<p>Consider a duopoly setting. Firm <span class="math-container">$1$</span> has the following profit equation:
<span class="math-container">$\pi_1(q_1,q_2,v_1)=(a+v_1-q_1-\gamma q_2)q_1 .$</span>
Where <span class="math-container">$a&gt;0$</span> is a utility parameter, <span class="math-container">$q_i\geq0$</span> is the quantity and <span class="math-container">$v_i\geq0$</span> is the quality the of firm <span class="math-container">$i\in1,2$</span>, and <span class="math-container">$\gamma\in(1,0)$</span> is the degree of product differentiation.(assume no cost - irrelevant).</p>
<p>Now, I first want to find the change in <span class="math-container">$\pi_1$</span> when firm <span class="math-container">$2$</span> increases its quantity. Then I calculate
<span class="math-container">$\frac{\partial\pi_1}{\partial q_2}=-\gamma q_1&lt;0$</span>.
Obviously, this means that <span class="math-container">$\pi_1$</span> decreases when <span class="math-container">$q_2$</span> increases.</p>
<p>Arriving to my question:</p>
<p>I want to find the change in <span class="math-container">$\frac{\partial\pi_1}{\partial q_2}$</span> when <span class="math-container">$\gamma$</span> increases. So I take the cross-partial derivative,
<span class="math-container">$\frac{\partial^2\pi_1}{\partial q_2\partial\gamma}=-q_1&lt;0$</span>.</p>
<p>How do I interpret this result?
Does this mean that &quot;lower differentiation (higher <span class="math-container">$\gamma$</span>) weakens the effect of a larger quantity of firm 2 (higher <span class="math-container">$q_2$</span>) on the profits of firm 1 (<span class="math-container">$\pi_1$</span>)?&quot;</p>
<p>This makes no sense to me because just by looking at the profit equation above I can see that higher <span class="math-container">$\gamma$</span> means a stronger effect of <span class="math-container">$q_2$</span> on <span class="math-container">$\pi_1$</span>.</p>
<p>P.S.
Is there a problem with taking the partial derivative w.r.t <span class="math-container">$\gamma$</span> when <span class="math-container">$\gamma$</span> is in an open set?</p>
<p>Thank you.</p>

CC BY-SA 4.0
2021-07-03T23:49:47.040Z
That has to do with the meaning of "large." Higher $\gamma$ does indeed mean that the effect is stronger. But since the effect on profits is clearly negative, a stronger effect must be more negative. So the absolute value of the effec gets larger, but the effect becomes smaller as it becomes more negative. An taking derivatives is usually a problem on domains that are not open.
Michael Greinecker
CC BY-SA 4.0
2021-07-04T13:08:51.580Z
It is actually more commun to inteprete $\partial^2\pi_1/\partial q_2\partial q_1$ whose sign defines whether the goods are startegic substitutes or complements.
Bertrand
<industrial-organisation>
Interpreting cross-partial derivatives
Ohad Osterreicher
# Interpreting cross-partial derivatives

(Asked by: username_0 on 2021-07-03)

Consider a duopoly setting. Firm $1$ has the following profit equation:
$\pi\_1(q\_1,q\_2,v\_1)=(a+v\_1-q\_1-\gamma q\_2)q\_1 .$
Where $a>0$ is a utility parameter, $q\_i\geq0$ is the quantity and $v\_i\geq0$ is the quality the of firm $i\in1,2$, and $\gamma\in(1,0)$ is the degree of product differentiation.(assume no cost - irrelevant).

Now, I first want to find the change in $\pi\_1$ when firm $2$ increases its quantity. Then I calculate
$\frac{\partial\pi\_1}{\partial q\_2}=-\gamma q\_1<0$.
Obviously, this means that $\pi\_1$ decreases when $q\_2$ increases.

Arriving to my question:

I want to find the change in $\frac{\partial\pi\_1}{\partial q\_2}$ when $\gamma$ increases. So I take the cross-partial derivative,
$\frac{\partial^2\pi\_1}{\partial q\_2\partial\gamma}=-q\_1<0$.

How do I interpret this result?
Does this mean that "lower differentiation (higher $\gamma$) weakens the effect of a larger quantity of firm 2 (higher $q\_2$) on the profits of firm 1 ($\pi\_1$)?"

This makes no sense to me because just by looking at the profit equation above I can see that higher $\gamma$ means a stronger effect of $q\_2$ on $\pi\_1$.

P.S.
Is there a problem with taking the partial derivative w.r.t $\gamma$ when $\gamma$ is in an open set?

Thank you.

**Tags:** industrial-organisation
> **Comments:**
> **username_1**: That has to do with the meaning of "large." Higher $\gamma$ does indeed mean that the effect is stronger. But since the effect on profits is clearly negative, a stronger effect must be more negative. So the absolute value of the effec gets larger, but the effect becomes smaller as it becomes more negative. An taking derivatives is usually a problem on domains that are not open.  
> **username_2**: It is actually more commun to inteprete $\partial^2\pi_1/\partial q_2\partial q_1$ whose sign defines whether the goods are startegic substitutes or complements.  

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## 0 answers

