# MWG_3D4_C, why the solution seems in reverse?

Economics Asked by a.shirazian on January 4, 2021

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.