Compensability  and  the  meaning  of  weights

In order to see how the arithmetic and geometric aggregation rules affect the assumed com-pensability, we can derive the rate of substitution for (1) and (2). The formal derivation is shown in appendix 3. The rate of substitution 𝑆_!,! is defined as the marginal utility of 𝐼_! divid-ed by the marginal utility of 𝐼_!. It describes how much 𝐼_! needs to change in order to compen-sate for a given change in 𝐼_!. For the weighted arithmetic mean, the rate of substitution 𝑆_!,!

between any two indicators 𝐼_! and 𝐼_! is given by:

𝑆_!,! = 𝑤_!

𝑤_!      (3) For the weighted arithmetic mean, 𝑆_!,! is equal to the ratio of the weights. This is the reason why the weights have the meaning of trade-off ratios (Munda & Nardo, 2005). Further, the rate of substitution is constant and assumes an infinite elasticity when one indicator is ap-proaching zero. Therefore, deviations from the average by some indicators can be compen-sated by other indicators without losses.

Table 5: Exemplary application of aggregation rules.

Company 𝐼_! 𝐼_! 𝐼_! 𝐶𝐼   𝐶𝐼

………Min-Max……… weighted

arithmetic mean

weighted geometric mean

𝑎_! 1 0 0 .33 0

𝑎_! 0 1 .44 .48 0

𝑎_! .19 .98 1 .72 .57

𝑎_! .4 .4 .4 .4 .4

𝑎_! .15 .15 .9 .4 .27

Own presentation.

In the case of the weighted geometric mean, the rate of substitution is given by:

𝑆_!,! =𝑤_! 𝑤_!

𝐼_!

𝐼_!      (4) The geometric aggregation leads to a rate of substitution proportional to the inverse ratio of the indicators. Subsequently, the elasticity is constantly 1 because the ratio of relative changes does not change. If the 𝐼_! =𝐼_!, the weights are interpreted as trade-off ratios as well. Howev-er, the ratio changes. In the extreme case, if an indicator approaches zero, the rate of substitu-tion approaches infinity. Therefore, deviasubstitu-tions from the average by some indicators lead to lower scores compared to a set of indicator values that is collectively close to the mean. This is also the reason why, if an indicator approaches zero, the value of 𝑆𝑅_! approaches zero as well, if it is defined at all¹⁰. This illustrates the weighted geometric mean’s assumption about the origins of indicators. It requires data measured on the ratio level. After an interval trans-formation, this requires a new interpretation of the origin.

Both functions show, that the weights influence the rate of substitution (Munda & Nardo, 2005). Reversely, this affects the derivation of weights. First, methods need to be used to de-rive trade-offs. Second, these trade-offs need to apply to the indicators. In section 3.4 about the derivation of weights, I referred to variables. However, the real trade-off happens between indicators. The normalization is interposed between the selection of variables and their aggre-gation. In consequence, the weights apply to indicators and this meaning of weights needs to be reflected during the derivation of weights.

The following example illustrates the implications for the derivation of weights. Consider a weighted arithmetic mean after min-max normalization for CO2e and wastewater. If partici-pants assign 𝑤_!!!! = .15 and 𝑤_!! = .3, the rate of substitution would be 𝑆_!!!!,!!= 2 ac-cording to (3). If we take into account the normalization method, this means that being 1%

closer to the best performing company in with respect to CO2e compensates for being .5%

farther away from the best performing company with respect to wastewater. The trade-off ratio does not apply to absolute amounts of CO2e and wastewater anymore. In practice, the required abstractive power of participants leads to difficulties regarding the correct derivation

10 Different conventions exist. Occasionally the geometric mean is not defined for zero.

of weights (Munda, 2008, p. 91). A “theoretical inconsistency” exists, if weights are derived as importance coefficients and are applied in compensatory aggregation rules (Munda &

Nardo, 2005, p. 7).

To conclude the comparison of the weighted arithmetic mean and the weighted geometric mean, the main difference concerns the compensability. While an additive aggregation as-sumes full and constant compensability at all levels, the multiplicative aggregation limits the compensation by reflecting relative changes. It penalizes companies with differences between dimensions, e.g. 𝑎_! in the introductory example, compared to companies like 𝑎_! that perform equally well in all dimensions. Thus, companies with weak performance in some dimensions should be interested in the use of the weighted arithmetic mean. At the same time, marginal increases at a low level are rewarded more than marginal increases at a high level. Conse-quently, the use of the weighted geometric mean implies that companies are supposed to take equal care of all sustainability dimensions and leadership in some dimensions is not sufficient.

One could argue that if the prediction of financial performance is the primary objective, then the arithmetic mean should be used, as financial risks are completely compensatory. If the evaluation of welfare effects is the primary objective, then one may tend to use the geometric mean, particularly if low performance in some dimensions outweighs high performance in other dimensions.

There is a strong argument to assume compensability of environmental indicators in the case of companies. Their economic activities are a manifestation of the division of labor. Division of labor is necessary to use comparative advantages and economies of scale. At the same time, the division of labor leads to unbalanced environmental impacts because the specialization of a company determines the its impacts. For example, along one value chain, forestry operations cause very different environmental impacts than the pulp and paper industry and printing.

This is an inevitable consequence of the division of labor. Non-compensatory approach do not account for that. Thus, it seems justified to use compensatory approaches for the assessment of corporate environmental sustainability.

The discussion is less clear concerning social aspects of sustainability. Can a company with great support mechanisms for minorities at their European headquarters be lax about child labor in their supply chain? A normatively oriented consideration may deny. A purely

eco-nomic approach may estimate the reputation risks of child labor. The same difficulties apply when social and environmental aspects are considered together.

There are three options to account for concerns about compensability. One is the implementa-tion of exclusion criteria, as menimplementa-tioned in chapter 2. These work as lexicographic filters out-side of the composite indicator framework (Munda, 2008, p. 4) and exclude any compensability for those criteria. The second option is increasing the relative weight of the dimensions that are considered more important. This makes it more costly to compensate for one dimension. Lastly, one may choose a geometric aggregation, e.g. the weighted geometric mean. This effectively excludes compensability for dimensions where a company scores zero.

To sum things up, for the methodical choice of the aggregation rule, there is no correct way.

Full compensability is not self-evident. However, restricting compensability by using the weighted geometric mean brings along other implications. For example, the origin is inter-preted as a true zero, which assumes a ratio level of measurement. Both aggregation rules are applicable and the choice of one of them affects the results of the SR.

3.7 Remarks

It is tempting to split the reviewed literature on composite indicators into two general groups:

practitioners (Esty et al., 2005; Galli et al., 2008; Hsu et al., 2013; Krajnc & Glavic, 2005;

Nardo et al., 2005; OECD, 2008) and theorists (Böhringer & Jochem, 2007; Booysen, 2002;

Ebert & Welsch, 2004; Gasparatos et al., 2009; Munda, 2005, 2008, 2012; Parris & Kates, 2003). While theorists insist on a theoretically sound methodology, practitioners have (implic-itly) accepted that many compromises need to be made when multi-dimensional phenomena are aggregated into one measure. The two groups differ concerning the extent to which they accept methodical decisions to be mere value judgments where there is no right or wrong. Of course, this is a rough differentiation. Still, it can be noted that the literature does not agree on acceptable compromises in the construction of composite indicators.

Yet, all authors agree that the methods used and decisions made during the construction of composite indicators need to be transparent to enable the interpretation of the composite indi-cator. Based on this, the next chapter derives transparency criteria by summarizing the crucial parts of the construction.