# On Demand Functions and Engel Curves

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!

• en.wikipedia.org/wiki/Engel_curve – heh Feb 3 '20 at 17:43
• 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 Feb 4 '20 at 7:22
• 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 Feb 4 '20 at 15:20
• 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 Feb 4 '20 at 18:39