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!

  • $\begingroup$ en.wikipedia.org/wiki/Engel_curve $\endgroup$
    – heh
    Feb 3, 2020 at 17:43
  • $\begingroup$ 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. $\endgroup$
    – Ethan Mark
    Feb 4, 2020 at 7:22
  • $\begingroup$ 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. $\endgroup$
    – heh
    Feb 4, 2020 at 15:20
  • $\begingroup$ 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. $\endgroup$
    – Ethan Mark
    Feb 4, 2020 at 18:39


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