# Effect of price on utility

A consumer has an endowment vector $$w$$; at prices $$p$$ his demand for the first good exceeds his endowment; $$x_1^+(p; pw)>w_1$$ 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.

• 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 Nov 30 '20 at 20:45
• So, it shouldn't be based on Walras law you mean? – Ana Ellis Nov 30 '20 at 21:52
• 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 Nov 30 '20 at 21:58
• 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 Dec 1 '20 at 4:38

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.