<p>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...).</p>
<p>It is about exercise 3.D.4, the full statement of the exercise is as follows:</p>
<p>Let <span class="math-container">$(−∞,∞)×R^{(L−1)}_+$</span> denote the consumption set, and assume that preferences are strictly convex and quasilinear. Normalize <span class="math-container">$p_1=1$</span>.</p>
<p>(a) Show that the Walrasian demand functions for goods <span class="math-container">$2,...,L$</span> are independent of wealth. What does this imply about the wealth effect of demand for good 1?</p>
<p>(b) Argue that the indirect utility function can be written in the form <span class="math-container">$v(p,w)=w+\phi(p)$</span> for some function <span class="math-container">$\phi(⋅)$</span>.</p>
<p>(c) Suppose, for simplicity, that <span class="math-container">$L=2$</span>, and write the consumer's utility function as <span class="math-container">$u(x_1,x_2)=x_1+η(x_2)$</span>. Now, however, let the consumption set be <span class="math-container">$R^2_+$</span>, so that there is a nonnegative constraint on the consumption of the numeraire <span class="math-container">$x_1$</span>. Fix prices <span class="math-container">$p$</span>, and examine how the consumer's Walrasian demand changes as wealth w vary. When is the nonnegativity constraint on the numeraire irrelevant?</p>
<p><strong>Question</strong> In the solution in part c we reach to the conclusion that the consumer spends all the wealth on <span class="math-container">$x_2$</span> and spend what's left on <span class="math-container">$x_1$</span>. Shouldn't this be in reverse?</p>
<p><strong>p.s.</strong> 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.</p>

CC BY-SA 4.0
2020-09-02T00:52:40.500Z
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.
Walrasian Auctioneer
CC BY-SA 4.0
2020-09-02T10:47:47.223Z
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)?
a.shirazian
<microeconomics><utility><preferences>
MWG_3D4_C, why the solution seems in reverse?
a.shirazian
# MWG_3D4_C, why the solution seems in reverse?

(Asked by: username_0 on 2020-09-01)

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.

**Tags:** microeconomics, utility, preferences
> **Comments:**
> **username_1**: 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.  
> **username_0**: 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)?  

---

## 0 answers

