensional metric that we are interested in tracking the throughput of coal measures the physical scale of the economy in the flow in terms of carbon atoms rather than as coal and dimension for a forestry based industry. The economic carbon-di-oxide. However if we are interested in the throughput consists of two parts: on the source-side, qualitative transformation of the throughput, M is no we have the flow of timber from the forest into the longer a dimensionless scalar – indeed M now becomes economy (x ˙i), and on the sink-side we have the flow of a metric that characterises the transformation efficiency discarded forest products back to the ecosystem (x ˙o). of the economic process. Let Yˆso be the rate at which timber regenerates in the forest. 3.1. Dimensionless Quantities A simple scale metric that measures the physical We formalise the above discussion of dimensioned size of this model economy relative to the size of the quantities in the fund-flux space by defining three dif- x˙i ecosystem that contains it is S so = . Now suppose Yˆso ferent categories of dimensionless measures: we have S so measured for two different places: S so1 in Definition 3.1.1. Dimensionless Quantity: (Stock- a tropical forest, and S so2 in a temperate forest. Further, Flow Space) say S so1 = 0.8 and S so2 = 0.7. Can we automatically conclude that the forestry industry in the temperate forest is more sustainable than the one in the tropics A quantity is dimensionless if that quantity can be expressed as a scalar with no physical dimensions in because S so2 < S so1 ? The answer is a ‘no’, because we lack information about the fund-flux space. S so the stock-flow space. is a metric only in the stock-flow space. To be able to compare across regions, and across time, we need The metric M in the discussion above for example is metrics that are “dimensionless” in both the stock-flow dimensionless. space and the fund-flux space.13 Definition 3.1.2. Qualitatively Dimensionless Quan- tity: (Fund-Flux Space) To motivate the need for dimensionless measures consider the simple model of economy-ecosystem In addition to being dimensionless, qualitatively interaction again. While an elementary mass-balance dimensionless quantities can be expressed as a accounting identity is sufficient to track the flow scalar with no physical dimensions even after after ac- of carbon and hydrogen atoms that make up tim- counting for the qualitative differences between the con- ber, the three stocks that hold wood in our model stituent elements of the metric. are dramatically different funds. A forest cannot be more different from a stock of paper products in A qualitatively dimensionless quantity is di- the economy or certainly discarded artifacts in a landfill. mensionless in the fund-flux space, in addition to being dimensionless in the stock-flow space. A metric that is ‘dimensionless’ in the stock-flow space is not necessarily dimensionless in the fund-flux Definition 3.1.3. Strictly Dimensionless Quantity: space. Consider, for example, a measure that is (Fund-Flux Space) a ratio of two components of the throughput (Say M = x˙i ). The metric M is dimensionless in the x˙o Strictly dimensionless are quantities derived stock-flow space – bothx ˙i andx ˙o are measured in from two or more qualitatively dimensionless the same flow units, tons/year of timber for example. quantities that can be expressed as a scalar with no physical dimensions even after accounting for qualita- tive differences. 13 Indeed, throughput being less than regeneration (S so ≤ Yˆso) does not automatically imply a sustainable throughput. For example con- To motivate the need for strictly dimensionless quan- sider a degraded forest that is freshly regenerating. During the initial tities consider the following metric: C = S so(t2)−S so(t1) regeneration phase, a ‘sustainable’ throughput could might as well be S so(t1) S so = 0 [19]. where t1 and t2 are two different time periods at which 8 the flow measure of scale, S so was measured for some the source-side as well as sink-side into an area required throughput,x ˙i. The metric C simply computes the to support that throughput. The footprint methodology percentage change in scale S so between time t1 to time indirectly measures the amount of photosynthesis prod- t2. To see why C is not qualitatively dimensionless, we ucts needed to support a given throughput. The area only need to note that between time t1 and t2, the fund required to support a given throughput is a function of underlying the stock from which the throughputx ˙i is bioproductivity. The productivity is determined among derived could have changed. other things by the extant technology and can change with technological progress. Productivity is ultimately What is indeed comparable across time periods (and a function of ecosystem health, and past damages to across different regions) is departure from optimal scale the ecosystem undermines productivity. The footprint (or in some cases other benchmark measures). Consider methodology compares the land area available with δ(t2)−δ(t1) the metric Cδ = where δ(t) is the departure the land area required to support a given quantum of δ(t1) from optimal scale at time — δ(t) = S (t) − S ∗(t) throughput. where S ∗(t) is the optimal scale at time t. The metric Cδ is strictly dimensionless because departures from The area used in footprint accounting is measured in optimal scale are comparable even in the qualitative ‘global hectares.’ This abstract unit of area represents dimension.14 a weighted average of the different biologically pro- ductive areas on the surface of the earth. In figure -3, the ecological footprint methodology ‘translates’ any throughput, x˙(t) into a corresponding area measured in 4. Dimensional Analysis of Ecological Footprint abstract global hectares, a(t) where t is a simple time subscript to keep track of the temporal dimension. An Having developed a framework for dimensioned important point to note is that the transfer function metrics in the fund-flux space, we show that this heuristic that we have used here applies to both the framework is useful in evaluating the power of various source-side and the sink-side but with an important biophysical metrics. In this paper, we identify specific difference. “Sustainability” is built into how the transfer lacunae in the footprint methodology. In particular we show that ecological footprint does not consistently account for the relationship between the stock-flow space and the fund-flux space. x˙(t) Ecological a(t) Footprint The concept of ecological footprint, first developed by [25] and further refined by [28], measures “how much biologically productive land and water area a given population occupies to produce all the resources Figure 3: Ecological Footprint as a Transfer Function it consumes and to absorb its waste, using prevailing technology.” Much has been written about ecological function of figure - 3 computes a(t) on the sink-side. footprint and we do not propose to present a review of For example in computing the land area required for ab- footprint the literature but instead directly delve into sorbing the combustion products of burning fossil fuels, the conceptual apparatus of ecological footprint that is it is assumed that all the emitted carbon is terrestrially 15 most relevant for our purposes here. In figure - 3, we absorbed.16 On the source-side, no such assumption is present the central idea behind ecological footprint. The possible. For example, continuing with the fossil fuel ecological footprint methodology is best understood as burning example, consider the case of oil. There is a a transfer function. The basic step in calculating eco- finite stock of oil and increasing the land-area (global logical footprint is to convert every throughput on both hectares) set aside to drill oil out of the ground is not going to alter the stock of oil that can be recovered. While the footprint accounting includes non-renewable 14 For a detailed discussion on optimal scale and deviations from sources, it excludes waste throughput for which no sink optimal scale, see [17, 18]. [19] develops a general framework for benchmark measures of which optimal scale is a particular example. 15For example see [2], [5], [12], and [24]. [30] looks at how the footprint addresses the “scale imperative.” The widely cited [31] used 16In this section, we use ‘sustainability’ in the very limited stock- the footprint accounting framework to determine if the global econ- flow sense. In particular, the ecological footprint assumes that sink- omy is in overshoot. side throughput equals absorption (˙xo = Yˆsi). 9 capacity exists. For example toxic waste stream for We are now ready to define the global hectare, Gi(t) cor- which there are no known biophysical sinks (Yˆsi = 0) responding to the actual productive area, Pi(t) are excluded from the footprint accounting framework. The footprint calculations currently only include only Gi(t) = Pi(t)Ei(t) (25) those components of the aggregate throughput that is It is important to note that among other things Ei and potentially sustainable at least on the sink-side. Fossil Pi are driven by the extant technology. A technologi- fuel burning that contributes to nearly half of the global cal progress could make a new land or water area to be footprint is assumed to be potentially sustainable — all counted in Pi that influences yields, and thus Ei. For the carbon from fossil fuel burning can potentially be the earth as a whole, the global hectares are normal- absorbed by terrestrial sinks. ized such that the sum-total of of global hectares is the same as the total hectares of actual productive area. This We begin by looking at how the output of the transfer scaled global hectare, A is defined such that function, global hectares or gha is defined. We follow the notation used by the latest footprint methodology k k presented in [32]. The surface of the earth is divided Ai(t) = Pi(t) = H(t) (26) into k (for 2004, k = 9) broad aggregate categories Xi=1 Xi=1 of land/water area (crop land, pasture, forest, fisheries, and for each productive category, Ai(t) is given by: etc.) Let P (t) be the total “bioproductive” land area i k available of type i in time period t. Pi(t) is measured Pi(t) in standard hectares (not the abstract global hectare).  