In this problem set, we will walk through the steps to show the few equations we have in the lecture.

In the lecture $($Lecture $2,$ p$14),$ we have set up two utilities of the money holder $(U_M)$ and producer $(U_p)$ as:

$U_M=(1-M)xA(u+U_p-U_M)dots(1)$

$U_p=(1-M)xy(u-c)+(M)xA(U_M-U_p-c)dots(2)$

where $M$ is the share of people with money, $x$ is the probability of person i want person $j\text{'}$s item. $y$ is the probability of person $j$ want person $i\text{'}$s item, given i want $j.u$ is the utility of consuming a good one likes, $c$ is the cost of transporting goods $($except money$)$

Question $1$

Assume $c=0,$ subtract off eq$(2)$ from eq$(1),$ show that $U_M-U_P=\frac{(1-M)x(A-y)u}{1+xA}.$

$($Tip: Take the difference $U_M-U_P,$ break down the $(u+U_p-U_M)$ in eq$(1)$ into $(u)+(U_p-U_M),$ then regroup all the terms with $U_M-U_p$ and take them to the left$-$hand$-$side of the equation$)$ Help me solve this step by step so i can learn