On Absolute and Relative Change
OON ABSOLUTE AND RELATIVE CHANGE
SILVAN BRAUEN, PHILIPP ERPF AND MICHA WASEM
Abstract.
Based on an axiomatic approach we propose two related novel one-parameter families of indicators of change which put in a relation classical indica-tors of change such as absolute change, relative change and the log-ratio.
Keywords:
Absolute Change, Relative Change, Indicator of Change, Log-ratio
JEL Classification:
C02 1.
Introduction
One of the most basic concept in statistics is change (take e.g. the change of a quan-tity in time). Usually, change is either expressed in absolute or relative terms. Aswe will show below, already the interpretation of such most basic indicators mightbe difficult: For a fixed choice of a measurement unit, absolute and relative changemight be limited in comparing changes at different scales (see Example 2.1 below)and it could be desirable to find a suitable indicator of change which combines theinformation obtained from absolute and relative change. Based on a relaxation of anaxiomatic characterization of relative change introduced by [T¨ornqvist et al., 1985,p. 43] (see Section 3), we will introduce a one-parameter family f λ of indicators ofchange which is particularly well-behaved with respect to a change of measurementunits and which includes absolute and relative change as special cases.In another direction it is revealed that relative change has certain conceptual de-ficiencies like its lack of additivity and antisymmetry (see [T¨ornqvist et al., 1985],[Wetherell, 1986] and [Schurer, 1990]), which can be remedied by replacing it witha log-ratio which seems to appear for the first time in [McAlister, 1879] (see also[Aitchison, 1982] and [Aitchison, 1983]). We will introduce a new antisymmetricand additive one-parameter family of indicators F λ which interpolates between ab-solute change and the log-ratio and we will show how f λ and F λ are interlinked thus Date : December 1, 2020. a r X i v : . [ ec on . T H ] N ov S. BRAUEN, P. ERPF & M. WASEM providing a general framework for absolute change, relative change and the log-ratio.As applications we show how f λ and F λ allow to meaningfully compare changes atdifferent scales (see Example 2.1) and how they uncover a simple relationship be-tween marginal functions and economic elasticity (see Example 5.1).Throughout the article, will denote the set of (strictly) positive real numbers by R + ,absolute change by abs( x, y ) := y − x and relative change by rel( x, y ) := ( y − x ) /x ,( x (cid:54) = 0). 2. Exhibition of the Problem
We will now illustrate possible limitations of absolute and relative change whenanalyzed alone.
Example
Take the example case of a company selling a good through five different sales chan-nels I to V. The amount of units sold at a past time (past value) is denoted by x and the one at present time (present value) by y . Suppose the company measuresthe following numbers: Channel Past Value x Present Value y abs( x, y ) rel( x, y )I 10 20 10 100%II 500 570 70 14%III 140 210 70 50%IV 35 70 35 100%V 80 135 55 68.75% (1) Considering absolute change, it is impossible to distinguish between II andIII although III is better than II in relative terms. In absolute terms, thechannels I and IV have rather low growth while their relative growth iscomparably large in contrast to II and III with a high absolute growth evenif their relative growth is comparably small.(2) Considering relative change, I and IV perform equally well, although IV hasa higher absolute change.(3) Neither absolute nor relative change allow to meaningfully compare changesat different scales (i.e. is I better than II? Or: which of the sales channels isbest?) How could one compare channel V to the other channels?The indicator f λ we construct below depends on a number λ ∈ R and – as we willshow below – is an interpolation of absolute and relative change provided λ ∈ [0 , λ = (see Remark 3.2 (2)). In this case theindicator is given by f / : R → R , f / ( x, y ) = ( y − x ) / √ x. The values of f / forall sales channels are listed below: N ABSOLUTE AND RELATIVE CHANGE 3
Sales Channel Past Value x Present Value y f / ( x, y )I 10 20 3.16II 500 570 3.13III 140 210 5.92IV 35 70 5.92V 80 135 6.15 The upshot is that all the values become comparable using a single number (III andIV are equally good) and V is rated best without being absolutely nor relativelybest. 3.
Construction of f λ In [T¨ornqvist et al., 1985, p. 43], an indicator of relative change is characterized bya function r : R → R satisfying(1) r ( x, y ) = 0 ⇐⇒ x = y ,(2) r ( x, y ) ≶ ⇐⇒ x ≶ y ,(3) r is continuous and y (cid:55)→ r ( x, y ) is increasing.(4) r ( Cx, Cy ) = r ( x, y ) for all C > r ( x, y ) = abs( x, y ). Ouridea is to replace the scaling invariance (4) by a relaxed relative scaling invariance (see below). Furthermore, since y (cid:55)→ abs( x, y ) and y (cid:55)→ rel( x, y ) are affine linear,we include affine linearity as an axiom too. As we will show in the sequel, our setof axioms will determine a family of indicators which is unique up to a positivemultiplicative constant.3.1. Axioms. (1)
Affine Linearity in the Second Argument.
