Proof involving a convex set

Question

So, the problem is actually from a microeconomics class. The problem is this:

If preferences are represented by a utility function $u(x,y)=xy$, show that these preferences are convex.

Now in case you don't know, in economics, "convex preferences" means preferences such that the set of preferences that are at least as preferred to some bundle is convex. So basically what this means is I need to show this:

Let $0\le t\le 1$

if $u(x_1,y_1)=u(x_2,y_2)$, then $u(tx_1+(1-t)x_2,ty_1+(1-t)y_2)\ge u(x_1,y_1)$.

so $x_1y_1\le (tx_1+(1-t)x_2)(ty_1+(1-t)y_2)$.

Now, I have tried expanding this out and factoring all kinds of different ways, and I feel like I'm not getting anywhere. Am I going about this incorrectly by trying to expand this? Is there some simpler way? If anyone could give me some kind of hint that would be amazing.

Thanks!

Answer 1

Since $x_1y_1=x_2y_2$, what you want to prove is equivalent to $$tx_1y_1+(1-t)x_2y_2\leqslant(tx_1+(1-t)x_2)(ty_1+(1-t)y_2). $$ Expanding the RHS, one sees that the RHS minus the LHS is $$t(1-t)(x_1y_2+x_2y_1-x_1y_1-x_2y_2)=t(1-t)(x_1-x_2)(y_2-y_1). $$ Using $x_1y_1=x_2y_2$ once again, you know that $x_1>x_2$ implies $y_1<y_2$ and that $x_1<x_2$ implies $y_1>y_2$ hence the last product is always nonnegative. Done.