# What are cross price effects for the case of perfect substitutes?

U$(x_1,x_2)$ = $x_1 +x_2$ in the case of perfect substitutes

If $p_1 < p_2$ then $(x_1^*,x_2^*)$ = $(m/p_1,0)$, if $p_1 > p_2$ then $(x_1^*,x_2^*)$ = $(0,m/p_2)$

Trying to figure out how a change in the price of $p_2$ will affect the demand of of $x_1^*$ when $p_2$ increases but when $p_1 < p_2$, then $dx_1^*/dp_2$ = $d/dp_2(m/p_1) = 0$

And when $p_1 > p_2$ then $dx_1^*/dp_2$ = $d/dp_2(0) = 0$

Wondering how to show using calculus that this increase in $p_2$ so increase the demand of $x_1^*$ so $dx_1^*/dp_2$ should be greater then 0

• Your calculation seems correct. Perhaps you forgot that the demand functions you specify are not differentiable in $p_1 = p_2$ and hence you cannot apply calculus if the increase takes $p_2$ from one side of $p_1$ to the other. – Giskard Oct 21 '15 at 19:43