<p>Note that:
<span class="math-container">$$
f(Ax) = f(a_{11} x_1 + a_{12} x_2, a_{21} x_1 + a_{22} x_2).
$$</span>
So the derivative with respect to <span class="math-container">$x_1$</span> is given by:
<span class="math-container">$$
f_1 a_{11} + f_{2} a_{21}.
$$</span>
Taking the derivative of this with respect to <span class="math-container">$x_2$</span> gives:
<span class="math-container">$$
f_{11} a_{11} a_{12} + f_{12} a_{11} a_{22} + f_{21} a_{21} a_{12} + f_{22} a_{21} a_{22}.
$$</span>
Supermodularity means that this is non-negative. So:
<span class="math-container">$$
\sum_{i = 1}^2 \sum_{j = 1}^2 f_{ij} a_{i1} a_{j2} \ge 0 = \begin{bmatrix} a_{11} &amp; a_{21}\end{bmatrix} \begin{bmatrix} f_{11} &amp; f_{12} \\ f_{21} &amp; f_{22} \end{bmatrix} \begin{bmatrix} a_{12} \\ a_{22} \end{bmatrix}.
$$</span></p>

CC BY-SA 4.0
2024-02-18T09:37:23.193Z
2024-02-18T09:37:23.193Z
tdm
<p>Suppose <span class="math-container">$A$</span> is a <span class="math-container">$2x2$</span> matrix, e.g., <span class="math-container">$A=\begin{vmatrix}
a_{11} &amp; a_{12}  \\
a_{21} &amp; a_{22}  \\
\end{vmatrix}$</span>, and <span class="math-container">${\bf x}=(x_1, x_2)$</span>. Suppose <span class="math-container">$f()$</span> is continuous and twice differentiable.</p>
<p>What does &quot;<span class="math-container">$f(Ax)$</span> is supermodular&quot; mean?</p>
<p>For example, if <span class="math-container">$f(x)$</span> is supermodular, we have the cross partials of <span class="math-container">$f()$</span> with respect to <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> nonnegative.</p>
<p>How can we similarly represent <span class="math-container">$f(Ax)$</span> supermodular in terms of the second partial and cross partial derivatives?</p>
<p>Note: I am interested in the expression for the continuous case and not the general lattice definition if possible.</p>
<p>Thank you for all your help!</p>

<microeconomics><mathematical-economics><optimization><real-analysis>
Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean?
opre
# Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean?

(Asked by: username_0 on 2024-02-16)

Suppose $A$ is a $2x2$ matrix, e.g., $A=\begin{vmatrix}
a\_{11} & a\_{12} \\
a\_{21} & a\_{22} \\
\end{vmatrix}$, and ${\bf x}=(x\_1, x\_2)$. Suppose $f()$ is continuous and twice differentiable.

What does "$f(Ax)$ is supermodular" mean?

For example, if $f(x)$ is supermodular, we have the cross partials of $f()$ with respect to $x\_1$ and $x\_2$ nonnegative.

How can we similarly represent $f(Ax)$ supermodular in terms of the second partial and cross partial derivatives?

Note: I am interested in the expression for the continuous case and not the general lattice definition if possible.

Thank you for all your help!

**Tags:** microeconomics, mathematical-economics, optimization, real-analysis


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## 1 answers

### Answer by username_1 on 2024-02-18. Score: 2 | Accepted answer

Note that:
$$
f(Ax) = f(a\_{11} x\_1 + a\_{12} x\_2, a\_{21} x\_1 + a\_{22} x\_2).
$$
So the derivative with respect to $x\_1$ is given by:
$$
f\_1 a\_{11} + f\_{2} a\_{21}.
$$
Taking the derivative of this with respect to $x\_2$ gives:
$$
f\_{11} a\_{11} a\_{12} + f\_{12} a\_{11} a\_{22} + f\_{21} a\_{21} a\_{12} + f\_{22} a\_{21} a\_{22}.
$$
Supermodularity means that this is non-negative. So:
$$
\sum\_{i = 1}^2 \sum\_{j = 1}^2 f\_{ij} a\_{i1} a\_{j2} \ge 0 = \begin{bmatrix} a\_{11} & a\_{21}\end{bmatrix} \begin{bmatrix} f\_{11} & f\_{12} \\ f\_{21} & f\_{22} \end{bmatrix} \begin{bmatrix} a\_{12} \\ a\_{22} \end{bmatrix}.
$$


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