i=1  Ai(t) = Gi(t) P (27)  k  “Bioproductive land” has sustainability as well as an an-    Gi(t) thropocentric bias built into the definition of Pi. Only  i=1  the “usable portion of biomass [or equivalent] that can  P  An important thing to recollect is that in our discussion be renewably harvested and is valuable to people” is of the ecological footprint transfer function, each of the considered [32, emphasis added]. H(t) is the sum-total quantities measured has a time subscript, t to indicate of bioproductive land available at a given time, t such that the total productive area and therefore the standard- that: ized global hectare changes every year but for any given k year, t these two are the same for the earth as a whole H(t) = P (t) (22) i and is illustrated in figure - 4, reproduced from [32]. Xi=1 H(t) is measured in hectares but we need to convert hectares into global-heactares. This is achieved by defining a “equivalence factor,” Ei(t) corresponding to each of the k categories of productive areas.

bi(t) Ei(t) = (23) b¯(t) where bi is the productivity of the area of type i, cur- rently measured by the “suitability index” published by FAO and b¯(t) is the average productivity for a given year17: k bi(t)Pi(t) = b¯(t) = i 1 (24) P k Pi(t) i=1 Figure 4: Quantity of global hectares (Ai) and actual hectares (Pi) P for 2001. Area is measured in billions of hectares. Note that while , 17The suitability index is a discrete ranking of of different land Ai Pi, Ai = Pi = H. Chart reproduced from [32, p.11] types based on the physical characteristic of the land as well as the P P end use of the land. It takes into account physical characteristics of the land such as soil-type, slope of the land, climatic conditions, etc. 4.0.1. Dimensional Analysis For a detailed account of this standard methodology to determine land productivity, see [6, 7, 8] and FAO’s 1998 report titled World reference The definition of Gi(t) is dimensionally consistent base for soil resources. in the stock-flow space but not in the fund-flux space. 10 Equation - 25 can be written as:

k Pi(t)  i=1  Gi(t) = bi(t)Pi(t) P (28)  k     bi(t)Pi(t)  i=1   P  k   The sum i=1 Pi is defined in the stock-flow space but not inP the fund-flux space. In the fund-flux space, each of the individual bioproductive areas, Pi k have different dimensions, say acrei. Thus i=1 Pi involves adding k items with different dimensionsP – acre1 + acre2 + ... acrek. Each of the terms in this sum is a fund, and arithmetic summation of funds and has no real physical meaning, unlike an arithmetic addition of stocks. Thus the global hectare of type - i defined by equation - 25 is an abstract fund corresponding to an Figure 5: Six Components of Ecological Demand. On the vertical abstract stock – abstract because there is no physical axis, global hectares use year-specific global average bioproductivity. Chart reproduced from [31, p.9270] Gi(t). Defining the central fund in abstract terms helps comparison across regions with vastly varying real physical funds (Pi). While contributing to the power of the footprint methodology, this abstract aggregation space. In terms of stock-flow space, overshoot signi- is also, as we will see, the methodology’s primary fies a stock that has grown beyond the carrying capacity. drawback as well. To illustrate the problem with how the footprint framework aggregates different human activities, 4.1. Overshoot in the Fund-Flux Space consider the largest contributor to the total ecological As we discussed above, the primary effect of eco- demand – fossil fuel burning. Fossil fuel burning logical overshoot that is of economic interest is the results in increased concentration of atmospheric reduction in the regenerative capacity of natural capital, carbon that drives the greenhouse effect driven climate and even a possible collapse. The pertinent question to change. Estimates about what concentration should be ask here is: what specific overshoot is captured by the considered an “overshoot” vary from 300 ppm to 450 footprint methodology? Policy response to any over- ppm. Beyond this overshoot concentration, carbon in shoot is contingent on being able to identify specific the atmosphere is over the ‘carrying capacity’ and the ecosystem funds that are in overshoot. To motivate fund (which regulates global climate) suffers damage. this exercise, consider the pictorial representation of While empirically the abstract overshoot measured in how the footprint methodology calculates ecological terms of global hectares in the