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.

  • 2
    $\begingroup$ 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. $\endgroup$ – Walrasian Auctioneer Sep 2 '20 at 0:52
  • $\begingroup$ 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)? $\endgroup$ – a.shirazian Sep 2 '20 at 10:47

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