A map f : R → R is saidto be affine linear in the second argument if f ( x, y ) = m ( x ) y + b ( x ) , where m and b are values which may still depend on x .(2) Naturality.
We call f natural, if f is continuous and moreover f ( x, y ) ≶ x ≷ y and f ( x, x ) = 0 for all x > f assigns a positive (negative)number to growth (decrease) and zero to stagnation).(3) Relative Scaling Invariance. f is relative scaling invariant, if it behavesrelatively invariant under a change of scale, this is, for all C > x, y ) , (¯ x, ¯ y ) ∈ R the equation(3.1) f ( x, y ) · f ( C ¯ x, C ¯ y ) = f (¯ x, ¯ y ) · f ( Cx, Cy )should hold true. This means that the measurement units in which x and y are measured do not matter in a relative sense.3.2. Construction.
We have the following theorem:
S. BRAUEN, P. ERPF & M. WASEM
THEOREM 3.1
If a map f : R → R satisfies the axioms (1)-(3), then it holds that f ( x, y ) = C · y − xx λ for some C > and λ ∈ R .Proof. The affine linearity condition together with the zero assignment for stagnationimplies that f ( x, x ) = m ( x ) x + b ( x ) = 0 , which forces b ( x ) = − m ( x ) x . Hence, thegeneral form of f will be f ( x, y ) = m ( x )( y − x ) . Since (3.1) holds for all
C > x, y ) and (¯ x, ¯ y ), it follows that(3.2) f ( Cx, Cy ) = g ( C ) f ( x, y )for some continuous function g : R + → R . Hence (see [Efthimiou, C., 2010, p. 163]) g ( C ) is a solution to the Power Cauchy Equation which has the only continuoussolutions g ( C ) = C µ , µ ∈ R . Replacing this solution in equation (3.2), we deducethat m ( Cx ) = C µ − m ( x ), which implies that m is a homogeneous function of degree µ −
1. According to
Euler’s Homogeneous Function Theorem , m satisfies the ordinarydifferential equation m (cid:48) ( x ) − µ − x m ( x ) = 0 with the solution m ( x ) = C x − λ for someconstant C and λ := 1 − µ . Putting everything together we obtain f ( x, y ) = C · y − xx λ . and the sign of the constant, C >
0, is determined by the naturality assumption.This finishes the proof. (cid:3)
We will henceforth set C = 1 (see Remark 3.2 (2) below) and define f λ ( x, y ) := y − xx λ . Remark
We now list a few properties of f λ :(1) Formal Structure of a Cobb-Douglas Function.
Observe that we mightwrite f λ ( x, y ) = rel( x, y ) λ · abs( x, y ) − λ and therefore interpret f λ formally asa Cobb-Douglas function Y ( L, K ) = AL β K α with constant returns to scale,where A = 1, L = rel( x, y ), K = abs( x, y ), α = 1 − λ and β = λ .(2) Generalizing Absolute/Relative Change and Calibration.
The nor-malization choice C = 1 is justified by the observation that f ( x, y ) = abs( x, y ) and f ( x, y ) = rel( x, y ) . This indicates in how far f λ is a generalization of absolute and relativechange. The number λ allows to intentionally weigh between the two andtherefore serves as a calibration. If one assigns the same value to two pairs N ABSOLUTE AND RELATIVE CHANGE 5 ( x, y ) and (¯ x, ¯ y ) such that x (cid:54) = y , ¯ x (cid:54) = ¯ y and x (cid:54) = ¯ x , the value of λ can bedetermined using the formula λ = ln (cid:16) ¯ y − ¯ xy − x (cid:17) ln (cid:0) ¯ xx (cid:1) . A natural choice for being in the middle between absolute and relative is λ = , the value chosen in Example 2.1.(3) Relative Scaling Invariance.
If the unit in which x and y are measuredis u, then the unit of f λ ( x, y ) is u − λ . In this way f λ cannot directly beinterpreted but it useful in comparing different pairs ( x, y ) and (¯ x, ¯ y ) sincethe unit-free quotient f λ (¯ x, ¯ y ) /f λ ( x, y ) will be independent of the choice ofthe unit u according to the relative scaling invariance (3.1).(4) From Differences to Quantities.
