The property of compensation can be studied when a process (or a portion of it) is represented by sub-indicators that are aggregated into a derived indicator. In a nutshell, if variations in two or more sub-indicators may compensate each other — without any variation in the derived indicator—then the derived indicator fulfils the property of compensation.
The property can be formalized as follows. Let us consider two sub-indicators (I1 and I2), which are aggregated into a derived indicator (D).
If (first condition) a variation in I1 (i.e., ΔI1) always causes a variation in D (i.e., ΔD), ceteris paribus,
and if (second condition) there exist a variation in I2 (i.e., ΔI2) that compensates for the previous ΔD (i.e., ΔD ¼ 0),
then D fulfills the property of compensation and a substitution rate between I1 and I2 can be univocally determined.
Compensation is a typical property of additive and multiplicative aggregation models.
Returning to the formal definition, we note that the first condition is automatically satisfied if D is strictly monotonic with respect to I1 (sufficient but not necessary condition).
As for the substitution rate, it is defined as the portion (ΔI1) of the first sub-indicator (I1) that one should give up in order to obtain a variation (ΔI2) in the second sub-indicator (I2), keeping the derived indicator (D) constant. The substitution rate can be also seen as an analytic function connecting ΔI1 to ΔI2. Depending on the type of aggregation of sub-indicators, the substitution rate may be constant or it may depend on the so-called operating point, i.e., the initial values of sub-indicators I1 and I2. Examples 4.29 and 4.30 show how to calculate the substitution rate in an exact or approximate way. These examples also show that constant substitution rates are generally preferable as they make the mutual compensation of sub-indicators more controllable.
Further comments on the property of compensation follow:
• For simplicity, the previous definition of compensation is referred to two sub-indicators only. With appropriate adjustments, the same property can be extended to three or more sub-indicators.
• There may be derived indicators with more relaxed forms of compensation, than the one formalized previously. In general, we can define as weak compensation a form of compensation in which variations in two (or more) sub-indicators may compensate each other, not necessarily producing any variation in the derived indicator.
• The fact that an indicator follows the property of (weak) compensation or it does not follow it at all may depend on (i) the aggregation model of the derived indicator or (ii) any constraints in the sub-indicator scales. These concepts are clarified by the following four examples.
