MWG_3D4_C, why the solution seems in reverse?

I'm doing exercises of Chapter3 of MWG, there's a problem that I don't understand (I didn't figure out the solution manual either...).

It is about exercise 3.D.4, the full statement of the exercise is as follows:

Let $$(−∞,∞)×R^{(L−1)}_+$$ denote the consumption set, and assume that preferences are strictly convex and quasilinear. Normalize $$p_1=1$$.

(a) Show that the Walrasian demand functions for goods $$2,...,L$$ are independent of wealth. What does this imply about the wealth effect of demand for good 1?

(b) Argue that the indirect utility function can be written in the form $$v(p,w)=w+\phi(p)$$ for some function $$\phi(⋅)$$.

(c) Suppose, for simplicity, that $$L=2$$, and write the consumer's utility function as $$u(x_1,x_2)=x_1+η(x_2)$$. Now, however, let the consumption set be $$R^2_+$$, so that there is a nonnegative constraint on the consumption of the numeraire $$x_1$$. Fix prices $$p$$, and examine how the consumer's Walrasian demand changes as wealth w vary. When is the nonnegativity constraint on the numeraire irrelevant?

Question In the solution in part c we reach to the conclusion that the consumer spends all the wealth on $$x_2$$ and spend what's left on $$x_1$$. Shouldn't this be in reverse?

p.s. I'm thinking about the graph that maximum utility is achieved when we spend all the money on good 1 on nothing on good 2.

• Notice by (a) that the consumer wants good 2 up to a certain amount regardless of wealth, so she always spends money on good 2 first. Sep 2 '20 at 0:52
• I can't see how (a) implies that consumer buys positive amounts of $x_2$? And also why we should buy a zero amount of $x_1$ according to part (c)? Sep 2 '20 at 10:47