In this paper after this introduction we deal

In this paper, after this introduction, we deal with the oversight just mentioned. An alternative treatment is given to the existence of a proper solution involving a Hypercube Graph, which we call, by analogy with the magic square, the magic hypercube. In order for this paper to be self-contained, Section 2 revisits the configuration of the magic square. Section 3 uncovers the oversight regarding the concept of revised MS and its consequences. Section 4 introduces the concept of Magic Hypercube Graph and Section 5 shows some numerical results of the new formulation using the same data in Medrano-B and Teixeira (2013) for Brazil and Chile. Section 6 offers some concluding thoughts.
The original magic square re-formalization As we have mentioned in the introduction of this article, Medrano-B and Teixeira (2013) revised the original version of Magical Square correcting a mathematical problem this version presented. The diagram is conceived in its four cardinal directions (N, E, S and W) indicated by γ, τ, φ and ζ. All four variables (axes) are originally drawn at different scales expressed in percentages and the adjacent vertices are joined by straight lines. The fact that the scales of these variables are not uniform implies that the original area of such figure has no useful meaning. To construct the revised MS all four scales will be normalized so that each of the new variables assumes values between and a constant value β and the maximum (the wonderland or ideal economy) area of the MS is 1. From these conditions it nisoldipine follows immediately that the maximum value β of the normalized variables is . Given the original boundaries on our four variables, as stated in the previous section, it follows that the new normalized variables (identified with a prime superscript) arewhere the new “primed” variables will obey the following restrictions: In summary the Medrano-B & Teixeira magic square index is given by the expression: In other words the area of the inner magic square enclosed by the full square, the latter corresponding to the wonderland economy where all variables take their maximum value, equal to (see Fig. 2 below). Some observations are worthwhile at this juncture. Firstly, let us note that the term “wonderland economy” is a rhetorical expression and does not imply that we consider, for instance, a zero inflation rate (or, for that matter, a zero rate of unemployment) as ideal. We are well aware that a zero inflation rate is not a healthy feature of a modern economy as widely accepted by most economists (see, for instance Mankiw, 2009). Secondly, the choice of extreme values for the four variables considered is somewhat arbitrary but, nevertheless, as Kaldor and Schiller did in their time, we need to select some values in order to construct the magic square index. Notice in particular that an inappropriate choice of extreme values may lead to a misleading comparison of the performance of different countries. For instance, any country with a trade balance at its minimum value would have an index equal to zero and thus a worse performance than any other country with a strictly positive value for its index, regardless of how the other macroeconomic variables compare. Moreover, its performance could not be distinguished from that of any other country with identical (minimum) trade balance. These objections can, of course, be easily lifted by an appropriate choice of extreme values.
A not inconsequential oversight The main interest of the quantitative index introduced by Medrano-B and Teixeira (2013), the magic square, and its advantage over the simple geometrical illustration that preceded it, reside in its ability to proceed with inter-temporal explorations and cross country comparisons. The cited paper indeed compares Brazil and Chile over the period 2004–2011, claiming that, albeit this index showed Chile in a better light than Brazil, the dynamic examination revealed a better situation for Brazil than for Chile. On the other hand, one would expect any such comparison, based as it is on measurements of a collection of variables, to be independent of the particular ordering of those variables or, alternatively, the places they have on the planar axes. And here lies the oversight of the paper in question. As we shall see in a moment, the quantitative index introduced therein depends crucially on the order in which the variables are listed. More importantly, this order has also an effect on the results of comparison exercises, leading to undesirable inconsistencies.