<p>The stability conditions are from <a href="https://doi.org/10.2307/2296310" rel="nofollow noreferrer">Hahn (1962)</a>. They ensure that, under a specific adjustment process, the firms' outputs will converge to the Cournot-Nash equilibrium. The assumed adjustment process (using the notation from the paper in the question) is:</p>
<p><span class="math-container">$$\frac{dq_i(t)}{dt}=k_i\big(q_i^*-q_i(t)\big)$$</span></p>
<p>where <span class="math-container">$q_i^*$</span> denotes the equilibrium output of firm <span class="math-container">$i$</span> and where <span class="math-container">$k_i&gt;0$</span> for all <span class="math-container">$i=1,\ldots,n$</span>. The assumed adjustment process means that if a firm's current output is below (above) its equilibrium output, then output is increasing (decreasing).</p>
<hr />
<p>Here <span class="math-container">$q(q_0,n)$</span> is just the Cournot-Nash equilibrium output of each of the private firms, given the output of the public firm <span class="math-container">$q_0$</span> and given the number of private firms <span class="math-container">$n$</span>.</p>
<p>Each firm <span class="math-container">$i$</span> has first-order condition</p>
<p><span class="math-container">$$p(Q)+p'(Q)\cdot q_i-c'(q_i)=0$$</span></p>
<p>Summing up over <span class="math-container">$i=1,\ldots,n$</span> gives</p>
<p><span class="math-container">$$np(Q)+p'(Q)\cdot \sum_{i=1}^nq_i-\sum_{i=1}^nc'(q_i)=0$$</span></p>
<p>At a symmetric equilibrium (<span class="math-container">$q_i=q$</span> for all <span class="math-container">$i$</span>), we have</p>
<p><span class="math-container">$$np(q_0+nq)+p'(q_0+nq)\cdot (nq)-nc'(q)=0$$</span></p>
<p>The solution to this equation is <span class="math-container">$q(n,q_0)$</span>. The partial derivatives of <span class="math-container">$q$</span> with respect to <span class="math-container">$n$</span> and <span class="math-container">$q_0$</span> are then found using the implicit function theorem.</p>

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<p>I have two quick question regarding basic oligopoly models:</p>
<ol>
<li>What is meant by we impose the assumptions to  <span class="math-container">$p'+q_ip''&lt;0$</span> and <span class="math-container">$p'-c''&lt;0$</span> to ensure the stability of the Nash equilibrium among private firms?</li>
<li>What is the <span class="math-container">$q$</span> function that is partial differentiated?</li>
</ol>

<microeconomics><nash-equilibrium><firm><oligopoly>
How do the assumptions $p'+q_ip''<0$ and $p'-c''<0$ ensure the stability of the Nash equilibrium among private firms in basic mixed oligopoly model?
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# How do the assumptions $p'+q_ip''<0$ and $p'-c''<0$ ensure the stability of the Nash equilibrium among private firms in basic mixed oligopoly model?

(Asked by: username_0 on 2023-09-09)

[![enter image description here](https://i.sstatic.net/gWFRA.png)](https://i.sstatic.net/gWFRA.png)

I have two quick question regarding basic oligopoly models:

1. What is meant by we impose the assumptions to $p'+q\_ip''<0$ and $p'-c''<0$ to ensure the stability of the Nash equilibrium among private firms?
2. What is the $q$ function that is partial differentiated?

**Tags:** microeconomics, nash-equilibrium, firm, oligopoly

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## 1 answers

### Answer by username_1 on 2023-09-09. Score: 2

The stability conditions are from [Hahn (1962)](https://doi.org/10.2307/2296310). They ensure that, under a specific adjustment process, the firms' outputs will converge to the Cournot-Nash equilibrium. The assumed adjustment process (using the notation from the paper in the question) is:

$$\frac{dq\_i(t)}{dt}=k\_i\big(q\_i^\*-q\_i(t)\big)$$

where $q\_i^\*$ denotes the equilibrium output of firm $i$ and where $k\_i>0$ for all $i=1,\ldots,n$. The assumed adjustment process means that if a firm's current output is below (above) its equilibrium output, then output is increasing (decreasing).

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Here $q(q\_0,n)$ is just the Cournot-Nash equilibrium output of each of the private firms, given the output of the public firm $q\_0$ and given the number of private firms $n$.

Each firm $i$ has first-order condition

$$p(Q)+p'(Q)\cdot q\_i-c'(q\_i)=0$$

Summing up over $i=1,\ldots,n$ gives

$$np(Q)+p'(Q)\cdot \sum\_{i=1}^nq\_i-\sum\_{i=1}^nc'(q\_i)=0$$

At a symmetric equilibrium ($q\_i=q$ for all $i$), we have

$$np(q\_0+nq)+p'(q\_0+nq)\cdot (nq)-nc'(q)=0$$

The solution to this equation is $q(n,q\_0)$. The partial derivatives of $q$ with respect to $n$ and $q\_0$ are then found using the implicit function theorem.


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