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 $v(p_1, y) ≥ u(x_0) = v(p_0, y)$.

  • $\begingroup$ You can take a look at the envelope theorem(and its applications) for an alternate proof. $\endgroup$ – 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.

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