Preferences exhibiting local nonsatiation

Question

I'm self studying my way through Varian's Microeconomic Analysis right now and am having trouble answering this question:

Consider preferences defined over the nonnegative orthant by $(x_1, x_2) \succ (y_1, y_2)$ if $x_1+ x_2 < y_1+ y_2$. Do these preferences exhibit local nonsatiation? If these are the only two consumption goods and the consumer faces positive prices, will the consumer spend all of his income? Explain.

The solution manual says the preferences are locally non satiable except at (0,0) but I don't know why that is.

any help would be greatly appreciated.

Answer 1

Since we are working in $\mathbb{R}_+$ you can't go below (0,0). So consider the point (0,0). Are we guaranteed a point around it that is preferred? So using the notation you used, let $(x_1,x_2)=(0,0)$, then any other point $(y_1,y_2)$ where $y_1$ or $y_2$ is positive will be worse than $(0,0)$. So we have a satiation point at $(0,0)$.

Now consider anywhere else. We can also go lower by reducing either coordinate so there is always a better point until we reach both points =0.