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A Framework for Dimensional Analysis of Biophysical Metrics ✩,✩✩ 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. Ecological Footprint 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 Herman Daly, 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 sustainability, 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).
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