# I'm trying to understand a proof for First order stochastic dominance

Here's the theorem, consisting of 2 statements:

The equivalence is proven with the aid of this:

There are 2 things I don't understand about it. Firstly, why $$U(x).H(x)|\infty, 0 = 0$$. And secondly, to get to $$[F(x) - G(x)] \leq 0$$ from that proof. I know the definition of integration from parts, but I'm no hero with it and I haven't been able to find a way to statement 2 on my own. I figure part of it must consist of replacing $$\int U(x)h(x)dx$$ with $$-\int U'(x)h(x)dx$$, which would give me $$\int U'(x)h(x)dx \leq 0$$, but that's as far as I'm able to get. Please to try to plain in your answer, I don't find it easy to deal with jargon and deep abstraction.

[...] integration by parts [...]

Recall the formula for integration by parts: $$\int_a^b u\,\mathrm dv = uv\vert_a^b - \int_a^b v\,\mathrm du$$. Now let $$u=U$$ and $$v = H$$ (so that $$\mathrm dv = \mathrm dH = h\,\mathrm dx$$. Then \begin{align} \int_0^\infty \underbrace{U(x)}_{u}\,\underbrace{h(x)\mathrm dx}_{\mathrm dv} &= \underbrace{U(x)}_u\,\underbrace{H(x)}_{v}\bigg\vert_0^\infty - \int_0^\infty \underbrace{H(x)}_{v}\,\underbrace{\mathrm dU(x)}_{\mathrm du} \\ &=U(x)H(x)\bigg\vert_0^\infty - \int_0^\infty H(x)\, U'(x) \mathrm dx \end{align} This yields the middle line in your second picture.

why $$U(x)H(x)\vert_0^\infty=0$$?

Note that $$H(x)=[F(x)-G(x)]$$, and that $$F(0)=G(0)=0$$ and $$\lim_{x\to\infty}F(x)=\lim_{x\to\infty}G(x)=1$$ because $$F$$ and $$G$$ are cumulative distribution functions with support on $$[0,\infty)$$. Therefore $$H(0)=\lim_{x\to\infty}H(x)=0$$.

to get to $$[F(x)−G(x)]≤0$$

Given the assumption that utility is non-decreasing, i.e. $$U'(x)\ge0$$, it follows that $$H(x)=F(x)-G(x)\le0$$ for every $$x$$ ensures $$-\int U'(x)H(x)\mathrm dx \ge 0$$ thereby proving that statement (2) implies statement (1).

• Does that last line completely follow? I can think of functions that are negative in places whose integrals are positive
– dm63
Commented Oct 18, 2022 at 10:20
• @dm63: Thanks for the catch. The direction of implication should have been reversed. Commented Oct 18, 2022 at 14:25