Using the structure of the Cobb-Douglasfunction above and replacing the differences abs( x, y ) and rel( x, y ) by (ab-solute and relative) quantities, one obtains a generalization of absolute andrelative quantities: For an absolute quantity y (cid:62) x > x, y ) (cid:55)→ (cid:16) yx (cid:17) λ y − λ = yx λ , which recovers the absolute quantity provided λ = 0 and the relative one if λ = 1. This function is linear in the second argument and it satisfies therelative scaling invariance (axiom (3) above).4. An Antisymmetric, Additive Variant of f λ Common criticism on relative change includes its failure of antisymmetry , i.e. gen-erally rel( x, y ) (cid:54) = − rel( y, x ) and its failure of additivity i.e. generallyrel( x, y ) + rel( y, z ) (cid:54) = rel( x, z )(see [T¨ornqvist et al., 1985], [Wetherell, 1986] and [Schurer, 1990]). It is observed in[T¨ornqvist et al., 1985], that the log-ratio r ( x, y ) = ln( y/x ) is the unique indicatorof relative change that is antisymmetric, additive and normed , i.e. the linearizationof y (cid:55)→ r ( x, y ) around x is given by rel( x, y ). It is therefore natural to ask if anindicator F λ : R → R exists which is antisymmetric, additive and normed in thesense that the linearization of y (cid:55)→ F λ ( x, y ) around x is given by f λ ( x, y ). Thefollowing theorem answers this question affirmatively: THEOREM 4.1
The indicator F λ ( x, y ) = y − λ − x − λ − λ , if λ (cid:54) = 1ln( y/x ) , if λ = 1 S. BRAUEN, P. ERPF & M. WASEM is the unique antisymmetric and additive indicator such that the linearization of y (cid:55)→ F λ ( x, y ) around x is given by f λ ( x, y ) .Proof. We start by deducing a certain structure of F λ from differentiating the addi-tivity property F λ ( x , x ) + F λ ( x , t ) = F λ ( x , t ) ∀ x , x , t ∈ R + with respect to t .Hence ∂ y F λ ( x , t ) = ∂ y F λ ( x , t ) for all x , x , t ∈ R + which implies that ∂ y F λ onlydepends on t and in particular it holds that(4.1) ∂ y F λ ( x, t ) = ∂ y F λ ( t, t )for all ( x, t ) ∈ R . It follows that F λ ( x, y ) is additively separable (it decompoasesinto a sum of two single variable functions depending on x and y respectively). Thelinearization of y (cid:55)→ F λ ( x, y ) around x should equal f λ , so we obtain F λ ( x, x ) + ∂ y F λ ( x, x )( y − x ) = y − xx λ and since F λ ( x, x ) = 0 by antisymmetry of F λ , it follows that(4.2) ∂ y F λ ( x, x ) = 1 x λ and hence F λ ( x, y ) = (cid:90) yx ∂ y F λ ( x, t ) d t (4.1) = (cid:90) yx ∂ y F λ ( t, t ) d t (4.2) = (cid:90) yx t λ d t = y − λ − x − λ − λ , if λ (cid:54) = 1ln( y/x ) , if λ = 1 . It is readily checked that F λ ( x, y ) + F λ ( y, z ) = F λ ( x, z ) ∀ x, y, z ∈ R + . (cid:3) Remark
We now list a few properties of F λ :(1) Observe that F λ puts absolute change ( λ = 0) and the log-ratio ( λ = 1) intoa natural relation sincelim λ → y − λ − x − λ − λ = ln( y/x ) . Furthermore, F λ inherits the naturality and the relative scaling invariancefrom f λ (but not the affine linearity in the second argument).(2) Using a Taylor expansion of y (cid:55)→ F λ ( x, y ) around x we obtain F λ ( x, y ) = f λ ( x, y ) + n (cid:88) k =2 ( − k +1 Γ( λ + k − k ! x k + λ − Γ( λ ) ( y − x ) k + O (( x − y ) n +1 ) , where Γ denotes Euler’s Gamma function and we note that the truncatedseries also satisfies the relative scaling invariance and the naturality property.Truncation after the linear term and and using the Lagrange form of the N ABSOLUTE AND RELATIVE CHANGE 7 remainder, we obtain a quadratic global bound on the difference between F λ and f λ . | F λ ( x, y ) − f λ ( x, y ) | (cid:54) λ · ( y − x ) min { x, y } λ . (3) If x = 1 (which corresponds to a choice of measurement units), then thefunction y (cid:55)→ F λ (1 , y ) equals a Box-Cox transformation of parameter 1 − λ (see [Box & Cox, 1964]).In this case we have F (1 , y ) = y, F (1 , y ) = ( (cid:112) y − , F (1 , y ) = 2( √ y − F (1 , y ) = ln y (see Figure 1). Figure 1.
