Consider the continuous model p(t) = 20 − 3Q(t)

Q(t) = q1(t) + q2(t)

TC1(t) = 4q1(t)

TC2(t) = 4q2(t)

q˙1(t) = 0.2(x1(t) − q1(t))

q˙2(t) = 0.2(x2(t) − q2(t))

where xi(t) i = 1, 2 is the desired output level that maximises profits under the assumption that the other firm does not alter its output level.

(i) Find the Cournot solution.

(ii) Is the system dynamically stable?

(iii) Construct a phase diagram which includes the direction field and trajectories for initial conditions:

(a) firm 1 a monopolist

(b) firm 2 a monopolist

(c) (0,0)