As mentioned Kaldor s pioneering

As mentioned, Kaldor’s pioneering analysis did not benefit from quantitative nor graphical instruments. This absence was remedied by the introduction of the so-called “Magic Square” (MS), a graphical representation of Kaldor’s approach. According to Dickhaus (2004, p. 354) and others, the credit for this corresponds to Karl Schiller, a German politician and leader of the Social Democratic Party (from 1966 to 1972) who was also Economics Minister of the Federal Republic of Germany. Since the 1970s, economists at OECD began using this instrument, with minor modifications, to deal with the performance of a single country or the comparative performance among a set of nations or regions. Fig. 1 presents a diagram of the MS as it Go 6976 was conventionally used in the 1980s (Bernard et al., 1988). The annual variables considered in this Cartesian plane were: rate of GDP growth (%), trade balance (as percentage of GDP), rate of unemployment (%), and rate of inflation (%). Notice that, as measured from the origin, growth rate is supposed to take values from 0 to 10, trade balance values from −2 to 4, inflation from 10 to 0, and unemployment from 12 to 0 (the latter two variables on an inverted scale, given that higher values are less desirable than lower values). Alas, in such simple representation the authors did not bother with the different scales of the variables and they simply joined the four variables according to the axes. The ranges assigned to the macroeconomic variables are somewhat arbitrary but, for a magic square to be built, clearly some ranges had to be chosen. In addition, the correlation existing between some variables is recognized (e.g. Okun’s Law – unemployment versus real GDP; Phillips Curve – inflation versus unemployment). Medrano-B and Teixeira (2013) realized that such formulation contained a basic mistake since the original area of such figure has no useful meaning due to the non-uniform scales of the axes. To construct an adequate MS all four scales must be redefined to be uniform by normalizing the figure to a unit area. They also pointed out that the performance of any country, given by an area inside the unit square, is drawn not as a square but a diamond shape figure. Such geometric construction allows to quantify the inside figure as a proportion of the unitary MS. As a result, this work introduced a formal indicator, called Index of Economic Welfare. As an application the authors compared the macroeconomic impact of the recent global financial crisis in Brazil and Chile before (2004–2007) and during the crisis (2008–2011). In the last few years a number of articles, presented in international seminars, colloquia and published in journals, took Medrano-B & Teixeira approach as theoretical framework. Thus, Firme and Teixeira (2014) published a macro-econometric analysis focusing on Brazil and another set of countries using this approach; Teixeira et al. (2015) employs the same analytical framework to provide a composite index to measure the overall performance involving both socioeconomic activities and environmental sustainability for China and the USA. Furthermore, Kucera (2012) presents an alternative mathematical formulation of the magic square. During a recent series of seminars at the University of Brasilia, some of our colleagues noticed an important oversight related to the ordering of the variables along the vertices of the Medrano-B & Teixeira MS formulation. Numerical calculations show that alternative orderings of the four variables may produce different values for the index of welfare. We concluded that the previous result based on the original MS were valid only under special circumstances (symmetrical axes). This points out to the need for correction of the formulae upon which the index rests. This oversight is due to a not fully thought-out appreciation of ordering in the geometric transformation of a graph. Actually, as pointed out by Bergson (1959), the scientist tries to fit in all facts in a certain order, but forgets that other orderings are possible and may lead to different outcomes.