# Why is Engel curve a straight ray through the origin if $D_wx$ = x(p, 1)$? 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? • Hi, could you please clarify what the symbols mean and which textbook you are referring to? – Student Sep 25 '19 at 5:36 ## 2 Answers 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, $$w\sum_ix_i(p,1)p_i+a\sum_jp_j=w$$ 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 $$x(p,1)$$. • In fact there is no reason why$a$should be constant. It can be a column vector, which varies with$p$and can be denoted by$a(p)$. Adding up then implies$p'a(p)=0\$. – Bertrand Sep 25 '19 at 16:14
• Thanks! I think I was also confused because examples always show the Engel curve on (x2,x1) co-ordinate plane as it traces how x(p,w) changes in response to a change in w, so I was confused as to why if x(p,w) is a linear function of w, the curve is also straight, but now I remember the whole parametric equations of lines. – Rainroad Sep 25 '19 at 20:21

The function $$x(p,1)$$ is constant in $$w$$. If you integrate it wrt $$w$$, you end up with a straight line (wrt $$w$$), whose equation is ...