<p>A consumer has utility function <span class="math-container">$U(x,y)=(x−2)y$</span>, where <span class="math-container">$x≥2$</span> and <span class="math-container">$y≥0$</span>. The price of <span class="math-container">$x$</span> is <span class="math-container">$P_x$</span>, the price of <span class="math-container">$y$</span> is <span class="math-container">$P_y$</span> and the consumer's income is <span class="math-container">$I&gt;2P_x$</span>. (<span class="math-container">$x$</span> and <span class="math-container">$y$</span> do not have to be integers.)</p>

<p>I ended up with the demand functions for <span class="math-container">$x$</span> and <span class="math-container">$y$</span> to be <span class="math-container">$\frac I {2P_x}+1$</span> and <span class="math-container">$\frac I {2P_y}−\frac {P_x}{P_y}$</span> respectively.</p>

<p>The question then gives <span class="math-container">$P_x=1$</span> and <span class="math-container">$P_y=2$</span> and asks me to draw the Engel curve of <span class="math-container">$x$</span>.</p>

<p>Assuming my demand functions are correct (I am still new to the concept of utility maximisation and derivation of demand functions), I have found three points to plot the curve: <span class="math-container">$(3,4)$</span>, <span class="math-container">$(4,6)$</span> and <span class="math-container">$(5,8)$</span>. This would result in a linear Engel curve.</p>

<p>I would like to know if I have gone wrong anywhere along the way, as I have always thought Engel curves to be, as the name suggests, curves. If my answer is correct, may I know what kind of good would have a linear Engel curve?</p>

<p>For example, I do know that Engel curves have a positive and negative gradient for normal and inferior goods respectively. Moreover, if the curve tends towards the x or y-axis, the good is a necessity and a luxury respectively.</p>

<p>If I could get examples for linear Engel curves with positive and negative gradients respectively, that will be great!</p>

CC BY-SA 4.0
2020-02-03T17:43:23.500Z
https://en.wikipedia.org/wiki/Engel_curve
heh
CC BY-SA 4.0
2020-02-04T07:22:03.150Z
I have actually seen that article and note that it mentions something about linear Engel curves, but I do not understand the context it gives, hence my question here.
Ethan Mark
CC BY-SA 4.0
2020-02-04T15:20:06.873Z
Perhaps you can frame your questions with specific reference to some of the points that are giving you trouble.  That may make it easier for people to answer.  One thing I will say though, is that in the context of mathematics, "curves" don't have to be "curved".  What people colloquially know as "lines" are "curves" with "zero curvature", a property that results from having a second derivative that is everywhere zero.
heh
CC BY-SA 4.0
2020-02-04T18:39:25.433Z
Well, actually I am new to this topic of Economics, so while I know that curves do not actually have to be "curves", I am wondering if this applies to Engel curves as well and if so, in what kind of situations would I find Engel curves that are actually lines? The reason why I put this question up is because, as mentioned in my post, when I plot the 3 points that I found, I actually end up with a linear Engel curve, but since I do not know if Engel curves can actually be linear, I am here hoping that someone can verify my work/thought process for me.
Ethan Mark
<utility><consumer-theory><optimization><consumption><demand>
On demand functions and Engel curves
Ethan Mark
# On demand functions and Engel curves

(Asked by: username_0 on 2020-02-03)

A consumer has utility function $U(x,y)=(x−2)y$, where $x≥2$ and $y≥0$. The price of $x$ is $P\_x$, the price of $y$ is $P\_y$ and the consumer's income is $I>2P\_x$. ($x$ and $y$ do not have to be integers.)

I ended up with the demand functions for $x$ and $y$ to be $\frac I {2P\_x}+1$ and $\frac I {2P\_y}−\frac {P\_x}{P\_y}$ respectively.

The question then gives $P\_x=1$ and $P\_y=2$ and asks me to draw the Engel curve of $x$.

Assuming my demand functions are correct (I am still new to the concept of utility maximisation and derivation of demand functions), I have found three points to plot the curve: $(3,4)$, $(4,6)$ and $(5,8)$. This would result in a linear Engel curve.

I would like to know if I have gone wrong anywhere along the way, as I have always thought Engel curves to be, as the name suggests, curves. If my answer is correct, may I know what kind of good would have a linear Engel curve?

For example, I do know that Engel curves have a positive and negative gradient for normal and inferior goods respectively. Moreover, if the curve tends towards the x or y-axis, the good is a necessity and a luxury respectively.

If I could get examples for linear Engel curves with positive and negative gradients respectively, that will be great!

**Tags:** utility, consumer-theory, optimization, consumption, demand

> **Comments:**
> **username_1**: https://en.wikipedia.org/wiki/Engel_curve  
> **username_0**: I have actually seen that article and note that it mentions something about linear Engel curves, but I do not understand the context it gives, hence my question here.  
> **username_1**: Perhaps you can frame your questions with specific reference to some of the points that are giving you trouble.  That may make it easier for people to answer.  One thing I will say though, is that in the context of mathematics, "curves" don't have to be "curved".  What people colloquially know as "lines" are "curves" with "zero curvature", a property that results from having a second derivative that is everywhere zero.  
> **username_0**: Well, actually I am new to this topic of Economics, so while I know that curves do not actually have to be "curves", I am wondering if this applies to Engel curves as well and if so, in what kind of situations would I find Engel curves that are actually lines? The reason why I put this question up is because, as mentioned in my post, when I plot the 3 points that I found, I actually end up with a linear Engel curve, but since I do not know if Engel curves can actually be linear, I am here hoping that someone can verify my work/thought process for me.  

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## 0 answers

