What does '$a$' represent in this equation? I know what the rest means but cannot think what '$a$' represents?
Consider a Cournot duopoly market with demand curve $ =a −$,where $ = _1 +_2$. The firm’s respective marginal costs are $_1$, $_2$.
Also does anyone have any tips on how to the reaction functions on a graph please? Thanks
The parameter $a$ is just a positive number (which needs to be greater than $c_1$ and $c_2$ for both firms to produce in equilibrium). Mathematically it is the vertical intercept of the demand curve (when $P$ is measured on the vertical axis) because when $Q=0$, $P=a$. Economically, it is the "choke price", i.e. the price where quantity demanded just becomes zero.
To find the reaction function of firm $1$, you need to find the profit-maximizing choice of $q_1$, which requires first writing down the profit function for firm $1$: $$\pi_1(q_1,q_2)=[\underbrace{(a-q_2)-q_1}_{P}-c_1]q_1$$
The reaction function plots firm $1$'s profit-maximizing choice of $q_1$ against the quantity of the firm, $q_2$. Since the function above is quadratic in $q_1$, you can find the profit-maximizing value of $q_1$ by finding the profit function's horizontal intercepts, i.e. the $q_1$ where $\pi_1(q_1,q_2)=0$. The profit-maximizing $q_1$ will be halfway between the two intercepts (so long as the firm can make a positive profit). This is firm $1$'s reaction function. Firm $2's$ will be identical except for a change of indices.