# 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?

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?

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.