<p>I'm not sure there are enough assumptions here to move forward, in particular I would think you would need some conditions on your function $h$, at the very least. It's pretty easy to show that $g(\cdot,\cdot)$ will have increasing differences, which implies the single-crossing property for $g$ (this can be quickly seen by letting $h$ be identically zero). But I don't see how this is enough, since single-crossing doesn't imply supermodularity. Maybe I'm missing something, though.</p>

CC BY-SA 3.0
2016-10-08T06:49:33.123Z
2016-10-09T02:16:21.410Z
2016-10-09T02:16:21.410Z
Jay
<p>Suppose I denote $$f(x,t)=g(x,t)+h(x)$$ where $X\in \mathbb{R}$, $t\in [0,1]$ and $h(x)$ is any function.</p>

<p>I want to prove that if the <a href="https://en.wikipedia.org/wiki/Single_crossing_condition" rel="nofollow">single-crossing property</a> holds in $f(x,t)$, then $g(x,t)$ must be <a href="https://en.wikipedia.org/wiki/Supermodular_function" rel="nofollow">supermodular</a> over $X\times T$.</p>

<p>I know in general supermodular means single-crossing but the converse is not true, but my guess adding a random function $h(x)$ should guarantee single-crossing property will show supermodularity. </p>

<mathematical-economics><comparative-statics>
Converse in supermodularity and single-crossing
Jacky
# Converse in supermodularity and single-crossing

(Asked by: username_0 on 2016-10-05)

Suppose I denote $$f(x,t)=g(x,t)+h(x)$$ where $X\in \mathbb{R}$, $t\in [0,1]$ and $h(x)$ is any function.

I want to prove that if the [single-crossing property](https://en.wikipedia.org/wiki/Single_crossing_condition) holds in $f(x,t)$, then $g(x,t)$ must be [supermodular](https://en.wikipedia.org/wiki/Supermodular_function) over $X\times T$.

I know in general supermodular means single-crossing but the converse is not true, but my guess adding a random function $h(x)$ should guarantee single-crossing property will show supermodularity.

**Tags:** mathematical-economics, comparative-statics


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## 1 answers

### Answer by username_1 on 2016-10-08. Score: 2

I'm not sure there are enough assumptions here to move forward, in particular I would think you would need some conditions on your function $h$, at the very least. It's pretty easy to show that $g(\cdot,\cdot)$ will have increasing differences, which implies the single-crossing property for $g$ (this can be quickly seen by letting $h$ be identically zero). But I don't see how this is enough, since single-crossing doesn't imply supermodularity. Maybe I'm missing something, though.


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