I'm puzzled why, in the following optimal-choice problem, good $x$ depends on price $p_y$ but the opposite is not true. Should not the sign of $\frac{\partial x}{\partial p_y}$ be the same of $\frac{\partial y}{\partial p_x}$?
Optimal-choice problem: given prices $p_x$ and $p_y$ and income $m$ find the optimal demand function for $x$ and for $y$: \begin{equation} \max_{x,y} \{x\cdot(y+6)\} \qquad \text{s.t. } p_x x+p_y y \leq m \tag{1} \end{equation} after imposing $MRS_{xy}=p_x/p_y$ and substituting back in the budget constraint I find: \begin{equation} x = \frac{m + 6p_y}{2p_x} \qquad\qquad y = \frac{m}{2p_y}-3 \tag{2} \end{equation} so that $sign\left(\frac{\partial x}{\partial p_y}\right) = sign\left( \frac{6}{2p_x}\right)\neq sign\left(\frac{\partial y}{\partial p_x}\right) = 0$.
What I would like to understand better:
- How can this happen? I remember having read somewhere in MWG that the price-derivative matrix should be (even!) symmetric. Can you provide me with some explanation and/or some reference? Alternatively, is there any mistake in the calculations?
- Provided the situation described above can happen [i.e. $sign\left(\frac{\partial x}{\partial p_y}\right)\neq sign\left(\frac{\partial y}{\partial p_x}\right)$], I would be very happy if you can come up with some "real world" examples of that.