Editing Complementary good (section)

===Proof===
The standard [[John Hicks|Hicks]] decomposition of the effect on the ordinary demand for a good <math>x</math> of a simple price change in a good <math>y</math>, utility level <math>\tau^*</math> and chosen bundle <math>z^* = (x^*, y^*, \dots)</math> is

<math>\frac{\partial f_x(p, \omega)}{\partial p_y} = \frac{\partial h_x (p, \tau^*)}{\partial p_y} - y^* \frac{\partial f_x(p, \omega)}{\partial \omega}</math>

If <math>x</math> is a gross substitute for <math>y</math>, the left-hand side of the equation and the first term of right-hand side are positive. By the symmetry of Mosak's perspective, evaluating the equation with respect to <math>x^*</math>, the first term of right-hand side stays the same while some extreme cases exist where <math>x^*</math> is large enough to make the whole right-hand-side negative. In this case, <math>y</math> is a gross complement of <math>x</math>. Overall, <math>x</math> and <math>y</math> are not symmetrical.