<p>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.</p>
<p>Let's make this formal: So let <span class="math-container">$p=(p_1,p_2,\ldots,p_n)$</span> and <span class="math-container">$p'=(p_1+\epsilon,p_2,\ldots,p_n)$</span> with <span class="math-container">$\epsilon&gt;0$</span>
and we assume that <span class="math-container">$x_1^*(p,pw)&gt;w_1$</span> and <span class="math-container">$x_1^*(p',p'w)&gt;w_1$</span>.
Let <span class="math-container">$x=(x_1,\ldots,x_n)=x^*(p',p'w)$</span>. We first show that
<span class="math-container">$px&lt;pw.$</span> Indeed,
<span class="math-container">$$p'x=(p_1+\epsilon)x_1+ p_2 x_2+\cdots+p_nx_n\leq(p_1+\epsilon)w_1+p_2 w_2+\cdots+p_nw_n.$$</span>
<span class="math-container">$$(p_1+\epsilon)(x_1-w_1)+ p_2 x_2+\cdots+p_nx_n\leq p_2w_2+\cdots+p_nw_n.$$</span>
<span class="math-container">$$p_1(x_1-w_1)+\epsilon(x_1-w_1)+ p_2 x_2+\cdots+p_nx_n\leq p_2w_2+\cdots+p_nw_n.$$</span>
Therefore,
<span class="math-container">$$px=p_1x_1+ p_2 x_2+\cdots+p_nx_n\leq p_1 w_1+p_2w_2+\cdots+p_nw_n-\epsilon~\underbrace{(x_1-w_1)}_{&gt;0}&lt;pw.$$</span>
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 <span class="math-container">$x^*(p,pw)$</span> in which they would spend their whole income of <span class="math-container">$pw$</span> instead of choosing the bundle <span class="math-container">$x$</span>, which costs only <span class="math-container">$pw-\epsilon(x_1-w_1)$</span>. An assumption that guarantees it is optimal to spend the whole income is exactly what guarantees Walras' law.</p>

CC BY-SA 4.0
2020-12-01T04:43:35.243Z
"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?
Dayne
CC BY-SA 4.0
2020-12-01T05:53:58.727Z
"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](https://en.wikipedia.org/wiki/Local_nonsatiation) is sufficient to ensure that consumer exhausts her income.
Dayne
CC BY-SA 4.0
2020-12-01T07:16:13.287Z
See [this](https://economics.stackexchange.com/a/18725/24967) for the above comment.
Dayne
CC BY-SA 4.0
2020-12-01T08:08:30.630Z
@Dayne Thanks for pointing out the typo.  Lns implies Walras' law.
Michael Greinecker
CC BY-SA 4.0
2020-12-01T08:22:46.643Z
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 :)
Dayne
CC BY-SA 4.0
2020-11-30T22:19:56.553Z
2020-12-01T08:03:57.370Z
2020-12-01T08:03:57.370Z
Michael Greinecker
<p>A consumer has an endowment vector <span class="math-container">$w$</span>; at prices <span class="math-container">$p$</span> his demand for the first good exceeds his endowment; <span class="math-container">$x_1^+(p; pw)&gt;w_1$</span> then a small increase of <span class="math-container">$p_1$</span> will lower his utility.</p>
<p>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 <span class="math-container">$z(p)$</span> 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)</p>
<p>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.</p>

CC BY-SA 4.0
2020-11-30T20:45:21.067Z
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.
Michael Greinecker
CC BY-SA 4.0
2020-11-30T21:52:02.090Z
So, it shouldn't be based on Walras law you mean?
Ana Ellis
CC BY-SA 4.0
2020-11-30T21:58:29.403Z
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.
Michael Greinecker
CC BY-SA 4.0
2020-12-01T04:38:25.233Z
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?
Dayne
<microeconomics><utility><demand><walrasian>
Effect of price on utility
Ana Ellis
# Effect of price on utility

(Asked by: username_0 on 2020-11-30)

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.

**Tags:** microeconomics, utility, demand, walrasian
> **Comments:**
> **username_1**: 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.  
> **username_0**: So, it shouldn't be based on Walras law you mean?  
> **username_1**: 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.  
> **username_2**: 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?  

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## 1 answers

### Answer by username_1 on 2020-11-30. Score: 6 | Accepted answer

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 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}
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.

> **Comments:**
> **username_2**: "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?  
> **username_2**: "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](https://en.wikipedia.org/wiki/Local_nonsatiation) is sufficient to ensure that consumer exhausts her income.  
> **username_2**: See [this](https://economics.stackexchange.com/a/18725/24967) for the above comment.  
> **username_1**: @Dayne Thanks for pointing out the typo.  Lns implies Walras' law.  
> **username_2**: 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 :)  

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