Von Bortkiewicz Formulae

No doubt one of the most remarkable achievements in index-number theory is the linking of the relative gap between the Laspeyres and Paasche volume and price index numbers for a commodity aggregate, with the correlation between the quantity and price relatives of the single commodities comprised in that aggregate. As is so often the case with remarkable achievements, the underlying idea of this particular one, which is due to von Bortkiewicz (1923), is extraordinarily simple. The essential link between the index-number gap and the correlation between price and quantity relatives is the weighted covariance of the latter. Quite generally, the covariance of two associated variables x and y is defined as the mean-cross-product of the associated values of those variables measured in deviation from their means x ¯ $$\bar x$$ and ȳ. But in case each pair of associated values of the variables x and y is assigned a weight w i the covariance of x and y should be weighted as well. As such it becomes C o v w ( x , y ) = ∑ w i ( x i − x ¯ ) ( y i − y ¯ ) ∑ w i = ∑ w i x i y i − x ¯ ∑ w i y i − y ¯ ∑ w i x i + x ¯ y ¯ ∑ w i ∑ w i = ∑ w i x i y i ∑ w i − ∑ w i x i ∑ w i ⋅ ∑ w i y i ∑ w i $$\begin{gathered} Co{v_w}\left( {x,y} \right) = \frac{{\sum {{w^i}\left( {{x^i} - \bar x} \right)\left( {{y^i} - \bar y} \right)} }}{{\sum {{w^i}} }} \hfill \\ \quad \quad \quad \quad \;\, = \frac{{\sum {{w^i}{x^i}{y^i}} - \bar x\sum {{w^i}{y^i}} - \bar y\sum {{w^i}{x^i}} + \bar x\,\bar y\sum {{w^i}} }}{{\sum {{w^i}} }} \hfill \\ \quad \quad \quad \quad \;\, = \frac{{\sum {{w^i}{x^i}{y^i}} }}{{\sum {{w^i}} }} - \frac{{\sum {{w^i}{x^i}} }}{{\sum {{w^i}} }} \cdot \frac{{\sum {{w^i}{y^i}} }}{{\sum {{w^i}} }} \hfill \\ \end{gathered}$$