Price and cross price Elasticity of demand The question is: Prove or disprove, if the price elasticity of a demand of a good is unitary then the cross price Elasticity of demand of that good is zero.

Any hints on how to approach this?
$P_XX+P_YY=M$

Differentiating w.r.t. $P_X$,

$\frac{\partial X }{\partial P_X}P_X+X+\frac{\partial Y}{\partial P_X}P_Y=0$

Divide the whole term by $X$, considering $X\neq 0$

$\varepsilon _X+1+\frac{\partial Y}{\partial P_X}\frac{P_Y}{X}=0$

$\varepsilon _X=1$(Given) 

$2+\frac{\partial Y}{\partial P_X}\frac{P_X}{Y}(\frac{P_Y}{X}\frac{Y}{P_X})=0$

Here, $$P_X,P_Y,X,Y&gt; 0$$(ASSUMPTION)

Therefore, Cross- Price elasticity is $-2$ (some positive no.) and not $0.
So, this statement is not true.