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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?

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    $\begingroup$ Hi, could you please clarify what the symbols mean and which textbook you are referring to? $\endgroup$ – Student Sep 25 '19 at 5:36
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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 $x(p,1)$.

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    $\begingroup$ 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 $p'a(p)=0$. $\endgroup$ – Bertrand Sep 25 '19 at 16:14
  • $\begingroup$ 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. $\endgroup$ – Rainroad Sep 25 '19 at 20:21
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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 ...

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