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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?
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The stability conditions are from Hahn (1962). 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).


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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