<p>The only function whose derivative is constant is an affine function. Hence, <span class="math-container">$x(p,w)$</span> is of the form <span class="math-container">$wx(p,1)+a$</span> for some constant <span class="math-container">$a$</span>. We also know that <span class="math-container">$x(p,w)\cdot p=w$</span>. Hence,</p>

<p><span class="math-container">$$w\sum_ix_i(p,1)p_i+a\sum_jp_j=w$$</span></p>

<p>As <span class="math-container">$\sum_ix_i(p,1)p_i=1$</span> it must be that <span class="math-container">$a=0$</span>. </p>

<p>The Engles curve fixes <span class="math-container">$p$</span> and plots <span class="math-container">$x(p,w)$</span> as a function of <span class="math-container">$w$</span>. In the case of <span class="math-container">$x(p,w)=w\cdot x(p,1)$</span> its just a linear function in <span class="math-container">$w$</span> with slope <span class="math-container">$x(p,1)$</span>.</p>

CC BY-SA 4.0
2019-09-25T16:14:17.487Z
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
CC BY-SA 4.0
2019-09-25T20:21:05.787Z
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
CC BY-SA 4.0
2019-09-25T12:54:49.717Z
2019-09-25T12:54:49.717Z
guest12382
<p>The function <span class="math-container">$x(p,1)$</span> is constant in <span class="math-container">$w$</span>. If you integrate it wrt <span class="math-container">$w$</span>, you end up with a straight line (wrt <span class="math-container">$w$</span>), whose equation is ...</p>

CC BY-SA 4.0
2019-09-25T08:06:13.537Z
2019-09-25T08:18:11.070Z
2019-09-25T08:18:11.070Z
Bertrand
<p>I see in the textbook that the Engel curve will be straight if <span class="math-container">$D_wx(p,w) = x(p,1)$</span> but it's not immediately clear to me why this is the case. Could someone kindly explain to me? </p>

CC BY-SA 4.0
2019-09-25T05:36:55.297Z
Hi, could you please clarify what the symbols mean and which textbook you are referring to?
Student
<microeconomics>
Why is Engel curve a straight ray through the origin if $D_wx$ = x(p, 1)$?
Rainroad
# Why is Engel curve a straight ray through the origin if $D_wx$ = x(p, 1)$?

(Asked by: username_0 on 2019-09-25)

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?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Hi, could you please clarify what the symbols mean and which textbook you are referring to?  

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## 2 answers

### Answer by username_2 on 2019-09-25. Score: 0 | Accepted answer

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 $x(p,1)$.

> **Comments:**
> **username_3**: 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 $p'a(p)=0$.  
> **username_0**: 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.  

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### Answer by username_3 on 2019-09-25. Score: 1

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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