# Proof that utility is nonincreasing in prices

The following is a proof that the indirect utility function is nonincreasing in prices, but I can't understand the last step. How do they conclude that $$v(p_1, y) \ge$$ from the previous reasoning?

Consider $$p_0\ge p_1$$ and let $$x_0$$ solve the utility maximisation problem when $$p = p_0$$. Because $$x_0\ge 0$$, $$(p_0 − p_1) · x_0 ≥ 0$$. Hence, $$p_1·x_0 ≤ p_0·x_0 ≤ y$$, so that $$x_0$$ is feasible for the utility maximisation problem when $$p = p_1$$. We conclude that $$v(p_1, y) ≥ u(x_0) = v(p_0, y)$$.

• You can take a look at the envelope theorem(and its applications) for an alternate proof. – superhulk Oct 6 '18 at 11:20

That follows $$v(p_1,y)$$ being the highest utility one can get from a bundle affordable given prices $$p_1$$ and income $$y$$. Let $$x_1$$ be such a utility maximizing bundle. Then $$p_1\cdot x_1\le y$$ and $$u(x_1)\ge u(x)$$ for all $$x$$ such that $$p_1\cdot x\le y$$. In particular, $$u(x_1)\ge u(x_0)$$ since $$p_1\cdot x_0\le y$$ (this is what the argument you have shows.) Therefore, $$v(p_1,y)=u(x_1)\geq u(x_0)=v(p_0,y).$$ Intuitively, if prices are lower, one can buy from a richer set of options, and with more options one can always get at least as high utility.