footprint framework may demand on the global scale. Figure - 5 presents coincide with an actual overshoot in physical terms, the time-series for the six principal components that (measured in atmospheric concentration of carbon in make up the global ecological demand. The energy ppm), there is no theoretical reason to support such component is obtained by calculating the land area a correlation. The footprint methodology asks the that would be needed to terrestrially absorb all the question: how much land area is required to terrestrially carbon emitted from fossil fuel burning. The sum absorb all the carbon from fossil fuel burning? The total of the productive area demanded by all six major answer to the preceding question is then compared with categories of human activity currently exceeds the total the bioproductive area actually available. Thus all that available global productive area by about 20%. The the footprint exercise can conclude is that a system is footprint methodology therefore concludes that the in overshoot in terms of bioproductive area availability global ecosystem is in overshoot. The aggregate global that may empirically coincide with some real physical biophysical system is meaningful only in the stock-flow overshoot. Even with this theoretical problem, the space. As we showed earlier, this aggregation is not footprint only answers a yes/no question: is the system dimensionally consistent in the fund-flux space, and yet in overshoot? The footprint does not say anything overshoot is essentially a phenomenon of the fund-flux about the exact nature of the overshoot. Reducing 11 carbon concentration from 550 ppm to 500 ppm will determines the number of elemental throughput that get still leave the system in overshoot but the nature of the counted in Tn. The choice of n impacts the construction overshoot at those two concentrations is likely going to of the fund, Nn. be different. The footprint methodology cannot discern between the two states of the world. Biophysical metrics can continue to make a strong contribution to the sustainability discourse. However, As a more dramatic illustration of the problem, the sotck-fund representation developed here suggests consider a thought experiment where the five other that dimensional properties of aggregate metrics like components of footprint, are reduced to zero. The the ecological footprint ought to be an important theo- footprint methodology will conclude that there is retical consideration. Any aggregation of information enough biocapacity for terrestrial absorption of carbon involves abstraction – and aggregating on the complex from fossil fuel burning. The effect of this thought ex- fund-flux space is no exception. As a matter of praxis periment on the actual physical ecosystem is contingent the analysis here underscores the need for sustainability on the state of the world before all other components science to work at the local-level where aggregation of footprint were reduced to zero. If the actual physical in the fund-flux space would be theoretically more system was already in overshoot, we know that it will meaningful. take over a century before the actual climate system driven by atmospheric carbon concentration is below An important consequence of the fund-flux rep- overshoot. The snapshot view (and as we discussed resentation developed here is that it lends itself to a earlier the footprint only gives a snapshot view) of more democratic allocation of ecosystem services. the footprint methodology would have erroneously Traditional methods of contingent valuation have concluded that the climate system was below over- relied on determining shadow prices for each of the shoot. This is only a thought experiment as it is clearly constituent elements of Nn. Not only is this cum- impossible in reality to burn any fossil fuel without bersome and fraught with technical difficulties, it some additional throughput. However this illustration amounts to allocating the fund-flux space (which is a nevertheless points to the theoretical problems with pure public good in most cases) on the basis of ability aggregation in the fund-flux space. Similar arguments to pay (one-dollar one-vote) rather than on a more could be made for other components of the footprint as democratic (one-person one-vote) basis [18]. The well. explicit analytical representation of the fund developed here can be used to modify extant contingent valuation techniques. Instead of consumers ‘voting’ on the value Nn 5. Object Lessons of individual constituents of (with the complex task of aggregating individual shadow prices being The framework for dimensional analysis in the fund- relegated to a central authority), citizens can decide the flux space described above inspires a certain humility value of the entire fund in one single social-political when dealing with the fund-flux space. While aggregate process. 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