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 $(-\infty,\infty)\times R_+^{L-1}$ 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 $u(x_1,x_2)=x_1+\eta(x_2)$. Now, however, let the consumption set be $R_+^2$, so that there is a nonnegative constraint on consumption of the numeraire $x_1$. Fix prices $p$, and examine how the consumer's Walrasian demand changes as wealth $w$ varies. When is the nonnegativity constraint on the numeraire irrelevant?
My question is about the (c) part: 1. what does it mean? 2. Can anyone explain the solution to it?
The solution to (c) (from the solution manual) is:
The non-negativity constraint is binding if and only if $p_2x_2(p,0)>w$. Note that $x_2(p,0)=(\eta')^{-1}(p_2)$, because $p_1=1$. Hence the constraint is binding if and only if $p_2(\eta')^{-1}(p_2)>w$. If so, the Walrasian demand is given by $x(p,w)=(0,w/p_2)$. Thus, as $w$ changes, the consumption level of the first good is unchanged and the consumption of the second good changes at rate $1/p_2$ with $w$ until the non-negativity constraint no longer binds.