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A consumer has an endowment vector $w$; at prices $p$ his demand for the first good exceeds his endowment; $x_1^+(p; pw)>w_1$ then a small increase of $p_1$ will lower his utility.

I was discussing this with a friend and he believes that it's based on the walras law. Prices go up, demand goes down and converges to walras law equilibrium in order to find the $z(p)$ vector which makes excess demand 0, price goes up, excess demand goes down, walras eq. will be satisfied and this is optimum so the utility cannot decrease. (something along those lines as far as I remember)

But I'm more concerned that it has to do with the Income and Substitution effect more than the Walras law. But we both agree that it won't lower the utility level. However, I'm kind of stuck on the math proof behind it.

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    $\begingroup$ Walras' law says that excess demand is zero for every relevant (say, strictly positive) price, not just an equilibrium price. It is equivalent to consumers always spending their whole income. $\endgroup$ – Michael Greinecker Nov 30 '20 at 20:45
  • $\begingroup$ So, it shouldn't be based on Walras law you mean? $\endgroup$ – Ana Ellis Nov 30 '20 at 21:52
  • $\begingroup$ It should, as far as Walras law amounts to a consumer speding their whole income. If a consumer does not spend their whole income to begin with, the can still consume the same bundle if one price is slightly raised. So they would not be worse off then. $\endgroup$ – Michael Greinecker Nov 30 '20 at 21:58
  • $\begingroup$ This question was posted earlier also that was closed. There you sought proof that small increase in $p_1$ would lower utility. Here it is unclear what mathematical proof you are looking for? That the utility will not decrease? $\endgroup$ – Dayne Dec 1 '20 at 4:38
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That was tricky. The idea is as follows: First, under standard assumptions, demand is continuous. If you change prices a little bit, demand will not change a lot. In particular, if your excess demand for the first good was initially strictly positive, it will still be strictly positive for small changes in price. Now, the new bundle after the price was slightly raised was affordable at the old price with some money left to spare on more stuff. So the old bundle must be at least as good as the new bundle with some additional stuff on top and, therefore, be better than the new bundle.

Let's make this formal: So let $p=(p_1,p_2,\ldots,p_n)$ and $p'=(p_1+\epsilon,p_2,\ldots,p_n)$ with $\epsilon>0$ and we assume that $x_1^*(p,pw)>w_1$ and $x_1^*(p',p'w)>w_1$. Let $x=(x_1,\ldots,x_n)=x^*(p',p'w)$. We first show that $px<pw.$ Indeed, $$p'x=(p_1+\epsilon)x_1+ p_2 x_2+\cdots+p_nx_n\leq(p_1+\epsilon)w_1+p_2 w_2+\cdots+p_nw_n.$$ $$(p_1+\epsilon)(x_1-w_1)+ p_2 x_2+\cdots+p_nx_n\leq p_2w_2+\cdots+p_nw_n.$$ $$p_1(x_1-w_1)+\epsilon(x_1-w_1)+ p_2 x_2+\cdots+p_nx_n\leq p_2w_2+\cdots+p_nw_n.$$ Therefore, $$px=p_1x_1+ p_2 x_2+\cdots+p_nx_n\leq p_1 w_1+p_2w_2+\cdots+p_nw_n-\epsilon~\underbrace{(x_1-w_1)}_{>0}<pw.$$ Now, we need an assumption that guarantees that a consumer is better off if they spend all their income. Then we know that the consumer is better off under $x^*(p,pw)$ in which they would spend their whole income of $pw$ instead of choosing the bundle $x$, which costs only $pw-\epsilon(x_1-w_1)$. An assumption that guarantees it is optimal to spend the whole income is exactly what guarantees Walras' law.

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  • $\begingroup$ "So the old bundle must be at least as good as the new bundle with some additional stuff on top and, therefore, be better than the old bundle." I think you meant new in the end of the sentence, right? $\endgroup$ – Dayne Dec 1 '20 at 4:43
  • $\begingroup$ "An assumption that guarantees it is optimal to spend the whole income is exactly what guarantees Walras' law.". Here we don't really need to invoke Walras' law (which is more suitable when there are more than one consumers trading). Assumption of local nonsatiation is sufficient to ensure that consumer exhausts her income. $\endgroup$ – Dayne Dec 1 '20 at 5:53
  • $\begingroup$ See this for the above comment. $\endgroup$ – Dayne Dec 1 '20 at 7:16
  • $\begingroup$ @Dayne Thanks for pointing out the typo. Lns implies Walras' law. $\endgroup$ – Michael Greinecker Dec 1 '20 at 8:08
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    $\begingroup$ Yes that is right. But since you mentioned that we need "An assumption that guarantees it is optimal to spend the whole income..". Lns is the assumption, Walras' law is the derived result based on it. BTW, +1 for the brilliant solution!! I have bookmarked it for reference :) $\endgroup$ – Dayne Dec 1 '20 at 8:22

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