<p>In problems like this, complementarity and substitutability are often defined with respect to the cross-derivatives.</p>

<p>Write $v_i(\cdot)$ for the derivative of $v$ with respect to its $i$th argument. Thus, $v_i(a_1,\ldots,a_n)$ measures how much output increases if $i$ increases his effort slightly.</p>

<p>Now let's think about the derivative of $v_i$ with respect to $a_j$: $v_{ij}(a_1,\ldots,a_n)$. If $v_{ij}&gt;0$ then an extra unit of effort from $i$ yields more output when $a_j$ is high (and vice-versa). We say that $i$'s and $j$'s efforts are complements because extra effort from $j$ makes $i$'s effort more effective (and vice versa).</p>

<p>Conversely, if $v_{ij}&lt;0$ then extra effort from $j$ makes $i$'s effort less <em>effective</em>, and this is often taken as a definition of substitutability.</p>

CC BY-SA 3.0
2016-04-06T19:33:42.677Z
2016-04-06T19:33:42.677Z
Ubiquitous
<p>Suppose $n$ players are in joint production to produce a common output $v(a_1,a_2,...,a_n)$ with their individual actions $a_i$. Does the convexity or concavity of $v(a_1,...,a_n)$ in the actions say something about the substitutability or complementarity of their actions? </p>

CC BY-SA 3.0
2016-04-06T18:45:04.680Z
You mean $v(\cdot)$ is con-something in all players' actions individually, or is it just con-something alltogether? (The second one is stronger.) Also, could you please include your definition for substitutability and complementarity.
Giskard
CC BY-SA 3.0
2016-04-06T19:02:46.873Z
Con-something altogether. As for substitutability and complementarity, for example, if $v = v(a_1 + a_2 + ... + a_n)$ that would be perfectly substitutable efforts and $v = v(a_1a_2...a_n)$ would be complementary. I understand I have still not defined them precisely but I am only interested in the relation between con-something and the notions of substitutability or complementarity.
esperanto
<microeconomics><mathematical-economics>
Substitutability vs complementarity of efforts
esperanto
# Substitutability vs complementarity of efforts

(Asked by: username_0 on 2016-04-06)

Suppose $n$ players are in joint production to produce a common output $v(a\_1,a\_2,...,a\_n)$ with their individual actions $a\_i$. Does the convexity or concavity of $v(a\_1,...,a\_n)$ in the actions say something about the substitutability or complementarity of their actions?

**Tags:** microeconomics, mathematical-economics
> **Comments:**
> **username_1**: You mean $v(\cdot)$ is con-something in all players' actions individually, or is it just con-something alltogether? (The second one is stronger.) Also, could you please include your definition for substitutability and complementarity.  
> **username_0**: Con-something altogether. As for substitutability and complementarity, for example, if $v = v(a_1 + a_2 + ... + a_n)$ that would be perfectly substitutable efforts and $v = v(a_1a_2...a_n)$ would be complementary. I understand I have still not defined them precisely but I am only interested in the relation between con-something and the notions of substitutability or complementarity.  

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## 1 answers

### Answer by username_2 on 2016-04-06. Score: 4 | Accepted answer

In problems like this, complementarity and substitutability are often defined with respect to the cross-derivatives.

Write $v\_i(\cdot)$ for the derivative of $v$ with respect to its $i$th argument. Thus, $v\_i(a\_1,\ldots,a\_n)$ measures how much output increases if $i$ increases his effort slightly.

Now let's think about the derivative of $v\_i$ with respect to $a\_j$: $v\_{ij}(a\_1,\ldots,a\_n)$. If $v\_{ij}>0$ then an extra unit of effort from $i$ yields more output when $a\_j$ is high (and vice-versa). We say that $i$'s and $j$'s efforts are complements because extra effort from $j$ makes $i$'s effort more effective (and vice versa).

Conversely, if $v\_{ij}<0$ then extra effort from $j$ makes $i$'s effort less *effective*, and this is often taken as a definition of substitutability.


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