Effect of price on utility A consumer has an endowment vector $w$; at prices $p$ his demand for the first good exceeds his endowment; $x_1^+(p; pw)&gt;w_1$ 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.
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 $\epsilon&gt;0$
and we assume that $x_1^*(p,pw)&gt;w_1$ and $x_1^*(p',p'w)&gt;w_1$.
Let $x=(x_1,\ldots,x_n)=x^*(p',p'w)$. We first show that
$px&lt;pw.$ 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,
$$px=p_1x_1+ p_2 x_2+\cdots+p_nx_n\leq p_1 w_1+p_2w_2+\cdots+p_nw_n-\epsilon~\underbrace{(x_1-w_1)}_{&gt;0}&lt;pw.$$
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.