The plot shows the graphs of F (1 , y ), F (1 , y ) (solid), F (1 , y ) (dotted) and F (1 , y ) (dashed) for y ∈ (0 , Application
Every quantity which involves either absolute or relative changes can naturally begeneralized using f λ (or F λ ): Application (Relation between Marginal Functions and Elasticity)The indicators f λ and F λ uncover a relationship between marginal functions andelasticity: For a differentiable economic function g ( x ), the associated marginal func-tion is given by its derivative g (cid:48) ( x ). The corresponding elasticity function is definedby ε g ( x ) = lim y → x ( g ( y ) − g ( x )) /g ( x )( y − x ) /x = g (cid:48) ( x ) · xg ( x ) , which is the limit of a quotient of relative changes. Replacing the relative changesin this definition by f λ , we obtain a generalized elasticity function ε λg ( x ) = lim y → x ( g ( y ) − g ( x )) /g ( x ) λ ( y − x ) /x λ = g (cid:48) ( x ) · (cid:18) xg ( x ) (cid:19) λ S. BRAUEN, P. ERPF & M. WASEM which puts the marginal function ( λ = 0) and the classical elasticity ( λ = 1) into anatural relation. 6. The Choice of λ The choice of λ depends heavily on the context. If one wishes to interpolate “sym-metrically” between absolute and relative change, the following conceptual evidencereveals λ = as a good choice: In order to fix λ we require that a scaling ofthe absolute change by a factor C > x, y ) and (
Cx, Cy ) satisfy abs(
Cx, Cy ) = C abs( x, y ) andrel( Cx, Cy ) = rel( x, y ), hence the absolute change scales with C whereas the relativechange is preserved. The pairs ( x, y ) and (cid:0) xC , y − x + xC (cid:1) satisfyabs (cid:16) xC , y − x + xC (cid:17) = abs( x, y )rel (cid:16) xC , y − x + xC (cid:17) = C rel( x, y ) , hence the relative change scales with C whereas the absolute change is preserved.The parameter λ will now be chosen such that f λ satisfies f λ ( Cx, Cy ) = f λ (cid:16) xC , y − x + xC (cid:17) . This equation reduces to C − λ = C λ and hence λ = . We will add a concreteexample to illustrate the choice. As a reference pair we choose (1 ,
2) which satisfies f λ (1 ,
2) = 1 for every choice of λ . The pair (2 ,
4) has an absolute change which istwice as big as the one of (1 ,
2) but rel(1 ,
2) = rel(2 , ,
2) but twice its relative change is given by (cid:0) , (cid:1) .Then f λ (2 ,
4) = 2 − λ , f λ (cid:0) , (cid:1) = 2 λ and 2 − λ = 2 λ iff λ = . Hence λ is chosen insuch a way that doubling either the absolute or the relative change amounts to thesame result.In the perspective of f as a Cobb-Douglas function we note that the choice λ = has the following consequence: The marginal rate of substitution at a point ( L, K )of a Cobb-Douglas function Y ( L, K ) = AL β K α is given by(6.1) ∂ L Y ( L, K ) ∂ K Y ( L, K ) = βL β − K α αL β K α − = βα · KL .
In our case, whenever L = rel( x, y ), K = abs( x, y ), α = 1 − λ and β = λ , the latterexpression in (6.1) equals λ/ (1 − λ ) · x . This quantity equals the past value x exactlywhen λ = . N ABSOLUTE AND RELATIVE CHANGE 9 Conclusion
We have shown that the indicator of change f λ can be singled out from some simpleand natural axioms. In this way, the basic concepts of absolute and relative changeare identified as special cases of a more general quantity. Building upon this indi-cator, a new antisymmetric and additive indicator F λ is constructed, which relatesabsolute change to the log-ratio. Our analysis therefore contains the main result of[T¨ornqvist et al., 1985] as a special case.Moreover, we have obtained a novel generalized elasticity function which uncovers arelationship between two classical concepts in economics – marginal functions andelasticity. Acknowledgements.
The authors would like to thank Thomas Mettler for pointingout the formal structure of f λ as a Cobb-Douglas function. References [Aitchison, 1982] John Aitchison, (1982).
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Rivella AG,Neue Industriestrasse 10, CH-4852 Rothrist [email protected]
Institute for Research on Management of Associations,Foundations and Co-operatives (VMI),University of Fribourg,Boulevard de P´erolles 90, CH-1700 Fribourg [email protected]