I see in the textbook that the Engel curve will be straight if $D_wx(p,w) = x(p,1)$ but it's not immediately clear to me why this is the case. Could someone kindly explain to me?
The only function whose derivative is constant is an affine function. Hence, $x(p,w)$ is of the form $wx(p,1)+a$ for some constant $a$. We also know that $x(p,w)\cdot p=w$. Hence,
As $\sum_ix_i(p,1)p_i=1$ it must be that $a=0$.
The Engles curve fixes $p$ and plots $x(p,w)$ as a function of $w$. In the case of $x(p,w)=w\cdot x(p,1)$ its just a linear function in $w$ with slope $x(p,1)$.