<p>I don't really have a background in economics. I'm trying to understand the relationship between these two terms, if they are related at all. The definitions that I understand are here:</p>
<p><strong>Substitutable preferences</strong> (generally see this in matching theory, eg in <a href="https://pubsonline.informs.org/doi/abs/10.1287/moor.13.4.619" rel="nofollow noreferrer">Blair 1988</a>, <a href="https://www.jstor.org/stable/1911460?casa_token=UQshCLgj938AAAAA%3AfNFuu2VzcQ02ARHG3qyVkcWi2fFFdTM4f9XwBy0jg4qIfAVIadddvCmNbpyJ8pCQCuk7uZoBSNiyfNU09SLsrbwB2ljciCUpF0hUB523tQ4tvFksNZBx" rel="nofollow noreferrer">Roth 1984</a>): Let <span class="math-container">$S$</span> be a base set of items. If <span class="math-container">$C:2^S\to2^S$</span> is a substitutable choice function, then if <span class="math-container">$a\in C(T)$</span>, we also have <span class="math-container">$a\in C(T'\cup \left\{a\right\})$</span> for all <span class="math-container">$a\in T'$</span>.</p>
<p><strong>Gross substitutes</strong> (<a href="https://www.jstor.org/stable/1913392?casa_token=MgX5NDRX8jIAAAAA%3AfyH-q4dwfhYA6zJRT7RdZQvYbpKe1jORwopfVPMDv-ow90zzLgXtw302mzcaewiGUBTeLy9ND6n4cbYPL2GpPFZeOJmhYqJug15DYAjnvwuGHmbEKZtA" rel="nofollow noreferrer">Kelso and Crawford 1982</a>): many definitions, one of which is: Let <span class="math-container">$S$</span> be a base set of items. If <span class="math-container">$v:2^S\to\mathbb R$</span> is a valuation function that satisfies gross substitutes and <span class="math-container">$D$</span> is the demand correspondence, then if <span class="math-container">$T\in D(p)$</span> and <span class="math-container">$p'\ge p$</span>, then <span class="math-container">$\left\{a\mid p'_a=p_a\right\}\cap T\subseteq T'$</span> for some <span class="math-container">$T'\in D(p')$</span>.</p>
<p>Are these supposed to be the same thing, in different models? Like, I can sort of see how substitutable preferences are kind of like saying, if <span class="math-container">$T'\subseteq C(T)$</span> and you increase the prices of items in <span class="math-container">$C(T)\setminus T'$</span> to infinity (which effectively removes them from the offered set of items <span class="math-container">$T$</span>), then you still demand <span class="math-container">$T'$</span>. Are gross substitutes valuations a subset of substitutable preferences? As in, when you fix a gross substitutes valuation function <span class="math-container">$v$</span> and a price vector <span class="math-container">$p$</span>, this induces a choice function <span class="math-container">$C$</span>, which satisfies substitutability?</p>
<p>Edit: also, where does net substitutes come into all of this? I'm not familiar with the term at all really, but I've seen it mentioned occasionally. It also seems to be a demand theory concept like gross substitutes? Is it comparable to substitutable preferences at all?</p>

CC BY-SA 4.0
2024-01-09T16:35:23.063Z
curly braces $\left\{ \right\}$ in MathJax are \left\{ and \right\},
Giskard
CC BY-SA 4.0
2024-01-09T16:37:59.737Z
Also, can you provide a bit of context, like where are you getting these definitions from?
Giskard
<mathematical-economics><utility><consumer-theory>
Substitutable preferences vs. gross substitutes over indivisible items
AWhite
# Substitutable preferences vs. gross substitutes over indivisible items

(Asked by: username_0 on 2024-01-09)

I don't really have a background in economics. I'm trying to understand the relationship between these two terms, if they are related at all. The definitions that I understand are here:

**Substitutable preferences** (generally see this in matching theory, eg in [Blair 1988](https://pubsonline.informs.org/doi/abs/10.1287/moor.13.4.619), [Roth 1984](https://www.jstor.org/stable/1911460?casa_token=UQshCLgj938AAAAA%3AfNFuu2VzcQ02ARHG3qyVkcWi2fFFdTM4f9XwBy0jg4qIfAVIadddvCmNbpyJ8pCQCuk7uZoBSNiyfNU09SLsrbwB2ljciCUpF0hUB523tQ4tvFksNZBx)): Let $S$ be a base set of items. If $C:2^S\to2^S$ is a substitutable choice function, then if $a\in C(T)$, we also have $a\in C(T'\cup \left\{a\right\})$ for all $a\in T'$.

**Gross substitutes** ([Kelso and Crawford 1982](https://www.jstor.org/stable/1913392?casa_token=MgX5NDRX8jIAAAAA%3AfyH-q4dwfhYA6zJRT7RdZQvYbpKe1jORwopfVPMDv-ow90zzLgXtw302mzcaewiGUBTeLy9ND6n4cbYPL2GpPFZeOJmhYqJug15DYAjnvwuGHmbEKZtA)): many definitions, one of which is: Let $S$ be a base set of items. If $v:2^S\to\mathbb R$ is a valuation function that satisfies gross substitutes and $D$ is the demand correspondence, then if $T\in D(p)$ and $p'\ge p$, then $\left\{a\mid p'\_a=p\_a\right\}\cap T\subseteq T'$ for some $T'\in D(p')$.

Are these supposed to be the same thing, in different models? Like, I can sort of see how substitutable preferences are kind of like saying, if $T'\subseteq C(T)$ and you increase the prices of items in $C(T)\setminus T'$ to infinity (which effectively removes them from the offered set of items $T$), then you still demand $T'$. Are gross substitutes valuations a subset of substitutable preferences? As in, when you fix a gross substitutes valuation function $v$ and a price vector $p$, this induces a choice function $C$, which satisfies substitutability?

Edit: also, where does net substitutes come into all of this? I'm not familiar with the term at all really, but I've seen it mentioned occasionally. It also seems to be a demand theory concept like gross substitutes? Is it comparable to substitutable preferences at all?

**Tags:** mathematical-economics, utility, consumer-theory

> **Comments:**
> **username_1**: curly braces $\left\{ \right\}$ in MathJax are \left\{ and \right\},  
> **username_1**: Also, can you provide a bit of context, like where are you getting these definitions from?  

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## 0 answers

